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STRUCTURAL SYNTHESIS OF PARALLEL ROBOTS
SOLID MECHANICS AND ITS APPLICATIONSVolume 149
Series Editor: G.M.L. GLADWELLDepartment of Civil EngineeringUniversity of WaterlooWaterloo, Ontario, Canada N2L 3GI
Aims and Scope of the SeriesThe fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchersgiving vision and insight in answering these questions on the subject of mechanics as itrelates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it includesthe foundation of mechanics; variational formulations; computational mechanics;statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids andstructures; dynamical systems and chaos; the theories of elasticity, plasticity andviscoelasticity; composite materials; rods, beams, shells and membranes; structuralcontrol and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts aremonographs defining the current state of the field; others are accessible to final yearundergraduates; but essentially the emphasis is on readability and clarity.
For a list of related mechanics titles, see final pages.
Structural Synthesis ofParallel Robots
Part 1: Methodology
By
GRIGORE GOGU
Mechanical Engineering Research Group,French Institute of Advanced Mechanics andBlaise Pascal University, Clermont-Ferrand, France
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Contents
Preface ..................................................................................................... IX Acknowledgements ...........................................................................XIV List of abbreviations and notations..................................................... XV
1 Introduction.............................................................................................1 1.1 Robot ................................................................................................1 1.2 Robotics ............................................................................................7 1.3 Parallel Robot .................................................................................10 1.4 Terminology ...................................................................................10 1.5 Structural synthesis.........................................................................23 1.6 The objectives and originality of this book ....................................25
2 Structural parameters ..........................................................................31 2.1 Critical review of mobility calculation ...........................................32
2.1.1 Chebychev’s contribution........................................................35 2.1.2 Sylvester’s contribution...........................................................36 2.1.3 Grübler’s contribution .............................................................36 2.1.4 Somov’s contribution ..............................................................37 2.1.5 Hochman’s contribution ..........................................................37 2.1.6 Somov-Malytsheff’s formula ..................................................38 2.1.7 Koenigs’ formula.....................................................................39 2.1.8 Kutzbach’s mobility equation..................................................39 2.1.9 Dobrovolski’s mobility equation .............................................40 2.1.10 Contribution of Y.F. Moroskine ............................................40 2.1.11 Contribution of R. Voinea and M. Atanasiu..........................41 2.1.12 Kolchin’s mobility equation ..................................................42 2.1.13 Rössner’s contribution...........................................................42 2.1.14 Boden’s mobility equation.....................................................42 2.1.15 Manafu’s formula ..................................................................43 2.1.16 Ozol’s formula........................................................................ 43 2.1.17 Contribution of K. J. Waldron ...............................................44 2.1.18 Contribution of N. Manolescu ...............................................44 2.1.19 Contribution of C. Bagci .......................................................45
VI Contents
2.1.20 Contribution of P. Antonescu ................................................ 46 2.1.21 Contribution of F. Freudenstein and R. Alizade.................... 47 2.1.22 Hunt’s contribution................................................................ 48 2.1.23 Hervé’s contribution .............................................................. 49 2.1.24 Gronowicz’s contribution ...................................................... 50 2.1.25 Baker’s contribution .............................................................. 50 2.1.26 Davies’s contribution............................................................. 51 2.1.27 Contribution of V.P. Agrawal and J.S. Rao........................... 52 2.1.28 Contribution of J. Angeles and C. Gosselin .......................... 52 2.1.29 Contribution of F. Dudiţă and D. Diaconescu ....................... 53 2.1.30 Contribution of P. Fanghella and C. Galletti ......................... 55 2.1.31 Fayet’s contribution............................................................... 55 2.1.32 Tsai’s formula........................................................................ 56 2.1.33 McCarthy’s formula .............................................................. 56 2.1.34 Contribution of Z. Huang, L.F. Kong and Y.F. Fang ............ 57 2.1.35 Contribution of J.M. Rico, J. Gallardo and B. Ravani........... 57
2.2 Chebychev-Grübler-Kutzbach mobility formulae .......................... 58 2.2.1 The original Chebychev-Grübler-Kutzbach formula............... 58 2.2.2 The extended Chebychev-Grübler-Kutzbach formula............. 61 2.2.3 Limits of applicability of CGK formulae ................................ 62
2.3 Mobility and connectivity of parallel robots .................................. 78 2.3.1 General definitions and formulae for mobility and connectivity of mechanisms ............................................................. 79 2.3.2 Mobility and connectivity of simple open kinematic chains ... 82 2.3.3 Mobility and connectivity of single-loop kinematic chains..... 88 2.3.4 Connectivity between two elements of a single-loop kinematic chain................................................................................. 96 2.3.5. Mobility and connectivity of parallel robots with simple limbs ............................................................................................... 100 2.3.6. Mobility and connectivity of parallel robots with complex limbs ............................................................................................... 109 2.3.7. General formulae for robot mobility and connectivity ......... 114
2.4 Overconstraints in parallel robots................................................. 120 2.5 Redundancy in parallel robots ...................................................... 125 2.6 General formulae for structural parameters .................................. 127
3 Structural analysis .............................................................................. 131 3.1 Simple open kinematic chains ...................................................... 131 3.2 Single-loop kinematic chains........................................................ 137 3.3 Parallel mechanisms with simple limbs........................................ 148 3.4 Parallel mechanisms with complex limbs..................................... 168 3.5 Other multi-loop kinematic chains ............................................... 228
VII
4 Kinematic analysis ..............................................................................235 4.1. Decoupling in axiomatic design ..................................................236 4.2. Geometric modeling ....................................................................238
4.3 Kinematic modeling .................................................................241 4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation.......................................................................................242 4.3.2 Design and conventional Jacobian matrices ..........................243
4.4 Types of workspaces and singularities .........................................248 4.4.1 Types of workspaces .............................................................248 4.4.2 Types of singularities.............................................................249
4.5. Kinetostatic performance indices ................................................253 4.5.1 Cross-coupling indices ..........................................................256 4.5.2 Indices of input-output propensity.........................................262 4.5.3 Kinetostatic indices defined in connection with manipulability ellipsoids and polytops ...........................................266
4.6 Design Jacobian and motion decoupling ......................................274 4.6.1. Parallel robots with coupled motions ...................................276 4.6.2. Parallel robots with decoupled motions................................285 4.6.3. Parallel robots with uncoupled motions ...............................293 4.6.4. Maximally regular parallel robots ........................................296
5 Structural synthesis ............................................................................299 5.1 Structural synthesis: a systematic approach in mechanism design299 5.2. Morphological and evolutionary approaches...............................304
5.2.1 Morphological approaches ....................................................305 5.2.2 Evolutionary algorithms ........................................................307
5.3 Evolutionary morphology.............................................................310 5.3.1 Design objectives...................................................................310 5.3.2 Constituent elements..............................................................311 5.3.3 Morphological operators........................................................313 5.3.4 Set of solutions ......................................................................314 5.3.5 General structure of the evolutionary morphology................314
5.4 General approach to structural synthesis of parallel robots ..........317 5.4.1 General conditions for structural synthesis of parallel robots via theory of linear transformations.....................................317 5.4.2 General approach to structural synthesis of parallel robots via evolutionary morphology ...............................................................319
6 Limbs with two degrees of connectivity ............................................329 6.1 Limbs with two translational motions ..........................................329 6.2 Limbs with two rotational motions...............................................333 6.3 Limbs with one translational and one rotational motion ..............334
Contents
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6.4 Other limbs with two degrees of connectivity .............................. 335 6.5 Redundant limbs with two degrees of connectivity...................... 338
7 Limbs with three degrees of connectivity ......................................... 341 7.1 Limbs with three translational motions ........................................ 341 7.2 Planar limbs with one rotational and two translational motions... 352 7.3 Non planar limbs with one rotational and two translational motions ............................................................................................... 359 7.4 Limbs with one translational and two rotational motions............. 364 7.5 Limbs with three rotational motions............................................. 364 7.6 Other limbs with three degrees of connectivity ............................ 365 7.7 Redundant limbs with three degrees of connectivity.................... 375
8 Limbs with four degrees of connectivity........................................... 377 8.1 Limbs with Schönflies motion...................................................... 377 8.2 Limbs with two translational and two rotational motions ............ 425 8.3 Limbs with one translational and three rotational motions........... 435 8.4 Other limbs with four degrees of connectivity ............................. 436 8.5 Redundant limbs with four degrees of connectivity ..................... 442
9 Limbs with five degrees of connectivity ............................................ 445 9.1 Limbs with two rotational and three translational motions .......... 445 9.2 Limbs with two translational and three rotational motions .......... 637 9.3 Other limbs with five degrees of connectivity.............................. 644 9.4 Redundant limbs with five degrees of connectivity...................... 650
10 Limbs with six degrees of connectivity ........................................... 653 10.1 Limbs with three translational and three rotational motions ...... 653 10.2 Redundant limbs with six degrees of connectivity ..................... 661
References............................................................................................... 665
Index ....................................................................................................... 693
Preface
“…if the new theory is to lay claim to general interest, it must be capable of producing something new; it must make problems solvable which before could not be solved in a systematic way."
Reuleaux, F., Theoretische Kinematik, Braunschweig: Vieweg, 1875
Reuleaux, F., The Kinematics of Machinery, London: Macmil lan, 1876 and New York: Dover, 1963 (translated by B.W. Kennedy)
Parallel robotic manipulators can be considered a well-established op-
tion for many different applications of manipulation, machining, guiding, testing, control, tracking, haptic force feed-back, etc. A typical parallel ro-botic manipulator (PM) consists of a mobile platform connected to the base (fixed platform) by at least two kinematic chains called limbs. The mobile platform can achieve between one and three independent transla-tions (T) and one to three independent rotations (R).
Parallel manipulators have been the subject of study of much robotic re-search during the last two decades. Early research on parallel manipulators has concentrated primarily on six degrees of freedom (DoFs) Gough-Stewart-type PMs introduced by Gough for a tire-testing device, and by Stewart for flight simulators. In the last decade, PMs with fewer than 6-DoFs attracted researchers’ attention. Lower mobility PMs are suitable for many tasks requiring less than six DoFs.
The motion freedoms of the end-effector are usually coupled together due to the multi-loop kinematic structure of the parallel manipulator. Hence, motion planning and control of the end-effector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, load-to-weight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncou-pled, fully-isotropic and maximally regular PMs can overcome these dis-advantages.
X Preface
Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robotic manipulator is fully-isotropic if its Jacobian matrix is isotropic throughout the entire workspace, i.e., the con-dition number of the Jacobian matrix is equal to one. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the ac-tuated joint velocity space on the end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator.
We distinguish five types of PMs (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fully-isotropic PMs, if the Jacobian J is a diagonal matrix with identical diago-nal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maxi-mally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space.
The first solution for a fully-isotropic T3-type translational parallel robot was developed at the same time and independently by Carricato and Par-enti-Castelli at University of Genoa, Kim and Tsai at University of Cali-fornia, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works.
The general methods used for structural synthesis of parallel mecha-nisms can be divided into three approaches: the method based on dis-placement group theory, the methods based on screw algebra, and the method based on the theory of linear transformations. The method pro-posed in this book is based on the theory of linear transformations and the evolutionary morphology and allows us to obtain the structural solutions of decoupled, uncoupled, fully-isotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, con-nectivity (spatiality), redundancy and overconstraint of PMs proposed re-cently by the author are integrated into the synthesis approach developed in this book.
Various solutions of TaRb-type PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel ro-bots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we
XI
are aware, this is the first book on robotics presenting solutions of uncou-pled, fully-isotropic and maximally regular PMs.
Non-redundant/redundant, overconstrained/isostatic solutions of uncou-pled and fully-isotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthe-sis. These solutions are derived from families of PMs with decoupled mo-tions. A serial kinematic chain is associated with each elementary limb and at least one closed loop is integrated in each complex limb.
The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the au-thor in the last four years in the framework of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the National Center for Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature:
a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of over-constraints of parallel robotic manipulators that overcome the drawbacks of the classical Chebychev-Grübler-Kutzbach formulae,
b) a new approach to systematic innovation in engineering design called evolutionary morphology,
c) solutions to follow fully-isotropic parallel robots: parallel wrists with two and three degrees of mobility, planar parallel robots, parallel robots of types T2R1 and T3R2, parallel robots with Schönflies motions, hexapods with six degrees of freedom.
The various solutions proposed by the author belong to a modular family called IsoglideN-TxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots can have any combination of n inde-pendent translations (T) and rotations (R). The Isogliden-TaRb modular family was developed by the author and his research team at the French In-stitute of Advanced Mechanics (IFMA) in Clermont-Ferrand.
This work is organized in two parts published in two distinct books. Part 1 presents the methodology proposed for structural synthesis and Part 2 the various topologies of parallel robots systematically generated by the struc-tural synthesis approach. The originality of this work resides in combining the new formulae for mobility connectivity, redundancy and overcon-straints, and the evolutionary morphology in a unified approach of struc-tural synthesis giving interesting innovative solutions for parallel robots.
Part 1 is organized in ten chapters. The first chapter is intended to intro-duce the main concepts, definitions and components of the mechanical ro-botic system. Chapter 2 reviews the contributions in mobility calculation
Preface
XII Preface
systematized in the so called Chebychev-Grübler-Kutzbach mobility for-mulae. The drawbacks and the limitations of these formulae are discussed, and the new formulae for mobility, connectivity, redundancy and overcon-straint are demonstrated via an original approach based on the theory of linear transformations. These formulae are applied in chapter 3 for the structural analysis of parallel robots with simple and complex limbs. The new formulae are also applied to calculate the mobility and other structural parameters of single and multi-loop mechanisms that do not obey the clas-sical Chebychev-Grübler-Kutzbach formulae, such as the mechanisms pro-posed by De Roberval, Sarrus, Bennett, Bricard and other so called “paradoxical mechanisms”. We have shown that these mechanisms com-pletely obey the definitions, the theorems and the formulae proposed in the previous chapter. There is no reason to continue to consider them as “para-doxical”. Chapter 4 presents the main models and performance indices used in parallel robots. We emphasize the Jacobian matrix, which is the main issue in defining robot kinematics, singularities and performance in-dices. New kinetostatic performance indices are introduced in this section to define the motion decoupling and input-output propensity in parallel ro-bots. Structural parameters introduced in the second chapter are integrated in the structural synthesis approach founded on the evolutionary morphol-ogy (EM) presented in chapter 5. The main paradigms of EM are presented in a closed relation with the biological background of morphological ap-proaches and the synthetic theory of evolution. The main difference be-tween the evolutionary algorithms and the EM are also discussed. The evo-lutionary algorithms are methods for solving optimization-oriented problems, and are not suited to solving conceptual design-oriented prob-lems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions.
The first stage in structural synthesis of parallel robots is the generation of the kinematic chains called limbs used to give some constrained or un-constrained motion to the moving platform. The constrained motion of the mobile platform is obtained by using limbs with less than six degrees of connectivity. The various solutions of simple and complex limbs with two to six degrees of connectivity are systematically generated by the structural synthesis approach and presented in chapters 6-10. We focus on the solu-tions with a unique basis of the operational velocity space that are useful for generating various topologies of decoupled, uncoupled, fully-isotropic and maximally regular parallel robots presented in Part 2. Limbs with mul-tiple bases of the operational velocity space and redundant limbs are also presented in these chapters. These limb solutions are systematized with re-spect to various combinations of independent motions of the distal link. They are defined by symbolic notations and illustrated in about 250 figures
XIII
containing more than 1500 structural diagrams. Special attention was paid to graphic quality of structural diagrams to ensure a clear correspondence between the symbolic and graphic notation of joints and the relative posi-tion of their axes. The graphic illustration of these solutions is associated with the author’s conviction that a good structural diagram really “is worth a thousand words”, especially when you are trying to disseminate the re-sult of the structural synthesis of kinematic chains. The kinematic chains presented in chapters 6-10 are useful as innovative solutions of limbs in parallel, serial and hybrid robots. In fact, serial and hybrid robots may be considered as a particular case of parallel robots with only one limb which can be a simple, complex or hybrid kinematic chain. Many serial robots actually combine closed loops in their kinematic structure.
The various types of kinematic chains generated in chapters 6-10 are combined in Part 2 to set up innovative solutions of parallel robots with two to six degrees of mobility and various sets of independent motions of the moving platform.
Part 2 is organized in 16 chapters presenting the following: translational PMs T2-type (chapter 1), T1R1-type PMs with screw motion (chapter 2), other T1R1-type PMs (chapter 3), R2-type spherical parallel wrists (chap-ter 4), translational PMs T3-type (chapter 5), T2R1-type PMs with planar motion (chapter 6), other T2R1-type PMs (chapter 7), T1R2-type PMs (chapter 8), R3-type spherical parallel wrists (chapter 9), T3R1-type PMs with Schönflies motion (chapter 10), other T3R1-type PMs (chapter 11), T2R2-type PMs (chapter 12), T1R3-type PMs (chapter 13), T3R2-type PMs (chapter 14), T2R3-type PMs (chapter 15) and PMs with six degrees of freedom (chapter 16).
Parallel robots with coupled, decoupled and uncoupled motions, along with fully-isotropic and maximally regular solutions, are systematically presented in each chapter. Innovative solutions of overconstrained or non-overconstrained, non-redundant or redundantly actuated parallel robots with simple or complex limbs actuated by linear or rotating motors are set up in Part 2 by applying the structural synthesis methodology presented in this first part. The writing of part 2 is still in progress and will soon be fi-nalized.
Many solutions for parallel robots obtained through this systematic ap-proach of structural synthesis are presented here for the first time in the lit-erature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.
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XIV Preface
Acknowledgements
The scientific environment of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the CNRS was the main source of encouragement and motivation to pursue the research on the structural synthesis of parallel robots and to write this book. Deep grati-tude is expressed here to Dr. François Pierrot, Deputy Director of LIRMM and coordinator of both ROBEA projects, and also to all colleagues in-volved in these projects from the research laboratories LIRMM, INRIA, IRCCyN LASMEA and LaMI for the valuable scientific exchanges during the joint work in these projects.
Moreover, financial support from the CNRS, FR TIMS and IFMA for developing the innovative Isoglide-family of parallel robots are duly ac-knowledged.
Furthermore, Mrs. Nathalie Jacobs, Springer Dordrecht Engineering Editor is gratefully acknowledged for her availability and encouragement in pursuing this book project.
Ms. Sarah Davey and Prof. Graham M.L. Gladwell are also gratefully acknowledged for linguistic reviewing of the manuscript. Prof. Gladwell not only took great care of the book language but also made numerous suggestions for improving and clarifying some fine points of its content.
May I also acknowledge the excellent facilities and research environ-ment provided by LaMI and IFMA which contributed actively to the com-pletion of this project.
To conclude, I cannot forget my wife Iléana and my son Christian for their love, affection and encouragement, providing the fertile ambience for this sustained work very often prolonged late into the evening and mostly during week-ends and holidays.
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List of abbreviations and notations
ai – axial offset of the ith-link A (1A-2A-…-nA) and A (1-2-…-n) – simple open kinematic chain and its
links A (0≡1A-2A-3A-4A-5A) simple open kinematic chain with link 1A fixed to
base 0 A (1A≡0-2A-…nA≡1A) – simple closed kinematic chain with link 1A fixed
to base 0 A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) simple open kinematic chain
obtained by serial concatenation of two simple chains A1 and A2 AI – artificial intelligence bi – length of the i-link B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) single-loop kinematic
chain obtained by connected the distal links (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2)
c – condition number C ← A1-A2-…Ak parallel robot with k simple limbs Ai (i=1, 2,…,k) C – cylindrical pair CNRS - Centre National de la Recherche Scientifique (National Center
for Scientific Research) CGK - Chebychev-Grübler-Kutzbach Ci - instantaneous degree of motion coupling C - global degree of motion coupling d - linear dispalcement D ← A1-A2-…Ak1-E1-E2-…-Ek2 parallel robot with k1 simple limbs Ai (i=1, 2,…,k1) and k2 complex limbs Ei (i=1, 2,…,k2) DoF - degree-of-freedom DPs – design parameters eA and eA1 - link of A1-limb (e=1,2,3,…,n) eB and eA2 - link of A2-limb (e=1,2,3,…,n) eC and eA3 - link of A3-limb (e=1,2,3,…,n) eD and eA4 - link of A4-limb (e=1,2,3,…,n)
1( ,..., )nE ε ε= - set of constituent elements EA – evolutionary algorithm EM – evolutionary morphology ES - Evolution strategies F - linear transformation f - degree of mobility of a joint in the general motion space with six dimensions
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XVI Preface
F ← G1-G2-…Gk general notation for the kinematic chain associated to a serial, parallel or hybrid robot with k simple and/or complex limbs Gi (i=1, 2,…,k)
FRs – functional requirements FR TIMS - Fédération de Recherche Technologies de l’Information et
de la Mobilité, de la Sûreté GA – genetic algorithm Gi (1Gi-2Gi-…nGi) the kinematic chain associated to the ith limb (i=1,2,…,k). H – characteristic point of the distal link/end-effector H- helical pair IFMA - Institut Français de Mécanique Avancée (French Institute of
Advanced Mechanics) INRIA - Institut National de Recherche en Informatique et en Automa-
tique (The French National Institute for Research in Computer Science and Control)
IRCCyN - Institut de Recherche en Communications et Cybernétique de Nantes
n nI × - n×n identity matrix J - Jacobian matrix JT – transpose of the Jacobian matrix J-1 – inverse of the Jacobian matrix JS - direct kinematics matrix/Jacobian JP - inverse kinematics matrix/Jacobian k - total number of limbs of the parallel manipulator k1 - number of simple limbs of a parallel manipulator k2 - number of complex limbs of a parallel manipulator KP - kinematic pair KC - kinematic chain Q - generic notation for a kinematic chain or a mechanism LaMI - Laboratoire de Mécanique et Ingénieries (Mechanical Engineer-
ing Research Group) LASMEA - Laboratoire des Sciences et Matériaux pour l'Electronique,
et d'Automatique (Laboratory of Sciences and Materials for Elec- tronic, and of Automatic)
LIRMM - Laboratoire d'Informatique, de Robotique et de Microélec- tronique de Montpellier (Montpellier Laboratory of Computer Sci- ence, Robotics, and Microelectronics)
Lc - characteristic length m - total number of links (mechanism elements) including the fixed base mQ - number of joints in kinematic chain Q
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M - mobility MQ - mobility of kinematic chain/mechanism Q iM - instantaneous mobility iMQ - instantaneous mobility of kinematic chain/mechanism Q N – number of overconstraints n – number of kinematic elements (links) excepting the fixed link
t 1 n( ,..., )Ο ο ο= - set of morphological operators applicable at each generation t O0x0y0z0 - reference frame O0On - position vector of point On in fixed reference frame p - number of joints pp - number of passive joints p - operational velocity vector P - prismatic joint/pair P - actuated prismatic joint Pa - parallelogram loop Pat - telescopic parallelogram loop Pn2 - planar close loop with two degrees of mobility Pn4 - planar close loop with three degrees of mobility PM - parallel manipulator q - number of independent closed loops q - joint velocity vector qi - finite displacement in the ith actuated joint r - number of joint parameters that lost their independence rQ - number of joint parameters that lost their independence in kinematic chain/mechanism B rl - number of joint parameters that lost their independence in the closed
loops integrated in the complex limbs of a parallel robot Gi
lr number of joint parameters that lost their independence in the closed loops which may exist in ith limb Gi of a parallel robot
R – revolute joint/pair R – actuated revolute joint Rb – rhombus loop RF - the range of linear transformation F RQ - vector space of operational velocities of the kinematic
chain/mechanism Q Qa / bR - vector space of relative velocities between links a and b in
kinematic chain/mechanism Q ( Q
a / bR ) - base of the vector space Qa / bR
R - reangularity
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XVIII Preface
Qa / bS - connectivity between links a and b in a kinematic
ckain/mechanism Q SQ - connectivity of kinematic chain/mechanism Q iSQ - instantaneous connectivity of kinematic chain/mechanism Q S . - semangularity T - structural redundancy
1t n( ,..., )Τ τ τ= - set of evolution criteria from generation t to generation t+1
U - vector space of joint velocities UQ - vector space of joint velocities of kinematic chain/mechanism Q v - translational velocity vector w - general velocity vector of the moving platform w1, w2, …, w6 - general coordinates W - vector space of external velocities WQ - vector space of external velocities of kinematic chain/mechanism
Q xP, yP, zP - coordinates of point P in the reference frame
, , ,α β δ ϕ - rotation angles , , ,α β δ ϕ - time derivatives of the rotation angles
∆q - vector of the infinitesimal motions of joints ∆w - vector of infinitesimal motions of the end-effector
{ }: ,i true falseι ι → - termination criterion for evolutionary morphology
1υ - conditioning index
2υ - minimum conditioning index
3υ and 4υ - manipulability indexes
iξ - square root of the eigenvalues of (JJT)-1 λ - scalar / proportionality factor µ - vector of the operational forces τ - vector of the joint forces ψ - velocity transmission factor ω - angular velocity
1( ,..., )t nσ σΣ = - set of solutions at each generation t
1 n( ,..., )Φ φ φ= - set of design objectives 0 - fixed base of a kinematic chain/mechanism 1-2-…-n - links (elements) of a kinematic chain/mechanism 1Q-2Q-…-nQ - links (elements) of kinematic chain/mechanism
1 Introduction
This section introduces the terminology and presents the main objectives of the book. The family of parallel robots is placed in the general field of robots and the structural synthesis in the general context of robotics.
1.1 Robot
The origin of the term robot can be found in a science fiction play written by Karel Čapek (1920). This play titled R.U.R. was published in Czech in 1920 as an abbreviation of Rozum's Universal Robots, a collective drama in a comic prologue and three acts first performed in National Theatre in Prague on January 25, 1921. We note that in Czech “rozum” means wis-dom and “robota” means servitude/forced labour/drudgery. Since the word robot was apparently coined by the author's brother, cubist painter and writer Joseph Čapek, the term is used to refer to a wide range of man-made things from devices to programs that are used to do automatically and autonomously various tasks by replacing humans.
Human-made analogs of living systems could be found long ago in Antiquity despite the recent useage of the word robot. The pigeon conceived by the Greek mathematician Archytas of Tarentum in 425 BC and chronicled by Favorinus and by Aulus Gellius in his Attic Nights could be considered the first “flying robot” and the south-seeking chariot, possibly used by the Chinesse Yellow Emperor Hunag Ti, the first “field robot”. Other legendary solutions such as the “walking machines” called wooden ox and gliding horse of Chu-ko Liang were invented to transport army food supplies across rough terrain during the period of China’s Three Kingdoms in 209 AC (Yan 1999). The automatic machines (Al-Jazary 1973) constructed by Al-Jazari (1136-1206), the designs of mechanical knight (Rosheim 2006) and flying machines (Da Vinci) of Leonardo da Vinci (1452-1519), the mechanical duck and the flute player – the first automatons (Balpe and Drye 1994) constructed by Jacques de Vaucanson (1709-1782) and the androids (the draughtsman, the musicians and the writer) constructed in 1774 by Pierre Jacquet-Droz (1771-1790) and his
2 1 Introduction
son Henri-Louis (1753-1791) could also be considered as precursors of the current robots from the pre-digital era.
Hundreds of robot definitions are known today and hundreds of millions of web sites could be referenced today by a simple search on the word ro-bot. According to the American Heritage Dictionary a robot is a mechani-cal device that is capable of performing a variety of often complex human tasks on command or by being programmed in advance. Other definitions of the word robot proposed by on-line dictionaries are systematized in Ta-ble 1.1.
Robots can be found today in the manufacturing industry, agricultural, military and domestic applications, space exploration, medicine, education, information and communication technologies, entertainment, etc.
We can see that the word robot is mainly used to refer to a wide range of mechanical devices or mechanisms, the common feature of which is that they are all capable of movement and can be used to perform physical tasks. Robots take on many different forms, ranging from humanoid, which mimic the human form and mode of movement, to industrial, whose appearance is dictated by the function they are to perform. Robots can be categorized as robotic manipulators, wheeled robots, legged robots swimming robots, flying robots, androids and self reconfigurable robots which can conform themselves to a given task. This book focuses on the parallel robotic manipulators which are the counterparts of the serial robots.
We also note that the term robot has recently morphed to also refer to “bots”, which are automated programs used in several online functions in computer science and artificial intelligence. The term has also entered into the informal language to define certain forms of human behaviour or even a traffic light in South Africa.
1.1 Robot 3
Table 1.1. Various definitions of the term robot
Dictionary Definition 1. Programmable machine for performing tasks: a mechani-cal device that can be programmed to carry out instructions and perform complicated tasks usually done by people. 2. Imaginary machine: a machine that resembles a human in appearance and can function like a human, especially in science fiction. 3. Person like a machine: somebody who works or behaves mechanically and emotionlessly.
Encarta® World English Dictionary, North American Edition1
4. South Africa traffic light: a set of automatic traffic lights (informal).
Compact Oxford English Dictionary2
A machine capable of carrying out a complex series of ac-tions automatically, especially one programmable by a computer.
Merriam-Webster's Online Dictionary, 10th Edition3
1a. A machine that looks like a human being and performs various complex acts (as walking or talking) of a human being; also: a similar but fictional machine whose lack of capacity for human emotions is often emphasized. 1b. An efficient insensitive person who functions automati-cally. 2. A device that automatically performs complicated often repetitive tasks. 3. A mechanism guided by automatic controls.
Cambridge Interna-tional Dictionary of English4
1. A machine used to perform jobs automatically, which is controlled by a computer. 2. Disapproving someone who does things in a very quick and effective way but never shows their emotions. 3. A traffic light in South Africa.
Cambridge Dictionary of American English5
A mechanical device that works automatically or by com-puter control.
Columbia Encyclope-dia, Sixth Edition6
Mechanical device designed to perform the work generally done by a human being.
1http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx 2http://www.AskOxford.com 3http://www.m-w.com/ 4http://dictionary.cambridge.org/ 5,6http://dictionary.cambridge.org/
4 1 Introduction
Table 1.1. (cont.)
The American Heritage® Dictionary of the English Language7
1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in ad-vance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others. 1. A mechanical device, esp. one controlled electronically, that can perform physical tasks of a human being, in indus-trial assembly or the like. 2. A person who moves or acts mechanically, without thought or feeling.
The Wordsmyth English Dictionary-Thesaurus8
Infoplease Dictionary9
1. A machine that resembles a human and does mechanical, routine tasks on command. 2. A person who acts and responds in a mechanical, routine manner, usually subject to another's will; automaton. 3. Any machine or mechanical device that operates auto-matically with humanlike skill.
Dictionary.com10 1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in ad-vance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others.
Wiktionary11
1. A machine that resembles humans in shape or scope of function. 2. A machine that operates automatically. 3. A machine controlled by a fundamentally ingrained com-puter.
7http://www.bartleby.com/61/s0.html
8http://www.wordsmyth.net/
9http://www.infoplease.com/dictionary.html
10http://www.dictionary.com/
11http://www.wiktionary.org
1.1 Robot 5
Table 1.1. (cont.)
Rhymezone12 LookWAYup Translating Dictioary /Thesaurus13
WordNet 1.7 Vocabulary Helper14
A mechanism that can move automatically.
Wikipedia, the Free Encyclopedia15
1. A robot is a type of device. 2. Robot can also refer to Internet bot, an automated software program used on the Internet. 1. Especially in science-fiction: a machine that vaguely re-sembles a human being and which can be programmed to carry out tasks. Compare android. 2. An automatic machine that can be programmed to per-form specific tasks. 3. Someone who works efficiently but who lacks human warmth or sensitivity. 4. An automatic traffic signal.
AllWords.com Multi-Lingual Dictionary16
WebmasterWorld Webmaster and Search Engine Glossary17
Program that automatically does "some action" without user intervention. In the context of search engines, it usu-ally refers to a program that mimics a browser to download web pages automatically. A spider is a type of robot. Some times referred to as Webbots.
Netlingo18 Traditionally, it's a device that can move and react to sen-sory input… The term "robot" has morphed to also refer to bots, which are automated programs used in several online functions.
Webopedia19
1. A device that responds to sensory input. 2. A program that runs automatically without human inter-vention.
12http://www.rhymezone.com/ 13http://lookwayup.com/free
14http://poets.notredame.ac.jp/cgi-bin/wn 15http://www.wikipedia.org
16http://www.allwords.com/
17http://www.webmasterworld.com/glossary/ 18http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/
6 1 Introduction
Table 1.1. (cont.)
CCI Computer20 1. A mechanical device that performs a task that would otherwise be done by a human. Robots can be useful for jobs that are boring or dangerous for humans to perform. The simplest robots are capable only of repeating a pro-grammed motion; the most sophisticated models can use sensors and artificial intelligence to distinguish between objects, understand natural language, and make decisions. Robots can be programmed or operated by remote control. 2. A computer program that performs intelligent tasks such as retrieving World Wide Web documents and indexing references. 3. A program that performs a programmed communication function such as automated email answering, responding to newsgroup message, or regulatory functions in IRC, graphical chat, and other online environments 1. A device that responds to sensory input. They are often just called "Bots". 2. A program that runs automatically without human inter-vention. Typically, a robot is endowed with some artificial intelligence so that it can react to different situations it may encounter. Two common types of robots are agents and spiders.
Computer Telephony & Electronics Dictionary and Glossary21
Internet Terms22 A robot is a program that is designed to automatically go
out and explore the Internet for a specific purpose. Robots that record and index all of the contents of the network to create searchable databases are sometimes called spiders or Worms. WebCrawler and Lycos are popular examples of robots.
TECHNICAL23
Robot - Besides being a mechanical device used to mimic human form, usually to accomplish some repetitive task, this term refers to a computer program that scans Web pages and links. Like a similar spider program, robots are used to scan Web pages and index them.
20http://www.currents.net/resources/dictionary/index.html
21http://www.csgnetwork.com/glossary.html
22http://www.learnthenet.com/english/glossary/glossary.htm
23http://www.ugeek.com/glossary/glossary_search.htm
1.2 Robotics 7
1.2 Robotics
The science of robotics has undergone an explosive growth since the word robotics was coined by the writer Isaac Asimov in 1950. In his science fic-tion book “I, Robot” he presented the famous three laws of robotics: (i) a robot may not injure a human being, or, through inaction, allow a human being to come to harm; (ii) a robot must obey the orders given it by human beings except where such orders would conflict with the first law and (iii) a robot must protect its own existence as long as such protection does not conflict with the first or second law. According to the American Heritage Dictionary, robotics is the science or study of the technology associated with the design, fabrication, theory, and application of robots. This tech-nology combines mechanical devices, actuators, sensors, controllers, soft-ware and computers.
Today robots get closer to humans and new issues rise in matters related to human robot interaction, in advances and experiences of robots and automation at home, at work, for education, as well as in other emerging areas. The next generation of robots will be capable of interacting with humans on a sophisticated level, making decisions, helping, caring and giving companionship.
Recent research by the Japan Robotics Association (JPA), United Na-tions Economic Commission (UNEC) and the International Federation of Robotics (IFR) indicates that the nascent personal and service robotics market will exhibit exceptional near term growth. The service and personal robotics marketplaces together will equal the size of the industrial robotics market (the combination of manufacturing and bio-industrial) by 2005, and will be twice the size of the industrial robotics market by 2010, and almost 4X its size by 2025.
Many of the technologies required to build functional personal and ser-vice robots already exist, and markets for these products are already in place. Some of the most salient enabling technologies include advances in microprocessor technology, wireless technology, image processing, speech recognition, motion sensor technology, and embedded systems develop-ment tools.
Hundreds of universities worldwide have research programs in robotics and many are awarding degrees in robotics.
Various definitions of the word robotics are systematized in Table 1.2. We can see that they converge towards the integration of the design and the end use in the studies related to robotics. This book focuses on the con-ceptual design of parallel robots.
8 1 Introduction
Table 1.2. Various definitions of the term robotics
Dictionary Definition Encarta® World English Dictionary, North American Edition1
Design and use of robots: the science and technology relating to computer-controlled mechanical devices such as the automated tools commonly found on automobile assembly lines.
Compact Oxford English Dictionary2
The branch of technology concerned with the design, construction, and application of robots.
Merriam-Webster's Online Dictionary, 10th Edition3
Technology dealing with the design, construction, and operation of robots in automation.
Cambridge International Dictionary of English4
The science of making and using robots.
Cambridge Dictionary of American English5
The science of designing and operating robots.
Columbia Encyclopedia, Sixth Edition6
Science and technology of general purpose, pro-grammable machine systems.
The American Heritage® Dictionary of the English Language7,10
The science or study of the technology associated with the design, fabrication, theory, and application of robots.
The Wordsmyth English Dictionary-Thesaurus8
The technology of designing and using robots for various, usually industrial tasks.
Infoplease Dictionary9
The use of computer-controlled robots to perform manual tasks, especially on an assembly line.
Wiktionary11,15 The science and technology of robots, their design, manufacture, and application.
1http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx 2http://www.AskOxford.com 3http://www.m-w.com/ 4http://dictionary.cambridge.org/ 5,6http://dictionary.cambridge.org/
7http://www.bartleby.com/61/s0.html
8http://www.wordsmyth.net/
9http://www.infoplease.com/dictionary.html
10http://www.dictionary.com/
11http://www.wiktionary.org
1.2 Robotics 9
Table 1.2. (cont.)
Rhymezone12 LookWAYup Translating Dictioary/ Thesaurus13
WordNet 1.7 Vocabulary Helper14
The area of AI concerned with the practical use of ro-bots 1. The branch of engineering concerned with the de-sign, construction, operation and use of industrial ro-bots, and which incorporates many of the concepts used in artificial intelligence. 2. A form of dancing in which dancers imitate the stiff jerky movements of robots.
AllWords.com Multi-Lingual Dictionary16
The On-line Medical Dictionary17
The application of electronic, computerised control systems to mechanical devices designed to perform human functions. Formerly restricted to industry, but nowadays applied to artificial organs controlled by bionic (bioelectronic) devices, like automated insulin pumps and other prostheses.
Netlingo18,19 The field of computer science and engineering con-cerned with creating robots. Robotics is a branch of artificial intelligence.
CCI Computer20 The design, manufacture, and use of robots.
Computer Telephony & Electronics Dictionary and Glossary21
The field of computer science and engineering con-cerned with creating robots or bots, devices that can move and react to sensory input. Robotics is one branch of artificial intelligence.
12http://www.rhymezone.com/ 13http://lookwayup.com/free
14http://poets.notredame.ac.jp/cgi-bin/wn 15http://www.wikipedia.org
16http://www.allwords.com/
17http://cancerweb.ncl.ac.uk/omd/ 18http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/
20http://www.currents.net/resources/dictionary/index.html
21http://www.csgnetwork.com/glossary.html
10 1 Introduction
1.3 Parallel Robot
Although the appearance and capabilities of robots vary vastly, all robots share the features of a mechanical, movable structure under some form of control. The structure of a robot is usually mostly mechanical and takes the form of a mechanism having as constituent elements the links connected by kinematic joints. Serial or parallel kinematic chains are concatenated in the robot mechanism. The serial kinematic chain is formed by links connected sequentially by joints. Links are connected in series as well as in parallel making one or more closed-loops in a parallel mechanism. The members of a mechanism may take various forms: bars, plates, platforms, cams, gears, sliders, sliding blocks, etc.
The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a mobile platform is connected to a reference element by at least two limbs that can be elementary (simple) or complex. The robot actuators are integrated in the limbs (also called legs) usually closed to the fixed member, also called the base or the fixed plat-form. The mobile platform positions the robot end-effector in space and could have between two and six degrees of freedom. Usually the number of actuators coincides with the degrees of freedom of the mobile platform, exceeding them only in the case of redundantly-actuated parallel robots.
The paradigm of parallel robots is the hexapod-type robot, which has six degrees of freedom, but recently the machine industry has discovered the potential applications of lower-mobility parallel robots with just 2, 3, 4 or 5 degrees of freedom. Indeed, the study of this type of parallel manipula-tors is very important. They exhibit interesting features if compared to hexapods, such as simpler architecture, simpler control system, high-speed performance, low manufacturing and operating costs. Furthermore, for several parallel manipulators with decoupled or uncoupled motions the ki-nematic model can be easily solved to obtain algebraic expressions, which are well suited for an implementation in optimum design problems. Paral-lel mechanisms can be considered a well-established solution for many dif-ferent applications of manipulation, machining, guiding, testing, control, etc.
1.4 Terminology
The terminology used in this book and defined in this section is mainly es-tablished in accordance with the terminology adopted by the International Federation for the Promotion of Mechanism and Machine Science
1.4 Terminology 11
(IFToMM) and published in (Ionescu 2003). The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.3-1.5. They are completed by some complementary remarks, notations and symbols used in this book.
In the standard terminology, a kinematic chain is an assembly of links (mechanism elements) and joints, and a mechanism is a kinematic chain in which one of its links is taken as a “frame”. In this definition the “frame” is a mechanism element deemed to be fixed. In this book, we use the no-tion of reference element to define the “frame” element. The reference element can be fixed or may merely be deemed to be fixed with respect to other mobile elements. The fixed base is denoted in this book by 0. A mo-bile element in a kinematic chain Q is denoted by nQ (n=1, 2, …). Two or more links connected together in the same link such that no relative motion can occur between them are considered as one link. The identity symbol “≡” is used between the links to indicate that they are welded together in the same link. For example, the notation 1Q≡0 is used to indicate that the first link 1Q is connected to the fixed base by a common link. A kinematic chain Q is denoted by the sequence of its links. The notation Q (1Q≡0-2Q-…nQ) indicates a kinematic chain in which the first member is fixed and the notation Q (1Q-2Q-…nQ) a kinematic chain with no fixed member.
We will use the notion of mechanism to qualify the whole mechanical system, and the notion of kinematic chain to qualify the sub-systems of a mechanism. So, in this book, the same assembly of links and joints Q1 is considered to be a kinematic chain when it is integrated as a sub-system in another assembly of links and joints Q2, and it is considered a mechanism when Q1 represents the whole system. The systematization, the definitions and the formulae presented in this book are also valuable for mechanisms and kinematic chains.
We use the term mechanism element or link to name a component (member) of a mechanism with any form (bar, gear, cam, etc.). In this text, unless otherwise stated, we consider all links to be rigid. We distinguish the following types of links:
a) monary link - a mechanism element connected in the kinematic chain by only one joint (a link which carries only one kinematic pairing ele-ment),
b) binary link - a mechanism element connected in the kinematic chain by two joints (a link connected to two other links),
c) polinary link - a mechanism element connected in the kinematic chain by more than two joints (ternary link - if the link is connected by three joints, quaternary link if the link is connected by four joints,…).
12 1 Introduction
Table 1.3. IFToMM terminology with respect to the kinematic pairs
Term Definition Pairing element Assembly of surfaces, lines or points of a solid body through
which it may contact with another solid body.
Mechanism element
Solid body or fluid component of a mechanism.
Frame Mechanism element deemed to be fixed.
Base Body (link) of a robot that carries the first joint(s) of the ki-
nematic chain of a manipulator or pedipulator.
Rigid body Theoretical model of a solid body in which the distances be-tween particles are considered to be constant, regardless ofany forces acting upon the body.
Link 1. Mechanism element (component) carrying kinematic pair-ing elements. 2. Element of a linkage.
Limit position of a link
Position of a link for which a coordinate which describes itsposition relative to an adjacent link is a maximum or a mini-mum.
Input (driving) link
Link whereby motion and force are imparted to a mechanism.
Output (driven) link (follower)
Link from which required forces and motions are obtained.
Bar Link that carries only revolute joints.
Crank Link able to rotate completely about a fixed axis.
Rocker Link that oscillates within a limited angle of rotation about afixed axis.
Coupler (floating link)
Link that is not connected directly to the frame.
Slider Link that forms a prismatic pair with one link and a revolute pair with another link.
1.4 Terminology 13
Table 1.3. (cont.)
Sliding block Compact element of a prismatic pair which slides along aguiding element (e.g. in a slot).
Guide Element of a prismatic pair that is fixed to the frame and con-strains the motion of a sliding block.
Crosshead Component between a piston and a connecting rod which, byforming a prismatic joint with the frame, provides a reaction to the component of force in the connecting rod normal to theline of stroke of the piston.
Connecting rod Coupler between a piston or a crosshead and a crank shaft.
Cam Component with a curved profile or surface whereby it im-parts a displacement either by point or line contact with acam follower.
Kinematic pair Mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of free-dom. Note: see synonym at joint.
Joint 1. Physical realization of a kinematic pair, including connec-tion via intermediate mechanism elements. 2. Kinematic pair.
Degree of freedom (connectivity) of a kinematic pair
The number of independent coordinates needed to describe the relative positions of pairing elements.
Closure of a kinematic pair
Process of constraining two rigid bodies to form a kinematic pair by force (force closure), geometric shape (form closure), or flexible materials (material closure).
Force-closed (open) pair
Kinematic pair the elements of which are held in contact bymeans of external forces.
Form-closed pair Kinematic pair the elements of which are constrained to con-tact each other by means of particular geometric shapes.
Lower pair Kinematic pair that is formed by surface contact between itselements.
14 1 Introduction
Table 1.3. (cont.)
Higher pair Kinematic pair that is formed by point or line contact be-tween its elements.
Revolute pair (hinge)
Pair that allows only a rotary motion between two links.
Prismatic pair Pair that allows only a rectilinear translation between twolinks.
Helical (screw) pair
Pair that allows only a screw motion between two links.
Cylindrical pair Pair for which the degree of freedom is two and that allows a rotation about a particular axis together with an independent translation in the direction of this axis.
Spherical pair Pair for which the degree of freedom is three and that allows independent relative rotations about three mutually orthogo-nal axes.
Planar contact (sandwich) pair
Pair for which the degree of freedom is three and that allows relative motion in parallel planes.
Universal (Cardan, Hooke’s) joint (uni-versal coupling)
Kinematic joint connecting two shafts with intersecting axes.
Pin joint Joint using a pin as the connecting component between tworigid bodies.
Joint variable Quantity that describes the relative motion between two con-secutive links of a manipulator.
Joint space Space defined by a vector whose components are joint vari-ables.
1.4 Terminology 15
Table 1.4. IFToMM terminology with respect to the kinematic chains
Term Definition Kinematic chain Assembly of links and joints.
Closed kinematic chain
Kinematic chain each link of which is connected with at leasttwo other links.
Open kinematic chain
Kinematic chain in which there is at least one link which car-ries only one kinematic pairing element.
Loop Subset of links that forms a closed circuit.
Tree (mobile) Kinematic chain that contains no loops.
Linkage Kinematic chain whose joints are equivalent to lower pairsonly.
Four-bar linkage Linkage with four binary links.
Kinematic joint Kinematic chain whose kinematical properties are equivalent in some respects to those of a kinematic pair.
Mechanism 1. System of bodies designed to convert motions of, andforces on, one or several bodies into constrained motions of,and forces on, other bodies. 2. Kinematic chain with one of its components (links) taken as a frame.
Kinematic inversion
Transformation of one mechanism into another by choosing a different member to be the frame.
Limit position of a mechanism
Configuration of a mechanism in which one of its links is in alimit position
Planar mechanism Mechanism in which all points of its links describe paths lo-cated in parallel planes.
Spherical mechanism
Mechanism in which all points of its links describe paths lo-cated on concentric spheres.
16 1 Introduction
Table 1.4. (cont.)
Spatial mechanism
Mechanism in which some points of some of its links de-scribe non-planar paths, or paths located in non-parallel planes.
Guidance mechanism
Mechanism that guides a link through a prescribed sequence of positions.
Four-bar mechanism
Mechanism with four binary links.
Structure (of a mechanism)
Number and kinds of elements in a mechanism (members andjoints) and the sequence of their contact.
Isomorphism Equality of structures in respect of the numbers of members and joints, and the sequence of their interconnections.
Equivalent mechanism
Mechanism whose kinematical properties are equivalent insome respects to those of another mechanism with a different structure.
Degree of freedom (mobility) of a kinematic chain or mechanism
Number of independent coordinates needed to define the con-figuration of a kinematic chain or mechanism.
Constraint Any condition that reduces the degree of freedom of a sys-tem.
Drive System of mutually connected devices for setting in motion one or several parts of a machine or a mechanism.
Actuator Device that creates direct physical action on the process,other mechanical devices or the surrounding environment toperform some useful function.
1.4 Terminology 17
Table 1.5. IFToMM terminology with respect to the robot kinematics
Term Definition Robot
Mechanical system under automatic control that performs operations such as handling and locomotion.
Manipulator Device for gripping and the controlled movement of objects.
Anthropomorphic robot
Robot with a manipulator containing rotary joints similar to those of a human arm.
End-effector Device attached to the distal end of a robot arm by which objects can be grasped or acted upon.
Gripper End-effector which grasps, holds and releases objects.
Module Self-contained assembly of robot elements which can be connected in various ways with other, not necessarily identi-cal, modules to form a robot.
Parallel manipulator
Manipulator that controls the motion of its end-effector by means of at least two kinematic chains joining the end-effector to the frame.
Walking machine Robot that performs locomotion functions similar to human beings or animals.
Pedipulator Articulated leg of a walking machine.
Proximal Close to the body (away from the end-effector) of a robot arm.
Distal Away from the base (towards the end-effector) of a robot.
Motion Changing position of a body relative to a frame of reference.
Absolute motion Motion with respect to a fixed frame of reference.
Relative motion Motion with respect to a moving frame of reference.
Inverse motion Motion of a frame of reference relative to a moving body.
Frame motion (transportation)
Motion of a moving frame of reference.
18 1 Introduction
Table 1.5. (cont.)
Displacement Change of position of a body with respect to a fixed frame of reference.
Path (trajectory) Line that a moving point describes in a given system of refer-ence.
Axis of rotation Straight line in a rotating rigid body whose points have zero displacement either in a finite time interval, or in an infini-tesimal time interval, with respect to a frame of reference.
Screw axis Straight line in rigid body whose points are displaced relative to a frame of reference, either in a finite or an infinitesimal time interval, coaxial with the line itself.
Translation Motion (or component of the motion) of a rigid body in which each straight line rigidly connected with the body re-mains parallel to its initial direction.
Rectilinear transla-tion
Translation in which the paths of points of a rigid body are straight lines.
Rotation Motion (or a component of the motion of) a rigid body in which all its points move on circular arcs centred on the same axis.
Angular displace-ment
Displacement of a rigid body in rotation.
Angle of rotation Angle turned through by any line rigidly connected to a rotat-ing body and perpendicular to the axis of rotation.
Precession Rotation of a rigid body about an axis fixed in space when combined with rotation about an axis fixed in the body, the two axes being concurrent.
Nutation Motion of a rigid body occurring simultaneously with a pre-cession, when the angle between the axes of rotation and pre-cession varies in time.
Orientation Movement or manipulation of a rigid body into a pre-determined attitude.
Pose Combination of position and attitude.
1.4 Terminology 19
Table 1.5. (cont.)
Reference point Point chosen for position reference in defining a pose.
Planar (plane) motion
Motion of a rigid body in which its points describe curves lo-cated in parallel planes.
Spatial motion Motion of a body in which at least one of its points describes a spatial curve.
Screw motion Motion consisting of a rotation combined with a translationthat is parallel to the rotation.
Spherical motion Spatial motion of a body in which all points of the bodymove on concentric spheres.
Limit position Configuration of a mechanism in which the position of a par-ticular link, such as the output, is in some significant respecta maximum or a minimum.
Dead point Configuration of a mechanism in which the input cannot be moved without assistance to the motion of another link.
Velocity Rate of displacement with respect to time.
Absolute velocity Velocity with respect to a fixed frame of reference.
Relative velocity Velocity with respect to a moving frame of reference.
Frame (transporta-tion) velocity
Absolute velocity of a particular point of a moving frame ofreference.
Angular velocity Rate of angular displacement with respect to time.
Instantaneous screw axis
Locus of the points in a rigid body in general spatial motion, whose linear velocity is parallel to the angular velocity vectorof the body at the given instant.
Working range Range of any variable for normal operation of a robot.
Working space (volume)
Totality of points that can be reached by the reference point of a robot arm.
20 1 Introduction
Table 1.5. (cont.)
Singular configura-tion
Special position of the robot links which implies a reduction of mobility of the end-effector.
Direct task Computation of the pose, motion and forces at the end-effector of a robot arm from given actuator forces, displace-ments, velocities and accelerations.
Inverse task Computation of actuator forces, displacements, velocities andaccelerations from given forces, pose and motion of the end-effector of a robot.
Redundant mobility (of a robot)
Amount by which the degree of freedom of a robot exceeds the number of independent variables that are necessary to de-fine the task to be performed.
Redundancy Existence of more than one means to accomplish a givenfunction.
Manoeuvrability Ability of a robot with redundant mobility to solve a task byusing various combinations of movements of its links.
Kinematics Branch of theoretical mechanics dealing with the geometry of motion, irrespective of the causes that produce the motion.
Kinematic analysis Analysis of the kinematic aspects of mechanisms.
The IFToMM terminology defines open/closed kinematic chains and
mechanisms, but it does not introduce the notions of simple (elementary) and complex kinematic chains and mechanisms. A closed kinematic chain is a kinematic chain in which each link is connected with at least two other links, and an open kinematic chain is a kinematic chain in which there is at least one link which carries only one kinematic pairing element (see Ta-bles 1.3 and 1.4). In a simple open kinematic chain (open-loop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one ternary element exists. We designate in each mechanism two extreme elements called reference element and final element. They are also called distal links. In an open kinematic chain, these elements are situated at the extremities of the chain. In a single-loop kinematic chain, the final element can be any element of the chain except the reference element. In a parallel mechanism, the two extreme elements are the mobile and the ref-erence platform. The two platforms are connected by at least two elemen-
1.4 Terminology 21
tary or complex kinematic chains called limbs. Each limb contains at least one joint. An elementary limb is composed of a simple open kinematic chain in which the final element is the mobile platform. A complex limb is composed of a complex kinematic chain in which the final element is also the mobile platform.
As we can see in Table 1.3, IFToMM terminology (Ionescu 2003) use the term kinematic pair to define the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. The word joint is used as a synonym for the kinematic pair and also to define the physical realization of a kinematic pair, including con-nection via intermediate mechanism elements. Both synonymous terms are used in this text.
Usually, in parallel robots, lower pairs are used: revolute R, prismatic P, helical H, cylindrical C, spherical S and planar pair E. The definitions of these kinematic pairs are presented in Table 1.3. The graphical representa-tions used in this book for the lower pairs are presented in Fig. 1.1(a)-(f). Universal joints and homokinetic joints are also currently used in the me-chanical structure of the parallel robots to transmit the rotation motion be-tween two shafts with intersecting axes. If the instantaneous velocities of the two shafts are always the same, the kinematic joint is homokinetic (from the Greek “homos” and “kinesis” meaning “same” and “move-ment”). We know that the universal joint (Cardan joint or Hooke’s joint) are heterokinetic joints. Various types of homokinetic joints are known to-day: Tracta, Weiss, Bendix, Dunlop, Rzeppa, Birfield, Glaenzer, Thomp-son, Triplan, Tripode, UF (undercut-free) ball joint, AC (angular contact) ball joint, VL plunge ball joint, DO (double offset) plunge ball joint, AAR (angular adjusted roller), helical flexure U-joints, etc. (Dudiţǎ et al. 2001a,
Fig. 1.1. Symbols used to represent the lower kinematic pairs and the kinematic joints: (a) revolute pair, (b) prismatic pair, (c) helical pair, (d) cylindrical pair, (e) spherical pair, (f) planar contact pair, (g) universal joint and (h) homokinetic joint
22 1 Introduction
Fig. 1.2. Various representations of a Gough-Stewart type parallel robot: physical implementation (a), CAD model (b), structural diagram (c) and its associated graph (d), A-limb (e) and its associated graph (f)
1.5 Structural synthesis 23
2001b). The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1(g)-(h).
A parallel robot can be illustrated by a physical implementation or by an abstract representation. The physical implementation is usually illustrated by robot photography and the abstract representation by CAD model, structural diagram and structural graph. Figure 1.2 gives an example of the various representations of a Gough-Stewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.2a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.2b) the links and the joints are represented as being as close as possible to the physical imple-mentation (Fig. 1.2a). In a structural diagram (Fig. 1.2c) they are repre-sented by simplified symbols, such as those introduced in Fig. 1.1, respect-ing the geometric relations defined by the relative positions of joint axes. A structural graph (Fig. 1.2d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the Gough-Stewart type parallel robot has six identical limbs denoted in Fig. 1.2c by A, B, C, D , E and F. The final link is the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F and the reference member is the fixed platform 1A≡1B≡1C≡1D≡1E≡1F≡0. Each limb is connected to both platforms by spherical pairs. A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The mechanism associated with the Gough-Stewart type parallel robot is a complex mechanism with a multi-loop as-sociated graph (Fig. 1.2d). The simple open kinematic chain associated with A-limb is denoted by A (1A≡0-2A-3A-4A≡4) – Fig. 1.2e and its associ-ated graph is tree-type (Fig. 1.2f).
1.5 Structural synthesis
Recent advances in research on parallel robots have contributed mainly to expanding their potential use to both terrestrial and space applications in-cluding areas such as high speed manipulation, material handling, motion platforms, machine tools, medical applications, planetary and underwater exploration. Therefore, the need for methodologies devoted to the system-atic design of highly performing parallel robots is continually increasing. Structural synthesis is directly related to the conceptual phase of robot de-sign, and represents one of the highly challenging subjects in recent robot-ics research. One of the most important activities in the invention and the
24 1 Introduction
design of parallel robots is to propose the most suitable solutions to in-crease the performance characteristics.
The challenging and difficult objective of structural synthesis is to find a method to set up the mechanical architecture achieving the required struc-tural parameters. The mechanical architecture is defined by number, type and relative position of joint axes in the parallel robot. The structural pa-rameters are mobility, connectivity, redundancy and degree of overcon-straint. They define the number of actuators, the degrees of freedom and the motion-type of the moving platform.
In general, parallel manipulators’ performances are highly dependent on their mechanical architecture, so that structural synthesis becomes the cen-tral problem in the conceptual design phase, but only a few works can be found in the literature on this topic. We note that this is the first book fo-cusing on the structural synthesis of the mechanical architecture of parallel robots. The topic of this book addresses the problem of structural, also called topological, synthesis of parallel robotic manipulators in a system-atic way. This is an “urgent need” put forward by robot manufacturers and scientists.
Torgny Brogårdh (2002) from ABB Automation Technology Products / Robotics, in his conference paper entitled “PKM Research - Important Is-sues, presented as seen from a Product Development Perspective at ABB Robotics” delivered in the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, said:
“This paper will address some of the most important PKM research is-sues as seen from a robot manufacturer’s point of view. In section 2 the importance of linking research activities to the application requirements is discussed and in section 3, which is the most important part of this paper, the urgent need for a systematic topology synthesis is put forward. The rest of the paper is structured according to a proposed development process for industrial parallel kinematic robots.”
In his keynote speech entitled “Still a long way to go on the road for parallel mechanisms” presented in ASME 2002 DETC Conference, Jean-Pierre Merlet from INRIA Sophia Antipolis, France, the author of the first books on parallel robots (Merlet 1990, 1997, 2006) has stated (Merlet 2002): “Synthesis of parallel robot is an open field (there is a very limited number of papers addressing this issue) and, in my opinion, one of the main issue for the development of parallel robots in practice”.
The importance of design synthesis defined by structural and dimen-sional synthesis is also reiterated by J-P. Merlet (2005): “ Synthesis of robots may be decomposed into two processes: structural synthesis (determine the general arrangement of the mechanical struc-
1.6 The objectives and originality of this book 25
ture such as the type and number of joints and the way they will be con-nected) and dimensional synthesis (determine the length of the links, the axis and location of the joints, the necessary maximal joint forces/torques). The performances that may be obtained for a robot are drastically de-pendent on both synthesis.”
In this paper J-P. Merlet has put the emphasis on dimensional synthesis. In this book, we look mainly at structural synthesis which we consider as an essential step in robot design and optimization.
1.6 The objectives and originality of this book
Early research on parallel manipulators (PMs) concentrated on the six de-grees of freedom Gough-Stewart-type mechanism introduced by Gough for a tire-testing device (Gough and Whitehall, 1962) and by Stewart for flight simulators (Stewart, 1965). In the last decade, parallel manipulators with fewer than six degrees of freedom have attracted researchers’ atten-tion. Lower mobility solutions are suitable for many tasks requiring less than six independent motions of the end-effector.
Parallel mechanisms have been the subject of study of much robotic re-search during the last two decades. Some important studies are now avail-able to support the design of parallel manipulators, such as the pioneering works of Hunt (1973, 1978, 1983) that showed various kinematic architec-tures of parallel robots, the comprehensive enumerations presented by Merlet (1990, 1997, 2000, 2006) and Tsai (1999, 2001), and the various methods for systematic synthesis of PMs presented in the latest editions of Advances in Robot Kinematics (Lenarčič J, Parenti-Castelli V 1996; Lenarčič and Husty 1998; Lenarčič J, Stanišić MM 2000; Lenarčič and Thomas 2002; Lenarčič and Galletti, 2004; Lenarčič and Roth, 2006).
The motion freedoms of the end-effector are usually coupled together due to the multi-loop kinematic structure of the parallel manipulator(PM). Hence, motion planning and control of the end-effector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, load-to-weight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncou-pled, fully-isotropic and maximally regular PMs can overcome these dis-advantages.
Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Mer-
26 1 Introduction
let 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to mini-mize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to in-crease dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kine-matic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electro-mechanical systems (Mukherjee and Murlidhar 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b).
For the first time in the literature, the author recently presented new formulae for calculating the degree of redundancy of parallel robots, along with the use of redundancy for motion decoupling of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs beyond the minimum necessary for actuating the manipu-lator.
Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robot manipulator is said to be isotropic if the condition number of its Jacobian matrix is equal to one. A robot ma-nipulator is said to be fully-isotropic if it is isotropic throughout the entire workspace. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space on the end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. Thus, the condition number of the Jacobian matrix is a useful performance indicator characterizing the distortion of a unit hy-persphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salisbury and Craig (1982) to design mechanical fingers and developed by Angeles (1997) as a kinetostatic performance in-dex of robotic mechanical systems. The isotropic design aims at ideal ki-nematic and dynamic performance of the manipulator (Zanganeh and An-geles 1997; Takeda and Funabashi 1998; Baron and Barnier 2001; Baron et al. 2002; Mohammadi Daniali and Zsombor-Murray 1999; Badescu et al. 2002; Fattah and Ghasemi 2002; Krut et al. 2002; Caro et al. 2003; Fassi et al. 2003, 2005; Tsai and Hunag 2003; Carricato and Parenti-Castelli 2002; Carricato 2005; Liu et al. 2005; Pashkevich et al. 2005).
The author has proposed a systematization of PMs based on the form of the Jacobian matrix J. We distinguish five types of PMs (Gogu, 2004a; Gogu, 2007a): (i) maximally regular PMs, if the Jacobian is an identity
1.6 The objectives and originality of this book 27
matrix throughout the entire workspace, (ii) fully-isotropic PMs, if the Jacobian is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space.
The first solution for a fully-isotropic T3-type translational parallel robot was developed at the same time and independently by Carricato and Par-enti-Castelli at University of Genoa, Kim and Tsai at University of Cali-fornia, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works (Carricato and Parenti-Castelli 2002; Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2002b). Each of the last three groups has built a prototype of this robot in their research laboratories and called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3-T3 (Gogu 2004b). The first practical implementation of this robot was the CPM developed at the University of California.
The systematic synthesis of spatial mechanisms used in parallel robots is mainly founded on four approaches: group theory, screw theory, velocity-loop equations and theory of linear transformations (Table 1.6).
The method proposed in this book is founded on the theory of linear transformations and evolutionary morphology (Gogu 2005a) and allows us to obtain the structural solutions of decoupled, uncoupled, fully-isotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and over-constraint of PMs proposed recently by the author, Gogu (2005a, 2005b, 2005c, 2005d) are integrated into the synthesis approach developed in this book.
Various solutions of TaRb-type PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel ro-bots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we are aware, this is the first book of robotics presenting solutions of uncou-pled, fully-isotropic and maximally regular PMs.
28 1 Introduction
Table 1.6. Approaches used in the systematic synthesis of spatial mechanisms and parallel robots
Approach References Group theory
Hervé 1978, 1995, 1999, 2004; Hervé and Sparacino 1991; Karouia and Hervé 2000, 2002, 2004; Angeles 2004; Li et al. 2004; Huynh and Hervé 2005; Lee and Hervé 2006; Rico et al. 2006.
Screw theory Tsai 1999; Fang and Tsai 2002; Frisoli et al. 2000; Kong and Gosselin 2001, 2004a, 2004b, 2004c, 2004d, 2006; Huang and Li 2002, 2003; Carricato 2005
Velocity-loop equations Di Gregorio and Parenti-Castelli 1998; Di Gregorio 2002, Carricato and Parenti-Castelli 2002, 2003
Theory of linear transformations
Gogu 2002, 2003, 2004a, 2004b, 2004c, 2004d, 2005e, 2005f, 2005g, 2005h
Non-redundant/redundant, overconstrained/isostatic solutions of uncou-
pled and fully-isotropic PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. These solutions are derived from families of PMs with decoupled motions. A serial kine-matic chain is associated with each elementary limb, and at least one closed loop is integrated in each complex limb.
The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the au-thor in the last four years in the framework of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the French Na-tional Council of Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature:
a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of over-
1.6 The objectives and originality of this book 29
constraints of parallel robotic manipulators that overcome the drawbacks of the formulae founded on the Chebychev-Grübler-Kutzbach (CGK) mo-bility formula (Gogu 2005b-e),
b) a new approach to systematic innovation in engineering design called evolutionary morphology (Gogu 2005a),
c) solutions to follow fully-isotropic parallel robots: parallel wrists with two (Gogu 2005f) and three (Gogu 2007c) degrees of mobility, planar par-allel robots (Gogu 2004c), parallel robots of T1R2-type (Gogu 2005i), par-allel robots with Schönflies motions (Gogu 2004a, 2005g, 2006c, 2007a), parallel robots of T3R2 type (Gogu 2006a,b,d), hexapods with six degrees of freedom (Gogu 2006e).
The various solutions proposed by the author belong to a modular family called IsoglideN-TxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots (Gogu 2007b) can have any combina-tion of n independent translations (T) and rotations (R).The Isogliden-TaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics (IFMA) in Clermont-Ferrand.
The two parts of this work gather the author’s contributions, and de-velop them in a general and organic treatment of structural synthesis of parallel robotic manipulators, including new solutions presented for the first time in the literature. The main objectives of this book are as follows:
• to present an original method of defining and calculating the struc-tural parameters of parallel robots (mobility, connectivity, redun-dancy, degree of overconstraint) founded on the theory of linear transformations,
• to present an original method for structural synthesis of parallel robots founded on the theory of linear transformations and evolu-tionary morphology,
• to propose innovative solutions of parallel manipulators with 2 to 6 degrees of freedom and various degrees of motion coupling, with the emphasis on decoupled, uncoupled, fully-isotropic and maxi-mally regular solutions.
The innovative solutions presented in this book are useful in the design of new parallel robots for many applications: haptic devices, machine-tools called parallel-kinematics-machines (PKM), robot-assisted surgery, sur-veillance, telescope and pointing devices, motion simulation, testing ma-chines, etc. The book will be useful for students, engineers and researchers working in associated fields of engineering.
Many solutions of parallel robots obtained through this systematic ap-proach of structural synthesis are presented here for the first time in the lit-
30 1 Introduction
erature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.
2 Structural parameters
This section introduces the main structural parameters used in the struc-tural synthesis of parallel robots: connectivity, mobility, redundancy, de-gree of overconstraint. We recall that IFToMM definitions of mobility, connectivity, redundancy and constraint are presented in Tables 1.3-1.5. In this chapter, we present an original approach based on the theory of lin-ear transformations to define and to demonstrate new formulae for the cal-culation of these parameters along with the state of the art before the first publication of this new approach by the author of this work (Gogu 2005b-e).
We will show that the classical Chebychev-Grübler-Kutzbach formula for global mobility calculation of multi-loop mechanisms does not fit many parallel manipulators. We know that a mobility formula is an ex-plicit relationship between mobility and the structural parameters of the mechanism. Usually, these structural parameters are easily determined by inspection without the need to develop the set of constraint equations used in instantaneous mobility calculation. We introduce a mathematical framework for defining and calculating the global mobility of parallel ro-bots. With this formal structure, combined with ideas from the theory of linear transformations, we define a new mobility formula for parallel ma-nipulators with simple or complex limbs. Limb connectivity and mobile platform connectivity are the key parameters of the new mobility formula. These parameters are used also to calculate the degree of redundancy and the degree of overconstraint of a parallel manipulator. They are also useful to determine the necessary conditions to obtain a non-overconstrained so-lution.
As far as we are aware, this is the first book to present the general rela-tions between mobility, redundancy and overconstraint of parallel manipu-lators and the connectivity of the mobile platform. These relations are demonstrated via the theory of linear transformations and are used mainly in the structural analysis and synthesis presented in the following chapters. Examples of practical applications of the new formulae in structural analy-sis of parallel robots and other single and multi-loop mechanisms are pre-sented in chapter 3.
32 2 Structural parameters
2.1 Critical review of mobility calculation
Mobility is the main structural parameter of a mechanism and also one of the most fundamental concepts in the kinematic and the dynamic model-ling of mechanisms. IFToMM terminology defines the mobility or the de-gree of freedom as the number of independent coordinates required to de-fine the configuration of a kinematic chain or mechanism (Ionescu 2003). Mobility M is used to verify the existence of a mechanism (M>0), to indi-cate the number of independent parameters in robot modelling and to de-termine the number of inputs needed to drive the mechanism.
Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century. During the twentieth century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulae and approaches were derived and presented in the literature. Contributions have continued to emerge in the last years. Mobility calculation still remains a central subject in the theory of mechanisms.
In the calculation of a mechanism’s mobility, the following key control-ling parameters are used: the number of joints, p; the number of mecha-nism elements n (n=m-1; m denotes the total number of elements, includ-ing the fixed base); the mobility f; the degree of constraint, c ,of the joint; the motion parameter,b; the constraint parameter d=6-b of the mechanism; and the number of independent closed loops of the mechanism, q.
The various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories:
a) approaches for mobility calculation based on setting up the kinematic constraint equations and calculating their rank for a given position of the mechanism with specific joint locations,
b) formulae for a quick calculation of mobility without the need to de-velop the set of constraint equations.
The approaches used for mobility calculation based on setting up the ki-nematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. There is no way to derive informa-tion about mechanism mobility without performing position, velocity or static analysis by using analytical tools (screw system theory, linear alge-bra, affine geometry, Lie algebra, etc). For this reason, the real and practi-cal value of these approaches is very limited in spite of their valuable theo-retical foundations. The rank of the constraint equations is calculated in a
2.1 Critical review of mobility calculation 33
given position of the mechanism with specific joint locations. The mobility calculated in relation to a given configuration of the mechanism is an in-stantaneous mobility which can be different from the general mobility (global mobility or gross mobility). Global mobility has a single value for a given mechanism; it is a global parameter characterizing the mechanism in all its configurations except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration in-cluding singular ones. In a singular configuration the instantaneous mobil-ity could be different from the global mobility.
The formula for a quick calculation of mobility is an explicit relation-ship between structural parameters of the mechanism: the number of links and joints, the motion/constraint parameters of joints and of the mecha-nism. Usually, these structural parameters are easily determined by inspec-tion without any need to develop the set of kinematic constraint equations.
We note that the classical formulae for a quick calculation of mobility do not fit many classical mechanisms such as the mechanisms proposed by De Roberval 1670; Sarrus 1853; Delassus 1900, 1902, 1922; Bennett 1902; Bricard 1927; Myard 1931a,b; Goldberg 1943; Altman 1952; Baker 1978; Baker et al. 1982, Waldron 1979; Wohlhart 1987, 1991, 1995; etc., or for many recent parallel robots. Special geometric conditions play a sig-nificant role in the determination of the mobility of these mechanisms. When these mechanisms were limited to special examples, considered as “curiosities”, they were called by various names: paradoxical mechanisms (Bricard, 1927), paradoxical chains (Hervé 1999, 1978a,b; Norton 1999), exceptions (Myszka 2001), special cases (Mabie 1987); linkage with a paradox between “practical degrees of freedom” and “computed degrees of freedom” (Eckhardt 1998), overconstrained yet mobile linkage (Phillips 1984, 1990; Waldron and Kinzel 1999), linkages with anomalous mobility (Waldron and Kinzel 1999).
The formulae known in the literature for the determination of mobility are not applicable to many types of recent parallel robots, see for example the robots Delta (Clavel 1988, 1990), Star (Hervé and Sparacino 1992, 1993), H4 (Pierrot and Company 1999), Orthoglide (Wenger and Chablat 2000), CPM (Kim and Tsai 2002), Tripteron (Gosselin et al. 2004) and other parallel manipulators presented in the literature (Kong and Gosselin 2002, Carricato and Parenti-Castelli 2002) or recently proposed by the au-thor belonging to Isoglide family (Gogu 2002b, 2004a-d, 2005f-j, 2006a-e, 2007a-c). We cannot continue to consider the mechanisms that do not fit the various methods or equations for determining mobility as having struc-tural flaws. We have to review the formulae and consider the flaw to be that of the formula rather than of the mechanism. In this chapter we will
34 2 Structural parameters
establish limits of these formulae, and propose a new formula for quick calculation of the mobility of parallel robots.
Many formulae/approaches proposed in the last 150 years for the calcu-lation of the mechanism mobility can be reduced to the same original for-mula. Previous works on mobility calculation (Wittenbauer 1923; Feder-hofer 1932; Artobolevskii 1953; Boden 1962; Bogolyubov 1964; Mruthyunjaya 2003) have mentioned various formulae but they have not analysed their genesis and similarities. In this section, we focus on a brief presentation and a critical analysis of these formulae/approaches to clearly identify the different contributions in this very important subject of the the-ory of mechanisms. Thirty five contributions are reviewed. To ensure unity of presentation, the original notations of some of the authors were not maintained; this also made it easier to distinguish between the different formulae. These contributions had been published before the first publica-tion, in 2005, of the new approach proposed by the author (Gogu 2005b-e).
We apply each formula to determine the degree of mobility of a recent parallel manipulator with simple limbs, the parallel Cartesian robotic ma-nipulator CPM (Kim and Tsai 2002; Kong and Gosselin 2002, Carricato and Parenti-Castelli 2002, Gogu 2002b) presented in Fig. 2.1, which only contains revolute (R) and prismatic (P) joints. A serial kinematic chain is associated with each simple limb linking the mobile platform to the fixed base. The mechanism has n=10 kinematic elements (links), p0=3 prismatic joints adjacent to the fixed element, pn=9 revolute joints non-adjacent to the fixed element and q=2 independent closed loops. Each loop contains the same number and the same type of joints. The three limbs are identical
Fig. 2.1. Parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associ-ated graph
2.1 Critical review of mobility calculation 35
from a structural point of view. In each limb the direction of the prismatic joint and the axes of the revolute joints are parallel (P||R||R||R). Kim and Tsai (2002) have mentioned that this mechanism has 3 degrees of freedom, in spite of the fact that the general degrees-of-freedom equation presented by Tsai (2000) predicts that the mechanism has zero degrees of freedom.
We analyse the results obtained by using the different formulae to de-termine the degree of mobility of the CPM robotic manipulator, and we draw conclusions on the limits to the various formulae.
2.1.1 Chebychev’s contribution
In the second half of the nineteenth century, the mathematician P. A. Che-bychev published an article (concerning a 4-bar linkage) in two parts in the proceedings of the Russian Royal Academy of Science of Saint-Petersburg. Chebychev was the first scientist who proposed a mathemati-cal formalisation for the calculation of mechanism mobility. The first part of the article (Chebychev 1854) was published under the impulse of Watt’s invention, known at the end of the eighteenth century. In this article, Che-bychev applied his method of polynomial function approximation to find the lengths of the elements of the 4-bar mechanism so that it could de-scribe a given couple curve with the least error. In the second part of the article, Chebychev (1869) proposed the first formula for the calculation of the number of independent variables in a mechanism. This contribution was quickly integrated into the first treatises on the theory of mechanisms published at the end of the nineteenth century (Dwelshauvres-Dery 1876, Laboulaye 1878).
Chebychev expressed his formula in the form:
3n -2(p0+pn)=1 (2.1)
where 3n represents the number of variables required to describe the posi-tion and the orientation of the n kinematic (mobile) bars in the plane and 2(p0+pn) is the number of constraint equations imposed by the p=(p0+pn) revolute joints of the mechanism that can be adjacent (p0) or non-adjacent (pn) to the fixed base. It is known that each revolute joint introduces two constraint equations in a planar mechanism. Chebychev applied this for-mula to simple planar mechanisms with p0=2 and to complex planar mechanisms (p0=3) having only revolute joints and one degree of mobility. He did not extend the use of Eq. (2.1) to other types of mechanisms. This formula could be extended in the form
M= 3n -2(p0+pn) (2.2)
36 2 Structural parameters
to any planar mechanism with one degree of freedom joints (helical, pris-matic and revolute joints).
The well-known λ 4-bar mechanisms were obtained by Chebychev to describe couple curves containing a straight line segment by using Eq. (2.1).
Supposing that we try to extrapolate Eq. (2.2), we obtain M=6 for the mechanism presented in Fig. 2.1 (with n=10, p0=3 and pn=9). This result is obviously erroneous, Chebychev’s formula (2.2) being limited to the planar mechanisms (mechanisms in which the axes of all the revolute joints are parallel and perpendicular to the plane containing the directions of the prismatic pairs).
2.1.2 Sylvester’s contribution
J. J. Sylvester (1874) presented a modified form of Eq. (2.1) as a structural condition for one degree of freedom pin-connected planar mechanisms:
3m-2p-4=0. (2.3)
In Eq. (2.3), m is the total number of links of a mechanism including the fixed element. Sylvester stated Eq. (2.3) in the following words: “In order for a combination of links to fulfil this so to say fatalistic condition of con-strained motion and to entitle it to the name of a linkage in the speaker’s sense, a numerical relation must be satisfied between the number of links and the number of joints, viz., three times the number of links must be four greater than twice the number of joints. In applying this rule it must be un-derstood that, if three links are jointed together, the junction counts for two joints; if four are jointed together, for three joints; and so on.” We can see that Eq. (2.3) can be obtained by simply replacing n by m-1 in the Cheby-chev’s formula and, in the general case, joints connecting a links count as a-1 joints.
2.1.3 Grübler’s contribution
M. Grübler (1883, 1885) presented a structural condition for one degree of freedom planar mechanisms identical to Eq. (2.3). Later, Grübler proposed a structural condition for one degree of freedom spatial complex mecha-nisms with helical joints (Grübler 1916, 1917):
5h-6m+7=0 (2.4)
were h is the total number of helical joints.
2.1 Critical review of mobility calculation 37
2.1.4 Somov’s contribution
P. O. Somov (1887) proposed the following structural condition for one degree of freedom mechanisms:
m-q(b-1)=2. (2.5)
Somov introduced for the first time the motion parameter (b), called mobility number, for planar (b=3) and for spatial mechanisms (b=6).
2.1.5 Hochman’s contribution
Hochman made an important contribution to the structural analysis of mechanisms in his book on the kinematics of machinery (Hochman 1890). A detailed presentation of Hochman’s contribution can be found in (Bo-golyubov 1964; Dudiţă and Diaconescu 1987).
He proposed a relation between the total number of links m, the total number of joints p and the number of independent closed loops q in a complex mechanism (m+q-p=1) .This is Euler’s formula in the theory of graphs.
He also proposed various structural conditions for the existence of one degree of freedom mechanisms:
b(m-1)-C=1 (2.6)
F- qb=1 (2.7)
b (p-q)-C=1 (2.8)
with b 1
bi
i 1C iC
−
=
=∑ (2.9)
b 1b
ii 1
F iF−
=
=∑ (2.10)
F+C=pb (2.11)
where C is the total number of constraints imposed by the joints and F – the total number of degrees of freedom of the joints. b
iC represents the number of joints having the degree of constraint b
ic i= , and biF is the
number of pairs having the degree of connectivity bif i= . Hochman de-
38 2 Structural parameters
fined the motion parameter b as the mechanism category given by the number of simple existing motions between any two links (b<6). He marked the difference between degree of constraint of a joint in the motion space with dimension b, denoted by b bc b f= − , and the degree of con-straint of a joint in the motion space with dimension b=6 , also called joint class and denoted by c=6-f.
We can see that Eq. (2.6) extends Eq. (2.1) proposed by Chebychev. For the general case of a mechanism with mobility M, Eqs. (2.6-2.8) become:
M=b(m-1)-C (2.12)
M=F- qb (2.13)
M=b(p-q)-C. (2.14)
Hochman considered that a mechanism with M=1 has determined mo-tions. If M<1 the mechanism is a fixed structure without relative motions and if M>1 the motions of the mechanisms are not determined. We can see that Hochman limited the mechanisms with determined motions to one de-gree of freedom mechanisms. He considered that Eqs. (2.12-2.14) are ap-plicable to any kind of mechanism having various types of joints and the same b for each independent loop (b=3 for planar and spherical mecha-nisms, b=6 for spatial mechanisms).
By applying Eqs. (2.12)-( 2.14) to the mobility calculation of the mechanism presented in Fig. 2.1, with m=11, q=2, p=12, we obtain: M=6 (if we presume b=3 and consequently C=12*2=24), M=4 (if b=4 and C=12*3=36), M=2 (if b=5 and C=12*4=48) and M=0 (if b=6 and C=12*5=60).
We can see that the results obtained for this mechanism are erroneous for all values presumed for b (b=3,…,6). It is obvious that the mechanism presented in Fig. 2.1 has the same b for all the closed loops, but the results obtained by using the formulae proposed by Hochman are erroneous. So, we can say that Eqs. (2.12-2.14) do not work for all mechanisms even if the independent loops have the same motion coefficient.
2.1.6 Somov-Malytsheff’s formula
Somov-Malytsheff formula (Somov, 1887; Malytsheff 1923) 5
ii 1
M=6n- iC=∑
(2.15)
2.1 Critical review of mobility calculation 39
represents a particular case of Eqs. (2.12) and (2.9) proposed by Hochman (with the notations n=m-1 and 6
iC =C). Eq. (2.15) is applicable only to spatial mechanisms with b=6.
2.1.7 Koenigs’ formula
Gabriel Koenigs (1905) published an introduction to a new theory of mechanisms in which he presented an equation for the calculation of the degrees of freedom of a mechanism
M=6(m-1)-C. (2.16)
where 6(m-1) is the number of parameters necessary to define the relative positions of the n=m-1 kinematic links of the mechanism and C is the number of independent constraint equations given by the joints. We can see that Eq. (2.16) represents the particular case of Eq. (2.12), proposed by Hochman, in which b=6.
2.1.8 Kutzbach’s mobility equation
The mobility equation proposed by Kutzbach (1929)
M=(6-d)(m-1)-p
ii 1
( 6 d f )=
− −∑ (2.17)
also represents the particular case of Eq. (2.12) for a spatial mechanism with b=6-d, 6
i if f= and
p
ii 1
C (6 d f )=
= − −∑ . (2.18)
In the literature, Eq. (2.17) is considered to be applicable to all mecha-nisms having the same d for each independent loop.
By applying Eqs. (2.17) to the mobility calculation of the mechanism presented in Fig. 2.1 (for which m=11, p=12 and fei=1, i=1,2,…,12), we obtain : M=6 (if we assume d=3), M=4 (if d=2), M=2 (if d=1) and M=0 (if d=0). This shows that the results obtained for the mechanism analysed are also erroneous for all values presumed for d (d=0,…,3). So, we can say that Eq. (2.17) does not work for all mechanisms even if the independent loops have the same motion coefficient.
40 2 Structural parameters
2.1.9 Dobrovolski’s mobility equation
Dobrovolski named the constraint parameter d as the mechanism family and indicated the following mobility equation (Dobrovolski 1949, 1951):
5
d ii d 1
M =(6-d)n- ( i d )C= +
−∑ (2.19)
where 6 di iC C −= represents the number of joints having the degree of con-
straint 6 dic i d− = − . We can see that Eqs. (2.19) and (2.12) are similar by
taking into consideration that n=m-1 and 5
ii d 1
C ( i d )C= +
= −∑ (2.20)
Dobrovolski considered d=6-b=0,…,4, but he did not indicate any method for setting up d.
Eq. (2.19) written in the form 6 d 1
d ii 1
M (6 d )( m 1) (6 d i )F− −
=
= − − − − −∑ (2.21)
is also known as the Artobolevski-Dobrovolski mobility equation (Bagci 1971). In Eq. (2.21), Fi= 6 d
iF − is the number of pairs having the degree of connectivity 6 d
if i− = . Eqs. (2.19) and (2.21) are also considered in the lit-erature to be applicable to any mechanism having the same d for each in-dependent loop.
By applying Eqs. (2.19) and (2.21) to the mobility calculation of the mechanism presented in Fig. 2.1 we obtain the same erroneous results that were obtained by using Eqs. (2.17) and (2.12): M=6 (if we presume d=3), M=4 (if d=2), M=2 (if d=1) and M=0 (if d=0). We can see that Eqs. (2.19) and (2.21) do not work for all mechanisms, even if the independent loops have the same motion coefficient, as with Eqs. (2.12) and (2.17) from which they can be derived.
2.1.10 Contribution of Y.F. Moroskine
To calculate the mobility of a planar or spatial mechanism, Y. F. Mo-roskine (1954, 1958) proposed the formulae:
M=F-r (2.22)
2.1 Critical review of mobility calculation 41
5
1i
iM iC r
=
= −∑ (2.23)
where F is the total number of kinematic parameters of the mechanism, r - the rank of the linear homogeneous set of equations defining the kinematic constraints, Ci - the number of joints of class i (joints with the degree of freedom 6-i). The two formulae proposed by Moroskine are valid without exception, but these formulae allow a quick calculation of mobility only after developing the kinematic equations and finding their rank. These cal-culations will be presented in Section 2.2.
2.1.11 Contribution of R. Voinea and M. Atanasiu
R. Voinea and M. Atanasiu (1960) proposed the following formulae for mobility calculation of single-loop closed mechanisms:
1 2 eM F r F ( r +r -w )= − = − (2.24)
where F is the total number of degrees of freedom in the joints given by the total number of kinematic pairs with a connectivity of one (revolute, prismatic and helical joints) equivalent to the joints of the mechanism, r is the rank of the screw system equivalent to the joints of the single-loop mechanism, r1 and r2 are the ranks of the screw systems equivalent to the two open sub-chains connecting a link e to the fixed base. In the single-loop mechanism the link e has the connectivity we with respect to the fixed base.
To calculate the mobility of a complex mechanism, R. Voinea and M. Atanasiu (1960) proposed the formula:
1
q
j pj
M F r p=
= − +∑ (2.25)
where rj is the rank of the screw system of joints with connectivity one equivalent to the joints of the jth independent closed loop of the mecha-nism, and pp is the total number of passive joints in the mechanism. To calculate the rank rj, R. Voinea and M. Atanasiu used the theory of the in-stantaneous screw axis proposed by Ball (1900). An important contribution was provided by R. Voinea and M. Atanasiu by identifying the geometrical configurations of screw systems with 1≤r≤6 and by proposing an analytical method for computing the motion parameter bj=rj (Voinea and Atanasiu 1959, 1960).
42 2 Structural parameters
By applying Eq. (2.25) to the mobility calculation of the mechanism presented in Fig. 2.1 (for which F=p=12, r1=r2=5 and pp=0) we obtain M=2, which is an erroneous result. We can see that all the closed loops of the mechanism are identical and have the same rank (rj=5).
Eq. (2.25) is the first mobility formula applicable to mechanisms having independent closed loops with different values for the motion coefficient bj=rj, but this formula does not work for all complex mechanisms, even if rj is the same for all independent loops.
2.1.12 Kolchin’s mobility equation
Kolchin (1960) modified the Artobolevski-Dobrovolski mobility equation (2.21) and expressed it in the form:
6 d 1
ii 1
M 6( m 1) (6 i )F dq− −
=
= − − − +∑ (2.26)
Kolchin’s contribution was also summarized in (Erdman 1993). By applying Eq. (2.26) to the mobility calculation of the mechanism
presented in Fig. 2.1, we obtain the same erroneous results that were ob-tained by using Eqs. (2.12), (2.17), (2.19) and (2.21).
2.1.13 Rössner’s contribution
The mobility equation proposed by W. Rössner (1961)
M=p
ii 1
f=∑ -6(p-m)-6
(2.27)
also represents a particular case of Eq. (2.13) for the spatial mechanism with b=6 (by taking into consideration Euler’s formula q=p-m+1).
Rössner also mentioned the existence of so called “half open” kinematic chain containing at least one link which must have at least four joints. The “half open” kinematic chain is also known in the literature as a kinematic chain with fractionated freedom.
2.1.14 Boden’s mobility equation
H. Boden (1962) extended Eq. (2.27) proposed by Rössner to make it ap-plicable to spatial mechanisms (b=6) containing planar loops (b=3):
2.1 Critical review of mobility calculation 43
M=p
ii 1
f=∑ -6(p-m)-6+D
(2.28)
where
[ ]z
Ek Ekk 1
D 3( p m ) 3=
= − +∑ . (2.29)
z is the number of distinct directions of the joint axes belonging to planar loops, pEk and mEk are the number of joints and the number of links in the planar loops having the axes of the revolute joints in kth direction. D takes into consideration the planar loops existing in the spatial mechanism.
2.1.15 Manafu’s formula
In order to take into consideration the fact that the degree of mobility of a complex kinematic chain depends on the choice of the independent closed loops, N. Manolescu and V. Manafu (1963) proposed the following mobil-ity equation, presented as Manafu’s mobility equation (Manolescu 1968):
1
q
i ci
M max( M M )=
= −∑ (2.30)
where Mi is the mobility of the simple closed loop i, and Mc is the mobility of the “connecting mechanism” (the mechanism formed with the common joints between different closed loops). To calculate the degree of mobility of an simple closed loop, Manolescu and Manafu used Dobrovolski’s mo-bility equation (2.19). The mobility of the connecting mechanism is given by the sum of the degrees of freedom of the common joints.
Each simple closed loop of the mechanism presented in Fig. 2.1 has d=1 and Mi=3 (i=1,2). We can see that any two simple closed loops have four common joints (Mc=4). So, by using Eq. (2.30) we obtain the erroneous value of M=2 for the mobility of the mechanism presented in Fig. 2.1.
2.1.16 Ozol’s formula
Ozol’s formula (Ozol 1963) 5
cc 1
M= (6 c )C 6q=
− −∑ (2.31)
44 2 Structural parameters
represents a particular case of Eqs. (2.13) and (2.10) proposed by Hochman for b=6 and the notation 6
iF =Fi and 5cc=1
(6 c )C−∑ = 5ii 1
iF=∑ ,
where cC represents the number of joints of class c and iF the number of pairs having the degree of connectivity if i= .
2.1.17 Contribution of K. J. Waldron
To calculate the mobility of a closed loop, Waldron (1966) proposed the formula
M=F b− (2.32)
which also represents a particular case of Eqs. (2.13) and (2.10) proposed by Hochman (for q=1). Waldron called b the order of the equivalent screw system of the closed loop and he defined it as the order of the screw sys-tem of a series chain containing the same joints as the loop in the same geometrical relationship. To calculate b, Waldron used the theory of the instantaneous screw axis proposed by Ball (1900), and developed later by Voinea and Atanasiu (1959, 1960), Phillips and Hunt (1964), Hunt and Phillips (1965).
Waldron delineates the series and parallel laws for screw system theory and shows how to determine both the relative freedom between particular links and the mobility of a certain class of multi-loop linkages in which some members (links) are shared by two or more loops. He shows that such linkages may be handled by considering one loop to be a complex joint in the other. In such cases, the values of b applicable to each closed loop may vary with the order in which the closed loops are considered. He emphasized the importance of relative freedom between two particular members of a linkage in mobility calculation. An example is given in (Waldron, 1966) for the mechanism known as the De Roberval Scale.
2.1.18 Contribution of N. Manolescu
N. Manolescu and V. Manafu (1963) also proposed the mobility equation q5
i ji 1 j 1
M=6n- iC max d= =
+∑ ∑ (2.33)
that extends formula (2.15), proposed by Somov-Malytsheff, by adding a supplementary term. In this term, dj is the family of the simple closed loop j. By applying Eq. (2.33) to the mobility calculation of the mechanism pre-
2.1 Critical review of mobility calculation 45
sented in Fig. 2.1, with n=10, q=2, 5ii 1
C iC=
= ∑ =12*5=60, we obtain: M=0 (if we presume d1= d2=0), M=2 (if d1= d2=1), M=4 (if d1= d2=2), M=6 (if d1= d2=3), M=8 (if d1= d2=4) and M=10 (if d1= d2=5).
The mechanism presented in Fig. 2.1 has the same dj for all the closed loops, but the results obtained by using the formula proposed by Manolescu are erroneous even if the independent loops have the same mo-tion coefficient.
Manolescu (1968) proposed the mobility formula 5
cc d 1
M= (6 c )C (6 d )q= +
− − −∑ (2.34)
that extends formula (2.31), proposed by Ozol. By taking into considera-tion the notation 6-d=b, 6
iF =Fi and 5 5c ic=1 i 1
(6 c )C iF=
− =∑ ∑ , where
cC represents the number of joints of class c and iF the number of pairs having the degree of connectivity if i= . We can note that Manolescu’s formula (2.34) is identical to Eq. (2.13) proposed by Hochman.
Manolescu (1964) described three kinds of kinematic chains having to-tal, partial and fractional mobility. A multi-loop kinematic chain has a total mobility M>0 if each of its loops have a mobility Mi≥M and a partial mo-bility if it has at least one loop of mobility Mi<M. The multi-loop kine-matic chain of mobility M is said to have a fractional mobility if it contains a so-called “separation link” which when cut into two split the chain into two separate sub-chains of mobilities M1 and M2 such that M=M1+M2.
2.1.19 Contribution of C. Bagci
Bagci (1971) proposed the following mobility equation to calculate the de-grees of freedom of motion in a mechanism of m links and q loops:
M=Mi+Mr=M0+D+Mc-Mp (2.35)
in which Mi is the number of inputs required to drive the mechanism, 1
1
mr j rj
M f−
==∑ is the total number of redundant freedoms in the mecha-
nism, frj is the number of redundant freedoms of the link j. r
0 ii 1M b( m 1) ( b i )F
== − − −∑ (2.36)
where r=b-dmin-1 is the maximum number of degrees of freedom that a joint can have in the motion space of the mechanism and dmin is the small-
46 2 Structural parameters
est number of general constraints in the loops of the mechanism. By taking into consideration the notation b=6-d, Eq. (2.36) represents Artobolevski-Dobrovolski’s mobility equation (2.21) when dmin=d.
qjj 1
D d=
=∑ is the total number of general constraints in the mechanism.
This term is similar to the supplementary term introduced by Manolescu and Manafu in Eq. (2.33) to extend the formula proposed by Somov-Malytsheff.
Ngc cjj 1
M ( q 1)=
= −∑ is the total number of overclosing constraints in the
mechanism, gN - the number of groups of loops introducing overclosing constraints, cjq - the number of loops in the group j introducing overclosing constraints.
pp pjj 1
M f=
=∑ is the total number of passive mobilities in the joints of the
mechanism, pjf - the number of passive mobilities in joint j. The mechanism presented in Fig. 2.1 have Mc=Mp=0. By applying Eq.
(2.35) to the mobility calculation of this mechanism, we obtain the same erroneous results that were obtained by using Eq. (2.33).
2.1.20 Contribution of P. Antonescu
P. Antonescu proposed two mobility formulae applicable to mechanisms having independent closed loops with different values for the motion coef-ficient bi. The first formula (Antonescu 1973)
5
da a a ii 1
M =(6-d )(m-1)- ( i d )C=
−∑ (2.37)
represents a generalization of Eq. (2.18) proposed by Dobrovolski, in which m-1=n and da is the apparent family calculated as the average of the constraint parameters di (i=1,2,..q) of the independent loops
q
a ii 1
1d dq =
= ∑ . (2.38)
By applying Eq. (2.37) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were ob-tained by using Eq. (2.33).
The second formula proposed by Antonescu (1999)
2.1 Critical review of mobility calculation 47
5 6
i ji 1 j 2
M= iF jR= =
−∑ ∑ (2.39)
represents a particular case of Eq. (2.25) proposed by Voinea and Atanasiu where pp=0. We see that 5
ii=1F= iF∑ and q 6
j jj=1 j 2r jR
==∑ ∑ , where Fi
represents the number of joints having the degree of connectivity if i= and Rj is the number of closed loops having the motion coefficient bj=j. By applying Eq. (2.39) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were obtained by us-ing Eq. (2.25) from which Eq. (2.39) can be derived.
2.1.21 Contribution of F. Freudenstein and R. Alizade
F. Freudenstein and R. Alizade (1975) proposed a mobility equation that “applies to mechanisms without exception” as the authors stated in their paper :
qE
i ii 1 k 1
M e b= =
= −∑ ∑ (2.40)
in which ei is ith independent, scalar, displacement variable of the mecha-nism (associated with the relative displacements of the joint), and E is the total number of independent, scalar, displacement variables. The authors called bi the “mobility number”; it is given by the number of independent, scalar, differential loop-closure equations for the ith independent loop. In the general case (absence of metric restrictions) this number is equal to the degree of freedom of the space within which the mechanism operates. If the value of bi is not identical in each independent loop, the mechanism is said to have mixed mobility.
Three special cases of Eq. (2.40) are also presented (Freudenstein and Alizade 1975):
a) the displacement variables are in 1:1 correspondence with the degrees of freedom of the relative motion in joints
p q
i ki 1 k 1
M f b= =
= −∑ ∑ (2.41)
b) the number of independent scalar loop-closure equations is identical in each independent loop
48 2 Structural parameters
1
E
ii
M e qb=
= −∑ , (2.42)
c) general kinematic chains satisfying simultaneously the two previous conditions
1
p
ii
M f qb=
= −∑ . (2.43)
We can see that Eqs. (2.40)-( 2.43) represent particular cases of Eq. (2.25) proposed by Voinea and Atanasiu where pp=0, E
ii=1F= e∑ and
rj=bk. Eq. (2.43) is identical to Eq. (2.13) proposed by Hochman, by taking into consideration that p
ii=1F= f∑ .
By applying Eqs. (2.40)-( 2.43) to the mobility calculation of the mecha-nism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eq. (2.25) from which these equations can be derived.
In a recent work, K. J. Waldron, G. L. Kinzel (1999) presented a for-mula similar to Eq. (2.41) for the mobility calculation that accommodates overconstrained closures of arbitrary type. They critically analysed the utility of this equation by mentioning: “Unfortunately, unless the value of bk associated with the different closures can be identified by inspection, such expressions have no value. The reason is that the mobility equation gives a quick check of the number of position variables and independent equations without the need to develop these equations. However, the only way to verify an overconstraint closure of a type not identifiable by inspec-tion is to develop the closure equations and analyze them for dependency. Therefore the quick-check advantage of the mobility equation disappears, and there is no way to derive information about the linkage without per-forming a complete position analysis.”
By applying Eqs. (2.41) and (2.43) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eq. (2.21) from which these equations can be derived.
2.1.22 Hunt’s contribution
The formula proposed by Hunt (1978)
2.1 Critical review of mobility calculation 49
p
ii 1
M b( m p 1) f=
= − − +∑ (2.44)
is identical to Eq. (2.13) proposed by Hochman, by taking into considera-tion that p
ii=1F= f∑ and q=p-m+1.
Hunt (1978) also presented a formula similar to Eq. (2.41). A warning is added that this equation poses further questions of how to be sure of identi-fying suitable independent loops and how to assign the correct values to bk.
2.1.23 Hervé’s contribution
J.M. Hervé (1978a,b) proposed a mobility formula based on the algebraic group structure of the displacement set
p
ii 1
M b( m 1) ( b f )=
= − − −∑ (2.45)
where b is given by the dimension of the displacement subgroup associated with the kinematic chain and fi by the dimension of the displacement subset associated with the kinematic pair between the links i and i+1. This for-mula is identical to Kutzbach’s mobility equation (2.17) by taking into consideration the fact that b=6-d. As with Eq. (2.17), Eq. (2.45) also represents a particular case of Eq. (2.12).
J.M. Hervé (1978a,b) considered the kinematic chains that obey to Eq. (2.45) as “trivial chains” and added a warning concerning the application of this formula to “exceptional chains” and to “paradoxical chains”. An “exceptional chain” is a kinematic chain which involves several intersect-ing subgroups with different dimensions. The dimension of the displace-ment subgroup associated with a “paradoxical chain” depends on specific geometric constraints defined by link-length and twist angle relationships. In a more recent paper J.M. Hervé (1999) considered that “paradoxical chains are still an enigma”.
By applying Eq. (2.45) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eqs. (2.12) and (2.17) from which Eq. (2.45) can be derived. We note that the mechanism presented in Fig. 2.1 does not fit Hervé’s definitions of exceptional and paradoxical chains.
50 2 Structural parameters
2.1.24 Gronowicz’s contribution
A. Gronowicz (1981) presented a method for identification of mobility properties for multi-loop kinematic chains and he proposed the following formula for mobility calculation:
1
1 1 1
q q q
i iji i j i
M M F−
= = = +
= −∑ ∑ ∑ (2.46)
where Mi is the mobility of the loop i and Fij is the mobility of the connect-ing mechanism made of the joints common between any two loops i and j (the mobility of the joints which have been counted twice in the q loops of the mechanism).
We can see that Eq. (2.46) can be derived from Eq. (2.30) proposed by Manafu, by taking into account that 1
1 1
q qiji j i
F−
= = +∑ ∑ = Mc and ignoring the
fact that the degree of mobility of a complex kinematic chain depends on the choice of the independent closed loops, as in (Manolescu and Manafu 1963).
By applying Eq. (2.46) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were ob-tained by using Eqs. (2.30), from which Eq. (2.46) can be derived.
2.1.25 Baker’s contribution
J.E. Baker (1980a) proposed an algorithm for determining the freedom be-tween links which are separated by cross-jointing. The algorithm incorpo-rates quantitative relationships and embraces earlier results, particularly the series and parallel laws of screw system theory presented by Waldron (1966) and Davies and Primrose (1971). Baker (1980b) used the motor no-tations of screw system algebra to represent all six co-ordinates of a screw.
Baker (1981) placed the notion of mobility in perspective with the algo-rithm developed previously (Baker 1980b) and he pointed out that the de-termination of gross mobility without regard to relative freedom between specific links is not possible by the known techniques. By placing the cal-culation of gross mobility as a secondary stage after the study of relative freedom, Baker shows that the number of independent parameters of the linkage is obtained by considering the relative motion alone between indi-vidual members, for any pair of which we can write down the appropriate screw motor. He did not present any explicit formulae for the calculation of mechanism mobility, but said that “we may find the chain’s mobility as a consequence simply by counting the number of independent parameters”.
2.1 Critical review of mobility calculation 51
2.1.26 Davies’s contribution
T.H. Davies (1981) adapted Kirchhoff’s circulation law for potential dif-ference to find a set of independent instantaneous screw motors associated with any two links in a kinematic chain when the configuration of the ki-nematic chain is given. The procedure leads to a constraint matrix formula-tion and the rank calculation and is applicable to any kinematic chain: it does not require special cases to be identified. Davies (1983a-c) used a procedure based on the constraint matrix to determine mobility, passivity and redundancy of mobile and immobile assemblies called mechanical networks.
Davies proposed the following formulae for the calculation of the de-gree of mobility (Davies 1983a,b)
M=e-r (2.47)
q
kk 1
M e b=
= −∑ (2.48)
where e represents the number of joints with one degree of freedom asso-ciated to the mechanical network, r is the rank of the coefficient matrix of constraint equations and bk is the order of the screw system of the loop (circuit) k.
As mentioned in (Davies 1983b), Eq. (2.48) is identical to Eq. (2.22) proposed by Moroskine (by taking into consideration that e=F) and Eq. (2.48) is identical to Eq (2.41) proposed by Freudenstein and Alizade (1975) and by Hunt (1978).
A warning is added in (Davies 1983b) that Eq. (2.48) poses further questions of how to assign the correct values to bk. The danger lies in the interpretation of the words “bk is the order of the screw system of the cir-cuit k”. Davies stated that the interpretation that most would assume, and evidently the one intended in (Hunt 1978), is that bk is the order of the screw system to which the joints of loop (circuit) k belong when no other loop is closed. With this interpretation, Eq. (2.48) underestimates the value of M for some mechanical networks, whichever set of independent loop is chosen. To give Eq. (2.48) universal applicability, Davies mentioned that bk must be defined as the order of the screw system of loop k when the constraints imposed by all previously closed loops have been taken into account. The loops can be considered in any sequence but the constraints imposed by loops closed earlier in the sequence must be taken into ac-count. He also noticed in (Davies 1983b) that this interpretation of bk is un-fortunately rather more difficult to apply but there may be circumstances in
52 2 Structural parameters
which this stepwise approach proves easier than finding the rank of the constraint matrix of the mechanical network in its entirety.
2.1.27 Contribution of V.P. Agrawal and J.S. Rao
V.P. Agrawal and J.S. Rao (1987a) developed analytical tests for fractional degree of freedom kinematic chains based on the path loop connectivity matrix. They used loop connectivity properties of multi-loop kinematic chains to develop a seven-step hierarchical classification scheme of kine-matic structures (Agrawal and Rao 1987b). A formula is also proposed for calculating the mobility of a multi-loop mechanism with simple or multi-ple joints in terms of the mobility of its independent loops and connecting mechanism (Agrawal and Rao 1987c)
1 22 2N Nq q 1 qi i i i
i ij mi nii 1 i 1 j i 1 i 1 i 1
( m m 2 ) ( n 3n 2 )M M F F F2 2
−
= = = + = =
− − − += − + +∑ ∑ ∑ ∑ ∑
(2.49)
where: mi is the number of links forming the ith internal multiple joint, ni - the number of links forming the ith external multiple joint, N1 - the total number of internal multiple joints, N2 - the total number of external multi-ple joints, Fmi - the mobility of simple joint forming the ith internal multi-ple joint, Fni - the mobility of simple joint forming the ith external multiple joint.
A multiple joint formed by mi links is equivalent to (mi-1) simple joints. An internal multiple joint is completely inside the mechanism, i.e. it is common to independent loops, and not to the peripheral loop. An external multiple joint is on the peripheral loop of the kinematic chain and hence it is common to both independent loops and the peripheral loop.
We can see that Eq. (2.49) extends Eq. (2.46) proposed by Gronowicz to multiple jointed mechanisms. Agrawal and Rao (1987c) considered that Eq. (2.49) is applicable to any general mechanism with constant or vari-able general constraints with simple or multiple joints. In spite of this statement, by applying Eq. (2.49) to the mobility calculation of the mecha-nisms presented in Fig. 2.1 (which has only simple joints) we obtain the same erroneous results that were obtained by using Eq. (2.46).
2.1.28 Contribution of J. Angeles and C. Gosselin
J. Angeles and C. Gosselin (1988) addressed the problem of finding the mobility of an overconstraint mechanism, which is done by using the Jacobian matrix of a simple or multi-loop closed kinematic chain. They
2.1 Critical review of mobility calculation 53
have shown that, once the Jacobian matrix J of a kinematic chain coupled by either revolute or prismatic pairs is suitably defined, the mobility of the chain can be uniquely computed as the dimension of the nullspace of J (the nullity of J):
M=nullity(J). (2.50)
The authors mentioned that this idea was originally suggested by Freu-denstein (1962) in a very general manner. We can also see that Eq. (2.50) can be derived from Eq. (2.22) proposed by Moroskine. We take into ac-count that in a linear transformation (with a Jacobian matrix J) of a finite-dimensional vector space V, the following equation can be written:
rank(J) + nullity(J)=dim(V) (2.51)
and the fact that r=rank(J) and N= dim(V). The Jacobian matrix produces the mobility of the kinematic chain, re-
gardless of its type (single or multi-loop). Five full examples, including the Bennet mechanism and a planar three degrees of mobility parallel robotic manipulator are given in (Angeles and Gosselin 1988). As with Mo-roskine’s equation, Eq. (2.50) is valid without exception, but it does not give us a quick calculation of mobility without the need to develop the ki-nematic equations and to calculate the nullity of the Jacobian matrix. These calculations will be more disused in Section 2.2.
2.1.29 Contribution of F. Dudiţă and D. Diaconescu
F. Dudiţă and D. Diaconescu (1987) proposed mobility formulae applica-ble to simple or complex mechanisms. They demonstrated that the passive mobilities in joints do not influence the mobility of a simple (single loop) mechanism; consequently the following two formulae give the same re-sults:
1
p
ii
M f b=
= −∑ (2.52)
1
pe e
ii
M f b=
= −∑ (2.53)
where eif denotes the effective (active) mobilities in the ith joint, and be is
the dimension of the active motion space of the mechanism (the motion parameter obtained by taking into consideration only the active mobilities of the joints). We can see that Eq. (2.52) also represents a particular case
54 2 Structural parameters
of Eqs. (2.13) and (2.10) proposed by Hochman (for q=1). Dudiţă and Di-aconescu called b the kinematic rank of the mechanism (the dimension of the motion space). To calculate the kinematic rank, they used the screw co-ordinates and the system of generators from the screw algebra.
To calculate the degree of mobility of a complex (multi-loop) mecha-nism, Dudiţă and Diaconescu (1987) proposed the following formulae:
1 1
p q
i ji j
M f b= =
= −∑ ∑ (2.54)
1 1
p qe e
i ji j
M f b= =
= −∑ ∑ (2.55)
1
q
i ci
M M M=
= −∑ (2.56)
with e
c j com. jj
M ( q 1) f= −∑ (2.57
were Mc is the mobility of the common joints between different closed loops, e
com. jf is the active degree of mobility of the jth common joint and qj is the number of loops to which the common joint j belongs.
We can see that Eq. (2.54) represents a particular case of Eq. (2.25) pro-posed by Voinea and Atanasiu when pp=0 ( p
ii=1F= f∑ and
q qj jj=1 j 1
r b=
=∑ ∑ ). By applying Eq. (2.54) to the mobility calculation of
the mechanisms presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.25) from which Eq. (2.54) can be de-rived. The same erroneous results are also obtained by using Eq. (2.55). In the mechanisms presented in Fig. 2.1 all the joint mobilities are effective ( p p e
i ii=1 i=1 f f=∑ ∑ and q q e
j jj 1 j 1b b
= ==∑ ∑ ). By applying Eq. (2.56) to the
mobility calculation of the mechanisms presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.30) proposed by Manafu.
2.1 Critical review of mobility calculation 55
2.1.30 Contribution of P. Fanghella and C. Galletti
P. Fanghella and C. Galletti (1994) developed a systematic approach to de-termining the mobility properties of single-loop kinematic chains based on displacement groups introduced by Hervé (1978a). Such properties are de-fined by the following: the mobility of the chain, the number of degrees of freedom of the relative motion between any two links in the chain (link connectivity), the type of relative displacements between any two links in the chain (the displacement group containing all their relative displace-ments), and the set of invariant properties (Fanghella 1988) of every dis-placement subgroup.
The algorithm uses the following equation for determining mobility of single-loop kinematic chains:
p
i iii 1
M f min( co ,i 1,...,m )=
= − =∑ (2.58)
where coii is the loop connectivity defined as the connectivity of the open chain obtained by cutting the link i of the closed loop. Any link i (i=1,…,m) of the closed chain can be cut, and, due to non-associative char-acteristics of the kinematic constraints composition, different values of the connectivities coii can be obtained. The minimum value coii gives the cor-rect loop connectivity.
The approach is applicable to single-loop kinematic chains which can be defined by the composition of the ten displacement groups defined in [46]. Any mechanism not complying with this scheme cannot be considered. The approach being independent of local singularities cannot be used to determine the local or permanent changes in mechanism mobility due to singular configurations.
When taking into account the fact that min(coii, i=1,…,m)=b, we can see that Eq. (2.58) also represents a particular case of Eq. (2.13 ) proposed by Hochman (for q=1). The main contribution of P. Fanghella and C Galletti (1994) is to give a systematic approach to determining the loop connec-tivity b based on the displacement groups.
2.1.31 Fayet’s contribution
M. Fayet (1995a) proposed a general iterative process to obtain the spaces of twists between any two bodies of a multi-loop mechanism. A linear ap-plication represented by a “generative matrix” is set up by a triangular pro-jective method to generalize the sum of dependent spaces in a multi-loop mechanism.
56 2 Structural parameters
The iterative method proposed to obtain the space of twists between any two bodies (Fayet 1995a) is used by Fayet and Bonnet (1995) in order to obtain the rank of constraint equations of a multi-loop mechanism. They bring out a general proof for the stepwise approach put forward by Davies (1983b) to determine the order of the screw system of loop k when the constraints imposed by all previously closed loops have been taken into account. Three examples are given in (Fayet and Bonnet 1995), including the mechanism analysed by Davies (1983b), to emphasize the applicability of the iterative approach. This method has universal applicability but it gives the instantaneous mobility (mobility in a given position of the mechanism). The rank of the constraint equations is obtained by the rank of the twists consistent with the iterative approach defined for the given position.
2.1.32 Tsai’s formula
To take into consideration the fact that passive (internal) degrees of free-dom cannot be used to transmit motion or torque about an axis, Tsai (1999) proposed subtracting the number of passive degrees of freedom, fp, from the degree of freedom equation:
p
i pi 1
M b( m p 1) f f=
= − − + −∑ . (2.59)
By applying Eq. (2.59) to the mobility calculation of the mechanisms presented in Fig. 2.1 having fp=0, we obtain the same erroneous results that were obtained by using Eq. (2.13) from which Eq. (2.59) can be de-rived.
2.1.33 McCarthy’s formula
To calculate the mobility of the platform linkages, in which the platform is connected to the ground by serial chains, McCarthy (2000) proposed the mobility formula
cpci
i 1M b ( b M )
=
= − −∑ (2.60)
where pc represents the number of serial open chains connecting the plat-form to the ground and c
iM is the mobility of the ith chain including its part of the platform. By applying the formula (2.60) to the mobility calcu-
2.1 Critical review of mobility calculation 57
lation of the mechanism presented in Fig. 2.1, having ciM 4= , we obtain
the erroneous results M=0 (if we consider the motion parameter b=6), M=2 (if we consider b=5), M=4 (if we consider b=4).
2.1.34 Contribution of Z. Huang, L.F. Kong and Y.F. Fang
Huang, Kong and Fang proposed the formula p
ii 1
M b( m p 1) f υ=
= − − + +∑ (2.61)
considered as a “correct application of the Grübler-Kutzbach formula” in the case of parallel mechanisms (Huang et al. 2003). In Eq. (2.61) b=6-d is called order of the mechanism and d is the number of independent com-mon constraints in the mechanism. A common constraint is defined in (Huang et al. 2003) as a screw reciprocal to the unit twists associated with all kinematic pairs in the mechanism. The corrective term υ represents the number of redundant constraints that are linearly dependent on other con-straints. Eq. (2.61) gives the instantaneous mobility (mobility in a given position of the mechanism). The order of the mechanism and the number of redundant constraints are calculated for each given position.
2.1.35 Contribution of J.M. Rico, J. Gallardo and B. Ravani
Rico and Ravani (2003) have extended previous works (Hervé 1978a,b, 1999; Fanghella 1988; Fanghella and Galletti 1994) in the application of group theory to analysis of kinematic chains. They proposed the following formula to calculate the mobility of a single-loop closed mechanism:
p
i c cc ai 1
M f dim( H ( i, j )) dim( H ( i, j )) dim( H ( i, j ))=
= − − +∑ (2.62)
where Hc(i,j) and Hcc(i,j) are the clockwise and the counterclockwise com-posite subgroups associated with the two open chains connecting two links i and j in the single-loop mechanism, and Ha(i,j) is the absolute composite subgroup between i and j (the connectivity between links i and j). Dim stands for the dimension of the respective subgroup of the Euclidean group. In a kinematic chain with n links, the clockwise kinematic chain is defined by i, i+1,…,j-1, j and the counterclockwise kinematic chain by i, i-1,…, 2, 1, n,n-1,…,j+1, j.
58 2 Structural parameters
We can see that Eq. (2.62) is equivalent to Eq. (2.24) proposed by Voinea and Atanasiu by taking into account that p
ii 1N f
==∑ ,
1 cr =dim(H (i,j)) , r2=dim(Hcc(i,j)), and we= dim(Ha(i,j)). Rico, Gallardo and Ravani (2003) demonstrated that mobility formula
can be obtained either by using arguments of finite kinematics or by using arguments of infinitesimal kinematics. The authors translated the mobility formula (2.62) from the finite kinematics language of group theory into the infinitesimal kinematics language of Lie algebra. The instantaneous form of the mobility formula presented here is based on the theory of subspaces and subalgebras of the Lie algebra of the Euclidean group and their possi-ble intersections.
2.2 Chebychev-Grübler-Kutzbach mobility formulae
The most widely known formulae, presented in the previous section, can be grouped together into a Chebychev-Grübler-Kutzbach (CGK) formula and the extensions of this formula proposed by Voinea and Atanasiu, and by Manafu. In this section we explain why these formulae for quick calcu-lation of mobility do not work for all mechanisms, and we demonstrate the limits of their applicability. We will point out that some motion parameters used in mobility formulae cannot be associated with a loop independently of any other closed loop. Loop independence is also revisited in this sec-tion.
2.2.1 The original Chebychev-Grübler-Kutzbach formula
As we have presented in Section 2.1, the development of a formula for mobility calculation can be dated back to about 150 years ago. Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century (Chebychev 1869; Sylvester 1874; Grübler 1883, 1885; Somov 1887; Hochman, 1890) and the beginning of the twentieth century (Koenigs 1905; Grübler 1917; Malytsheff 1923; Kutzbach 1929). At that time the main formulae defining what we call the CBK formula for multi-loop mechanisms were already set up. Versions of these formulae were proposed all through the twentieth century (Dobrovolski 1949, 1951; Ar-tobolevski 1953; Kolchin 1960; Rössner 1961; Boden 1962; Ozol 1963; Manolescu and Manafu 1963; Manolescu 1968; Bagci 1971; Hunt 1978; Hervé 1978a,b; Tsai 1999).
2.2 Chebychev-Grübler-Kutzbach mobility formulae 59
The original CGK formula gave the mobility M of a mechanism with q≥1 independent loops having the same motion parameter b for each loop, and could be expressed by
1
p
ii
M f qb=
= −∑ . (2.63)
The first term of Eq. (2.63) represents the number of independent mo-tion parameters of the joints before loop closures provide further con-straints, and the second term the number of joint parameters that lose their independence after closing the q loops. As presented in the previous sec-tion, the motion parameter b was introduced by Somov (1887) to charac-terize the motion space of the mechanism, and is called the mobility num-ber.
Table 2.1. Terminology used for parameter b in the CGK formula
Terminology References Motion parameter Somov 1887 Mechanism category Hochman 1890 Rank of linear set of screws Voinea and Atanasiu 1959 Link mobility Voinea and Atanasiu 1960 Relative infinitesimal displacements of two bodies
Hunt 1967
Connectivity of the complex joint between two bodies
Waldron 1966
Connectivity of the instantaneous screw system of two bodies
Davies and Primrose 1971
Degree of freedom of the complex joint between two bodies
Hervé, 1978a,b
Relative freedom between links Baker 1980b Complex connectivity Baker 1980b Internal freedom Baker 1981 Connectivity between a pair of members Phillips 1984 Freedom of the complex joint between a pair of members
Phillips 1984
Rank of the kinematic space of the mechanism Dudiţǎ and Diaconescu 1987 Kinematic constraints between two bodies Fanghella 1988 Connectivity Fanghella and Galletti 1994 Dimension of the space of twists between two bod-ies
Fayet 1995a
Link connectivity Shoham and Roth 1997 Degree of freedom of the mechanism with an out-put member
Zhao et al. 2004
Spatiality Gogu 2005b-e
60 2 Structural parameters
Table 2.2. Tools used to define and to calculate instantaneous values of parameter b in a given position of the mechanism
Tools References Aronhold-Kennedy theorem of three axes in planar motion of three bodies
Aronhold 1872; Kennedy 1886
Phillips-Hunt theorem of three axes in spatial mo-tion of three bodies Phillips and Hunt 1964
Chasles-Ball theory of the instantaneous screw axes Chasles 1830; Ball 1900; Waldron 1966; Davies and Primrose 1971; Dudiţǎ and Diaconescu 1987;
Schoenflies’s geometry of linear complex Schoenflies 1893; Hunt 1967, 1978
Klein’s projective space of linear set of screws Voinea and Atanasiu 1959; Rico and Ravani 2004b
Reciprocal screw theory Zhao et al. 2004 Screw motor Baker 1980b Twist theory and triangular projective method Fayet 1995a Mathematical group structure of the set of dis-placements
Hervé 1978a,b; Fanghella 1988; Fanghella and Galletti 1994; Rico and Ravani, 2003
Lie group of rigid body displacement Hervé 1999; Rico et al. 2003, 2004a
In the literature, various terminologies are used to define this parameter (Table 2.1). Various tools can be used to define and to calculate instanta-neous values of this parameter for a given position of the mechanism (Ta-ble 2.2). This book presents a development of these previous contributions. We determine general connectivity, also called spatiality in the author’s previous publications, without the calculation of instantaneous connec-tivity. We base our approach on the theory of linear transformations (Greenberg 1998; Gogu, 2002b).
Dobrovolski (1949, 1951) introduced “mechanism family” d=6-b to characterize the dimension of the constraint space of a mechanism (Manolescu and Manafu 1963; Manolescu 1968; Antonescu 1973). The constraint parameter of the mechanism represents the number of general motion components suppressed by the mechanism or the number of gen-eral constraints of the mechanism.
Various expressions of CGK mobility formula, known in the literature (Dobrovolski 1949, 1951; Artobolevski 1953; Kolchin 1960; Rössner 1961; Boden 1962; Ozol 1963; Manolescu and Manafu 1963; Manolescu 1968; Hunt 1978; Hervé 1978a) can be obtained from Eq. (2.63) by taking into consideration Euler’s formula, particular values of b (b=3 for planar
2.2 Chebychev-Grübler-Kutzbach mobility formulae 61
or spherical mechanisms, b=6 for spatial mechanisms) or the relations be-tween:
a) the degree of mobility f and the degree of constraint c of a joint (c=6-f),
b) the motion parameter b and the constraint parameter d of the mechanism (d=6-b),
c) the total number of kinematic links n and the total number of links m including the fixed base (n=m-1).
2.2.2 The extended Chebychev-Grübler-Kutzbach formula
A slightly different formula, that we call the extended CGK formula, was proposed in the middle of the twentieth century by Voinea and Atanasiu (1960) and by Manafu (Manolescu and Manafu 1963).
The original CGK mobility formula was extended by Voinea and Atanasiu (1960) to mechanisms having q≥1 independent loops with differ-ent motion parameters bk (k=1,…,q):
p q
i ki 1 k 1
M f b= =
= −∑ ∑ . (2.64)
We can see that Eq. (2.63) represents a particular case of Eq. (2.64) where bk=b (k=1,2,…,q).
This extended formula was developed later by Antonescu (1973), Freu-denstein and Alizade (1975), Dudiţă and Diaconescu (1987). Some addi-tional terms were introduced in the CGK mobility formula to take into consideration passive mobilities, redundant freedoms and overclosing con-straints existing in the mechanism (Voinea and Atanasiu 1960; Bagci 1971; Tsai 1999).
The extended formula in the form proposed by Manafu was developed later by Gronowicz (1981), Davies (1983b), Agrawal and Rao (1987c), Dudiţă and Diaconescu (1987), Rico and Ravani (2004a):
1
q
i ci
M M M=
= −∑ (2.65)
where Mi is the mobility of the ith simple closed loop and Mc is the mobil-ity of the common joints between different closed loops (the mobility of the joints that appears in mobility calculation of two or more closed loops). We can see that Eq. (2.65) is directly derived from Eq. (2.64) by taking into account the fact that
62 2 Structural parameters
1
ip
i j ij
M f b=
= −∑ (2.66)
and
1 1 1
ipq p
c j ji j j
M f f= = =
= −∑∑ ∑ (2.67)
where pi is the number of joints of ith simple closed loop. We can see that Eq. (2.66) represents a particular case of Eq. (2.64) when q=1 and pi=p.
For a given multi-loop mechanisms Eq. (2.65) can give different results depending of the choice of the independent closed loops. To overcome this flaw Manolescu and Manfu (1963) proposed taking into consideration only the maximum value of mobility given for the same mechanism by Eq. (2.65). Dudiţă and Diaconescu (1987) indicated that passive mobilities of common joints must not be taken into consideration in the calculation of Mc. Agrawal and Rao (1987c) introduced some additional terms in Eq. (2.65) to take into consideration the existence of multiple joints in the multi-loop mechanism.
From Fig. 2.1 we can simply determine by inspection the values of the structural parameters necessary to calculate the mobility of the parallel ro-bot CPM. This mechanism has p=12 joints (3 prismatic and 9 revolute joints) and q=2 independent loops. Each joint has one degree of mobility (fi=1). Passive mobilities do not exist in the joints of this mechanism. The two independent loops have the same motion parameter b=b1=b2=5. The CGK mobility formula, defined by Eqs. (2.63-2.65) or by any other de-rived formula proposed by different authors mentioned in this section, gives M=2. This is an erroneous result. Kim and Tsai (2002) have shown that by considering b=6 Eq. (2.63) gives M=0; that also represent an erro-neous result. In fact any other value of b gives an erroneous result. We can see that the CGK mobility formula is not applicable to this mechanism nor to many other multi-loop mechanisms. The main objective of the following section is to explain this flaw and to propose accurate limits of applicabil-ity of the original and the extended CGK mobility formulae.
2.2.3 Limits of applicability of CGK formulae
Let us consider a mechanism with p joints and q≥1 closed loops. Each joint i introduces fi independent motion parameters. The closed loops can-cel the independence of a part of the joint’s independent parameters. Eq. (2.63) shows that the number of independent coordinates (parameters)
2.2 Chebychev-Grübler-Kutzbach mobility formulae 63
needed to define the configuration of a mechanism with closed loops is the difference between the number of independent motion parameters of the joints (
1
pii
f=∑ ) before loop closures provide further constraints, and the
number r of joint parameters that lose their independence after loop clo-sures. So, Eq. (2.63) can be rewritten as:
1
p
ii
M f r=
= −∑ . (2.68)
Eq. (2.68) is valid without exception. Moroskine (1954) indicated that the number of joint parameters that lose their independence in the closed loops of the mechanism is given by the rank of the homogeneous linear set of velocity equations of the mechanism. To calculate this rank we need to set up the velocity constraint equations of the mechanism. These calcula-tions are not simple in the case of multi-loop mechanisms. The numerical calculation of the rank in the proximity of the singular positions introduces certain ambiguities in the interpretation of the results. It is therefore very difficult to tell the difference between a real zero and a very small numeri-cal value close to zero in the calculation of the determinants. This draw-back is specific to all methods for mobility calculation based on rank cal-culation of constraint equations in a given position of the mechanism when joint location is defined numerically. Symbolic calculation of the rank can overcome these limits if the constraint equations are defined symbolically in a generic position, as will be presented in this section.
The basic kinematic structure of the parallel Cartesian robotic manipula-tor presented in Fig. 2.1 is obtained by concatenating three limbs denoted by A (0≡1A-2A-3A-4A-5A), B (0≡1B-2B-3B-4B-5B) and C (0≡1C-2C-3C-4C-5C). The first link 0≡1X of each limb is the fixed platform and the final link is the moving platform 5≡5X (X=A,B,C). The first joint of each limb is actu-ated. We denote by d2X and 2 Xd (X=A,B,C) the finite displacements and the velocities of the actuated prismatic joints and by iXϕ and iXϕ (i=3,4,5 and X=A,B,C) the finite displacements and the angular velocities of the passive joints. We note that all revolute joints are passive joints. The clo-sure equations of the parallel Cartesian robotic manipulator presented in Fig. 2.1 can be established by the condition that the linear ( 0
Hv ) and angu-lar ( 0
Hω ) velocity of point H situated on the mobile platform (expressed in the reference system O0x0y0z0 – see Fig. 2.2) must be the same in the three limbs (H≡HA≡HB≡HC):
64 2 Structural parameters
00 0
00 0
HCHA HB
HCHA HB
vv v= =
ωω ω,
(2.69)
or
[ ] −
0 0HA HB
0 0HA HB
v v= 0
ω ω
(2.70)
and
[ ]=
−
00HCHA
00HCHA
vv0
ωω.
(2.71)
Eqs. (2.70) and (2.71) represents the closure equations of the closed loops 0≡1A-2A-…-5A≡5B-…-2B-1B≡0 and 0≡1A-2A-…-5A≡5C-…-2C-1C≡0 . These loops can be considered as two independent closed loops of the Car-tesian robotic manipulator, presented in Fig. 2.1. By calculating the veloc-ity of point H in the three limbs of the parallel robotic manipulator Eqs. (2.70) and (2.71) lead to the following sets of linear equations:
[ ] [ ]T
1 2 A 3 A 4 A 5 A 2B 3B 4 B 5B 8 1D d d 0ϕ ϕ ϕ ϕ ϕ ϕ
× = (2.72)
[ ] [ ]T
2 2 A 3 A 4 A 5 A 2C 3C 4C 5C 8 1D d d 0ϕ ϕ ϕ ϕ ϕ ϕ
× = (2.73)
[ ]T
2 A 3 A 4 A 5 A 2B 3B 4 B 5B 2C 3C 4C 5CD d d dϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ
[ ]12 10
×=
(2.74)
where
[ ]1
3B 3B 4B 43B 4B 43B
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A 3B 3B 4B 43B 4B 43B
D
1 0 0 0 0 -b C -b C -b C 00 -b S -b S -b S 0 1 0 0 00 b C +b C b C 0 0 b S +b S b S 00 1 1 1 0 0 0 00 0 0 0 0 1 1 10 0 0 0 0 0 0 0
=
− − − −
(2.75)
2.2 Chebychev-Grübler-Kutzbach mobility formulae 65
[ ]2
3C 3C 4C 43C 4C 43C
3A 3A 4A 43A 4A 43A 3C 3C 4C 43C 4C 43C
3A 3A 4A 43A 4A 43A
D
1 0 0 0 0 b S +b S b S 00 -b S -b S -b S 0 0 -b C -b C -b C 00 b C +b C b C 0 1 0 0 00 1 1 1 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 1 1 1
=
− − −
(2.76)
[ ] [ ]3 4D D D=
3B 3B 4B 43B
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A 3B 3B 4B 43B
3
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A
1 0 0 0 0 -b C -b C0 -b S -b S -b S 0 1 00 b C +b C b C 0 0 b S +b S0 1 1 1 0 00 0 0 0 0 10 0 0 0 0 0
D1 0 0 0 0 00 -b S -b S -b S 0 0 00 b C +b C b C 0 0 00 1 1 1 0 00 0 0 0 0 00 0 0 0 0 0
− −=
4B 43B
4B 43B
43C 3C 4C 43C 4C 43C
3C 3C 4C 43C 4C 43C
-b C 0 0 0 0 00 0 0 0 0 0
b S 0 0 0 0 00 0 0 0 0 01 1 0 0 0 0
0 0 0 0 0 0D
0 0 0 b S +b S b S 00 0 0 -b C -b C -b C 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 1 1
− − = − − −
(2.77)
66 2 Structural parameters
and 3 A 3 AS sin( )ϕ= , 3 A 3 AC cos( )ϕ= , 43 A 4 A 3 AS sin( )ϕ ϕ= + ,
43 A 4 A 3 AC cos( )ϕ ϕ= + , etc. Fig. 2.2 presents the notations associated with the three limbs. By sym-
bolic calculation of the rank of these matrices with MAPLE® we obtain: 1 A B 1r r rank( D ) 5−= = = , 1 A C 2r r rank( D ) 5−= = = and A B Cr r rank( D )− −= = =
9. We can see that 1 2r r r≠ + . This result indicates that the structurally in-dependent closed loops 0≡1A-2A-…-5A≡5B-…-2B-1B≡0 and 0≡1A-2A-…-5A≡5C-…-2C-1C≡0 are not independent from a kinematic point of view. In the general case, rank r of the linear set of velocity equations of a multi-loop mechanism cannot be obtained by summing up the ranks (rj, j=1,2,…,q) of the linear sets of velocity equations of the q structurally in-dependent closed loops.
We recall that in the theory of graphs, the number of independent closed loops q is given by Euler formula q=p-m+1, where p represents the num-ber of edges (arcs) and m the number of vertices (nodes). In section 2.1.5, we have seen that Hochman (1890) applied Euler formula to calculate the number of independent closed loops q in a multi-loop mechanism. In this case, p is associated with the total number of joints and m with the total number of links including the fixed base. In section 1.4, we have defined the structural graph associated with the multi-loop mechanism by network-ing the joints and links with no geometric relations.
We can establish the following properties in relation with the structural limitation of the independence of the closed loops.
Property 1. The independence of the q closed loops of a multi-loop mechanism given by Euler’s formula is restricted, in the general case, to the structural independence
Property 2. In the general case, q structural independent closed loops in a multi-loop mechanism are not implicitly kinematically independent.
We note that the structural independence is defined in the sense of the theory of graphs and the definition of the structural graph by ignoring any geometric and kinematic relation between the links and joints of the multi-loop mechanism. In the general case, the motions of the q closed-loops are not independent.
By using Eq. (2.68) we get the correct value of mobility of the parallel Cartesian robotic manipulator presented in Fig. 1 (M=12-9=3).
We recall that the rank of the matrices [D1], [D2] and [D] can be calcu-lated numerically or symbolically. The numerical calculation gives us the instantaneous rank in a given position of the mechanism defined by the numerical values of joint variables and geometric parameters. The sym-bolic calculation gives us the global rank in an arbitrary position of the mechanism without indicating the numerical values of the joint variables
2.2 Chebychev-Grübler-Kutzbach mobility formulae 67
Fig. 2.2. Simple open kinematic chains associated with the three limbs of parallel Cartesian robotic manipulator and the notations used in the kinematic constraint equations: (a) limb A (0A≡1A-2A-3A-4A-5A); (b) limb B (0B≡1B-2B-3B-4B-5B); (c) limb C (0C≡1C-2C-3C-4C-5C)
and the geometric parameters. We just use the symbolic definition of the matrices given by (2.75)-(2.76). The global rank represents the maximum value of the instantaneous ranks. The mobility calculation based on the in-stantaneous rank gives us the instantaneous mobility of the mechanism. By using symbolic calculation of the rank we can calculate global or full cycle mobility of mechanisms; by using numerical calculation of the rank we can calculate instantaneous mobility.
The following three theorems can be demonstrated in relation to the rank of the homogeneous linear set of kinematic constraint equations of a multi-loop mechanism.
68 2 Structural parameters
Let us consider the general case of a multi-loop mechanism with q struc-turally independent closed loops and N=F independent motion parameters of the joints, before loop closures provide further constraints (
1
pii
N F f=
= =∑ ), defined by the vector x=[X] N 1× . The homogeneous lin-ear set of kinematic constraint equations of the multi-loop mechanism can be written as
[ ] [ ] [ ]6 q N N 1 6 q 1A X 0
× × ×= . (2.78)
The matrix [ ]6 q NA
× from Eq. (2.78) can be partitioned in many forms:
[ ]
1
2
6 q N
q 6 q N
AA
A
A
×
×
⋅
= ⋅ ⋅
,
(2.79)
[ ]
1 2
3
6 q N
q 6 q N
AA
A
A
−
×
×
⋅
= ⋅ ⋅
(2.80)
[ ]
1 2 ... i
i 1
6 q N
q 6 q N
AA
A
A
− − −
+
×
×
⋅
= ⋅ ⋅
(2.81)
where
j 6 NA
× is the partition of [ ]6 q N
A×
corresponding to the loop jth,
2.2 Chebychev-Grübler-Kutzbach mobility formulae 69
[ ]
1
2
1 2 ... i 6 i N
i 6 i N
AA
A
A
− − − ×
×
⋅
= ⋅ ⋅
is the partition of [ ]6 q NA
× corresponding to the
first i loops (i=2,3,…, q-1),
[ ] 11 2 12 N
2 12 N
AA
A− ××
=
is the partition of [ ]6 q NA
× corresponding to the first
two loops. We notice that the loops can be considered in any sequence. We denote
by rj the rank of the matrix j 6 NA
× (j=1,2,…,q).
We consider :U W
( )→F
x F x
the matrix transformation from the finite dimensional vector space U (called joint vector space) into the finite dimensional vector space W (called external vector space). The linear transformation F is the transfor-mation implied by multiplication by the matrix [ ]6 q N
A×
, that is to each
vector x in the joint vector space U the linear transformation F assigns a uniquely determined vector F(x) in the external vector space W
F(x)=[A][X]. (2.82)
The vector F(x) in W is the image of vector x in U under transformation F. If SU is a sub-set of U, then the set of all vectors F(x) in W, for which x is in SU, represents the image of SU under F. Conversely, if SW is a sub-set of W, then the set of all vectors x in U, for which F(x) is in SW, represents the inverse image of SW. Since U and W are vector spaces, each must con-tain a zero vector. We will denote these vectors as 0U and 0W. The image of U in W represents the range RF of F and the inverse image of 0W in U is the nullspace or kernel KF of F, that is
KF { }WU ( )= ∈ =x F x 0 (2.83)
The rank of F is the dimension of the range RF, and the nullity of F is the dimension of kernel KF
dim(RF)=rank(F) (2.84)
70 2 Structural parameters
and
dim(KF)=nullity(F). (2.85)
As F is a matrix transformation, with matrix [A], then
dim(RF)=rank(A) (2.86)
and
dim(KF)= nullity(A). (2.87)
We know that in a matrix transformation F, with matrix [A], we have:
dim(RF)+dim(KF)=dim(U) (2.88)
rank(F)+nullity(F)=dim(U) (2.89)
rank(A)+nullity(A)=dim(U). (2.90)
Eqs. (2.88)-(2.90) express a sort of “conservation of dimension” in the ma-trix transformation F (Greenberg, 1989).
Similarly, we consider Fj: jU W→ (j=1,2,…,q) and F1-2-…-i:
1 2 ... iU W − − −→ (i=2,…,q-1) the matrix transformations with the matrices
j 6 NA
× and [ ]1 2 ... i 6 i N
A − − − × . We note the range and kernel of Fj by RF(j)
and KF(j), and the range and kernel of F1-2-…-i by RF(1-2-…-i ) and KF(1-2-…-i). Theorem 1. The rank r of the homogeneous linear set of kinematic con-
straint equations of a mechanism with q structurally independent closed loops is given by
( 1 2 ... i )
q 1
1 ( i 1 ) / Ki 1
r r dim( R )− − −
−
+=
= +∑ FF (2.91)
where r1 is the rank of [ ]1 6 NA
× and
( 1 2 ... i )( i 1 ) / KR− − −+ FF is the range of the re-
striction of Fi+1 to the kernel of F1-2-…-i (i=1,2,…,q-1): i 1 F( 1 2 ... i ) F ( 1 2 ... i ) i 1
i 1
/ : W
( )+ − − − − − − +
+
→F K K x F x
and
( 1 2 ... i )( i 1 ) / K
( 1 2 ... i ) ( 1 2 ... i ) ( i 1 )
dim( R )
dim( K ) dim( K K )− − −+
− − − − − − +
=
− ∩FF
F F F
(2.92)
Theorem 2. The rank r of the homogeneous linear set of kinematic con-straint equations of a mechanism with q structurally independent closed
2.2 Chebychev-Grübler-Kutzbach mobility formulae 71
loops is less than, or equal to, the sum of the ranks (ri , i=1,2,…,q) of the linear sets of kinematic constraint equations of the q loops
q
ii 1
r r=
≤∑ . (2.93)
Theorem 3. The rank r of the homogeneous linear set of kinematic con-straint equations of a mechanism with q structurally independent closed loops is equal to the sum of the ranks ri (i=1,2,…,q) of the linear sets of kinematic constraint equations of the q loops if and only if the rank of the linear set of kinematic constraint equations of (i+1)th loop is equal to the dimension of the range of the restriction of Fi+1 to the kernel of F1-2-…-i
F ( 1 2 ... i )i 1 F( i 1 ) / Kr dim( R ), i=1,2,..., q-1− − −+ += (2.94)
The proof of Eqs. (2.91) and (2.92) is obtained by recurrence on the ba-sis of the demonstration presented by Fayet (1995, b) for q=2.
Let us consider the matrix [ ]6 q NA
× from Eq. (2.78) partitioned as
[ ] 1 2 ... ( q 1 )6 q N
q 6 q N
AA
A− − − −
××
=
. (2.95)
Eq. (2.95) results from Eq. (2.81) with i=q-1. In this case, the rank of [ ]6 q FA
×is equal to the rank of 1 2 ... ( q 1 ) 6( q 1 ) N
A − − − − − × augmented by the di-
mension of the range of the restriction of Fq to the kernel of F1-2-…-(q-1), that is
( 1 2 ... ( q 1 ))1 2 ... ( q 1 ) ( q ) / Kr rank( A ) dim( R )− − − −− − − −= +
FF . (2.96)
The matrix 1 2 ... ( q 1 ) 6( q 1 ) NA − − − − − × can be partitioned as
[ ] 1 2 ... ( q 2 )
6( q 1 ) Nq 1 6( q 1 ) N
AA
A− − − −
− ×− − ×
=
(2.97)
and
( 1 2 ... ( q 2 ))1 2 ... ( q 1 ) 1 2 ... ( q 2 ) ( q 1 ) / Krank( A ) rank( A ) dim( R )− − − −− − − − − − − − −= +
FF (2.98)
We can see that by inverse recursion we obtain Eq. (2.91). Eq. (2.92) is obtained by taking into account the fact that Eq. (2.78) is
equivalent to the following equations:
72 2 Structural parameters
[ ] [ ]1 2 ... ( q 2 )
N 1 6( q 1 ) 1q 1 6( q 1 ) N
AX 0
A− − − −
× − ×− − ×
=
(2.99)
[ ] [ ][ ] [ ]
1 2 1 1 6 1 16 1
1 6 16
0
0
... ( q ) N ( q )( q ) N
q NN
A X
A X
− − − − × − ×− ×
× ××
= =
(2.100)
That is equivalent to saying that
KF = KF(1-2-…-(q-1))∩KFq. (2.101)
Let 'qF =
( 1 2 ... ( q 1 ))q / K − − − −FF be the restriction of Fq to the kernel of F1-2-…-(q-1).
Similarly with Eqs. (2.83) and (2.88), we can write
KF’(q) { }( 1 2 ... ( q 1 )) WqK ( )− − − −= ∈ =F qx F x 0 . (2.102)
The “conservation of dimension” in the matrix transformation 'qF gives
dim(KF(1-2-…-(q-1)))=dim(KF’(q))+dim(RF’(q)). (2.103)
We can also express
KF’q { }1 2 ... ( q 1 ) W ( 1 2 ... ( q 1 )) q WqU ( ) ( )− − − − − − − −= ∈ = =x F x 0 , F x 0 , (2.104)
that is
KF’(q) = KF(1-2-…-(q-1))∩KF(q)= KF . (2.105)
From Eqs. (2.103) and (2.105) we can write
dim(RF’(q))= dim(KF(1-2-…-(q-1)))-dim(KF(1-2-…-(q-1))∩KF(q)) (2.106)
Eq. (2.92) is equivalent to Eq. (2.106) with i=q-1. The proof of Eq. (2.93) can be derived directly from Eq. (2.91) by tak-
ing into consideration that
( 1 2 ... i )i 1 ( i 1 ) / Kr dim( R ), i 1,2,..., q - 1− − −+ +≥ =
FF . (2.107)
The proof of Eq. (2.94) results directly from Eqs. (2.91), (2.93) and (2.107).
As we have seen, for the parallel Cartesian robotic manipulator pre-sented in Fig. 2.1, the symbolic calculation of the rank of the matrix (2.77) using MAPLE® gives A B Cr r rank( A ) 9− −= = = . This result can also be ob-tained by using Eq. (2.91):
2.2 Chebychev-Grübler-Kutzbach mobility formulae 73
( 1 2 ... i ) ( 1 )
q 1
1 ( i 1 ) / K 1 ( 2 ) / Ki 1
1 ( 1 ) ( 1 ) ( 2 )
r r dim( R ) r dim( R )
r dim( K ) dim( K K )
− − −
−
+=
= + = + =
+ − ∩ =
∑ F FF F
F F F 5+7-3=9
where r1=5 is the rank of [ ]1 6 FA
× and
( 1 )( 2 ) / Kdim( R ) =FF
( 1 ) ( 1 ) ( 2 )dim( K ) dim( K K )− ∩F F F =7-3=4 is the dimension of the range of the restriction of F2 to the kernel of F1. The dimension of the kernel of F1 and the dimension of the intersections of the kernels of F1 and F2 are de-termined by using symbolic calculation with MAPLE®. The rank of [ ]1D gives the motion parameter b1 of the first loop (0≡1A-2A-…-5A≡5B-…-2B-1B≡0) before the closure of the second loop. The dimension of the range of the restriction of F2 to the kernel of F1 gives the number of independent constraints associated to the second loop by taking into account the closure of the first loop. The value of dim(
( 1 )( 2 ) / KRFF )=4 is different from the rank
of [ ]2D (r2=rank( [ ]2D =5). The parallel Cartesian robotic manipulator presented in Fig. 2.1 does not obey Eq. (2.94). For this reason, mobility calculation using the same motion parameter for both loops fails. The fol-lowing interpretation could be given to this result. The rank of [ ]2D gives the motion parameter b2=5 of the second loop (0≡1A-2A-…-5A≡5C-…-2C-1C≡0) before the closure of the first loop. In fact, we have to consider the number of independent constraints associated to the second loop by taking into account the closure of the first loop. We have to take into account the dimension of the range of the restriction of F2 to the kernel of F1 instead of the motion parameter b2=5. The motion parameters b1 and b2 are easily ob-tained by inspection with no need to calculate the rank of the matrices [ ]1D and [ ]2D . The motion parameter b1 is given by the number of inde-pendent motions between the extreme links 1B and 0 in the serial open ki-nematic chain 0≡1A-2A-…-5A≡5B-…-2B-1B associated with the first loop when no other loop is closed (Fig. 2.3). We can observe that five inde-pendent motions ( , , , ,x y z x yv v v ω ω ) exist between the extreme links 1B and 0 (Fig. 2.3). These velocities form the basis of RF(1). The motion pa-rameter b2 is given by the number of independent motions between the ex-treme links 1C and 0 in the serial open kinematic chain 0≡1A-2A-…-5A≡5C-…-2C-1C associated with the second loop when no other loop is closed (Fig. 2.4). Five independent motions ( , , , ,x y z x zv v v ω ω ) exist between the extreme links 1C and 0 (Fig. 2.4). These velocities form the basis of RF(2). The dimension of the range of the restriction of F2 to the kernel of F1 can
74 2 Structural parameters
Fig. 2.3. Simple open kinematic chain (0A≡1A-2A-…-5A≡5B-…2B-1B) associated with the loop A-B of parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
Fig. 2.4. Simple open kinematic chain (0A≡1A-2A-…-5A≡5C-…2C-1C) associated with the loop A-C of parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
2.2 Chebychev-Grübler-Kutzbach mobility formulae 75
Fig. 2.5. Complex open kinematic chain obtained by concatenating the closed loop A-B and the simple open kinematic chain associated with limb C of the parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
also be obtained by inspection. Dim(( 1 )( 2 ) / KR
FF ) is given by the number of
independent motions between the extreme links 1C and 0 in the complex open kinematic chain from Fig. 2.5 obtained by concatenating the closed loop A-B and the simple open kinematic chain associated with limb C. Only four independent motions ( , , ,x y z zv v v ω ) exist in this case between the extreme links 1C and 0 (Fig. 2.5). These velocities form the basis of
( 1 )( 2 ) / KRFF .
Two other examples of parallel mechanisms with translational motions and with Schönflies motions proposed by the author of this work (Gogu, 2002) are presented to illustrate the applicability of Eq. (2.91).
Fig. 2.6 presents a parallel robotic manipulator (Gogu 2002) with three limbs derived from the solution presented in Fig. 1.1 by eliminating the ac-tuated prismatic joint from limb C. In this case, limb C just has a guiding role, constraining the mobile platform to a planar motion. This mechanism has p=11 joints (2 prismatic and 9 revolute joints) and q=2 independent loops. Each joint has one degree of mobility (fi=1). The two independent loops also have the same motion parameter b1=b2=5 whichever set of in-dependent loops is chosen. The CGK mobility formula gives M=11-(5+5)=1. This is an erroneous result. In fact Eq. (2.91) gives r=5+4=9 and Eq. (2.68) indicates two degrees of mobility M=11-9=2. This is the correct result.
76 2 Structural parameters
Fig. 2.6. Translational robotic manipulator with two degrees of mobility: (a) ki-nematic chain; (b) associated graph
Fig. 2.7. Fully-parallel robotic manipulator with Schönflies motions: (a) kinematic chain; (b) associated graph
2.2 Chebychev-Grübler-Kutzbach mobility formulae 77
Fig. 2.7 presents a parallel robotic manipulator with decoupled Schön-flies motions (Gogu 2002). The mobile platform has four degrees of free-dom: three independent translations and one rotation about an axis of fixed direction. The parallel mechanism has four limbs integrating p=20 joints (4 prismatic and 16 revolute joints) and q=3 independent loops. The three independent loops have the same motion parameter b=b1=b2=6 whichever set of independent loops is chosen. The CGK mobility formula gives M=20-(6+6+6)=2. This is also an erroneous result. In fact Eq. (2.91) gives r=6+5+5=16 and Eq. (2.68) indicates the right value of the degrees of mobility: M=20-16=4.
We can conclude that original and extended CKG formulae for mobility calculation of multi-loop mechanisms are applicable just to the mecha-nisms that have the rank r of the homogeneous linear set of kinematic con-straint equations equal to the sum of the motion parameters bi associated with the q structurally independent closed loops (i=1,2,…,q) when no other loops are closed.
q
ii 1
r b=
=∑ . (2.108)
The CGK formulae are appropriate only for the mechanisms that obey Eqs. (2.94) or (2.108). Equation (2.108) provides a simple criterion under which the CGK formulae are applicable.
For example, the simple planar jointed systems presented in Fig. 2.8 has p=6 revolute joints and q=2 structuralal independent loops with b1=b2=3. The linkage in Fig. 2.8b is derived from Fig. 2.8a by intyroducing the fol-lowing geometric constraints: AB=CD=EF, AD=BC and DE=CF. The CGK formulae give M=6-6=0. This is a correct result for the jointed sys-tem in Fig. 2.8a which is an immobile structure with r=6, and an eroneous
Fig. 2.8. Planar jointed systems with M=0 (a) and M=1 (b), and their associated graph (c)
78 2 Structural parameters
result for the double parallelogram linkage in Fig 2.8b. In Chapter 3 exam-ple 39 we will see that r=5 and M=1 in the case of the double parallelo-gram linkage. The jointed system in Fig. 2.8a obeys Eq. (2.108) and the linkage in Fig. 2.8b fails to obey this criterion. The CGK formulae can not take into account the geometric constraints mentioned above.
Critical analysis of the CGK formula was previously discussed by Wal-dron (1966), Davies (1983b) and Fayet and Bonnet (1995), but they did not formalize the limits of applicability of this formula.
2.3 Mobility and connectivity of parallel robots
We have seen that in the last 150 years, sustained efforts have been made to find general mobility formulae for a quick calculation of mobility of any multi-loop mechanism, but the “magic formula” had not yet been found. We have shown in the previous sections that several dozen approaches proposed for the calculation of mechanism mobility can be reduced to the same original formula that we call the Chebychev-Grübler-Kutzbach for-mula in its original or extended forms. The CGK formula is still largely used for mobility calculation of multi-loop mechanisms. Even so, this for-mula does not fit many classical mechanisms or recent parallel robots. We have explained why this well-known formula does not work for some multi-loop mechanisms, and we have set up a criterion governing mecha-nisms to which this formula can be applied. We founded our demonstra-tion on two properties related to structural limitation of the independence of the closed loops and on three theorems related to the rank of the homo-geneous linear set of constraint equations of a multi-loop mechanism.
The lack of mobility formula for quick calculation of mobility of multi-loop mechanism with wider applicability represents a stumbling block in structural analysis and type synthesis of spatial mechanisms. To overcome this, the author has recently proposed new formulae (Gogu 2005b-e) to en-able quick calculation of the mobility of parallel mechanisms obeying Eq. (2.91). We have demonstrated new formulae to calculate r for various types of parallel mechanisms without calculating the rank of the homoge-neous linear set of constraint equations associated with loop closure or without calculating the rank of the complete screw system associated to the joints of the mechanism. The parameters used in these new formulae can be easily obtained by inspection. An analytical method to compute these parameters has also been developed just for verification and for a better understanding of the meaning of these parameters. This new approach
2.3 Mobility and connectivity of parallel robots 79
based on the theory of linear transformations is presented in the following sections.
2.3.1 General definitions and formulae for mobility and connectivity of mechanisms
In this section we present some general definitions and formulae for mobil-ity and connectivity of mechanisms by using the theory of linear transfor-mations. For a mechanism (simple or complex, open or closed) with holonomic joints, we can define the mobility from a geometric and from a kinematic point of view. We take into consideration the fact that the differ-ential joint constraint equations are linear and homogeneous. So, this set of equations can be integrated to obtain the finite displacements in joints.
Alternative definitions, theorems, properties and conditions are pre-sented in this chapter in order to illustrate the dual (infinitesimal and finite) nature of mobility and connectivity, their correlations and various ways to express them. They are formalized for various types of kinematic chains used in parallel robots in the logical order of increasing complexity.
It is common in scientific approaches to choose a number of hypotheses (postulates) that are assumed to be true within a given theory, and then the theory consists of all theorems and propositions provable using those hy-potheses as assumptions. The logic of the propositions presented in this section is:
If A is a given type of kinematic chain than various relations Bi exist and can be demonstrated between the structural parameters of the chain.
In this case the given theory is the theory of structural parameters of ki-nematic chains in which we assume, as a postulate, that any kinematic chain gives a linear transformation between joint velocity space and exter-nal velocity space. The various definitions, properties and propositions presented in this chapter with respect to mobility, connectivity, redundancy and overconstraint form the bases of this approach based on the theory of linear transformations.
Definition 2.1A (Geometric definition of mechanism mobility). Mo-bility represents the number of independent finite displacements in the joints needed to define the configuration of the mechanism.
Definition 2.1B (Kinematic definition of mechanism mobility). Mo-bility represents the number of independent infinitesimal displacements in the joints needed to define the infinitesimal displacements of any element of the mechanism.
We distinguish instantaneous mobility iM and general mobility M. The instantaneous mobility characterises the mechanism in a given position i.
80 2 Structural parameters
The general mobility represents the minimum value of the instantaneous mobility
M=min(iM) (2.109)
General mobility is also called full-cycle mobility. A given mechanism has an infinite number of positions and, for each position, we can calculate an instantaneous mobility. In these conditions, general mobility calculation is theoretically impossible by using Eq. (2.109). In this section we will pre-sent methods for calculating general mobility without the calculation of the instantaneous mobility. In the following sections, general mobility will be called simply mobility. Simple open mechanisms have M=iM in any po-sition of the mechanism. Complex open kinematic chains and sim-ple/complex closed mechanisms have M≤ iM.
Mobility proposition 1. The mobility of a mechanism with p joints and q≥1 structurally independent closed loops is given by the difference between the total number of independent motion parameters of the joints
1
pii
f=∑ before loop closure provides further constraints, and the r joint pa-
rameters that lose their independence after closing the loops. This proposition could be expressed by Eq. (2.68) and the proof is ob-
tained directly by taking into account the fact that: a) each joint i with a degree of mobility (connectivity) fi, introduces fi independent motion pa-rameters before all closure provides further constraints, and b) the closed loops cancel the independence of r joint parameters.
Eq. (2.68) was proposed by Moroskine (1954, 1958). He indicated that r is given by the rank of the homogeneous linear set of constraint equations of the mechanism.
Mechanism existence condition 1. A mechanism with q≥1 structur-ally independent closed loops exists if and only if the total number of in-dependent motion parameters of joints
1
pii
f=∑ before closing the loops is
higher than the number r of joint parameters that lose their independence in the closed loops:
1
p
ii
f r=
>∑ . (2.110)
Eq. (2.110) follows directly from Eq. (2.68) by taking into considera-tion that a jointed system is a mechanism if M>0. In this case, the jointed system is mobile. If the jointed system is immobile (M=0) it is called a jointed structure or simply a structure (see Fig. 2.8a).
In this section we will define M and r by using the theory of linear trans-formations. As we have seen in section 2.2.3, the kinematic equation of an
2.3 Mobility and connectivity of parallel robots 81
simple open chain can be described by a matrix transformation from the finite dimensional vector space U (called joint velocity space) into the fi-nite dimensional vector space W (called external velocity space) of basis (W)=( x y z x y zv ,v ,v , , ,ω ω ω ):
:U W
( )→F
q F q .
The linear transformation F is the transformation multiplication by the matrix [ ]6 N
J×
, where 1
pii
N f=
=∑ . That is to each vector q , in the joint vector space U, the linear transformation F assigns a uniquely determined vector ( )F q in the vector space W
F( q )= [ ] [ ]6 N N 1J
× ×q . (2.111)
Matrix [ ]6 NJ
×is usually called the Jacobian matrix. Vector q represents
the joint velocity vector and it is composed of the relative velocities in the joints of the mechanism. Vector ( )F q in W is the image of vector q in U under transformation F. The image of U in W represents the range RF of F and we call it the operational space. For a mechanism Q, its operational space will be denoted by QRF or simply by RQ or RQ.
As F is a matrix transformation, with matrix[ ]6 NJ
×, then we can write
1
p
ii
dim(U ) N f=
= =∑ (2.112)
dim(RF)=rank(F)=rank( [ ]6 NJ
×). (2.113)
For closed chains with 1
pii
N f=
=∑ freedoms in joints and q independ-ent loops, the closure equations of the q loops involve ( )F q =0 and the Jacobian becomes a 6q×N matrix.
For a closed simple (q=1) or complex (q>1) mechanism, the number of joint parameters that lose their independence after loop closures is
r= dim(RF)=rank(F)=rank( [ ]6 q NJ
×). (2.114)
Let us consider a general mechanism Q (that can be simple or complex, open or closed).
82 2 Structural parameters
Definition 2.2A (Geometric definition of connectivity between two links of a mechanism). Connectivity Q
a / bS between two links a and b in a mechanism Q represents the number of independent finite displacements allowed by the mechanism between the two links.
Definition 2.2B (Kinematic definition of connectivity between two links of a mechanism). Connectivity Q
a / bS between two links a and b in a mechanism Q represents the number of independent infinitesimal dis-placements allowed by the mechanism between the two links.
Connectivity proposition 1. Connectivity Qa / bS between two links (a
and b) in a mechanism Q is given by the dimension of the vector space Qa / bR of the relative velocities between links a and b in mechanism Q
Q Qa / b a / bS dim( R )= . (2.115)
The proof of Eq. (2.115) results directly from Definition 2.2 and the definition of the vector space Q
a / bR . We notice that Qa / bR is a sub-space of
QR and implicitly a sub-space of W. Any mechanism gives a linear transformation F and consequently Eq.
(2.68) is valid without exception. It is applicable to any type of mechanism (simple or complex, open or closed). In the following section we will pre-sent and demonstrate formulae for calculating the dimension of the range RF for simple open mechanisms without calculating the rank of the screw system associated with the joints of the mechanism. The new formulae, demonstrated by using linear transformations, are based on connectivity of a simple open kinematic chain. This important structural parameter is eas-ily determined by inspection. An analytical method will also be presented just for verification and for a better understanding of the meaning of this parameter.
2.3.2 Mobility and connectivity of simple open kinematic chains
A simple open kinematic chain A (1-2-…-n) is a serial kinematic chain with n links and p joints (Fig. 2.9). We denote the reference link by 1 and the final link by n. The reference link can be fixed (1≡0) or mobile. All in-termediate links (2, …,n-1) are binary links. We denote by M=MA the mo-bility of a simple open kinematic chain given by Eq. (2.68) with r=0. For a given simple open kinematic chain, MA has a constant positive value.
By analogy with the definition of mobility, we can define connectivity from both a geometrical and a kinematical point of view. Connectivity SA of a simple open kinematic chain A (1-2-…-n) is given by the connectivity
2.3 Mobility and connectivity of parallel robots 83
Fig. 2.9. General case of a simple open kinematic chain
between the final and reference links n and 1. Links n and 1 are the ex-treme links of the simple open kinematic chain.
Definition 2.3A. (Geometric definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (1-2-…-n) represents the number of independent finite displacements be-tween the extreme links n and 1 of A.
Definition 2.3B. (Kinematic definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (1-2-…-n) represents the number of independent infinitesimal displace-ments between the extreme links n and 1 of A
/1A
A nS S= . (2.116)
By analogy with mobility, we distinguish instantaneous connectivity iSA and general connectivity SA. The instantaneous connectivity characterizes the kinematic chain in a given position i. The general connectivity repre-sents the maximum value of the instantaneous connectivity
SA=max(iSA). (2.117)
A given kinematic chain has an infinite number of positions, and with each position we can associate an instantaneous connectivity. In these con-ditions, general connectivity calculation is theoretically impossible by us-ing Eq. (2.117). In this section we will present methods for calculating the general connectivity without calculating the instantaneous connectivity. General connectivity will be called simply connectivity. For the simple open kinematic chain, SA≥ iSA.
For a simple open kinematic chain A (1- 2- …-n), the linear transforma-tion FA and Eq. (1.111) becomes (Gogu 2002a):
v=
ω[ ]J [ ]q
(2.118)
where: v and ω are the translation and angular velocity vectors of the distal link n, q is the joint velocity vector composed of the relative veloci-ties in the joints of the mechanism and [ ] [ ]6 N
J J×
= is the Jacobian matrix.
84 2 Structural parameters
Vectors v and ω belong to the velocity vector space An / 1R . This space
defines the operational space of the simple open kinematic chain and
dim( An / 1R )=dim(RA) (2.119)
Mobility proposition 2. The mobility of the simple open kinematic chain A with p joints is given by the dimension of its joint space
MA=dim (UA) =1
p
ii
f=∑ .
(2.120)
Eq. (2.120) results directly from Eqs. (2.68). and (2.112). Connectivity proposition 2A. The connectivity of the simple open ki-
nematic chain A (1- 2- …-n) is given by the dimension of its operational space
SA=dim (RA). (2.121)
Connectivity proposition 2B. The connectivity of the simple open ki-nematic chain A (1- 2- …-n) is given by the rank of the Jacobian matrix of the linear transformation FA associated with A
SA=rank (JA). (2.122)
Eqs. (2.121) and (2.122) result directly from Eqs. (2.113) and (2.116). Connectivity indicates the number of independent velocities of the final
link related to the reference link. For a given simple open kinematic chain, SA depends on the geometric configuration of the chain, defined by the fi-nite displacements in the joints, and on the geometric parameters of the links. We recall that the rank of the Jacobian matrix can be calculated nu-merically or symbolically. The numerical calculation gives us the instanta-neous connectivity in a given position of the kinematic chain defined by numerical values of joint variables and geometric parameters. The sym-bolic calculation gives us the general connectivity in an arbitrary position of the kinematic chain without indicating numerical values of joint vari-ables and geometric parameters.
Connectivity proposition 3. The connectivity of the simple open kine-matic chain A is less than or equal to its mobility:
SA≤MA . (2.123)
This is a direct result of Eqs. (2.118), (2.120) and (2.122). The joint velocity space UA is a product space
2.3 Mobility and connectivity of parallel robots 85
p
A i 1 2 pi 1
U U U U U=
= = × ×⋅ ⋅ ⋅×∏ (2.124)
where Ui is the velocity space of the ith joint (the joint between links i and i-1).
Let us consider that the simple open kinematic chain A (1-2-…-i-…-n) is obtained by serial concatenation of two simple open kinematic chains A1 (1-2-…-i) and A2.(i-…-n). In this case, A is denoted by A≡A1-A2. We con-sider the sub-spaces UA1 and UA2 of the joint velocity space UA and their corresponding operational sub-spaces RA1 and RA2 of the operational space RA≡A1-A2 related to the linear transformation FA≡FA≡A1-A2. The sub-spaces UA1 and UA2 have the property { }A1 A2 UU U 0∩ = , that is the intersection of the two sub-spaces UA1 and UA2 is the zero vector of the linear space UA.
The operational space RA≡A1-A2 is an addition space
RA≡A1-A2=RA1+RA2 . (2.125)
The vector spaces UA, Ui, RA, RA≡A1-A2, RA1 and RA2 have finite dimen-sions. To determine the dimensions of the joint space UA and of the opera-tional spaces RA and RA≡A1-A2, the following properties of the vector spaces are useful:
Property 3 (Dimension of the joint space). The dimension of the joint space is given by the dimension of the product space UA
p
A Aii 1
dim(U ) dim(U )=
=∑ . (2.126)
From Eqs. (2.68) and (1.126) we can see, as a particular case, that the dimension of the velocity space of a joint is given by its degree of mobility (dim(Ui)= fi).
Property 4 (Dimension of the operational space). The dimension of the operational space RA≡A1-A2 is given by the dimension of the addition space:
A A1 A2 A1 A2
A1 A2 A1 A2
dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
= − = + =+ − ∩
(2.127)
where A1 A2R R∩ is the common range of the sub-spaces RA1 and RA2 of the operational space RA=A1-A2 associated with the simple open mechanism A≡A1-A2. We call A1 A2R R∩ the common operational space of A1 and A2.
Connectivity proposition 4A. The connectivity of a simple open kine-matic chain A≡A1-A2, obtained by serial concatenation of two open simple
86 2 Structural parameters
kinematic chains A1 and A2, is given by the dimension of its addition op-erational space
A A1 A2 A A1 A2 A1 A2
A1 A2 A1 A2
S dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
≡ − ≡ −= = + =+ − ∩
. (2.128)
Connectivity proposition 4B. The connectivity of a simple open kine-matic chain A≡A1-A2, obtained by serial concatenation of two open simple kinematic chains A1 and A2, is given by the rank of the Jacobian matrix of the linear transformation FA≡A1-A2 associated to A≡A1-A2
A A1 A2 A A1 A2S rank( J )≡ − ≡ −= . (2.129)
Eqs. (2.128) and (2.129) result directly from Eqs. (2.121), (2.122) and (2.127).
Singular configurations and redundancy of a simple open kinematic chain can be defined by using its mobility and connectivity.
Definition 2.4 (Structural singularity of a simple open kinematic chain). When the general connectivity of a simple open kinematic chain A is smaller than its general mobility (SA<MA), the kinematic chain A has a structural (architectural) singularity.
Definition 2.5 (Instantaneous singularity of a simple open kinematic chain). When the instantaneous connectivity of a simple open kinematic chain in a given position i is smaller than its general connectivity (iSA<SA)., the kinematic chain A is in an instantaneous (kinematic) singularity. The position i is called a singular configuration.
Definition 2.6. (Structural redundancy of a simple open kinematic chain). The structural redundancy TA of the simple open kinematic chain A represents the difference between the dimension of the joint space and the dimension of the operational space
TA=MA-SA (2.130)
Structural redundancy proposition 1A. The structural redundancy of a simple open kinematic chain A (1- 2- …-n) is given by the dimension of the kernel of the linear transformation FA associated with A
TA= dim(KF(A)) (2.131)
Structural redundancy proposition 1B. The structural redundancy of a simple open kinematic chain A (1- 2- …-n) is given by the nullity of the linear transformation FA associated with A
TA = nullity(FA)=nullity (JA). (2.132)
2.3 Mobility and connectivity of parallel robots 87
Eqs. (2.131) and (2.132) follow directly from Definition 2.6 and Eqs. (2.85) and (2.87).
Similar redundancy propositions can be set up for a simple open kine-matic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) obtained by connect-ing in the same link the extreme links (1A1≡1A2 and nA1≡nA2) of the two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2).
Structural redundancy proposition 2A. The structural redundancy of a simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) is given by the dimension of the kernel of the linear transformation FA≡A1-A2 associated with A≡A1-A2
TA= dim(KF(A≡A1-A2)) (2.133)
Structural redundancy proposition 2B. The structural redundancy of a simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) is given by the nullity of the linear transformation FA≡A1-A2 associated with A≡A1-A2
TA = nullity(FA≡A1-A2)=nullity (JA≡A1-A2). (2.134)
Eqs. (2.133) and (2.134) follows directly from Definition 2.6 and Eqs. (2.85) and (2.87).
Note 1. Formulae (2.122), (2.123), (2.131) and (2.132) apply to the sim-ple open kinematic chain A (1-2-…-n-(n+1)).
Note 2. Formulae (2.128), (2.129), (2.133) and (2.134) apply to the sim-ple open kinematic chain A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) obtained by serial concatenation of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) having the extreme elements joined in the same link nA1≡nA2.
Note 3. Formulae (2.122), (2.129), (2.131)-(2.134) involve the calcula-tion of the Jacobian matrices. Numerical calculation of the rank of the Jacobian matrix allows us to calculate only instantaneous values of the structural parameters. Symbolic calculation of the rank could overcome this drawback but both cases are very time-consuming. For this reason, these formulae have limited practical value.
Note 4. Formulae (2.121), (2.128) and (2.130) are based on the connec-tivity of the simple open chain. This parameter can be easily determined by inspection by observing the independent motions allowed by the kinematic chain between its extreme links.
Connectivity of a simple open kinematic chain is the main parameter used to calculate mobility of closed-loop kinematic chains as will be pre-sented in the following sections.
88 2 Structural parameters
2.3.3 Mobility and connectivity of single-loop kinematic chains
A single-loop kinematic chain is a simple closed kinematic chain with only binary links. We denote it by the letter B. There are two possibilities to ob-tain a single-loop kinematic chain B having p pairs and n links :
a) by connecting in the same link the extreme links 1 and n+1 of a sim-ple open kinematic chain A (1-2-…-n-(n+1)), we obtain the closed kine-matic chain B (1-2-…-n-(n+1)≡1) – see Fig. 2.10(a),
b) by connecting in the same link the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), we obtain the single-loop kinematic chain (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) which we denote by B ← A1-A2 – see Fig. 2.10(b).
In the first case, the closure condition indicates the fact that the velocity of link (n+1) related to reference element 1 in the single-loop kinematic chain B is zero:
B
vω
=
[0]. (2.135)
In the second case, the closure condition indicates the fact that velocity of the connecting point situated on the common link (nA1≡nA2) related to the reference link 1A1≡1A2 is the same in the two simple open kinematic chains A1 and A2
A1 A2
=
v vω ω
. (2.136)
The connecting point is denoted by H, and is called the characteristic point.
Eq. (2.136) can be written in the form
= =
A1 A2
v v ω ω
[0]. (2.137)
or
+ =
A1 A2 B
v v v=
ω ω ω[0].
(2.138)
If the open kinematic chain A (1-2-…-n-(n+1)) is identical to the open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2), we can see that the closure equations (2.135) and (2.138) are also identical.
2.3 Mobility and connectivity of parallel robots 89
Fig. 2.10. Simple closed kinematic chains obtained by: (a) connecting in the same link the extreme links 1 and (n+1) of a simple open kinematic chain A (1-2-…-n-(n+1)) and (b) connecting in the same links the extreme links (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2)
Reciprocally, to any single-loop kinematic chain B with n links, we can associate by splitting up a link:
a) a simple open kinematic chain A (1-2-…-n-(n+1)) having n+1 links, if we split up the reference element, Fig. 2.10(a), and we keep one part as a reference element and the other part becomes the (n+1)th link.
b) two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), having n+2 links , if we split up the links 1A1≡1A2 and nA≡nB, Fig 2.12(b).
The simple open kinematic chains A (1-2-…-n-(n+1)), A1 (1A1-2A1-…-nA1), A2 (1A2-2A2-…-nA2) and A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) are called simple open kinematic chains associated with the single-loop mechanism B.
We denote by /1A
A nS S= , A1A1 nA1 / 1A1S S= , A2
A2 nA2 / 1A2S S= , A A1 A2A A1 A2 1A2 / 1A1S S ≡ −≡ − =
the connectivity of the simple open kinematic chains A (1-2-…-n-(n+1)), A1 (1A1-2A1-…-nA1), A2 (1A2-2A2-…-nA2) and A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2). We also denote by UA, UA1, UA2, UA≡A1-A2 and by RA, RA1, RA2,
90 2 Structural parameters
RA≡A1-A2 the joint velocity vector spaces and the operational velocity vector spaces associated with the simple open kinematic chains A, A1, A2, A≡A1-A2.
Mobility proposition 3. The mobility of the single-loop mechanism B with p joints and one closed loop is given by the difference between the number of independent motion parameters of the joints
1
pii
f=∑ before
loop closure provide further constraints and the number of joint parameters rB that lose their independence after loop closure:
1
p
B i Bi
M f r=
= −∑ . (2.139)
Eq. (2.139) represents a particular case for single loop mechanisms of Eq. (2.68).
The following connectivity propositions can be set up for a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) obtained by connecting the extreme links 1 and (n+1) of a simple open kinematic chain A (1-2-…-n-(n+1)) in the same link.
Connectivity proposition 5A. The number of joint parameters that lose their independence in a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) is given by the connectivity of the open simple chain A (1-2-…-n-n+1) as-sociated with the single-loop chain B
B Ar S= (2.140)
Connectivity proposition 5B. The number of joint parameters that lose their independence in a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) is given by the dimension of the operational space of the simple open chain A (1-2-…-n-(n+1)) associated with B
B Ar dim( R )= . (2.141)
Connectivity proposition 5C. The number of joint parameters that lose their independence in a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) is given by the rank of the Jacobian matrix of the linear transformation FA corresponding to the simple open chain A (1-2-…-n-(n+1)) associated with B
B Ar rank( J )= (2.142)
Proof. The kinematic equation of the simple open kinematic chain A (1-2-…-n-n+1) is expressed by Eq. (2.118). By connecting the last link n+1
2.3 Mobility and connectivity of parallel robots 91
with the reference link 1, the relative velocity between links n+1 and 1 becomes zero, and (2.118) becomes a homogeneous linear set
[ ]AJ [ ]q =[0]. (2.143)
In the homogeneous linear set (2.143), the rank of the Jacobian matrix [ ]AJ indicates the number of dependent joint velocities in the single-loop kinematic chain B (1-2-…-n-(n+1)≡1). But at the same time the rank of the Jacobian matrix [ ]AJ indicates the connectivity and the dimension of the operational space of the simple open chain A (1-2-…-n-(n+1)) associated with single-loop chain B – see connectivity propositions 2A and 2B and Eqs. (2.121) and (2.122).
Similar connectivity propositions can be set up for a single-loop kine-matic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) in Fig. 2.10b, obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of the two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2).
Connectivity proposition 6A. The number of joint parameters that lose their independence in a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the connectivity of the open simple chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
rB=SA≡A1-A2 . (2.144)
Connectivity proposition 6B. The number of joint parameters that lose their independence in a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the dimension of the operational space of the open simple chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associ-ated with B ← A1-A2
B B A1 A2 A A1 A2 A1 A2
A1 A2 A1 A2
r dim( R ) dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
← − ≡ −= = = + =+ − ∩
(2.145)
Connectivity proposition 6C. The number of joint parameters that lose their independence in a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the rank of the Jacobian matrix of the linear transformation FA1-A2 corresponding to the simple open simple chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
B A A1 A2r rank( J )≡ −= . (2.146)
92 2 Structural parameters
Proof. By connecting the distal link 1A2 and the reference link 1A1, the relative velocity between 1A2 and 1A1 becomes zero in the single-loop ki-nematic chain B ← A1-A2 ,and Eq. (2.118) becomes
[ ]A1-A2J [ ] [ ]A1 A2q 0
−= . (2.147)
In the homogeneous linear set (2.147), the rank of the Jacobian matrix [ ]A A1-A2J ≡ indicates the number of dependent joint velocities in the closed simple kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1). At the same time the rank of the Jacobian matrix [ ]A A1-A2J ≡ indicates the con-nectivity and the dimension of the operational space of the open simple chain A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2 – see connectivity propositions 4A and 4B and Eqs. (2.128) and (2.129).
The following mobility propositions can be set up for a single-loop ki-nematic chain B (1-2-…-n-(n+1)≡1) obtained by connecting in the same link the extreme links 1 and (n+1) of a simple open kinematic chain A (1-2-…-n-(n+1)).
Mobility proposition 4A. The mobility of a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) is given by the difference between the mobility and the connectivity of the open simple chain A (1-2-…-n-(n+1)) associated with B:
B A AM M S= − (2.148)
Eq. (2.148) results directly from Eqs. (2.120), (2.139) and (2.140). Mobility proposition 4B. The mobility of a single-loop kinematic chain
B (1-2-…-n-(n+1)≡1) is given by the difference between the dimensions of joint and operational spaces of the simple open chain A (1-2-…-n-(n+1)) associated with B
B A AM dim(U ) dim( R )= − (2.149)
Eq. (2.149) results directly from Eqs. (2.120), (2.139) and (2.141). Mobility proposition 4C. The mobility of a single-loop kinematic chain
B (1-2-…-n-(n+1)≡1) is given by the difference between the dimension of the joint space and the rank of the Jacobian matrix of the linear transforma-tion FA corresponding to the simple open chain A (1-2-…-n-(n+1)) associ-ated with B
B A AM dim(U ) rank( J )= − . (2.150)
Eq. (2.150) results directly from Eqs. (2.120), (2.139) and (2.142). Mobility proposition 4D. The mobility of a single-loop kinematic chain
B (1-2-…-n-(n+1)≡1) is given by the nullity of the linear transformation
2.3 Mobility and connectivity of parallel robots 93
FA corresponding to the simple open chain A (1-2-…-n-(n+1)) associated with B
MB= nullity(FA)=nullity(JA) (2.151)
Eq. (2.151) results from Eq. (2.149) by taking into consideration the con-servation of the dimension in the matrix transformation FA
dim(RA)+nullity(FA)=dim(UA) (2.152)
and
rank(JA)+nullity(JA)=dim(UA). (2.153)
Mobility proposition 4E. The mobility of a single-loop kinematic chain B (1-2-…-n-(n+1)≡1) is given by the structural redundancy of the simple open chain A (1-2-…-n-(n+1)) associated with B
MB= TA . (2.154)
Eq. (2.154) results directly from Eqs. (2.132) and (2.151). Similar mobility propositions can be set up for a single-loop kinematic
chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) in Fig. 2.10b, obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of the two sim-ple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2).
Mobility proposition 5A. The mobility of a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the difference be-tween the mobility and the connectivity of the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2:
B A A1 A2 A A1 A2M M S≡ − ≡ −= − (2.155)
Eq. (2.155) follows directly from Eqs. (2.120), (2.139) and (2.144). Mobility proposition 5B. The mobility of a single-loop kinematic chain
B B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the difference between the dimensions of joint and operational spaces of the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
B A A1 A2 A A1 A2M dim(U ) dim( R )≡ − ≡ −= − (2.156)
Eq. (2.156) follows directly from Eqs. (2.120), (2.139) and (2.145). Mobility proposition 5C. The mobility of a single-loop kinematic chain
B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the difference be-tween the dimension of the joint space and the rank of the Jacobian matrix of the linear transformation FA corresponding to the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
94 2 Structural parameters
B A A1 A2 A A1 A2M dim(U ) rank( J )≡ − ≡ −= − (2.157)
Eq. (2.157) follows directly from Eqs. (2.120), (2.139) and (2.146). Mobility proposition 5D. The mobility of a single-loop kinematic chain
B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the nullity of the linear transformation FA≡A1-A2 corresponding to the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
MB= nullity(FA≡A1-A2)=nullity(J A≡A1-A2) (2.158)
Eq. (2.158) results from Eq. (2.156) by taking into consideration the con-servation of the dimension in the matrix transformation FA≡A1-A2.
Mobility proposition 5E. The mobility of a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the structural re-dundancy of the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2
MB= T A≡A1-A2 (2.159)
Eq. (2.159) follows from Eq. (2.134) and (2.158). Various combinations of Eqs. (2.139) and (2.140)-(2.142) can be used to
obtain equivalent formulae for mobility calculation of a single-loop kine-matic chain B (1-2-…-n-(n+1)≡1) obtained by connecting the extreme links 1 and (n+1) of a simple open kinematic chain A (1-2-…-n-(n+1)) – Fig. 2.10(a):
p
B i Ai 1
M f S=
= −∑ (2.160)
p
B i Ai 1
M f dim( R )=
= −∑ (2.161)
p
B i Ai 1
M f rank( J )=
= −∑ . (2.162)
Various combinations of Eqs. (2.139) and (2.144)-(2.146) can be used to obtain equivalent formulae for mobility calculation of a single-loop kine-matic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2). – Fig. 2.10(b):
2.3 Mobility and connectivity of parallel robots 95
p
B i A=A1-A2i 1
M f S=
= −∑ (2.163)
p
B i A A1 A2i 1
M f dim( R )≡ −=
= −∑ (2.164)
p
B i A1 A2 A1 A2i 1
M f dim( R ) dim( R ) dim( R R )=
= − − + ∩∑ (2.165)
p
B i A A1 A2i 1
M f rank( J ) ≡ −=
= −∑ . (2.166)
The following propositions can be set up concerning the existence of a single-loop mechanism by expressing the fact that MB>0.
Mechanism existence condition 2. A single-loop mechanism B exists if and only if the total number of independent motion parameters of joints
1
pii
f=∑ before loop closure is higher than the number of joint parameters
rB that lose their independence after loop closure:
1
p
i Bi
f r=
>∑ . (2.167)
Eq. (2.167) is a particular case of (2.110) when q=1. Mechanism existence condition 3A. A single-loop mechanism B (1-2-
…-n-(n+1)≡1) exists if and only if the mobility of the simple open chain A (1-2-…-n-(n+1)) associated with B is higher than the connectivity of A
MA>SA . (2.168)
Eq. (2.168) results directly from Eqs. (2.120), (2.140) and (2.167). Mechanism existence condition 3B. A single-loop mechanism B ← A1-
A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) exists if and only if the mobility of the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2 is higher than the connectivity of A≡A1-A2
MA>SA≡A1-A2 . (2.169
Eq. (2.169) results directly from Eqs. (2.120), (2.144) and (2.167). Mechanism existence condition 4A. A single-loop mechanism B (1-2-
…-n-(n+1)≡1) exists if and only if the simple open chain A (1-2-…-n-(n+1)) associated with B is structurally redundant
96 2 Structural parameters
TA>0 . (2.170)
Eq. (2.170) results directly from (2.154). Mechanism existence condition 4B. A single-loop mechanism B ← A1-
A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) exists if and only if the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2 is structurally redundant
TA≡A1-A2 >0 . (2.171)
Eq. (2.171) results directly from (2.159). Note 5. Formulae (2.140)-(2.142), (2.148)-(2.154), (2.160)-(2.162),
(2.168) and (2.170) apply to the single-loop kinematic chain B (1-2-…-n-(n+1)≡1) obtained by connecting in the same link the extreme links 1 and (n+1) of the simple open kinematic chain A (1-2-…-n-(n+1)) associated with B.
Note 6. Formulae (2.144)-(2.146), (2.155)-(2.159), (2.163)-(2.166), (2.169) and (2.171) apply to the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2). In this case the three chains A1, A2 and A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) are called simple open kine-matic chains associated with B ← A1-A2.
Note 7. Formulae (2.142), (2.146), (2.150)-(2.153), (2.157), (2.158), (2.162), (2.166) involve the calculation of the Jacobian matrices. Numeri-cal calculation of the rank and nullity of the Jacobian matrix allows us to calculate only instantaneous values of the structural parameters. Symbolic calculation of the rank and nullity could overcome this drawback but both cases are very time-consuming. For this reason, these formulae also have limited practical value for the quick calculation of mobility.
Note 8. Formulae (2.140), (2.141), (2.144), (2.145), (2.148), (2.149), (2.154)-(2.156), (2.159)-(2.161), (2.163)-(2.165) and (2.168)-(2.171) are based on the connectivity of the simple open chain associated with the closed mechanism. This parameter can be easily determined through in-spection by observing the independent motions given by the associated open chain between its extreme links.
2.3.4 Connectivity between two elements of a single-loop kinematic chain
Let us consider a single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements
2.3 Mobility and connectivity of parallel robots 97
(1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link - Fig. 2.10b. We con-sider UA1 and UA2 the joint velocity sub-spaces of the open simple kine-matic chains A1 and A2 and their corresponding operational spaces RA1 and RA2. The sub-spaces UA1 and UA2 have the property { }A1 A2 UU U 0∩ = . The common space of RA1 and RA2 is the vector space B
n / 1 A1 A2R R R= ∩ of rela-tive velocities between links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2.
Connectivity proposition 7A. The connectivity Bn / 1S between links
n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two open simple kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), to the same link, is given by the dimension of the common vector space of A1 and A2
Bn / 1 A1 A2S dim( R R )= ∩ . (2.172)
Proof. In the open simple kinematic chain A1, the number of relative independent infinitesimal displacements or velocities between extreme links nA1 and 1A1 is given by SA1=dim (RA1). The number of relative inde-pendent velocities between extreme links nA2 and 1A2 in the simple open kinematic chain A2 is given by SA2=dim (RA2). As we have considered 1A1≡1A2 and nA1≡nA2, the number of relative independent velocities between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 in the single-loop mecha-nism B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the dimension of the intersection of operational vector spaces RA1 and RA2.
Connectivity proposition 7B. The connectivity Bn / 1S between links
n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), to the same link is given by the difference between the sum of the connectivities of A1 and A2 and the con-nectivity of the simple open chain A=A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the closed chain B ←A1-A2
Bn / 1 A1 A2 A A1 A2S S S S ≡ −= + − . (2.173)
Proof. According to Eq. (2.172), Eq. (2.128) can be written as B
A A1 A2 A1 A2 n / 1S S S S≡ − = + − . (2.174)
98 2 Structural parameters
Eq. (2.174) expresses the fact that connectivity of the simple open chain A=A1-A2 associated with the closed chain B ←A1-A2 is given by the differ-ence between the sum of connectivities of A1 and A2 and the connectivity between links n≡nA1≡nA2 and 1≡1A1≡1A2 in single-loop kinematic chain B ←A1-A2.
Eq. (2.173) follows directly from (2.174). A similar equation to (2.173) was proposed by Voinea and Atanasiu (1960) to define the connectivity of a link in a single-loop mechanism by using screw theory.
Connectivity proposition 8A. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, have relative movements if the dimension of the common operational space of A1 and A2 is larger than zero
Bn / 1S >0 if A1 A2dim( R R ) 0∩ > . (2.175)
If
A1 A2dim( R R ) 0∩ = (2.176)
two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( Bn / 1S =0).
Eqs. (2.175) and (2.176) result directly from Eq. (2.172). Connectivity proposition 8B. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in
the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, have relative movements if the sum of connectivities of A1 and A2 is greater than the connectivity of the simple open chain A=A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the closed chain B ←A1-A2
Bn / 1S >0 if A1 A2 A A1 A2S S S ≡ −+ > . (2.177)
If
A1 A2 A A1 A2S S S ≡ −+ = (2.178)
two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( Bn / 1S =0).
Eqs. (2.177) and (2.178) result directly from Eq. (2.173). Connectivity proposition 9. Connectivity B
n / 1S between two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme
2.3 Mobility and connectivity of parallel robots 99
elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, is less than or equal to the mobility MB of the single-loop mechanism B ←A1-A2
Bn / 1 BS M≤ . (2.179)
The proof of Eq. (2.179) is founded on definitions 1 and 2 of mobility and connectivity. If B
n / 1 BS M> , MB independent parameters are not suffi-cient to describe the relative motion between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 of the single-loop kinematic chain B ←A1-A2 ;this contradicts the definition of mobility.
Connectivity proposition 10A. The basis of the vector space Bn / 1R of
relative velocities between the links n≡nA1≡nA2 and 1≡1A1≡1A2 in the sin-gle-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) does not vary with with the position of the characteristic point on link n≡nA1≡nA2.
Proof. We recall that the characteristic point is the point where link n≡nA1≡nA2 is split up to get associated open chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2). The independent velocities of this point define the ba-sis ( B
n / 1R ). Link n≡nA1≡nA2 has an infinite number of points and the veloc-ity of each point can be expressed by a linear combination of the inde-pendent velocities of the same basis. The basis ( B
n / 1R ) is valid for any point of the mobile platform including the points with the most restrictive mo-tion.
Connectivity proposition 10B. The connectivity Bn / 1S between the
links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, does not vary with the position of the characteristic point on link n≡nA1≡nA2.
Connectivity proposition 10B is a direct consequence of proposition 10A.
Connectivity proposition 11A. The basis of operational space RA≡A1-A2 of simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated to single-loop chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) does not vary with the choice of split up links 1 and n.
Proof. We have seen that the simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the single-loop kinematic chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) could be obtained by splitting up the link 1A1≡1A2 into 1A1 and 1A2. Two simple open chains A1
100 2 Structural parameters
(1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) can also be associated with A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) by splitting up the link nA1≡nA2 into nA1 and nA2. The operational vector space RA≡A1-A2 is an addition space RA≡A1-
A2=R1A+RA2 independently of the choice of 1A1≡1A2 and nA1≡nA2 to be split up.
Connectivity proposition 11B. The connectivity SA≡A1-A2 of the simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the single-loop chain B ←A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is does not vary with the choice of split up links 1 and n.
Connectivity proposition 11B is a direct consequence of proposition 11A.
The following remark is very important for the choice of the bases of operational vector spaces RA1 and RA2 when there are various ways to choose them.
Remark 1. (Choice of the bases of operational vector spaces RA1 and RA2). If there are different ways to choose the bases of the operational vec-tor spaces RA1 and/or RA2, the choice must be made by respecting the fol-lowing:
a) connectivity propositions 7-9 stating that Bn / 1 B0 S M< ≤ (2.180)
b) connectivity propositions 10A and B and 11A and B, c) propositions 2 and 3 of the existence of the mobile single-loop
mechanism and the corresponding conditions (2.167)-(2.171), d) definition of connectivity (in the sense of general connectivity) as the
maximum value of the instantaneous connectivity - Eq. (2.117), e) permanent constraints in the relative motions of the links generated
by the closed loop with special geometric conditions. The formalisation for mobility and connectivity of simple open kine-
matic chains and single-loop mechanisms will be used in the next sections to demonstrate new formulae for quick mobility calculation of parallel ro-bots with simple and complex limbs.
2.3.5. Mobility and connectivity of parallel robots with simple limbs
In this section we consider parallel robots with simple limbs. They are complex closed mechanisms composed of two polinary links and many bi-nary links. The two polinary links are the reference link and the final link. They represent the reference and mobile platforms.
2.3 Mobility and connectivity of parallel robots 101
Fig. 2.11. The parallel mechanism C=A1-A2-...-AK with k elementary limbs
Definition 2.7 (Parallel robot with simple limbs). A parallel robot with simple limbs is a complex mechanism C ← A1-A2-…Ak with two poli-nary elements 1C≡1Ai and nC≡nAi connected by k simple open chains Ai (1Ai-2Ai-…-nAi), i=1,2,…k (Fig. 2.11). The two polinary elements are the reference platform 1C and the mobile platform nC. The simple open kine-matic chain Ai is called a simple limb.
The complex mechanism C ← A1-A2-…Ak has p joints and k-1 structurally independent closed loops concatenated in-parallel. We denote by UAi and RAi the joint velocity space and the operational velocity space of the simple open kinematic chain Ai, and by UA1-Ai and RA1-Ai the joint velocity space and the operational velocity space of the simple open kinematic chain A1-Ai (1A1-2A1-…-nA1≡nAi-(n-1)Ai-(n-2)Ai-…-2Ai-1Ai) associated with each closed loop Bi ← A1-Ai (1A1-2A1-…-nA1≡nAi-(n-1)Ai-(n-2)Ai-…-2Ai-1Ai≡1A1), i=2,3,… ,k. We denote by r=rC the number of joint parameters that lose their inde-pendence in the k-1 structurally independent closed loops concatenated in-parallel in the complex mechanism C ← A1-A2-…Ak.
We note that the simple limbs Ai are detached from the parallel mecha-nism C ← A1-A2-…Ak when defining the associated parameters UAi, RAi and SAi. Todisconnect them the mobile platform is cut up in the proximity of each limb.
Definition 2.8. (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity C
n / 1S of the mobile platform n≡nAi in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k is given by the number of independent velocities be-
102 2 Structural parameters
tween the mobile platform n≡nAi and the reference platform 1=1Ai (i=1,2,…k) in the parallel mechanism C.
Connectivity proposition 12 (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity C
n / 1S of the mo-bile platform n≡nAi in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k is given by the dimension of the com-mon operational spaces of the simple limbs Ai associated with the parallel mechanism C
C Cn / 1 n / 1 A1 A2 AkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ (2.181)
where C A1 A2 ... Akn / 1 n / 1S S − − −= denotes the connectivity between the mobile platform
n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1-A2-…Ak,
C A1 A2 ... Akn / 1 n / 1R R − − −= - operational velocity vector space of parallel mecha-
nism C ← A1-A2-…Ak defined by the relative velocities between the mobile platform n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1-A2-…Ak.
The proof of the Eq. (2.181) is obtained by recurrence A1 A2 A1 A2n / 1 n / 1 A1 A2S dim( R ) dim( R R )− −= = ∩ (2.182)
A1 A2 A3 A1 A2 A3 A1 A2n / 1 n / 1 n / 1 A3
A1 A2 A3
S dim( R ) dim( R R ) dim( R R R )
− − − − −= = ∩ =∩ ∩
(2.183)
A1 A2 ... Ai A1 A2 ... Ai A1 A2 ... A( i 1 )n / 1 n / 1 n / 1 Ai
A1 A2 A( i 1 ) Ai
S dim( R ) dim( R R ) dim( R R ... R R )
− − − − − − − − − −
−
= = ∩ =∩ ∩ ∩ ∩
(2.184)
where A1 A2n / 1S − is the connectivity and the between the mobile platform n≡nAi
and the reference platform 1≡1Ai (i=1,2) in the parallel mechanism C ← A1-A2 with only two limbs,
A1 A2 A3n / 1S − − is the connectivity between the mobile platform n≡nAi and the
reference platform 1≡1Ai (i=1,2,3) in the parallel mechanism C ← A1-A2-A3 with only three limbs,
2.3 Mobility and connectivity of parallel robots 103
A1 A2 ... Ain / 1S − − − is the connectivity between the mobile platform n≡nAi and
the reference platform 1≡1Ai (i=1,2,…k) in the parallel mechanism C ← A1-A2-…Ai with i limbs,
A1 A2n / 1R − is the operational velocity space of the parallel mechanism
C ← A1-A2 with only two limbs, A1 A2 A3n / 1R − − - operational velocity space of the parallel mechanism C ← A1-
A2-A3 with only three limbs, A1 A2 ... Ain / 1R − − − - operational velocity space of the parallel mechanism
C ← A1-A2-…-Ai with i limbs. Conectivity proposition 13A. The number of joint parameters that lose
their independence in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi) is given by the difference between the sum of the dimensions of the operational spaces RAi of the simple open chains Ai (i=1,2,…,k) associated with the simple limbs and the connectivity of the mobile platform C
n / 1S in the parallel mechanism C ← A1-A2-…Ak k
C A1 A2 ... Ak Ai A1 A2 Aki 1
r r dim( R ) dim( R R ... R )− − −=
= = − ∩ ∩ ∩∑ . (2.185)
Proof. The proof of Eq. (2.185) is also obtained by recurrence. We ini-tially consider the complex open mechanism associated with the parallel mechanism by considering that the reference links 1Ai of the simple limbs Ai are not integrated into the reference platform 1Ai≠1Aj (i,j=1,2,…k, i≠j). At this initial step the reference platform does not exist. It is divided into small parts corresponding to the links 1Ai of each limb. Only the links nAi of the simple limbs Ai are connected in the mobile platform n=nAi (i=1,2,…k) - see Fig. 2.12. The complex open kinematic chain has no closed loops. All mobility parameters of the joints are independent.
In the first step, we connect only links 1A1 and 1A2 of limbs A1 and A2 to form the reference platform 1 (1≡1A1≡1A2). The complex kinematic chain in Fig. 2.13 has only one closed loop A1-A2. By taking into consideration connectivity proposition 6B, the number of joint parameters that lose their independence in the closed loop B ← A1-A2 (1A1-2A1-…-(n-1)A1-nA1≡nA2-(n-1)A2 -…-2A2-1A2≡1A1) is given by the connectivity of the simple open chain A1-A2 (1A1-2A1-…-(n-1)A1-nA1≡nA2-(n-1)A2 -…-2A2-1A2) associated with the closed loop B ← A1-A2
A1 A2 A1 A2 A1 A2
A1 A2 A1 A2
r dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
− −= = + =+ − ∩
. (2.186)
104 2 Structural parameters
In the second step, we connect link 1A3 of limb A3 with the reference platform 1 (1≡1A1≡1A2≡1A3) - see Fig. 2.14. The number of joint parame-ters that lose their independence in the closed loops A1-A2-A3 is
Fig. 2.12. Complex open kinematic chain with k branches
Fig. 2.13. Complex kinematic chain with one closed loop and k-2 open branches
2.3 Mobility and connectivity of parallel robots 105
A1 A2A1 A2 A3 A1 A2 ( A1 A2 ) A3 A1 A2 n / 1 A3
A1 A2 A1 A2A1 A2 n / 1 A3 n / 1 A3
A1 A2 A1 A2 A3 A1 A2 A3
A1 A2 A3 A1 A2 A3
r r S r dim( R R )
r dim( R ) dim( R ) dim( R R )r dim( R R ) dim( R ) dim( R R R )dim( R ) dim( R ) dim( R ) dim( R R R ).
−− − − − − −
− −−
−
= + = + + =
+ + − ∩ =+ ∩ + − ∩ ∩ =
+ + − ∩ ∩
(2.187)
In Eq. (2.187), we denoted by S(A1-A2)-A3 the connectivity of the complex open kinematic chain (A1-A2)-A3 obtained by serial concatenation of the simple closed kinematic chain B ← A1-A2 and the simple open kinematic chain A3 in Fig. 2.13. S(A1-A2)-A3 is given by the connectivity between the link 1A3 and the reference link 1≡1A1≡1A2 in the complex kinematic chain (A1-A2)-A3 by taking into consideration the existence of the closed loop B ← A1-A2
A1 A2( A1 A2 ) A3 n / 1 A3
A1 A2 A1 A2n / 1 A3 n / 1 A3
A1 A2 A3 A1 A2 A3
S dim( R R )
dim( R ) dim( R ) dim( R R )dim( R R ) dim( R ) dim( R R R ).
−− −
− −
= + =
+ − ∩ =∩ + − ∩ ∩
(2.188
In the third step, we connect link 1A4 of limb A4 with the reference plat-form 1≡1A1≡1A2≡1A3≡1A4 - see Fig. 2.15. The number of joint parameters that lose their independence in the closed loops A1-A2-A3-A4 is
A1 A2 A3 A4 A1 A2 A3 ( A1 A2 A3 ) A4
A1 A2 A3A1 A2 A3 n / 1 A4
A1 A2 A3 A1 A2 A3 A4A1 A2 A3 n / 1 A4 n / 1 n / 1
A1 A2 A3 A1 A2 A3 A4
A1 A2 A3 A4
A1
r r S
r dim( R R )
r dim( R ) dim( R ) dim( R R )r dim( R R R ) dim( R )dim( R R R R )dim( R ) di
− − − − − − − −
− −− −
− − − −− −
− −
= + =
+ + =
+ + − ∩ =+ ∩ ∩ + −∩ ∩ ∩ =+ A2 A3 A4
A1 A2 A3 A4
m( R ) dim( R ) dim( R )dim( R R R R ).
+ +− ∩ ∩ ∩
(2.189)
The connectivity of the complex open kinematic chain (A1-A2-A3)-A4 ob-tained by serial concatenation of the complex closed kinematic chain A1-A2-A3 and the simple open kinematic chain A4 in Fig. 2.14 is denoted by S(A1-A2-A3)-A4 in Eq. (2.189). S(A1-A2-A3)-A4 is given by the connectivity between link 1A4 and the reference link 1≡1A1≡1A2≡1A3 in the complex open kine-matic chain (A1-A2-A3)-A4 by taking into consideration the closed loops of the complex closed kinematic chain A1-A2-A3
106 2 Structural parameters
A1 A2 A3( A1 A2 A3 ) A4 n / 1 A4
A1 A2 A3 A1 A2 A3n / 1 A4 n / 1 A4
A1 A2 A3 A4 A1 A2 A3 A4
S dim( R R )
dim( R ) dim( R ) dim( R R )dim( R R R ) dim( R ) dim( R R R R )
− −− − −
− − − −
= + =
+ − ∩ =∩ ∩ + − ∩ ∩ ∩
.
(2.190)
Fig. 2.14. Complex kinematic chain with two structurally independent closed loops and k-3 open branches
Fig. 2.15. Complex open kinematic chain with three structurally independent closed loops and k-4 open branches
2.3 Mobility and connectivity of parallel robots 107
At the end of the recursive procedure we obtain Eq. (2.185). Connectivity proposition 13B. The number of joint parameters that
lose their independence in the parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi) is given by the difference between the sum of the connectivities SAi of the open simple chains Ai (i=1,2,…,k) associ-ated with the simple limbs Ai ,and the platform connectivity C
n / 1S in the parallel mechanism C ← A1-A2-…Ak
kC
C A1 A2 ... Ak Ai n / 1i 1
r r S S− − −=
= = −∑ (2.191
where SAi=dim(RAi), Cn / 1 A1 A2 AkS d im( R R ... R )= ∩ ∩ ∩ , RAi is the opera-
tional vector space of the simple open kinematic chain Ai associated with the simple limb Ai.
The proof of Eq. (2.191) is obtained directly from Eq. (2.185) and from the definition of the connectivity of a simple open chain as the relative connectivity between the extreme links nAi and 1 of the elementary open chain Ai (SAi= Ai
nAi / 1S ). We can see that Eq. (2.174) represents a particular case of Eq. (2.191) when k=2. In this case the parallel mechanism with two limbs becomes a single-loop mechanism.
Mobility proposition 6. The mobility of a parallel mechanism C ← A1-A2-…Ak with p joints and k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k, is given by
k k
C Ai Ai A1 A2 Aki 1 i 1
M dim(U ) dim( R ) dim( R R ... R )= =
= − + ∩ ∩ ∩∑ ∑ (2.192)
or p k
CC i Ai n / 1
i 1 i 1M f S S
= =
= − +∑ ∑ . (2.193)
where: UAi and RAi are the joint velocity space and the operational velocity
space of the simple open kinematic chain Ai (1Ai-2Ai-…-nAi) associated with Ai- limb,
fi is the mobility of the ith joint, SAi is the connectivity of simple open kinematic chain Ai associated with
Ai-limb, Cn / 1S is the connectivity of mobile platform in the parallel mechanism C.
Eqs. (2.192) and (2.193) are obtained directly by integrating Eqs. (2.126), (2.185) and (2.191) in Eq. (2.68).
108 2 Structural parameters
Connectivity proposition 14. Mobile and reference platforms n≡nAi and 1≡1Ai of a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k have relative movements if the dimension of the common operational space of Ai is greater than zero
Cn / 1S >0 if A1 A2 Akdim( R R ... R ) 0∩ ∩ ∩ > . (2.194)
If
A1 A2 Akdim( R R ... R ) 0∩ ∩ ∩ = (2.195)
two platforms are relatively immobile ( Cn / 1S =0).
Eqs. (2.194) and (2.195) result directly from Eq. (2.181). Connectivity proposition 15.The connectivity C
n / 1S between the mobile and reference platforms n≡nAi and 1≡1Ai of the parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k is less than or equal to the mobility MC of the parallel mechanism C ← A1-A2-…Ak
Cn / 1 CS M≤ . (2.196)
The proof of Eq. (2.196) is founded on definitions 1 and 2 of mobility and connectivity. If c
n / 1 cS M> , Mc independent parameters are not suffi-cient to describe the relative motion between 1≡1Ai and n≡nAi of the paral-lel mechanism C ← A1-A2-…Ak which is contradictory to the definition of mobility.
Connectivity proposition 16. The basis of the vector space Cn / 1R of
relative velocities between the mobile and reference platforms n≡nAi and 1≡1Ai in the parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k does not vary with the position of the characteristic point on the mobile platform n≡nAi.
Proof. The independent velocities of the characteristic point define the basis ( C
n / 1R ). The mobile platform n≡nAi has an infinite number of points, and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( C
n / 1R ) is valid for any point of the mobile platform including the points with the most restric-tive motion.
The following remark is very important for the choice of the bases of operational vector spaces RAi when multiple possibilities exist to define them.
Remark 2. (Choice of the bases of operational vector spaces RAi). If there are different ways to choose the bases of the operational vector spaces RAi ,the choice must be made by respecting the following:
2.3 Mobility and connectivity of parallel robots 109
a) connectivity propositions 14 and 15 stating that Cn / 1 C0 S M< ≤ (2.197)
b) connectivity proposition 16, c) definition of mobility (in the sense of general mobility) as the mini-
mum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.181) are se-
lected so that the minimum value of Cn / 1S is obtained. By this choice, Eqs.
(2.192) and (2.193) fit in with Eq. (2.109) setting up general mobility as the minimum value of instantaneous mobility.
2.3.6. Mobility and connectivity of parallel robots with complex limbs
In this section we consider parallel robots with complex limbs. They are complex closed mechanisms integrating more than two polynary links. Two polynary links are the mobile and the reference platforms. The re-maining polinary links represents intermediary platforms integrated into the complex limbs.
Definition 2.9 (Parallel robot with complex limbs). A parallel robot with complex limbs denoted by D ← A1-A2-…Ak1-E1-E2-…-Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is con-nected to the reference platform 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex.
A complex limb contains at least one closed loop. We denoted by Ai (i=1,2,…k1) the elementary limbs, and by Ej (j=1,2,…k2) the complex limbs (k1+k2=k).
We can see that parallel mechanisms with simple limbs represents a par-ticular case of D when k2=0. If k1=0 all limbs are complex. We denote by UAi, UEj and RAi, REj the joint velocity spaces and the operational velocity spaces associated with limbs Ai and Ej.
We also denote by r=rD the number of joint parameters that lose their independence in the closed loops of the complex mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2.
The joint velocity space UEj and the operational velocity space REj asso-ciated with the complex limb Ej can be introduced by analogy with the cor-responding parameters UAi and RAi defined for the simple open kinematic chains in section 2.3.2.
110 2 Structural parameters
Definition 2.10. (Joint velocity space associated with a complex limb). Joint velocity space UEj associated with complex limb Ej is the joint space of the complex open kinematic chain associated with the limb by opening all closed loops integrated in the limb.
Definition 2.11. (Operational velocity space associated with a com-plex limb). The operational velocity space REj associated with the complex limb Ej is defined by the relative velocities between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain as-sociated with limb Ej.
We note that the constraints generated by the closed loops integrated in the complex limb Ej are taken into account in the operational velocity space REj ,and are ignored in the joint velocity space UEj.
Definition 2.12. (Connectivity of a complex limb) The connectivity SEj of a complex limb is given by the number of independent motions between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain associated with limb Ej.
We note that simple and complex limbs Ai and Ej are detached from the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 when defining the asso-ciated parameters UAi, RAi, SAi and UEj, REj, SEj. To disconnect them, the mobile platform is cut up near each limb.
Definition 2.13. (Connectivity of the mobile platform in a parallel robot with complex limbs). The connectivity D
n / 1S of the mobile platform n≡nAi≡nEj in a parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k1 and and k2 complex limbs Ej (j=1,2,…k2) is defined as the number of independent motions between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D.
Connectivity proposition 17 (Connectivity of the mobile platform in a parallel mechanism with complex limbs). The connectivity of the mo-bile platform n≡nAi≡nEj in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is defined as the dimension of the common operational spaces of the simple and complex limbs associated with the parallel mechanism D
D Dn / 1 n / 1 A1 Ak1 E1 Ek 2S dim( R ) dim( R ... R R ... R )= = ∩ ∩ ∩ ∩ ∩ . (2.198)
where D A1 A2 ... Ak1 E1 E2 ... Ek 2n / 1 n / 1S S − − − − − − −= denotes the connectivity between the mo-
bile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the paral-lel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2,
2.3 Mobility and connectivity of parallel robots 111
D A1 A2 ... Ak1 E1 E 2 ... Ek 2n / 1 n / 1R R − − − − − − −= is the operational velocity vector space of
the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 defined by the rela-tive velocities between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2,
Eq. (2.198) is obtained by recurrence, as in Eq. (2.181). Connectivity proposition 18A. The number of joint parameters that
lose their independence in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by
k1 k 2D
D A1 ... Ak1 E1 ... Ek 2 Ai Ej n / 1 li 1 j 1
r r dim( R ) dim( R ) dim( R ) r− − − − −= =
= = + − +∑ ∑ (2.199)
Eq. (2.199) follows from Eq. (2.185) by taking into account the fact that rl=
2
1
k Ejlj
r=∑ additional joint parameters lose their independence in the
closed loops integrated in the complex limbs. By Ejlr we have denoted the
joint parameters that lose their independence in the closed loops integrated in complex limb Ej. We note that in complex limb Ej, the closed loops could be concatenated serially or in parallel. Two closed loops are serially concatenated if they do not have any common joint. One or more common joints could be shared by closed loops concatenated in parallel. The num-ber of joint parameters that lose their independence in a single closed loop or in a closed loop serially concatenated in Ej-limb is given by connectivity propositions 5A, B, C and 6A, B, C – see Eqs. (2.140)-(2.142) and (2.144)-(2.146). The number of joint pareameters that lose their independence in closed loops concatenated in parallel is given by connectivity propositions 13A and B – see Eqs. (2.185) and (2.191).
Connectivity proposition 18B. The number of joint parameters that lose their independence in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by
k1 k 2D
D A1 ... Ak1 E1 ... Ek 2 Ai Ej n / 1 li 1 j 1
r r S S S r− − − − −= =
= = + − +∑ ∑ (2.200)
where: SAi and SEj are the connectivities of limbs Ai and Ej. Eq. (2.200) is obtained directly from Eq. (2.199) and from the definition of the connec-tivity of simple and complex chains associated with the limbs, as the num-ber of independent motions between the extreme links (nAi and 1Ai of limb Ai and nEj and 1Ej of limb Ej). We can see that Eqs. (2.181), (2.185) and
112 2 Structural parameters
(2.191) represents particular cases of Eqs. (2.198), (2.199) and (2.200) when k2=0. In this case the parallel mechanism with complex limbs D ← A1-A2-…Ak1-E1-E2-…-Ek2 becomes a parallel mechanism with simple limbs C ← A1-A2-…Ak1.
Serially and parallel concatenated closed loops can be integrated in a limb. A closed loop is serially concatenated in a limb if the joints of the loop do not belong to other closed loops of the same limb. A group of closed loops are parallel concatenated in a limb if they have at least one common joint. The additional number of joint parameters that lose their independence in a closed loop serially concatenated in a complex limb is easily obtained by inspection by using the connectivity propositions 5A-C and 6A-C and their associated equations (2.140)-(2.142) and (2.144)-(2.146). The additional number of joint parameters losing their independ-ence in a group of closed loops that are parallel concatenated in a complex limb is also easily obtained by inspection by using the connectivity propo-sitions 13A-B and their associated equations (2.185) and (2.191). We note that the closed loops parallel concatenated form a parallel kinematic chain integrated in the complex limb.
Mobility proposition 7. The mobility of a parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 com-plex limbs Ej (j=1,2,…k2) is given by
k1 k 2 k1 k 2
D Ai Ej Ai Eji 1 j 1 i 1 j 1
Dn / 1 l
M dim(U ) dim(U ) dim( R ) dim( R )
dim( R ) r= = = =
= + − − +
−
∑ ∑ ∑ ∑ (2.201)
or p k1 k 2
DD i Ai Ej n / 1 l
i 1 i 1 j 1M f S S S r
= = =
= − − + −∑ ∑ ∑ (2.202)
where: UAi and RAi are the joint velocity space and the operational velocity
space of the simple kinematic chain Ai associated with Ai-limb, UEj and REj are the joint velocity space and the operational velocity
space of complex kinematic chain Ej associated with Ej-limb, fi is the mobility of the ith joint and p the total number of joints of the
parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 SAi is the connectivity of simple kinematic chain Ai associated with Ai-
limb, SEj is the connectivity of complex kinematic chain Ej associated with Ej-
limb,
2.3 Mobility and connectivity of parallel robots 113
Dn / 1S is the connectivity of the mobile platform in the parallel mecha-
nism D ← A1-A2-…Ak1-E1-E2-…-Ek2. Eqs. (2.201) and (2.202) are obtained directly by integrating Eqs.
(2.199) and (2.200) in Eq. (2.68). By analogy with connectivity propositions 14, 15 and 16 applicable to
paralle robots with simple limbs, the following propositions can be formu-lated for parallel robots with complex limbs.
Connectivity proposition 19. The mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) have relative motions if the dimension of the common opera-tional space of Ai and Ej is greater than zero
Dn / 1S >0 ⇔ A1 Ak1 E1 Ek 2dim( R ... R R ... R ) 0∩ ∩ ∩ ∩ ∩ > . (2.203)
If
A1 Ak1 E1 Ek 2dim( R ... R R ... R ) 0∩ ∩ ∩ ∩ ∩ = (2.204)
two platforms are relatively immobile ( Dn / 1S =0).
Eqs. (2.203) and (2.204) result directly from Eq. (2.198). Connectivity proposition 20. The connectivity D
n / 1S between the mo-bile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mecha-nism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is less than or equal to the mobility MD of the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2
Dn / 1 DS M≤ . (2.205)
The proof of Eq. (2.205) is founded on the definitions 1 and 2 of mobil-ity and connectivity. If Eq. (2.205) is not satisfied, MD independent pa-rameters are not enough to describe the relative motion between n≡nAi≡nEj and 1≡1Ai≡1Ej of parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 which contradicts the definition of mobility.
Connectivity proposition 21. The basis of vector space Dn / 1R of relative
velocities between the mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) does not vary with the position of the characteristic point on the mobile plat-form n≡nAi≡nEj.
Proof. The independent velocities of the characteristic point define the basis ( D
n / 1R ). The mobile platform n≡nAi has an infinite number of points,
114 2 Structural parameters
and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( D
n / 1R ) is valid for any point of the mobile platform including the points with the most restric-tive motion.
The following remark is also very important for the choice of the bases of operational vector spaces RAi and REj when when there are various ways to choose them.
Remark 3. (Choice of the bases of operational vector spaces RAi and REj). If there are different ways to choose the bases of the operational vec-tor spaces RAi and REj , the choice must be made by respecting the follow-ing:
a) connectivity propositions 19 and 20 stating that Dn / 1 D0 S M< ≤ (2.206)
b) connectivity proposition 21, c) definition of mobility (in the sense of general mobility) as the mini-
mum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.198) are se-
lected so that the minimum value of Dn / 1S is obtained. By making this
choice, Eqs. (2.201) and (2.202) fit in with Eq. (2.109) setting up the gen-eral mobility as the minimum value of the instantaneous mobility.
2.3.7. General formulae for robot mobility and connectivity
The formulae for mobility and connectivity demonstrated in the previous section for parallel robots with complex limbs can be easily applied to par-allel robots with simple limbs (k2=0), to single loop mechanisms (k2=0 and k1=2) or to simple open kinematic chains (k2=0 and k1=1). In this way, the corresponding formulae for mobility and connectivity introduced in sec-tions 2.3.2-2.3.5 can be obtained as particular cases of general formulae demonstrated in section 2.3.6. To facilitate the correspondences and to bring together the different notations, in this section we present the formu-lae demonstrated in section 2.3.6 for parallel robots with complex limbs in a unified approach applicable to any kind of serial, parallel or hybrid se-rial-parallel robot.
We consider the general case of a robot in which the end-effector is connected to the reference link by k≥1 kinematic chains. We recall that the end-effector is a polynary link called a mobile platform in the case of par-
2.3 Mobility and connectivity of parallel robots 115
allel robots, and a monary link for serial robots. The reference link may ei-ther be the fixed base or may be deemed to be fixed. The kinematic chains connecting the end-effector to the reference link can be simple or complex. They are called limbs or legs in the case of parallel robots. A serial robot can be considered to be a parallel robot with just one simple limb, and a hybrid robot a parallel robot with just one complex limb.
We denote by F ← G1-G2-…Gk the kinematic chain associated with a general serial, parallel or hybrid robot, and by Gi (1Gi-2Gi-…nGi) the kine-matic chain associated with the ith limb (i=1,2,…,k). The end effector is n≡nGi and the reference link 1≡1Gi. If the reference link is the fixed base, it is denoted by 1≡1Gi≡0. The total number of robot joints is denoted by p.
Definition 2.14 (Serial robot). A serial robot F ← G1 is a robot in which end-effector n≡nG1 is connected to reference link 1≡1G1 by just one simple open kinematic chain Gi (1Gi-2Gi-…nGi) called a serial kinematic chain.
Definition 2.15 (Parallel robot). A parallel robot F ← G1-G2-…Gk is a robot in which end-effector n≡nGi is connected in parallel to reference link 1≡1Gi by k≥2 kinematic chains Gi (1Gi-2Gi-…nGi) called limbs or legs.
Definition 2.16 (Hybrid serial-parallel robot). A hybrid serial-parallel robot F ← G1 is a robot in which end-effector n≡nG1 is connected to refer-ence link 1≡1G1 by just one complex kinematic chain Gi (1Gi-2Gi-…nGi) called complex limb or complex legs.
Definition 2.17 (Fully-parallel robot). A fully-parallel robot F ← G1-G2-…Gk is a parallel robot in which the number of limbs is equal to the ro-bot mobility (k=M≥2), and each limb integrates just one actuator.
Definition 2.18. (Joint velocity space associated with a kinematic chain). The joint velocity space UGi associated with a kinematic chain Gi is the joint velocity space of the open kinematic chain associated with Gi by opening all closed loops which may exist in Gi.
Definition 2.19. (Operational velocity space associated with a kine-matic chain). Operational velocity space RGi associated with a kinematic chain Gi (1Gi-2Gi-…nGi) is the vector space of relative velocities between extreme links nGi and 1Gi.
We note that the constraints generated by the closed loops which may exist in Gi are taken into account in the operational velocity space RGi and are ignored in the joint velocity space UGi.
Definition 2.20. (Operational velocity space associated with a robot). The operational velocity space RF associated with a robot is defined by the relative velocities between the end-effector and the reference link in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi).
116 2 Structural parameters
Definition 2.21. (Connectivity of a kinematic chain) Connectivity SGi of a kinematic chain Gi (1Gi-2Gi-…nGi) is given by the number of independ-ent motions between extreme links nGi and 1Gi.
We note that the kinematic chain Gi is detached from the mechanism F ← G1-G2-…Gk when defining the associated parameters UGi, RGi and SGi. To disconnect them the mobile platform is cut up near each limb. For se-rial robots Gi and F represent the same kinematic chain.
Definition 2.22. (Connectivity of robot end-effector). The connec-tivity SF of the robot end-effector in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), is defined as the number of independ-ent motions between the end-effector and the reference link in the mecha-nism F. The connectivity SF of the robot-end effector is also called the ro-bot connectivity, and represents the connectivity of the kinematic chain F ← G1-G2-…Gk.
Connectivity proposition 22. (Connectivity of robot end-effector). The connectivity SF of the robot end-effector in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), is given by the dimension of the common operational spaces of the kinematic chains Gi
F F G1 G2 GkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ . (2.207)
where RGi is the operational velocity space of the kinematic chain Gi ,and RF is the operational velocity space of the kinematic chain F.
Eq. (2.207) indicates that the constraints introduced by all k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) are taken into account when defining the connec-tivity of robot end-effector. Eq. (2.207) represents a generalization of Eqs. (2.121), (2.172), (2.181) and (2.198) demonstrated in the previous sec-tions.
Connectivity proposition 23A. The number of joint parameters that lose their independence in themechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by
k
Gi F li 1
r dim( R ) dim( R ) r=
= − +∑ (2.208)
where RGi and RF are the operational velocity spaces of the kinematic chains Gi and F ,and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi.
2.3 Mobility and connectivity of parallel robots 117
We note that k
Gil l
i 1r r
=
=∑ , where Gilr represents the additional joint pa-
rameters that lose their independence in the closed loops which may exist in ith kinematic chain Gi.
Eq. (2.208) represents a generalization of of Eqs. (2.145), (2.185) and (2.199) demonstrated in the previous sections.
Connectivity proposition 23B. The number of joint parameters that lose their independence in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kine-matic chains Gi (1Gi-2Gi-…nGi) is given by
k
Gi F li 1
r S S r=
= − +∑ (2.209)
where: SGi and SF are the connectivities of the kinematic chains Gi ,and F and rl represents the additional joint parameters that lose their independ-ence in the closed loops which may exist in the kinematic chains Gi.
Eq. (2.209) represents a generalization of Eqs. (2.140), (2.191) and (2.200) demonstrated in the previous sections.
We recall that serially and parallel concatenated closed loops can be in-tegrated in the kinematic chain Gi. A closed loop is serially concatenated in Gi if the joints of the loop do not belong to other closed loops of Gi. A group of closed loops are concatenated in parallel in Gi if they have at least one common joint. The additional number of joint parameters losing their independence in a closed loop serially concatenated in Gi is easily obtained by inspection by using the connectivity propositions 5A-C and 6A-C and their associated equations (2.140)-(2.142) and (2.144)-(2.146). The addi-tional number of joint parameters losing their independence in a group of closed loops that are concatenated in parallel in a complex limb is also eas-ily obtained by inspection by using the connectivity propositions 13A-B and their associated equations (2.185) and (2.191). We note that the closed loops concatenated in parallel form a parallel kinematic chain integrated in Gi.
Mobility proposition 8A. The mobility of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by
k k
Gi Gi F li 1 i 1
M dim(U ) dim( R ) dim( R ) r= =
= − + −∑ ∑ . (2.210)
118 2 Structural parameters
Mobility proposition 8A. The mobility of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by
p k
i Gi F li 1 i 1
M f S S r= =
= − + −∑ ∑ (2.211)
where: UGi and RGi are the joint velocity space and operational velocity space of
the kinematic chain Gi, RF is the operational velocity space of the kinematic chain F ← G1-G2-
…Gk, fi is the mobility of the ith joint, and p the total number of joints of the
mechanism F, SGi is the connectivity of the kinematic chain Gi, SF is the connectivity of the kinematic chain F ← G1-G2-…Gk, rl represents the additional joint parameters that lose their independence
in the closed loops which may exist in the kinematic chains Gi. Eqs. (2.210) and (2.211) represent the generalization of of Eqs. (2.120),
(2.160), (2.161), (2.192), (2.193), (2.201) and (2.202) demonstrated in the previous sections.
Connectivity proposition 24. A mechanism F ← G1-G2-…Gk in which the end-effector is connected to the reference link by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) permits relative motions between end-effector n≡nGi and reference link 1≡1Gi if the dimension of the common operational space of Gi (i=1,2,…,k) is greater than zero
SF>0 ⇔ G1 G2 Gkdim( R R ... R ) 0∩ ∩ ∩ > . (2.212)
If
G1 G2 Gkdim( R R ... R ) 0∩ ∩ ∩ = (2.213)
the end-effector and reference link are relatively immobile (SF=0). Eqs. (2.212) and (2.213) represent the generalization of Eqs. (2.175),
(2.176), (2.194), (2.195), (2.203) and (2.204) demonstrated in the previous sections.
Connectivity proposition 25. The connectivity SF of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the refer-ence link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is less than or equal to the mobility M of the mechanism F ← G1-G2-…Gk
FS M≤ . (2.214)
2.3 Mobility and connectivity of parallel robots 119
Eq. (2.214) represents the generalisation of Eqs. (2.179), (2.196) and (2.205) demonstrated in the previous sections.
Connectivity proposition 26. The basis of the vector space RF of rela-tive velocities between the end-effector and the reference link in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), does not vary with the position of the characteristic point on the end-effector.
This connectivity proposition represents a generalization of connectivity propositions 10A, 16 and 21 demonstrated in sections 2.3.4-2.3.6.
The following remark is also very important for the choice of the bases of operational vector spaces RGi when there are various ways to choose them.
Remark 4. (Choice of the bases of operational vector spaces RGi). If there are various ways to choose the bases of the operational vector spaces RGi ,the choice must be made by respecting the following:
a) connectivity propositions 24 and 25 stating that
F0 S M< ≤ (2.215)
b) connectivity proposition 26, c) definition of mobility (in the sense of general mobility) as the mini-
mum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.207) are se-
lected so that the minimum value of FS is obtained. By this choice, Eqs. (2.210) and (2.211) fit in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility.
Remark 4 represents a generalization of previous remarks presented in sections 2.3.4-2.3.6.
We note that the new formulae presented in this section are applicable to the following:
(i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 ,and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
(iii) Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(iv) Serial robots with closed loops and parallel hybrid serial-parallel robots, when k=1, rl≠0 and G1 is a complex open kine-
120 2 Structural parameters
matic chain. In this case F and G1 also represent the same kine-matic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of the new formulae proposed in this section will be il-lustrated in chapter 3 for the various types of robots. In the following chap-ters these formulae will be used for structural synthesis of parallel robots with simple and complex limbs. Along with the morphological approach, they represent the cornerstone of the structural synthesis method.
2.4 Overconstraints in parallel robots
Usually overconstrained mechanisms are defined as mechanisms that, without satisfying the Chebychev-Grübler-Kutzbach mobility criterion, have full cycle mobility. Overconstrained mechanisms have mobility over a finite range of motions even though they should be structures (M≤0) ac-cording to the CGK formula given in Eq. (2.64). This definition was intro-duced to designate single and multi-loop mechanisms that do not obey the well-known mobility formulae presented in section 2.2 (see for examples the mechanisms presented in Figs. 2.1, 2.6, 2.7 and 2.8b).
The first overconstrained multi-loop mechanism which appeared in the literature was proposed by the French mathematician Gilles Personne (or Personier) De Roberval and dates from 1669. The original drawings pre-sented by De Roberval in “Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699” are presented in Fig. 2.16 (De Roberval 1669). This mechanism is known as the De Roberval scale, and is used in the standard laboratory double-pan balance. The horizontal arms together with both vertical pan supports make a double parallelogram mechanism. In this way the pans will stay horizontal in the full range of movement. More im-portantly, it prevents the pans from rotating as they move up and down, and this is the principle that ensures that the position of weights on the pans has no effect on the balance.
De Roberval’s scale (Fig. 2.8b) could be considered both as the first overconstrained mechanism and as the first “parallel mechanism with one degree of mobility” and three simple limbs. We will analyse this mecha-nism in the next chapter.
Since than, other overconstrained mechanisms have been proposed (Sar-rus 1853; Delassus 1900, 1902, 1922; Bennett 1902, 1914; Bricard 1927; Myard 1931a,b; Goldberg1943; Altman 1952; Baker 1978; Waldron 1968,1969 1979; Wohlhart 1987, 1991, 1995; Dietmaier 1995; Fang and Tsai 2004a; Mavroidis and Roth 1995) et al . References can be found to
2.4 Overconstraints in parallel robots 121
Fig. 2.16. De Roberval scale published in 1669: front cover of «Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699» (a), original drawings (b) and (c)
122 2 Structural parameters
almost all known overconstrained mechanisms in (Baker 1980a, 1984; Waldron 1973a,b; Phillips 1990).
In this section we revisit the definition of overconstrained mechanisms independently of the CGK formulae. The author presented the new general formulae for the calculation of the number of overconstrains in parallel ro-bots for the first time in (Gogu 2005e).
We consider a mechanism with p joints, n mobile links (reference link not included) and q structurally independent loops.
Overconstraint proposition 1A. The number of overconstraints N of a mechanism with mobility M is given by the difference between the number of kinematic constraints introduced by the joints of the mechanism before and after closing the loops
1(6 )
p
ii
N c n M=
= − −∑ (2.216)
Proof. The number of kinematic constraints introduced by the joints of the mechanism before closing the loops is given by the sum of the degrees of joint constraints
1
pii
c=∑ . The total number of constraints existing in the
mechanism after closing the loops is given by the difference between the number of independent motions of the n mobile members considered free in the general kinematic space with six dimensions, and the number of in-dependent motion parameters remaining after connecting the members by joints and closing the q independent loops of the mechanism.
Eq. (2.216) could also be interpreted from the static point of view. In this case,
1
pii
c=∑ represents the number of independent forces acting in the
joints of the mechanism, 6n indicates the number of static equations for the n mobile members, and M indicates the number of forces in the actuated joints.
Overconstraint proposition 1B. The number of overconstraints N of a mechanism with mobility M is given by the difference between the maxi-mum number of joint kinematic parameters that could lose their independ-ence in the q closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops:
N=6q-r (2.217)
Proof. Eq. (2.217) is obtained from Eq. (2.216) by taking into consid-eration Eq. (2.68), the definition of q given by the theory of graphs (q=p-n) and the relation between the degree of mobility and the degree of con-straint of a joint (f+c=6).
2.4 Overconstraints in parallel robots 123
If the number of overconstraints is zero (N=0) the mechanism is called non-overconstrained or isostatic.
Eqs. (2.216) and (2.217) are applicable to single and multi-loop mecha-nisms.
For a single-loop mechanism (q=1) we obtain
N=6-r. (2.218)
We note that a simple open kinematic chain cannot be overconstrained. Let us consider the case of the mechanism F ← G1-G2-…Gk in which end-
effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi). The operational vector spaces of F and Gi are RGi and RF with dim(RF)=SF and dim(RGi)=SGi. We recall that RGi is the vector space of relative velocities between extreme links nGi and 1Gi in kinematic chain Gi, and RF the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in the mechanism F ← G1-G2-…Gk. The dimensions of these spaces give connectivities SF and SGi of F and Gi (see definitions 19-22 in section 2.3.7 for more details).
Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by
k
Gi F li 1
N 6q dim( R ) dim( R ) r=
= − + −∑ (2.219)
where RGi and RF are the operational velocity spaces of kinematic chains Gi and F and rl represent the additional joint parameters that lose their inde-pendence in the closed loops which may exist in kinematic chains Gi.
Eq. (2.219) results directly from Eqs. (2.217) and (2.185) by taking into consideration the fact that the number of structurally independent loops formed between k kinematic chains in-parallel concatenated is q=k-1. Ad-ditional loops may exist in kinematic chains Gi.
Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by
k
Gi F li 1
N 6q S S r=
= − + −∑ (2.220)
where: SGi and SF are the connectivities of kinematic chains Gi and F, and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in kinematic chains Gi.
124 2 Structural parameters
Eq. (2.220) follows directly from Eq. (2.219) by taking into account the definition of connectivities SGi and SF as the dimensions of velocity vector spaces SGi and SF.
The number of overconstraints gives the degree of overconstraint of the mechanism.
To obtain a non-overconstrained mechanism, idle mobilities are intro-duced in the joints of the mechanism.
Definition 2.23. (Idle mobility). Idle mobility is a joint mobility that does not affect a mechanism’s mobility.
Idle mobilities could change just SGi, SF, and consequently the number of overconstraints N.
In theoretical conditions, when no errors exist with respect to parallel and perpendicular positions of joint axes, motion amplitude in an idle mo-bility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, de-pending on the precision of the parallel robot.
We note that formulae (2.219) and (2.220) are applicable to the follow-ing:
(i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
(iii) Parallel robots with complex limbs when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(v) Serial robots with closed loops and parallel hybrid serial-parallel robots, when k=1, rl≠0 and G1 is a complex open kine-matic chain. In this case F and G1 also represent the same kine-matic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of these new formulae along with the selection of idle mobilities to obtain non-overconstrained mechanisms will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of overconstrained and non- overconstrained parallel robots with simple and complex limbs.
2.5 Redundancy in parallel robots 125
2.5 Redundancy in parallel robots
Redundant robots are mechanismas with more degrees of freedom than re-quired for doing the prescribed task in the task space. Redundant actuation in parallel manipulators can be achieved by: (i) actuating some of the pas-sive joints within the existing limbs, (ii) introducing additional actuated limbs beyond the minimum necessary to actuate the manipulator and (iii) introducing some additional actuated joints within the limbs beyond the minimum necessary to actuate the manipulator. We note that actuator re-dundancy does not affect the connectivity of the end-effector and only in-creases the number of actuators. In all cases, only M actuators have inde-pendent motion. The other actuators have dependent motions. In extreme cases, dependency consists in actuator locking. A selective choice of ac-tuators with independent motion could have many advantages.
Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Mer-let 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to mini-mize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to in-crease dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kine-matic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electro-mechanical systems (Mukherjee et al. 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b).
The author was the first to present new formulae for calculating the de-gree of redundancy of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs, be-yond the minimum necessary to actuate the manipulator. In this way we reduce motion coupling in parallel manipulators.
Let us consider again the case of the mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k simple or complex kinematic chains Gi (1Gi-2Gi-…nGi). The mechanism F ← G1-G2-…Gk is characterized by:
(i) UGi - the vector space of joint velocities in kinematic chain Gi, (ii) RGi - the vector space of relative velocities between extreme
links nGi and 1Gi in kinematic chain Gi,
126 2 Structural parameters
(iii) RF - the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in parallel mechanism F ← G1-G2-…Gk,
(iv) SGi=dim(RGi) - the connectivity of Gi, (v) SF=dim(RF) - the connectivity of F, (see definitions 18-22 in section 2.3.7 for more details). Definition 2.24. (Structural redundancy of a kinematic chain). The
structural redundancy of the kinematic chain represents the difference be-tween its mobility and connectivity.
Structural redundancy proposition 3A. The structural redundancy of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi-2Gi-…nGi) is given by
k k
Gi Gi li 1 i 1
T dim(U ) dim( R ) r= =
= − −∑ ∑ (2.221)
Structural redundancy proposition 3B. The structural redundancy of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi-2Gi-…nGi) is given by
p k
i Gi li 1 i 1
T f S r= =
= − −∑ ∑ (2.222)
where: UGi and RGi are the joint velocity space and operational velocity space of
kinematic chain Gi, fi is the mobility of the ith joint, and p the total number of joints of the
mechanism F, SGi is the connectivity of the kinematic chain Gi, rl represents the additional joint parameters losing their independence in
the closed loops which may exist in kinematic chains Gi. Eqs. (2.221) and (2.222) results directly from structural redundancy
definition and Eqs. (2.210) and (2.211). In a parallel robot, the structural redundancy gives the internal mobili-
ties in the limbs. We note that formulae (2.221) and (2.211) are applicable to the follow-
ing: (iv) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic
chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(v) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
2.6 General formulae for structural parameters 127
(vi) Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(vi) Serial robots with closed loops and parallel hybrid serial-parallel robots, when k=1, rl≠0 and G1 is a complex open kine-matic chain. In this case F and G1 also represent the same kine-matic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of these new formulae will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of redundant and non redundant parallel robots with simple and complex limbs.
2.6 General formulae for structural parameters
The propositions and formulae demonstrated in the previous sections for mobility, connectivity, overconstraint and redundancy of various types of robots can be extrapolated to single and multi-loop mechanisms. In this case the end-effector n≡nGi is replaced by a link called a characteristic link, and the reference link is the fixed base 1≡1Gi≡0. Theoretically any link except the reference one can be considered as a characteristic link. In this way, a mechanism open or closed, single or multi-loop can be defined as a kinematic chain F ← G1-G2-…Gk with p≥2 joints.
Definition 2.25. (Mechanism). A mechanism F ← G1-G2-…Gk is a ki-nematic chain in which a characteristic link denoted by n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥1 sub-kinematic chains Gi (1Gi-2Gi-…nGi).
To simplfy the terminology, sub-kinematic chains Gi that are parts of the mechanism F ← G1-G2-…Gk are simply called kinematic chains.
Definition 2.26. (Simple mechanism). A simple mechanism is a mechanism F ← G1-G2-…Gk with just monary and binary links.
Definition 2.27. (Complex mechanism). A complex mechanism is a mechanism F ← G1-G2-…-Gk with at least one ternary link.
We recall the following:(i) a monary link is a link which carries only one kinematic pairing element, (ii) a binary link is a link connected in the kinematic chain by two joints and (iii) a polynary link is a link connected in the kinematic chain by more than two joints –see section 1.4.
Definition 2.28. (Serial mechanism). A serial mechanism F ← G1 is a kinematic chain in which characteristic link n≡nG1 is connected to refer-ence link 1≡1G1≡0 by just one simple open kinematic chain G1 (1G1-2G1-…nG1).
128 2 Structural parameters
Definition 2.29. (Single-loop mechanism). A single-loop mechanism F ← G1-G2 is a kinematic chain in which characteristic link n≡nGi (i=1,2) is connected to reference link 1≡1Gi≡0 by k=2 serial kinematic chains Gi (1Gi-2Gi-…nGi).
Definition 2.30. (Multi-loop mechanism). A multi-loop mechanism F ← G1-G2-…Gk is a kinematic chain, containing at least one ternary link, in which characteristic link n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥2 kinematic chains Gi (1Gi-2Gi-…nGi).
The ternary link can be the characteristic link or any other link of the ki-nematic chain.
We can see that serial and single-loop mechanisms are simple, and multi-loop mechanisms are complex mechanisms.
We note that joint velocity space UGi, ,operational velocity space RGi , and connectivity SGi of the kinematic chain Gi (1Gi-2Gi-…nGi) are defined in the previous section (see definitions 18, 19 and 21). The kinematic chain Gi is detached from the mechanism F ← G1-G2-…Gk when defining associ-ated parameters UGi, RGi and SGi. To disconnect it the characteristic link is cut after the last joint of Gi.
Definition 2.31. (Connectivity of the characteristic link). Connec-tivity SF of characteristic link n≡nGi in the mechanism F ← G1-G2-…Gk is defined by the number of independent motions between n≡nGi and refer-ence link 1≡1Gi≡0 allowed by the mechanism F.
A mechanism F ← G1-G2-…Gk permits relative motions between end-effector n≡nGi and reference link 1≡1Gi≡0 when n≡nGi and 1≡1Gi≡0 are connected by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) if and only if the di-mension of the common operational space of Gi (i=1,2,…,k) is higher than zero – see Eq. (2.212).
The general formule demonstrated in this chapter for mobility M, degree of overconstraint N and redundancy T of parallel robots are also applicable to the mechanism F ← G1-G2-…Gk.
To conclude this chapter we recall these formulae and the meaning of the parameters used to define them.
p
ii 1
M f r=
= −∑
(2.223)
N=6q-r (2.224)
T=M-SF (2.225)
where
2.6 General formulae for structural parameters 129
k
Gi F li 1
r S S r=
= − +∑ (2.226)
F F G1 G2 GkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ (2.227)
and k
Gil l
i 1r r
=
=∑ . (2.228)
We recall that p represents the total number of joints, k the total number of independent closed loops in the sense of graph theory, fi the mobility of the ith joint, r the number of joint parameters that lose their independence in the mechanism F, Gi
lr the number of joint parameters that lose their in-dependence in the closed loops of limb Gi, rl the total number of joint pa-rameters that lose their independence in the closed loops that may exist in the limbs of mechanism F, SF connectivity of characteristic link n≡nGi in the mechanism F, SGi the connectivity of characteristic link n≡nGi in limb Gi disconnected from the mechanism F, RG operational velocity space of the mechanism F ← G1-G2-…Gk and RGi the operational velocity space of limb Gi disconnected from the mechanism F.
We note that the intersection in Eq. (2.227) is consistent if the opera-tional velocity spaces RGi are defined by the velocities of the same point situated on the characteristic link. This point is called the characteristic point, and denoted in this work by H. It is a point with the most restrictive motions of the characteristic link.
The connectivity SF of the characteristic link n≡nGi in the mechanism F ← G1-G2-…Gk is less than or equal to the mobility M of the mechanism F.
The basis of the vector space RF of relative velocities between the char-acteristic link n≡nGi and the reference link 1≡1Gi≡0 in the mechanism F ← G1-G2-…Gk dos not vary with the position of the characteristic point on n≡nGi.
When there are various ways to choose the basis of the operational spaces, the bases of RGi in Eq. (2.227) are selected such as the minimum value of FS is obtained. By this choice, the result of Eq. (2.223) fits in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility.
Formulae (2.223)-(2.228) are applicable to any kind of mechanism F ← G1-G2-…Gk:
130 2 Structural parameters
(i) Serial mechanisms, when k=1, q=0, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Single-loop mechanisms, when k=2, q=1, rl≠0 and Gi (i=1,2) are simple open kinematic chains.
(iii) Multi-loop mechanisms, when k≥2, q≥2, rl≠0 and mechanism F ← G1-G2-…Gk contains at least one ternary link.
Kinematic inversion and change of the distal link may be applied to as-sociate a parallel mechanism with any multi-loop mechanism. Although relative motions between components remain the same, structural parame-ters (mobility, degree of overconstraint and degree of structural redun-dancy) of mechanisms are invariant with respect to kinematic inversion or the change of the distal link. Kinematic inversion changes the reference link by repositionning the “observer”. The following two properties may be formulated:
Property 5 (Invariance of structural parameters with the change of the reference link). Mobility, overconstraints and redundancy of the mechanism are invariants under the change of the reference link.
Property 6 (Invariance of structural parameters with the change of the distal link). Mobility, overconstraints and redundancy of the mecha-nism are invariants under the change of the distal link.
The application of these new formulae will be illustrated in chapter 3 for the various types of mechanism to overcome the drawbacks of the CGK mobility formulae widely used for mobility calculation of multi-loop mechanisms. These new formulae do not involve setting up the instantane-ous constraint systems and are successfully applied in chapter 3 to various mechanisms including so called “paradoxical” mechanisms.
3 Structural analysis
Structural analysis of parallel robotic manipulators aims at determining the main structural parameters: connectivity, mobility, redundancy, degree of overconstraint. The following examples will illustrate the definitions, the theorems and the properties presented in the previous chapter. We empha-sise the difference between mobility M and connectivity S; these are often confused in the literature under the name of degree of freedom (DoF).
As we have seen, mobility and connectivity are the key structural pa-rameters also used to calculate the degree of redundancy and the degree of overconstraint. We begin with simple examples of open and closed-loop kinematic chains associated with the limbs of parallel robots and we finish with structural analysis of various types of parallel robots with simple or complex limbs. We illustrate the fact that the formulae demonstrated in the previous chapter for parallel robots are also applicable to single or multi-loop mechanisms that do not obey the classical Chebychev-Grübler-Kutzbach (CGK) formulae.
3.1 Simple open kinematic chains
We recall that an open kinematic chain is a chain in which there is at least one link which carries only one kinematic pairing element, called a mon-ary link. In a simple open kinematic chain (open-loop mechanism) a link is connected by one, or a maximum of two, joints, which means only monary and binary links exist. Simple open kinematic chains are associated with simple limbs of parallel robotic manipulators.
Example 1. The simple open kinematic chain A (1-2-…-n+1) presented in Fig. 3.1a has n+1 links, and p=n cylindrical joints with the same axis. The mobility of each joint is fi=2 (i=1,2,…,p). This kinematic chain has MA=2p=2n, SA=2 and TA=MA-SA=2p-2. The mobility is given by Eq.(2.120). The independent finite displacements in the joints necessary to define the configuration of the simple open kinematic chain are i 1,iϕ + and
i 1,id + (i=1,2,…,n,n+1) - see definition 1 in chapter 2. The independent ve-locities in the joints required to define the velocities of the distal link (n+1)
132 3 Structural analysis
are i 1,i i 1 ,iω ϕ+ += and i 1 ,i i 1,iv d+ += (i=1,2,…,n,n+1) - see definition 2. They form the basis of the joint space (UA)=( 21ϕ , 21d , …, i 1,iϕ + , i 1,id + ,…, n 1,nϕ + ,
n+1,nd ) having the cardinal 2m. The dimension of this joint space gives the
mobility of the mechanism mA A i 1
M dim(U ) f 2m=
= = =∑ . The connectivity is given by Eq.(2.121). The dimension of the operational space can easily be determined by inspection. We can see that the independent finite dis-placements between the distal links 1 and n+1 are x and xϕ (see definition 3). The independent velocities between the distal links are xv and xω (see definition 4). They form the basis of the operational space (RA)=( x xv ,ω ) having the cardinal 2. The dimension of this space gives the connectivity SA=dim(RA)=2. The kinematic chain achieves the addition of linear and angular velocities of the joints ( 1 21 32 1+= = + + +x n n ,nv v v v ... v and
1 21 32 1+= = + + +x n n ,n...ω ω ω ω ω ). Structural redundancy TA=2(m-1) is given by Eq. (2.130).
Figure 3.1b presents the particular case where the p cylindrical joints are replaced by p revolute joints with the same axis. The mobility of each joint
Fig. 3.1. Simple open kinematic chains with n+1 links and p=n joints: (a) MA=2p=2n, SA=2, TA=2(n-1); (b and c) MA=p=n, SA=1, TA=n-1
3.1 Simple open kinematic chains 133
is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, SA=1 and TA=MA-SA= n-1. The only independent velocity between the distal links is xω , and (RA)=( xω ), SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the revolute joints of the mechanism. The mechanism achieves the addition of joint angular velocities ( 1 21 32 1+= = + + +x n n ,n...ω ω ω ω ω ).
The kinematic chain presented in Fig. 3.1c represents another particular case in which the p cylindrical joints are replaced by p prismatic joints having the same direction. The mobility of each joint is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, S=1 and TA=MA-SA=n-1. The only relative independent velocity between the distal and proximal links is xv . We ob-tain (RA)=( xv ) and SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the prismatic joints of the mechanism. The mechanism achieves the addition of joint linear velocities ( 1 21 32 1+= = + + +x n n ,nv v v v ... v ).
Example 2. The simple open kinematic chain in Fig. 3.2 has n+1 links and p=n (n≥3) revolute joints R with parallel axes. The mobility of each joint is fi=1 (i=1,2,…,n). The mechanism has MA=p=n, SA=3 and TA=MA-SA=n-3. The relative independent velocities between the distal links 1 and n+1 are x y zv ,v ,ω , than (RA)=( x y zv ,v ,ω ) and SA=dim(RA)=3. To control these velocities we have to know MA=p=n independent velocities in the revolute joints of the mechanism.
Fig. 3.2. Simple open kinematic chain with n+1 links and p=n revolute joints with parallel axes: MA=p=n, SA=3, TA=n-3
134 3 Structural analysis
Example 3. Each simple open kinematic chain associated with the three limbs of the parallel Cartesian robotic manipulator presented in Fig. 2.1 has MA1=MA2=MA3=4, SA1=SA2=SA3=4 and TA1=TA2=TA3=0. The independ-ent finite displacements in the joints needed to define the configuration of the simple open kinematic chain associated with limb A1 (0A≡1A-2A-3A-4A-5A) are 2 Ad , 3 Aϕ , 4 Aϕ and 5 Aϕ - Fig. 2.2a. The independent velocities in the joints needed to define the velocities of any link of A1-limb are 2 A 3A 4 Ad , ,ϕ ϕ and 5 Aϕ . They form the basis of the joint velocity space
(UA1)=( 2A 3A 4 A 5 Ad , , ,ϕ ϕ ϕ ) having the cardinal 4. The dimension of this space gives the mobility of the simple open mechanism MA1=dim (UA1)=4. The relative independent velocities between the distal links (0A≡1A and 5) of the simple open kinematic chain associated with A1-limb are x y zv ,v ,v
and xω . They form the basis of the operational velocity space (RA1)=( x y z xv ,v ,v ,ω ) having the cardinal 4. The dimension of this opera-tional space gives the connectivity of the simple open kinematic chain as-sociated with A1-limb, SA1=dim (RA1)=4. Similar results are obtained for the A2-limb (0B≡1B-2B-3B-4B-5B) - Fig. 2.2b - and for A3-limb (0C≡1C-2C-3C-4C-5C) - Fig. 2.2c. We have (RA2)=( x y z yv ,v ,v ,ω ) and (RA3)=( x y z zv ,v ,v ,ω ) and SA1=SA2=SA3=4. These results can easily be veri-fied by calculating the rank of the Jacobian matrix associated with each limb.
If we express the translation velocity of point H and the angular veloc-ity of link 5A in the reference coordinate system O0x0y0z0 (Fig. 2.2a) we ob-tain the following kinematic equation for the chain associated with A1-limb
[ ]TT
x y z x y z 2 A 3 A 4 A 5 AA1v v v J dω ω ω ϕ ϕ ϕ = (3.1)
with
[ ]3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43AA1
1 0 0 00 -b S -b S -b S 00 b C +b C b C 0
J0 1 1 10 0 0 00 0 0 0
=
. (3.2)
3.1 Simple open kinematic chains 135
Similar equations can be written for the simple open kinematic chains associated with A2-limb (Fig. 2.2a) and with A3-limb (Fig. 2.2b):
[ ]TT
x y z x y z 2B 3B 4 B 5BA2v v v J dω ω ω ϕ ϕ ϕ = (3.3)
and
[ ]TT
x y z x y z 2C 3C 4C 5CA3v v v J dω ω ω ϕ ϕ ϕ = (3.4)
with the following Jacobian matrices:
[ ]
3B 3B 4B 43B 4B 43B
3B 3B 4B 43B 4B 43BA2
0 b C +b C b C 01 0 0 00 -b S -b S -b S 0
J0 0 0 00 1 1 10 0 0 0
=
(3.5)
and
[ ]
3C 3C 4C 43C 4C 43C
3C 3C 4C 43C 4C 43C
A3
0 -b S -b S -b S 00 b C +b C b C 01 0 0 0
J0 0 0 00 0 0 00 1 1 1
=
(3.6)
The connectivity of the simple open mechanism associated with each limb of the parallel Cartesian robotic manipulator in Fig. 2.2 can also be obtained by using the rank of the Jacobian matrices: SA1=rank ([ ]A1
J )=4, SA2=rank ( [ ]A2
J )=4, SA3=rank ( [ ]A3J )=4. By analyzing the Jacobian ma-
trices (3.2), (3.5) and (3.6) we can see that each simple kinematic chain in Fig. 2.2 has an operational space with a unique basis. After eliminating the two lines containing only zero terms, we can calculate the determinants of
136 3 Structural analysis
Fig. 3.3. Singular configurations of simple open kinematic chain A1: structural singularities (a), (b) and (c) and instantaneous singularity (d)
Jacobian matrices [ ]Ai( 4 4 )J
× (i=1,2,3). For example, the determinant of the
Jacobian matrix [ ]A1( 4 4 )J
× is d=b3Ab4AS4A. This simple open kinematic chain
has structural singularities when b3A=0 and/or b4A=0 (see Fig. 3.3a-c) and an instantaneous singularity when S4A=0, that is 4 A 0ϕ = (see Fig. 3.3d). These singularities do not affect the mobility that remains MA1=4 for all cases in Fig. 3.3. The singularities affect only the connectivity of this sim-ple open kinematic chain that become SA1=3 (Fig. 3.3a, b and d) and SA1=2 - Fig. 3.3c.
Example 4. The simple open kinematic chains presented in Fig. 2.3 are obtained by serial concatenation of two limbs of the parallel Cartesian ro-botic manipulator from Fig. 2.1. The independent velocities of distal link 1B with respect to reference link 1A≡0A in the simple open chain A≡A1-A2 (0A≡1A-2A-…-5A≡5A-…2B-1B) are: x y z xv ,v ,v ,ω , and yω . They form the basis of the operational space (RA≡A1-A2)=( x y z x yv ,v ,v , ,ω ω ) having the cardinal 5. The dimension of this operational space gives the connectivity of the open
3.2 Single-loop kinematic chains 137
simple chain A≡A1-A2: SA1-A2=dim (RA1-A2)=5. This result can also be ob-tained by using Eq. (2.127) :
SA≡A1-A2=dim(RA≡A1-A2)= dim(RA1)+dim(RA2)-dim(RA1∩RA2)=4+4-3=5. This kinematic chain has the following structural parameters: MA=8,
SA=5, TA=3. Similar results may be obtained for the simple open kinematic chain in
Fig. 2.4. The independent velocities of the distal link 1C relative to the ref-erence link 1A≡0A in the simple open chain A≡A1-A3 (0A≡1A-2A-…-5A≡5C-…2C-1C) are: x y z xv ,v ,v ,ω , and zω . They form the basis of the operational space (RA≡A1-A3)=( x y z x zv ,v ,v , ,ω ω ) also having the cardinal 5. Implicitly, the connectivity of the open chain A≡A1-A3 is SA≡A1-A3= dim(RA≡A1-A3)=5. This result can also be obtained as follows: SA≡A1-A3=dim(RA≡A1-A3)= dim(RA1)+dim(RA3)-dim(RA1∩RA3)=4+4-3=5.
We can see that, in general, the dimensions of the velocity joint space and the operational velocity space can easily be determined by inspection for the simple open kinematic chains. The calculation of the rank of the Jacobian matrix is useful just for verification, if necessary.
3.2 Single-loop kinematic chains
We recall that, in a single-loop kinematic chain each link is connected with two other links, which means only binary links exist in a single-loop mechanism.
Example 5. The single-loop kinematic chain in Fig. 3.4 has p=8 one-degree-of-freedom joints and m=8 links, including the fixed reference link (1A≡1B≡0). The single-loop kinematic chain B1 (1A≡0-2A-…-5A≡5B-…2B-1B≡0) can be obtained by fixing together the distal links (1A≡0 and 1B) of the simple open kinematic chain A≡A1-A2 (1A≡0-2A-…-5A≡5B-…2B-1B) from Fig. 2.3. For the single-loop kinematic chain B1, Eq. (2.140) gives rB1=SA≡A1-A2=dim(RA≡A1-A2)=5 (see example 4) and the mobility given by Eq. (2.160) is MB1=8-5=3. We may obtain the same result by considering that the single-loop kinematic chain B1 ←A1-A2 (1A≡0-2A-…-5A≡5B-…2B-1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 5A with 5B of simple open kinematic chains A1 (0A≡1A-2A-3A-4A-5A) and A2 (0B≡1B-2B-3B-4B-5B) in Fig. 2.2a-b and by applying Eqs. (2.144) and (2.164).
The unique basis of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( x y z xv ,v ,v ,ω ) and (RA2)=( x y z yv ,v ,v ,ω ). We re-call that (RA1) and (RA2) represent the bases of the vector spaces of the ve-
138 3 Structural analysis
locities of point H in open kinematic chains A1 and A2 in Fig. 2.2a-b. The connectivity between links 5≡5A≡5B and 1≡1A≡1B≡0 in the single-loop mechanism B1 ←A1-A2 given by Eq. (2.172) is B1
5 / 1 A1 A2S dim( R R ) 3= ∩ = . The relative independent velocities allowed by the single-loop mechanism B1 ←A1-A2 between links 5≡5A≡5B and 1≡1A≡1B≡0 are vx, vy and vz.
Similar results are obtained for the single-loop kinematic chain B2 (0A≡1A-2A-…-5A≡5C-…2C-1C≡0) in Fig. 3.5: rB2=SA≡A1-A3=5 and MB2=8-5=3. This kinematic chain may be obtained by fixing together the distal links (1A≡0 and 1C) of the simple open kinematic chain A≡A1-A3 (0A≡1A-2A-…-5A≡5C-…2C-1C) from Fig. 2.4 or by connecting together the distal
Fig. 3.4 Single-loop kinematic chain B1 ←A1-A2 associated with limbs A and B of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph
Fig. 3.5. Single-loop kinematic chain B1 ←A1-A3 associated with limbs A and C of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph
3.2 Single-loop kinematic chains 139
links 1A≡0 with 1C≡0 and 5A with 5C of simple open kinematic chains A1 (0A≡1A-2A-3A-4A-5A) and A2 (0B≡1B-2B-3B-4B-5B) in Fig. 2.2a and c (see ex-ample 4 for the calculation of SA≡A1-A3 and dim(RA≡A1-A3)).
The basis of the operational space of open simple kinematic chain A3 is (RA3)=( x y z zv ,v ,v ,ω ) and B2
5 / 1 A1 A3S dim( R R ) 3= ∩ = . The relative inde-pendent velocities allowed by single-loop mechanism B2 ←A1-A3 between the links 5 and 1≡0 are also the translational velocities vx, vy and vz.
The degree of overconstraint of these single-loop mechanisms given by Eq. (2.218) is N=5. Five joint parameters lose their independence in each closed loop (r=rB1=rB2=5).
Example 6. The single-loop mechanism B (1A≡0-2A-…-4A≡4B-…2B-1B≡0) in Fig. 3.6a has p=6 one-degree-of-freedom revolute joints and m=6 links including the fixed reference link (1A≡1B≡0). It represents an or-thogonal version of the Sarrus mechanism (Sarrus 1853), and may be ob-tained by eliminating the prismatic joints from the kinematic chain in Fig. 3.4. The simple open kinematic chain A≡A1-A2 (1A≡0-2A-…-4A≡4B-…2B-1B) associated with the closed loop mechanism B has rB=SA≡A1-
A2=dim(RA≡A1-A2)=5 (Fig. 3.6b). The same five relative independent mo-tions ( , , , ,x y z x yv v v ω ω ) exist between the distal links 1B and 1A≡0 in the simple open kinematic chains presented in Figs. 2.3 and 3.6b. The mobility of the single-loop mechanism B given by Eq. (2.160) is MB=6-5=1. We could obtain the same result by considering that the single-loop kinematic chain B ←A1-A2 (1A≡0-2A-…-4A≡4B-…2B-1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 4A with 4B of the simple open kinematic chains A1 (0A≡1A-2A-3A-4A) and A2 (0B≡1B-2B-3B-4B) in Fig. 3.6c-d. Three relative independent motions ( , ,y z xv v ω ) exist between distal links 4A and 1A≡0 in the simple open kinematic chain A1 (Fig. 3.6c) and ( , ,x z yv v ω ) between distal links 4B and 1B≡0 in the simple open ki-nematic chain A2 (Fig. 3.6d). Eq. (2.144) gives rB=SA≡A1-A2= dim(RA≡A1-
A2)=dim(RA1)+dim(RA2)-dim(RA1∩RA2)=3+3-1=5. We can see that, in general, the mobility of a simple closed-loop mecha-
nism can easily be calculated if we know the dimension of the velocity op-erational space of the simple open kinematic chain associated with the closed-loop mechanism.
The unique bases of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( y z xv ,v ,ω ) and (RA2)=( x z yv ,v ,ω ). The connec-tivity between links 4≡4A≡4B and 1≡1A≡1B≡0 in single-loop mechanism B ←A1-A2 given by Eq. (2.172) is B
4 / 1 A1 A2S dim( R R ) 1= ∩ = . The relative
140 3 Structural analysis
Fig. 3.6. Orthogonal version of Sarrus mechanism: structural diagram and graph of the single loop-mechanism (a), associated simple open chain A≡A1-A2 (b), as-sociated simple open chain A1 (c) and associated simple open chain A1 (d)
3.2 Single-loop kinematic chains 141
independent velocity allowed by the single-loop mechanism B ←A1-A2 be-tween links 4≡4A≡4B and 1≡1A≡1B≡0 is vz. The degree of overconstraint of this single-loop mechanism given by Eq. (2.218) is N=5. Five joint pa-rameters lose their independence in the closed loop (r=rB=5).
Note. The following examples illustrate the application of the formulae presented in this section by emphasizing the choice of the bases of opera-tional vector spaces RA1 and RA2 when there are various ways to choose them.
Example 7. Four-bar planar mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.7a, has p=4 revolute joints with parallel axes and m=4 links
Fig. 3.7. Four-bar mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open kinematic chain A (1≡0-2-3-4-5) (b)
Fig. 3.8. Four-bar planar mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 and A2 obtained by splitting up link 4 (b), link 3 (c) and link 2 (d)
142 3 Structural analysis
Fig. 3.9. Four-bar planar parallelogram mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 (1A≡0-2A-3A) and A2 (1B≡0-2B-3B) (b)
Table 3.1. Valid bases of operational vector spaces RA1 and RA2 for the four-bar planar mechanism B ←A1-A2
No. A1 and A2 (RA1) (RA2) (RA1 ∩ RA2) 1 Fig. 3.8b ( x y zv ,v ,ω ) ( zω ) ( zω ) 2 Fig. 3.8c ( x yv ,v ) ( y zv ,ω ) ( yv ) 3 Fig. 3.8c ( x yv ,v ) ( x zv ,ω ) ( xv ) 4 Fig. 3.8d ( zω ) ( x y zv ,v ,ω ) ( zω )
including the fixed reference link (1≡5≡0). It can be obtained by connect-ing in the same link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 3.7b. We can simply observe by inspection that rA=SA=3 and (RA)=( x y zv ,v ,ω ). The mobility given by Eq. (2.160) is MB=4-3=1. The same result can be obtained by considering that the single-loop four-bar mechanism B (1≡0-2-3-4-5≡1) can be obtained by connect-ing in the same links the distal links of the two simple open kinematic chains A1 and A2 (Fig. 3.8). The simple chains A1 and A2 can be obtained by splitting up links 2, 3 or 4. The bases of vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and B
n / 1S are systematized in Table 3.1. These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We note that these bases are also valid for a four-bar planar parallelo-gram mechanism (Fig. 3.9). Special geometric conditions concerning the lengths of opposite links do not allow link 3 to rotate with respect to refer-ence link 1≡0. In the parallelogram planar mechanism, link 3 makes a translational circular motion around the reference link with a constant ori-entation.
Example 8. The four-bar spatial mechanism B (1≡0-2-3-4-5≡1) pre-sented in Fig. 3.10a has p=4 spherical joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same
3.2 Single-loop kinematic chains 143
link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 3.10b. We can simply observe by inspection that six independent mo-tions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.10b). We thus obtain (RA)=( x y z x y zv ,v ,v , , ,ω ω ω ) and rB=SA=6. The mobility given by Eq. (2.160) is MB=12-6=6.
We may obtain the same result by considering that the single-loop four-bar spatial mechanism B (1≡0-2-3-4-1) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡0-2A-3A) and A2 (1B≡0-2B-3B). In Fig. 3.10c, the simple chains A1 and A2 are obtained by splitting up link 3. The bases of the vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and B
3 / 1S are: (RA1)=( x y z x yv ,v ,v , ,ω ω ) and (RA2)=( x z x y zv ,v , , ,ω ω ω ) or (RA2)= ( y z x y zv ,v , , ,ω ω ω ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We recall that (RA1) and (RA2) represent the bases of the vector spaces of the velocities of point H in the open kinematic chains A1 and A2. We thus obtain B
3 / 1S =dim(RA1∩RA2)=4. Eq. (2.144) gives rB=SA≡A1-A2=dim(RA≡A1-A2)= dim(RA1)+dim(RA2)-dim(RA1∩RA2)=5+5-4=6.
Fig. 3.10. Four-bar spatial mechanism B (1≡0-2-3-4-5≡1) (a), its associated sim-ple open chains (b) and (c) and the particular case of four-bar spatial parallelo-gram (d)
144 3 Structural analysis
Link 3 may have four independent motions in a four-bar spatial mecha-nism: two translations and two rotations: x z x yv ,v , ,ω ω or y z x yv ,v , ,ω ω . If we consider link 3 as the characteristic link we obtain SF= B
3 / 1S =4, and Eq. (2.225) indicates two degrees of redundancy for this mechanism. They are given by the rotations of links 2 and 4 about the axes passing through the centres of the spherical joints adjacent to each link. We may consider that this mechanism has two internal and four external mobilities. We note that these structural analyses are also valid for the four-bar spatial mecha-nism with equal lengths of the opposite links (Fig. 10c). This mechanism is also called a four-bar spatial parallelogram.
Example 9. The slider-crank spatial mechanism B (1≡0-2-3-4-5≡1) pre-sented in Fig. 3.11a has p=4 joints RSSP-type and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 2.11b. We can simply observe by inspection that six independent mo-tions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.11b) and (RA)=( x y z x y zv ,v ,v , , ,ω ω ω ) and rB=SA=6.
Fig. 3.11. The slider-crank spatial mechanism B (1≡0-2-3-4-5≡1) of type RSSP (a) and its associated simple open chains (b) and (c)
3.2 Single-loop kinematic chains 145
The mobility given by Eq. (2.160) is MB=8-6=2. We can obtain the same result by considering that the single-loop slider-crank spatial mecha-nism B (1≡0-2-3-4-1) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡0-2A-3A) and A2 (1B≡0-2B-3B). In Fig. 3.11c, the simple chains A1 and A2 are obtained by splitting up link 3. The vector space RA1 may have one of the following basis: (RA1)=( x y z xv ,v ,v ,ω ), (RA1)=( x y z yv ,v ,v ,ω ) or (RA1)=( x y z zv ,v ,v ,ω ). The vector space RA2 has the unique basis (RA1)=( y x y zv , , ,ω ω ω ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We obtain
B3 / 1S =dim(RA1∩RA2)=2. Equation (2.144) gives rB=SA≡A1-A2=dim(RA≡A1-
A2)= dim(RA1)+dim(RA2)-dim(RA1∩RA2)=4+4-2=6. Link 3 may have two independent motions in this spatial mechanism: one translation and one ro-tation. The translation is yv and the rotation may be any component ( xω ,
yω or zω ) of the angular velocity ω of link 3 about the axis passing through the centre of the two spherical joints. If we consider link 3 as the characteristic link, we obtain SF= B
3 / 1S =2 and Eq. (2.225) indicates zero degrees of redundancy. Two independent parameters have to be known to control the motion of all links. One of these parameters is the rotation of link 3 about the axis passing through the centre of the two spherical joints. The second parameter may be the translation of slider 4, or the rotation of crank 2. We can consider that this mechanism has one internal and one ex-ternal mobility.
Example 10. The Bennett mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.12a has p=4 joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting distal links 2A and 4B of the simple open kinematic chains A1 (1≡0-2A) and A2 (1B≡0-2B-3B-4B) in Fig. 3.12b. We can simply observe by inspection that SA1=1 and SA2=3. In ac-cordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Ben-nett four-link mechanism B can have relative motions with respect to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is B
2 / 1S 1= . We may consider that (RA1)=( zω ) and (RA2) is any basis of cardinal 3 obtained by combining zω with two other rotational and translational velocities. Eq. (2.174) gives SA=A1-A2 =1+3-1=3. The mobility of the Bennett mechanism given by Eq. (2.163) is MB=4-3=1. One degree of freedom motion of the Bennett mechanism is due to its special geometric configuration (see Fig. 3.12a): (i) the opposite links (i.e., links that are not concurrent) have the same length denoted by a and b, (ii) the twist angles α and β are equal on oppo-site sides but with different signs and (iii) the link length and twist angles
146 3 Structural analysis
Fig. 3.12. Bennett mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 (1A≡0-2A) and A2 (1B≡0-2B-3B-4B) (b)
3.2 Single-loop kinematic chains 147
Fig. 3.13. Bricard mechanism B (1≡0-2-3-4-5-6-7≡1) (a) and its associated simple open chains A1 (1A≡0-2A) and A2 (1B≡0-2B-3B-4B-5B-6B) (b)
148 3 Structural analysis
must satisfy the relation: asinβ =bsinα. We can see that the Bennett mechanism completely obeys the definitions, the theorems and the formu-lae presented in chapter 2 for quick mobility calculation of single-loop mechanisms. There is no reason to continue to consider this mechanism as “paradoxical” more than one hundred years after its invention (Bennett 1902).
Example 11. The general trihedral 6R single-loop mechanism of Bri-card B (1≡0-2-3-4-5-6-7≡1) in Fig. 3.13a has p=6 joints and m=6 links in-cluding the fixed reference link (1≡7≡0). It can be obtained by connecting distal links 2A and 6B of the open simple kinematic chains A1 (1≡0-2A) and A2 (1B≡0-2B-3B-4B-5B-6B) in Fig. 3.13b. We can simply observe by inspec-tion that SA1=1 and SA2=5. In accordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Bricard mechanism B can have relative motions related to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is B
2 / 1S 1= . We may con-sider that (RA1)=( zω ), and (RA2) is any basis of cardinal 5 obtained by combining three rotational and two translational velocities. Eq. (2.174) gives SA=A1-A2 =1+5-1=5. The mobility of the Bricard mechanism given by Eq. (2.163) is MB=6-5=1. One degree of freedom motion of the Bricard mechanism is due to its special geometric configuration defined by the ex-istence of two pairs of three joint axes intersecting in two different points. In this way, a common line intersects all joint axes. We can see that the Bricard mechanism completely obeys the definitions, the theorems and the formulae presented in this section for quick mobility calculation of single-loop mechanisms. As with the Bennett mechanism, there is no reason to continue to consider the Bricard mechanism as “paradoxical” for so long (Bricard 1927).
3.3 Parallel mechanisms with simple limbs
We recall that a parallel mechanism with simple limbs is a complex mechanism C ← A1-A2-…Ak with two polinary links 1≡1Ai and n≡nAi con-nected by k simple chains Ai (1Ai-2Ai-…-nAi), i=1,2,…k. The two polinary links are the reference platform 1 and the mobile platform n. The simple open kinematic chain Ai is called a simple limb. When the reference plat-form is fixed to the base, we use the notation 1≡1Ai≡0.
We also illustrate the calculation of the structural parameters of various complex closed mechanisms that can be considered as parallel mechanisms with simple limbs. We emphasize examples of robotic manipulators and
3.3 Parallel mechanisms with simple limbs 149
complex closed mechanisms that do not obey the classical criteria for quick calculation of mobility presented in sections 2.1 and 2.2.
Example 12. The parallel mechanism C ← A1-A2-A3 in Fig. 3.14a is ob-tained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 5≡5A≡5B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=10 links including the fixed base, p=11 joints, two independent loops (q=11-10+1=2), k1=3 and k2=0. The limbs are composed of revolute (R) and prismatic (P) joints. Limbs A1 and A2 are of type P||R||R||R and limb A3 of type R||R||R. The axes of the revolute pairs are parallel to the O0x0-axis in limb A1, O0y0-axis in limb A2 and O0z0-axis in limb A3.
Fig. 3.14. Translational parallel mechanism Isoglide2-T2 of type 2PRRR-RRR: structural diagram (a), associated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)
150 3 Structural analysis
Fig. 3.15. CAD model of Isoglide2-T2 of type 2PRRR-RRR
This robot belongs to a family of innovative and modular parallel ro-botic manipulators with 2-6 degrees of mobility and decoupled motions, proposed by the author (Gogu 2002b), and developed under the name of Isogliden-TaRb at the French Institute of Advanced Mechanics (IFMA). In this notation, a represents the number of independent translational motions and b the number of independent rotational motions of the end-effector. The integer numbers a, b and n satisfy the relation n=a+b with 0≤a≤3, 0≤b≤3 and 2≤n≤6. When a or b equal zero, the corresponding notation T or R is missing in Isogliden-TaRb. This is the case for Isoglide2-T2 with a= 2 and b=0. Figure 3.15 presents a CAD model of this parallel robotic ma-nipulator. Other solutions of parallel robots of this family will be analyzed in the following examples.
We simply observe by inspection that the simple limbs (Fig. 3.14b, c, d) have the following unique bases of their operational velocity spaces: (RA1)=( x y z xv ,v ,v ,ω ), (RA2)=( x y z yv ,v ,v ,ω ) and (RA3)=( x y zv ,v ,ω ). The dimensions of these three operational spaces give the connectivity of the simple open chains associated with each limb Ai of the parallel mechanism C ← A1-A2-A3 (SA1= SA2=4 and SA3=3). The connectivity of the mobile plat-form in parallel mechanism C ← A1-A2-A3, given by Eq. (2.181), is C
5 / 1S =
A1 A2 A3dim( R R R ) 2∩ ∩ = . Only two relative independent velocities ( xv and yv ) exist between mobile platform 5 and reference base 1≡0 in paral-lel mechanism C ← A1-A2-A3. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =11. The number of joint parameters that lose their inde-pendence in the parallel mechanism C ← A1-A2-A3 given by Eq. (2.185) is rC=11-2=9. The mobility of the parallel mechanism C ← A1-A2-A3 given by Eq. (2.193) is MC=11-11+2=2. Two independent parameters are necessary
3.3 Parallel mechanisms with simple limbs 151
to define the position of any link in the mechanism. In the parallel mecha-nism C ← A1-A2-A3 in Fig. 3.14a, these parameters are the two displace-ments q1 and q2 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classical CGK formulae but it perfectly complies with the formulae presented in this sec-tion for mobility calculation of parallel mechanisms with simple limbs.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-9=3 and the degree of structural redundancy given by Eq. (2.222) is TC=0. We note that in a parallel robot with simple limbs, rl=0.
Example 13. Planar parallel mechanism C ← A1-A2-A3 in Fig. 3.16a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A), A2
Fig. 3.16. Translational parallel mechanism 3PPR: structural diagram (a), associ-ated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)
152 3 Structural analysis
(1B≡0-2B-3B-4B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=8 links including the fixed base, p=9 joints, two independent loops (q=9-8+1=2), k1=3 and k2=0. The mo-bile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The limbs are composed of revolute R and pris-matic P joints. The three limbs are planar P⊥ P⊥ ||R-type. The axes of the revolute pairs are parallel to O0z0.
We simply observe by inspection that the simple open chains associated with each limb Ai of the parallel mechanism C ← A1-A2-A3 (Fig. 3.16c-e) have the same basis for their operational velocity spaces (RA1)=(RA2)=(RA3)= ( x y zv ,v ,ω ) and the same connectivity SA1=SA2= SA3=3. The connectivity of the mobile platform in parallel mechanism C ← A1-A2-A3, given by Eq. (2.181), is C
4 / 1S = A1 A2 A3dim( R R R ) 3∩ ∩ = . Three relative independent velocities x yv ,v and zω exist between mo-
bile platform 4 and the reference base 1≡0 in the parallel mechanism C ← A1-A2-A3. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =9. The number of joint parameters that lose their inde-pendence in the parallel mechanism C ← A1-A2-A3, given by Eq. (2.191) is rC=9-3=6.
The mobility of the parallel mechanism C ← A1-A2-A3, given by Eq. (2.193) is MC=9-9+3=3. Three independent parameters are necessary to define the position of any link in the mechanism. In the parallel mecha-nism C ← A1-A2-A3 in Fig. 3.16a, these parameters may be the three dis-placements q1, q2 and q3 in the linear actuators mounted on the fixed base. This parallel mechanism obeys theorem 3 and Eq. (2.94). For this reason, the mobility of this parallel mechanism can also be obtained by using the classical CGK formulae presented in section 2.2.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-6=6 and the degree of structural redundancy given by Eq. (2.222) is TC=9-9=0. We note that rl=0.
Example 14. Results similar to those in example 13 are also obtained for the planar parallel mechanism C ← A1-A2-A3 in Fig. 3.17. This mecha-nism has only revolute pairs with parallel axes. It is used in planar parallel robots actuated by three rotating motors usually mounted on the fixed base.
Example 15. The parallel Cartesian robotic manipulator in Fig. 2.1 is a parallel mechanism C ← A1-A2-A3 obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-
3.3 Parallel mechanisms with simple limbs 153
Fig. 3.17. Planar parallel mechanism with rotation actuators C ← A1-A2-A3 (a) and the simple open kinematic chains associated with its simple limbs A1 (b), A2 (c) and A3(d)
154 3 Structural analysis
Fig. 3.18. CAD model of Isoglide3-T3
2C-3C-4C-5C). As we have seen in example 3 – section 2.2, we simply ob-serve by inspection (Fig. 2.2) that (RA1)=( x y z xv ,v ,v ,ω ), (RA2)= ( x y z yv ,v ,v ,ω ), (RA3)=( x y z zv ,v ,v ,ω ) and SAi=4 (i=1,2,3). The connectivity of the mobile platform 5≡5A≡5B≡5B in the parallel mechanism C ← A1-A2-A3, given by Eq. (2.181), is C
5 / 1S = A1 A2 A3dim( R R R ) 3∩ ∩ = . Three rela-tive independent translational velocities ( xv , yv and zv ) exist between the mobile and fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =12. The number of joint parameters that lose their inde-pendence in the parallel mechanism C ← A1-A2-A3 given by Eq. (2.191) is rC=12-3=9. The mobility of parallel mechanism C ← A1-A2-A3 given by Eq. (2.193) is MC=12-12+3=3. Three independent parameters are necessary to define the position of any link in the mechanism, for example the dis-placements q1, q2 and q3 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classi-cal CGK formulae but it completely obeys to new formulae for mobility calculation of parallel mechanisms with simple limbs.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-9=3 and the degree of structural redundancy given by Eq. (2.222) is TC=12-12=0 with rl=0.
The parallel robotic manipulator in Fig. 2.1 also belongs to the Isoglide family. It represents Isoglide3-T3 of type T3 (Fig. 3.18).
Example 16. Four versions of the parallel robotic manipulators with simple limbs Isoglide4-T3R1 are analysed in this example. They are ob-tained from the solution in Fig. 3.19 by changing the type of joints
3.3 Parallel mechanisms with simple limbs 155
(a)
(b)
Fig. 3.19. The parallel manipulator Isoglide4-T3R1 developed at the French Insti-tute of Advanced Mechanics in a modular design approach: partial view (a) and general view (b)
156 3 Structural analysis
Fig. 3.20. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 3-PRRRR+1PRRR: structural diagram (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d), A3 (e) and A4 (f)
3.3 Parallel mechanisms with simple limbs 157
Fig. 3.21. CAD model of 3-PRRRR+1PRRR-type Isoglide4-T3R1 parallel robot
connecting the four limbs to the moving platform. We can see that the so-lutions in Figs. 3.15 and 3.18 are particular cases derived from this solu-tion.
In the four solutions of Isoglide4-T3R1, the mobile platform has Schön-flies motions that are three independent translations T and one rotation R about a fixed direction axis.
Isoglide4-T3R1 (3-PRRRR+1PRRR-type) is a parallel mechanism C ← A1-A2-A3-A4 obtained by parallel concatenation of 4 simple limbs A1 (1A≡0-2A-3A-4A-5A-6A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C-6C) and A4 (1D≡0-2D-3D-4D-5D-6D) - see Figs. 3.20 and 3.21. The mechanism has three limbs of type P||R||R||R⊥R and one limb of type P||R||R||R. In each limb, the axes of the first three revolute joints are parallel to the direc-tion of the prismatic joint. The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs A1, A2 and A3 have or-thogonal directions. The actuated prismatic joints of limbs A3 and A4 have parallel directions.
For ( / 2, / 2)yϕ π π∈ − + the simple open kinematic chains associated with the four limbs have (Fig. 3.20c-f): (RA1)=( x y z x yv ,v ,v , ,ω ω ), (RA3)=(RA4)= ( x y z y zv ,v ,v , ,ω ω ), (RA2)= ( x y z yv ,v ,v ,ω ), and SA1=SA3=SA4
=5, SA2=4. The connectivity of the platform 6≡6A≡5B≡6C≡6D in parallel mechanism C ← A1-A2-A3-A4 is C
6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = . Four relative independent velocities x yv ,v , zv and yω exist between the
158 3 Structural analysis
(a) (b)
(c) Fig. 3.22. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 4-PRRRR (a), associated graph (b), CAD model (c)
3.3 Parallel mechanisms with simple limbs 159
mobile and the fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =19. The number of joint parameters that lose their independence in the parallel ro-bot given by Eq. (2.191) is rC=19-4=15, and the mobility given by Eq. (2.193) is MC=19-15=4. Four independent joint parameters are necessary to completely describe the motion of the mechanism. In Isoglide4-T3R1 presented in Figs. 3.19-3.24 these parameters are the displacements of four linear actuators qi (i=1,2,…,4) mounted on the fixed base.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=18-15=3 and the degree of structural redundancy given by Eq. (2.222) is TC=19-19=0 with rl=0.
Figures 3.22 and 3.23 present solutions of Isoglide4-T3R1 with two and, respectively, one degree of overconstraint. The solution presented in Fig. 3.22 is not overconstrained (NC=0).
Isoglide4-T3R1 of type 4-PRRRR -see Fig. 3.22- has four limbs of type P||R||R||R⊥R and the following structural parameters: (RA1)=(RA2)= ( x y z x yv ,v ,v , ,ω ω ), (RA3)= (RA4)=( x y z y zv ,v ,v , ,ω ω ), SAi=5 (i=1,2,3,4),
C6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , r=5+5+5+5-4=16, MC=20-6=4,
NC=18-16=2. The last revolute joint of limb A2 has an idle mobility de-noted by R¤. This limb is of type PRRRR¤.
Fig. 3.23. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 2-RRS+1-PRRRR+1-PRRR (a) and associated graph (b)
160 3 Structural analysis
(a) (b)
(c) Fig. 3.24. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 3-PRRS+1-PRRR (a), associated graph (b), CAD model (c)
3.3 Parallel mechanisms with simple limbs 161
Isoglide4-T3R1 of type 2-PRRS+1-PRRRR+1-PRRR -see Fig. 3.23- has two idle mobilities (one in each spherical joint S=RRR¤) and the following structural parameters: (RA1)=( x y z x yv ,v ,v , ,ω ω ), (RA2)=( x y z yv ,v ,v ,ω ), (RA3)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA4)=( x y z x y zv ,v ,v , , ,ω ω ω ), SA1=5,
SA2=4, SAi=6 (i=3,4), C6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , rC=21-
4=17, MC=21-17=4, NC=18-17=1. Isoglide4-T3R1 of type 3-PRRS+1-PRRR - see Fig. 3.24- has three idle
mobilities (one in each spherical joint S=RRR¤) and the following struc-tural parameters: (RA1)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA2)=( x y z yv ,v ,v ,ω ), (RA3)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA4)=( x y z x y zv ,v ,v , , ,ω ω ω ), SA2=4, SAi=6
(i=1,3,4), C5 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , rC=22-4=18 MC=22-
18=4, NC=18-18=0. Example 17. The parallel robotic manipulator Isoglide5-T3R2 (Fig.
3.25) is another solution in the Isoglide family.
Fig. 3.25. Parallel mechanism with simple limbs Isoglide5-T3R2 (a) and its asso-ciated graph (b)
162 3 Structural analysis
Fig. 3.26. Simple open chains associated with limbs A1 (a), A2 (b), A3 (c), A4 (d) and A5 (e) of Isoglide5-T3R2
Parallel mechanism C ← A1-A2-A3-A4-A5 is obtained by parallel concate-nation of 5 simple limbs A1 (1A≡0-2A-3A-4A- 5A-6A-7A), A2 (1B≡0-2B-3B-4B-5B-6B), A3 (1C≡0-2C-3C-4C-5C-6C-7C), A4 (1D≡0-2D-3D-4D-5D-6D) and A5 (1E≡0-2E-3E-4E-5E-6E-7E).
3.3 Parallel mechanisms with simple limbs 163
For (0, / 2)α π∈ and ( / 2, / 2)β π π∈ − + the simple open kinematic chains associated with the five limbs have (Fig. 3.26c-e): (RA1)=( x y z xv ,v ,v , , ,α βω ω ω ), (RA2)=(RA4)=( x y zv ,v ,v , ,α βω ω ), (RA3)= (RA5)=( x y z zv ,v ,v , , ,α βω ω ω ) and SA1=SA3=SA5 =6, SA2= SA4=5.
The connectivity of the platform 7≡7A≡6B≡7C≡6D≡7E is C7 / 1 A1 A2 A3 A4 A4S dim( R R R R R ) 5= ∩ ∩ ∩ ∩ = . Five relative independent
velocities x y zv ,v ,v , αω and βω exist between the mobile and the fixed
platforms. The dimension of the joint vector space U is dim(U)= pii 1
f=∑
=28. The number of joint parameters that lose their independence in the parallel robot is rC=28-5=23 and the mobility MC=28-23=5. Five inde-pendent joint parameters are necessary to completely describe the motion of the mechanism. In the Isoglide5-T3R2 these parameters are the dis-placements of the five linear actuators qi (i=1,2,…,5) mounted on the fixed base.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=24-23=1 and the degree of structural redundancy given by Eq. (2.222) is TC=28-28=0 with rl=0.
Example 18. The Gough-Stewart parallel robot 6-UPS-type in Fig. 3.27a is a parallel mechanism C ← A1-A2-A3-A4-A5-A6 obtained by parallel concatenation of 6 simple identical limbs A1 (1A≡0-2A-3A-4A), A2 (1B≡0-2B-3B-4B), A3 (1C≡0-2C-3C-4C), A4 (1D≡0-2D-3D-4D), A5 (1E≡0-2E-3E-4E) and A6 (1F≡0-2F-3F-4F). Each telescopic limb is connected to the fixed platform by a universal joint U and to the mobile platform by the spherical S joint (Fig. 3.27c). We have (RAi)=( x y z x y zv ,v ,v , , ,ω ω ω ) and SAi=6 (i=1,2,…,6). The connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1-A2-A3-A4-A5-A6 is C
4 / 1S 6= . Six relative independent velocities x y z x yv ,v ,v , ,ω ω and zω exist between the mobile and the fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =36. The number of joint parame-ters that lose their independence in the parallel robot is rC=36-6=30 and the mobility M=36-30=6. Six independent joint parameters are necessary to completely describe the motion of the mechanism. In Gough-Stewart type parallel robots these parameters are the displacements of the six linear actuators qi (i=1,2,…,6) integrated in the prismatic joints. We can see that for this type of parallel robot the actuators are not mounted on the fixed base. This solution is not overconstrained (NA=30-30=0) and non redun-dant (TA=36-36=0).
164 3 Structural analysis
Fig. 3.27. Gough-Stewart parallel mechanism of type 6-UPS (a), associated graph (b), simple open chain associated with limbs A1 (c)
The Gough-Stewart parallel robot 6-SPS-type in Fig. 1.2 is obtained by parallel concatenation of 6 simple identical limbs SPS-type. Each tele-scopic limb is connected to the fixed and moving platforms by spherical joints. In this case, both the limb connectivity and the connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1-A2-A3-A4-A5-A6 are unchanged (RAi)=( x y z x y zv ,v ,v , , ,ω ω ω ), SAi=6
(i=1,2,…,6) and C4 / 1S 6= . The dimension of the joint vector space U is
dim(U)= pii 1
f=∑ =42. The number of joint parameters that lose their inde-
pendence in the parallel robot is rC=30 and the mobility MC=42-30=12. Twelve independent joint parameters are necessary to completely describe the motion of all links in this mechanism. We note that, among these 12 joint parameters, 6 represent internal limb mobilities. Each limb has an in-ternal mobility that is a rotation around the axis directed to the centers of its spherical joints. These internal mobilities do not influence the motion of the mobile platform. For this reason, just six joint parameters are in fact used to actuate the robot. This non overconstarined solution (NC=0) has six degrees of redundancy (TC=42-36=6).
Example 19. The parallel mechanism C ← A1-A2-A3 in Fig. 3.28a is ob-tained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A), A2 (1B≡0-2B- 3B-4B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0.
3.3 Parallel mechanisms with simple limbs 165
Fig. 3.28. General case of 3-RPS mechanism (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d) and A3 (e)
166 3 Structural analysis
Each telescopic limb is connected to the base by a revolute pair and to the mobile platform by a spherical pair. This parallel mechanism called 3-RPS was proposed by Hunt (1983) in his pioneering article on structural synthesis of parallel mechanisms. We can observe by inspection that RA1 has a unique basis (RA1)=( y z x y zv ,v , , ,ω ω ω ), and RA2 and RA3 have multi-ple bases. The basis of RA2 is formed by any combination of five independ-ent velocities of the point H in the open chain associated with limb A2. We may consider one of the following sets of bases for RA2: ( y z x y zv ,v , , ,ω ω ω ), ( x z x y zv ,v , , ,ω ω ω ), ( x y x y zv ,v , , ,ω ω ω ), ( x y z x zv ,v ,v , ,ω ω ), ( x y z y zv ,v ,v , ,ω ω ) or ( x y z x yv ,v ,v , ,ω ω ). The same five possibilities exist for the basis of RA3. In accordance with Remark 2 and Eq. (2.197), the bases of RA2 and RA3 are selected such that the mini-mum value of C
4 / 1S is obtained. We have C C4 / 1 4 / 1S dim( R )= = A1 A2 A3dim( R R R )∩ ∩ =3, SA1=SA2=SA3=5, rC=3
and MC=3. The complete set of possible bases for C4 / 1R is: ( x y z, ,ω ω ω ),
( z x yv , ,ω ω ), ( z x zv , ,ω ω ), ( z y zv , ,ω ω ), ( y x yv , ,ω ω ), ( y x zv , ,ω ω ), ( y y zv , ,ω ω ), ( y z xv ,v ,ω ), ( y z yv ,v ,ω ), ( y z zv ,v ,ω ). As we can see, the pro-posed method gives us, in a very simple way, a complete and correct result concerning the mobility and connectivity of a 3-RPS parallel manipulator without studying the instantaneous constraint system. As previously men-tioned, a given parallel mechanism has an infinite number of positions and for each position we can associate an instantaneous constraint system. In these conditions, an instantaneous constraint study could offer just some partial results on the instantaneous connectivity of the mobile platform de-pending on the configuration of the mechanism (Huang and Fang 1996; Huang et al. 1996; Fang and Huang 1997; Huynh 2004).
This parallel manipulator is not overconstrained (N=0) and is non re-dundant (T=0).
Example 20. The parallel mechanism C ← A1-A2-A3 in Fig. 3.29a is ob-tained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A-6A), A2 (1B≡0-2B-3B-4B-5B-6B) and A3 (1C≡0-2C-3C-4C-5C-6C). The mobile platform is 6≡6A≡6B≡6C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0. Each telescopic limb is connected to the base and to the mobile platform by two intersecting revolute pairs (Cardan joints). This parallel mechanism called 3-RRPRR or 3-UPU was proposed by (Tsai 1996; Tsai and Joshi 2000) as a translational parallel manipulator. In order to keep the moving platform from changing its orientation, Tsai pointed out the following two geometric conditions for the four revolute joint axes
3.3 Parallel mechanisms with simple limbs 167
Fig. 3.29. 3-UPU mechanism (a), associated graph (b), simple open chains associ-ated with limbs A1 (c), A2 (d) and A3 (e)
168 3 Structural analysis
of each limb: (i) the first revolute joint axis is parallel to the last revolute joint axis and (ii) the two intermediate revolute joint axes are parallel to one another. These geometric conditions do not imply the intersection of the first two revolute joint axes specific to the universal joint.
Hence, any 3-RRPRR parallel manipulator whose revolute joint axes sat-isfy the above conditions will be a translational manipulator (Tsai and Joshi 2000). These geometric conditions were generalized later by Di Gregorio and Parenti-Castelli (1998, 2002). This parallel manipulator has multiple bases for RA1, RA2 and RA3. We can observe by inspection (Fig. 3.22c-e) that RA1 could have two different bases, RA2 and RA3 three different bases each. The two possible bases of RA1 are: ( x y z x zv ,v ,v , ,ω ω ) or ( x y z x yv ,v ,v , ,ω ω ). The three bases of RA2 and RA3 are: ( x y z x zv ,v ,v , ,ω ω ), ( x y z x yv ,v ,v , ,ω ω ), ( x y z y zv ,v ,v , ,ω ω ). In accordance with remark 2 and Eq. (2.197), the bases of RA1, RA2 and RA3 are selected so that the minimum value of C
6 / 1S is obtained. We thus obtain SA1=SA2=SA3=5, C6 / 1( R )=
A1 A2 A3( R R R )∩ ∩ =( x y zv ,v ,v ), C C4 / 1 4 / 1S dim( R )= =3, rC=12, MC=3,
NC=0, TC=0. The three independent translations of the mobile platform are obtained by actuating the prismatic pairs integrated in the limbs.
3.4 Parallel mechanisms with complex limbs
We recall that a parallel robotic manipulator with complex limbs denoted by D ← A1-A2-…Ak1-E1-E2-…-Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is connected to the reference plat-form 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex. A complex limb contains at least one closed loop. The elementary limbs are denoted by Ai (i=1,2,…k1) and the complex limbs by Ej (j=1,2,…k2) with k1+k2=k. When the reference platform is fixed to the base, we use the nota-tion 1≡1Ai≡1Ej≡0. We have denoted by k1 the total number of simple limbs, and by k2 the total number of complex limbs.
We illustrate the calculation of the structural parameters of parallel ro-botic manipulators with complex limbs which contain various combina-tions of planar or spatial closed loops. We emphasize examples which do not comply with the classical CGK formulae.
Example 21. The parallel mechanism 2PPa-type in Fig 3.30a represents a maximally-regular and fully-isotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1-E2 in Fig. 3.30 is obtained by parallel concatenation of two complex
3.4 Parallel mechanisms with complex limbs 169
Fig. 3.30. Fully-isotropic overconstrained translational parallel mechanism 2PPa-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
170 3 Structural analysis
Fig. 3.31. Fully-isotropic non overconstrained translational parallel mechanism PPass-PCPass-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
3.4 Parallel mechanisms with complex limbs 171
limbs E1 (1A≡0-2A-3A-4A-5A) and E2 (1B≡0-2B-3B-4B-5B). A planar paral-lelogram closed loop Pa is included in each complex limb. The mechanism has the following structural characteristics: m=8 links including the fixed base, p=10 joints, three independent loops (q=10-8+1=3), k1=0 and k2=2. The mobile platform is 5≡5A≡5B and the reference platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type P⊥Pa are identical. Each limb is composed of a prismatic pair P and a planar parallelogram loop Pa. The axes of the revolute joints in the parallelogram loops are parallel to the O0z-axis, and the directions of the two prismatic pairs are parallel to the axes O0x and O0y.
We simply observe by inspection (Fig. 3.30c,d) that the bases of the op-erational velocity spaces of two limbs are (RE1)=(RE2)=( x yv ,v ). The di-mensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of parallel mechanism D ← E1-E2. The connectivity of the mobile platform in the parallel mecha-nism D ← E1-E2, given by Eq. (2.198), is D
5 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent translational velocities xv and yv exist between mobile platform 5 and reference base1≡1A≡1B≡0 in the parallel mechanism D ← E1-E2. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =10. Three joint parameters lose their independence in each planar parallelogram (rl=6).
The number of joint parameters that lose their independence in the par-allel mechanism given by Eq. (2.200) is rD=4-2+6=8. The mobility of the parallel mechanism D ← E1-E2 given by Eq. (2.202) is MD=10-4+2-6=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the prismatic actuators mounted on the fixed base (underlined joints).
Equations (2.217) and (2.222) indicate that the parallel mechanism has ten degrees of overconstraint (ND=18-4+2-6=10) and zero degrees of redundancy (TD=10-4-6=0). We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.31 presents a non overconstrained solution of type PPass-PCPass. Two spherical joints S are included in each parallelogram loop.
A cylindrical joint C is also integrated in the E2-limb. We have denoted by Pass the parallelogram loop containing two spherical joints. Six joint parameters lose their independence in a Pass-type parallelogram loop. This solution has the following structural parameters: m=9, p=11, q=3, k1=0, k2=2, (RE1)=( x y yv ,v ,ω ), (RE2)=( x y z x zv ,v ,v , ,ω ω ), SE1=3, SE2=5, D
5 / 1S = 2, dim(U)=20, rl=12, rD=18, MD=2, ND=0, TD=0.
172 3 Structural analysis
Example 22. The parallel mechanism 2PaPa-type in Fig 3.32a also represents a maximally-regular and fully-isotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1-E2 in Fig. 3.32a is obtained by parallel concatenation of two complex limbs E1 (1A≡0-2A-3A-4A-5A-6A-7A) and E2 (1B≡0-2B-3B-4B-5B-6A-7A). Two planar parallelogram loops Pa are included in each complex limb. The mechanism has the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=16-12+1=5), k1=0 and k2=2. The mobile platform is 7≡7A≡7B and the refer-ence platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type Pa||Pa are identical. Each limb is composed of two planar parallelogram loops Pa. The axes of the revolute joints in the parallelogram loops are parallel to parallel to O0z.
Fig. 3.32. Fully-isotropic overconstrained translational parallel mechanism 2PaPa-type: kinematic structure (a), associated graph (b), complex chains associ-ated with limbs A1 (c) and A2 (d)
3.4 Parallel mechanisms with complex limbs 173
We simply observe by inspection (Fig. 3.32c,d) that the bases of the op-erational velocity spaces of both limbs are (RE1)=(RE2)=( x yv ,v ). The di-mensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of the parallel mechanism D ← E1-E2. The connectivity of the mobile platform in the parallel mecha-nism D ← E1-E2, given by Eq. (2.198), is D
5 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent translational velocities xv and yv exist between mobile platform 7 and reference base 1≡1A≡1B≡0 in parallel mechanism D ← E1-E2. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =16.
Fig. 3.33. Fully-isotropic non overconstrained translational parallel mechanism Pa*Pa*C-Pa*Pa*S-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
174 3 Structural analysis
Three joint parameters lose their independence in each planar parallelo-gram (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=4-2+12=14. The mobility of the parallel mechanism D ← E1-E2 given by Eq. (2.202) is MD=16-4+2-12=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has 16 degrees of overconstraint (ND=30-4+2-12=16) and zero degrees of redun-dancy (TD=16-4-12=0).
We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.33 presents a non overconstrained solution Pa*Pa*C-Pa*Pa*S-type. One cylindrical and one spherical joint are introduced in each parallelogram loop denoted by Pa*. A cylindrical joint C is also inte-grated in the E1-limb, and a spherical joint in the E2-limb. Six joint pa-rameters lose their independence in a Pa*-type parallelogram loop. This solution has the following structural parameters: m=14, p=18, q=5, k1=0, k2=2, (RE1)=( x y zv ,v ,v ), (RE2)=( x y x y zv ,v , , ,ω ω ω ), SE1=3, SE2=5, D
5 / 1S = 2, dim(U)=32, rl=24, rD=30, MD=2, ND=0, TD=0.
Example 23. The parallel mechanism presented in Fig 3.34a represents an innovative maximally-regular and fully-isotropic spherical parallel wrist proposed by the author in (Gogu 2005f). The parallel mechanism D ← A1-E1
in Fig. 3.34 is obtained by parallel concatenation of a simple limb A1 (1A≡0-2A-3A) and a complex limb E1 (1B≡0-2B-3B-4B-5B-6B-7B-8B). A planar parallelogram closed loop Pa is included in the complex limb E1.
The mechanism has the following structural characteristics: m=9 links including the fixed base, p=10 joints, two independent loops (q=10-9+1=2), one simple (k1=1) and one complex (k2=1) limb. The mobile plat-form is 8≡3A≡8B and the reference platform is fixed to the base (1≡1A≡1B≡0). The simple limb A1 is composed of two revolute pairs with orthogonal and intersecting axes R⊥R-type. The complex limb E1 is of typePa⊥R||R||R⊥R. The axes of the last revolute joints of the two limbs are parallel. The axes of the revolute joints in the parallelogram loop are parallel to the x0y0-plane and the three revolute pairs have axes parallel to O0z. We simply observe by inspection that the limbs (Fig. 3.34c,d) have the following bases of their operational velocity spaces: (RA1)=( xy z,ω ω ) and (RE1)=( x y z xy zv ,v ,v , ,ω ω ). The rotation axis of velocity xyω lies in the x0y0-plane. The dimensions of these operational spaces SA1=2 and
3.4 Parallel mechanisms with complex limbs 175
Fig. 3.34. Fully-isotropic overconstrained parallel wrist RR-PaRRRR-type with two degrees of freedom: kinematic structure (a), associated graph (b), kinematic chains associated with limbs A1 (c) and A2 (d)
SE1=5 give the connectivity of the kinematic chains associated with each limb of parallel mechanism D ← A1-E1. The connectivity of mobile platform in the parallel mechanism D ← A1-E1, given by Eq. (2.198), is
D8 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent angular velocities xyω and zω exist between the
mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← A1-E1. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =10. Three joint parameters lose their independence in the planar parallelogram loop incorporated in limb E1 (rl=3). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=7-2+3=8. The mobility of parallel mechanism D ← A1-E1 given by Eq. (2.202) is MD=10-7+2-3=2. Two independent pa-rameters are necessary to define the position and orientation of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equa-tions (2.217) and (2.222) indicate that the parallel mechanism has four de-grees of overconstraint (ND=12-8=4) and zero degrees of redundancy (TD=10-7-3=0).
176 3 Structural analysis
Fig. 3.35. Fully-isotropic non overconstrained parallel wrist RR-Pa*RRS-type with two degrees of freedom: kinematic structure (a), associated graph (b), kine-matic chains associated with limbs A1 (c) and A2 (d)
We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.35 presents a non overconstrained solution RR-Pa*RRS-type. One cylindrical and one spherical joint are introduced in the parallelogram loop denoted by Pa*. The complex limb E2 of type Pa*RRS includes a Pa*-type parallelogram loop, two revolute joints with parallel axes and a spherical joint. This solution has the following structural parameters: m=8, p=9, q=2, k1=1, k2=1, (RE1)=( xy z,ω ω ), (RE2)=( x y x y zv ,v , , ,ω ω ω ), SA1=2,
SE1=6, D5 / 1S = 2, dim(U)=14, rl=6, rD=12, MD=2, ND=0, TD=0.
Example 24. Orthoglide (Fig. 3.36) is a translational parallel robot (Wanger and Chablat 2000; Chablat and Wanger 2003) developed by IRCCyN (Institut de Recherche en Communications et Cybernétique de Nantes). The parallel mechanism D ← E1-E2-E3 in Figs. 3.36a and 3.37a is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A), E2 (1B≡0-2B-3B-4B-5B) and E3 (1C≡0-2C-3C-4C-5C). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 revolute and prismatic joints, five independent loops (q=21-17+1=5), three complex limbs (k1=0, k2=3). The mobile platform is
3.4 Parallel mechanisms with complex limbs 177
5≡5A≡5B≡5C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb P⊥R⊥Pa⊥R-type is composed of one prismatic and two revolute pairs combined with a planar parallelogram loop. The axes of the revolute joints are perpendicular to the axes of the parallelogram loop. In each limb, the prismatic pair is actuated and its di-rection is perpendicular to the axis of the revolute pair. The directions of the three actuated prismatic pairs are also perpendicular to each other.
We simply observe by inspection that the complex limbs (Fig. 3.37c-e) have the following unique bases of their operational velocity spaces: (RE1)=( x y z yv ,v ,v ,ω ), (RE2)=( x y z zv ,v ,v ,ω ) and (RE3)=( x y z xv ,v ,v ,ω ). The dimensions of these three operational spaces SEi=4 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of the parallel mechanism D ← E1-E2-E3. The connectivity of mobile platform in
Fig. 3.36. Translational parallel robot Orthoglide (courtesy of CNRS-IRCCyN)
178 3 Structural analysis
Fig. 3.37. Parallel robot Orthoglide: structural diagram (a), associated graph (b), complex open chains associated with limbs A1 (c), A2 (d) and A3 (e)
3.4 Parallel mechanisms with complex limbs 179
the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is D7 / 1S =
E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and
zv exist between the mobile platform 7 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =21. Three
joint parameters lose their independence in each planar parallelogram loop (rl=9). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=12-3+9=18. The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=21-12+3-9=3. Three independent parameters are necessary to define the posi-tion of any link in the mechanism. In the Orthoglide robot, these parame-ters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the Orthoglide parallel mechanism has twelve degrees of overconstraint (ND=30-12+3-9=12) and Eq. (2.222) gives zero degrees of redundancy (TD=21-12-9=0).
Example 25. The fully-isotropic translational parallel mechanism 3-PPaPa-type (Gogu 2004b) in Fig. 3.38 is obtained by parallel concatena-tion of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A-6A-7A-8A), E2 (1B≡0-2B-3B-4B-5B-6B-7B-8B) and E3 (1C≡0-2C-3C-4C-5C-6C-7C-8C). The mechanism has the following structural characteristics: m=20 links including the fixed base, p=27 revolute and prismatic joints, eight independent loops (q=27-20+1=8) and three complex limbs (k1=0, k2=3). The mobile platform is 8≡8A≡8B≡8C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb P||Pa||R-type is composed of one prismatic joint and two planar parallelogram loops. The prismatic joints of the three limbs are actuated and their directions are mutually perpendicu-lar.
We simply observe by inspection that the complex limbs (Fig. 3.38c-e) have the following unique bases of their operational velocity spaces: (RE1)=( x y zv ,v ,v ), (RE2)=( x y zv ,v ,v ) and (RE3)=( x y zv ,v ,v ). The dimen-sions of these three operational spaces SEi=3 (i=1,2,3) give the connec-tivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3.
The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is D
7 / 1S = E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and zv exist between the mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3.
180 3 Structural analysis
Fig. 3.38. Fully-isotropic overconstrained translational parallel mechanism 3-PPaPa-type: kinematic structure (a), associated graph (b), complex chains associ-ated with limbs A1 (c), A2 (d) and A3 (e)
3.4 Parallel mechanisms with complex limbs 181
Fig. 3.39. Fully-isotropic non overconstrained translational parallel mechanism 3-PRPa*Pa*-type: kinematic structure (a), associated graph (b)
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =27. Three
joint parameters lose their independence in each planar parallelogram loop (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=9-3+18=24.
The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=27-9+3-18=3. Three independent parameters are necessary to define the position of any link in the mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.38 has 24 degrees of overconstraint (ND=48-9+3-18=24) and Eq. (2.222) gives zero degrees of redundancy (TD=27-9-18=0).
182 3 Structural analysis
Fig. 3.40. Complex chains associated with limbs A1 (a), A2 (b) and A3 (c) of the fully-isotropic non overconstrained translational parallel mechanism 3-PRPa*Pa*-type
3.4 Parallel mechanisms with complex limbs 183
Figure 3.39 presents a non overconstrained solution 3-PRPa*Pa*R-type. One cylindrical and one spherical joint are introduced in the paral-lelogram loop denoted by Pa*. Two revolute joints with perpendicular axes are introduced in each complex limb outside the parallelogram loops. The first revolute joints of the three limbs are mutually orthogonal. The last revolute joints of the three limbs are also mutually orthogonal. We simply observe by inspection that the complex limbs isolated from the par-allel mechanism (Fig. 3.40a-c) have the following bases of their opera-tional velocity spaces: (RE1)=( x y z x yv ,v ,v , ,ω ω ), (RE2)=( x y z y zv ,v ,v , ,ω ω ) and (RE3)= ( x y z x zv ,v ,v , ,ω ω ). Six joint parameters lose their independence in each parallelogram loop. This solution has the following structural pa-rameters: m=26, p=33, q=8, k1=0, k2=3, SE1= SE2=SE3=5, D
10 / 1S = 3, dim(U)=51, rl=36, rD=48, MD=3, ND=0, TD=0.
Example 26. The Delta parallel mechanism 3-RPa4s-type invented by Clavel (1990) is widely used today in translational parallel robots. The so-lution presented in Fig. 3.41a (Clavel 1990) represent a hybrid robot in which a translational mechanism Delta-type with rotating actuators (Fig. 3.41a) is serially concatenated with a revolute joint. The translational par-allel mechanism (Fig. 3.42a) is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A), E2 (1B≡0-2B-3B-4B-5B) and E3 (1C≡0-2C-3C-4C-5C). The mechanism has the following structural characteristics: m=11 links including the fixed base, p=15 joints, five independent loops (q=15-11+1=5) and three complex limbs (k1=0, k2=3). The mobile plat-form is 5≡5A≡5B≡5C and the reference platform is fixed to the base
Fig. 3.41. Hybrid mechanism Delta-type invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Delta-type with rotating actuators (b)
184 3 Structural analysis
Fig. 3.42. Translational Delta mechanism 3-RPa4s-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)
(1≡1A≡1B≡1C≡0). Each complex limb RPa4s-type is composed of one revolute and a spatial parallelogram loop containing 4 spherical joints. The revolute joints of the three limbs are actuated and the directions of their ro-tation axes lie in a plane. Moving platform 5 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.42c-e) may realize three transla-tion velocities x y zv ,v ,v and two angular velocities iω and i+1ω (i=1 for A-limb, i=3 for B-limb and i=5 for C-limb): (RE1)= ( x y z 1 2v ,v ,v , ,ω ω ) (RE2)=( x y z 3 4v ,v ,v , ,ω ω ) and (RE3)= ( x y z 5 6v ,v ,v , ,ω ω ),
3.4 Parallel mechanisms with complex limbs 185
The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is D
5 / 1S =
E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and
zv exist between the mobile platform 5 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =39. Six
joint parameters lose their independence in each parallelogram loop Pa4S-type (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=15-3+18=30. The mobility of parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=39-15+3-18=9. Nine independent parameters have to be known to de-fine the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=39-15-18=6) for the Delta parallel mechanism. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining three degrees of mo-bility are external mobilities that are transmitted from the input to the out-put of the parallel mechanism. They represent the independent parameters
Fig. 3.43. Parallel mechanism linear Delta-type invented by R. Clavel and pre-sented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Delta-type with linear actuators (b)
186 3 Structural analysis
Fig. 3.44. Translational Delta mechanism 3-PPa4s-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)
that are necessary to define the position of the moving platform. These pa-rameters are the three angular displacements q1, q2 and q3 of the rotating actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.42 is not overconstraint (ND=30-15+3-18=0).
The same structural analysis and parameters are applicable to the trans-lational parallel Delta-type mechanism with linear actuators in Figs. 3.43b and 3.44a. The only difference is that the three degrees of external mobili-ties are associated with three linear displacements in the linear actuators mounted on the fixed base instead of rotating actuators (Fig. 3.44).
3.4 Parallel mechanisms with complex limbs 187
The hybrid mechanism in Fig. 3.41a may be considered a complex open kinematic chain F with the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=15-11+1=5) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parame-ters that lose their independence in the five closed loops of this mechanism is r=rD=30, where rD was calculated above for the mechanism in Fig 3.42a. Equations (2.223)-(2.225) give the following structural parameters for the mechanism in Fig. 3.44: M=10, N=0, T=6. The six degrees of re-dundancy represent internal mobilities in the limbs and the remaining four degrees of mobility are external mobilities given by the angular displace-ments of the four revolute actuated joints.
Example 27. The Delta parallel mechanism in Fig. 3.45 represents an-other solution invented by R. Clavel [Clavel US patent 4976582]. This ro-bot may be considered as a parallel mechanism D ← E1-E2 obtained by par-allel concatenation of 2 complex limbs E1 and E2 (Fig. 3.46a). Limb E1 (Fig. 3.46b) is the hybrid Delta mechanism analysed in Example 26, and limb E2 of type RUPU (Fig. 3.46c) is composed of a homokinetic double universal joint U with a telescopic intermediary shaft (1D≡0-2D-3D-4D-5D-6D-7D). As we have seen in Example 26, limb E1 of type 3RPa4s-R is com-posed by serial concatenation of a translational parallel mechanism 3RPa4s and a revolute joint. Five independent closed loops exist in limb E1, and
Fig. 3.45. Parallel mechanism Delta-type with rotating actuators invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)
188 3 Structural analysis
Fig. 3.46. Parallel mechanism Delta-type with rotating actuators: kinematic struc-ture (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)
four in limb E2. We have seen in the previous example that 30 joint pa-rameters lose their independence in the closed loops of limb E1 ( E1
lr =30). The connectivity theorem 5A and Eq. (2.140) in chapter 2, and example 1 in this chapter also indicate that one joint parameter loses its independence
3.4 Parallel mechanisms with complex limbs 189
Fig. 3.47. Parallel mechanism Delta-type with linear and rotating actuators: kine-matic structure (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)
in each closed loop composed of two revolute joints of same axis of the universal joints ( E 2
lr =4). The parallel mechanism in Fig. 3.46a has the fol-lowing structural characteristics: m=17 links including the fixed base, p=26 joints, ten independent loops (q=26-17+1=10) and two complex limbs (k1=0, k2=2). The end-effector is 7≡6A≡6B≡6C≡7D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). The bases and the di-mensions of the operational velocities spaces associated with the two com-
190 3 Structural analysis
plex limbs are: (RE1)=( x y z zv ,v ,v ,ω ), (RE2)=( x y z x y zv ,v ,v , , ,ω ω ω ), SE1=4 and SE2=6. The connectivity of the end-effector in parallel mechanism D ← E1-E2, given by Eq. (2.198), is D
7 / 1S = E1 E 2dim( R R ) 4∩ = . Three translational velocities xv , yv and zv and one angular velocity zω exist be-tween the end-effector 7 and the reference base 1≡0 in the parallel mecha-nism D ← E1-E2.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =50. The
number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=10-4+34=40. The mobility of par-allel mechanism D ← E1-E2 in Fig. 3.46a given by Eq. (2.202) is MD=50-10+4-34=10. Ten joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=50-10-34=6) for Delta paral-lel mechanism in Fig. 3.46a. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four de-grees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position of the moving plat-form. These parameters are the three angular displacements qi (i=1,…,4) in the rotating actuators mounted on the fixed base.
Equation (2.217) indicates that the parallel mechanism in Fig. 3.46a has ND=60-10+4-34=20. Five overconstraints are introduced in each closed loop with two revolute joints of same axis of the universal joints. Usually these overconstraints in the universal joints are eliminated by a precise alignment of rotation axes and the technical solutions used for the revolute joints. To avoid these overconstraints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. In this case, the mechanism in Fig. 46a may be considered as a parallel mechanism D ← E1-A1 with one complex limb E1 as in the previous analy-sis, and a simple limb A1 (1D≡0-2D-3D-4D-5D-6D-7D). The following struc-tural parameters are associated with mechanism D ← E1-A1: m=17, p=22, q=6, k1=1, k2=1, (RE1)=( x y z zv ,v ,v ,ω ), (RE2)=( x y z x y zv ,v ,v , , ,ω ω ω ),
SE1=4, SE2=6, D7 / 1S = 4 dim(U)= p
ii 1f
=∑ =46, rl=30, rD=36, MD=10, TD=6
3.4 Parallel mechanisms with complex limbs 191
and ND=0. With this hypothesis, mechanism D ← E1-A1 in Fig. 46a may be considered non overconstrained.
The same structural analysis and parameters are applicable to the paral-lel Delta-type mechanism in Figs. 3.43a and 3.47a. The only difference is that the four degrees of external mobilities are associated with three linear displacements in the linear actuators and one angular displacement in the rotating motor mounted on the fixed base (Fig. 3.47).
Example 28. The mechanism in Fig. 3.48 represents a translational Delta-type parallel robot with universal joints also invented by Clavel (1990). This translational parallel mechanism D ← E1-E2-E3 is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A-6A), E2 (1B≡0-2B-3B-4B-5B-6B) and E3 (1C≡0-2C-3C-4C-5C-6C) – Fig. 3.49a. The mechanism has the following structural characteristics: m=14 links includ-ing the fixed base, p=27 revolute joints, 14 independent closed loops (q=27-14+1=14) and three complex limbs (k1=0, k2=3). The mobile plat-form is 6≡6A≡6B≡6C and the reference platform is
Fig. 3.48. Translational Delta parallel mechanism 3RUU-type invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)
Table 3.2. Valid bases of operational vector spaces RE1, RE2 and RE3 of complex limbs of the Delta parallel mechanism
No. (RE1) (RE2) (RE3) (3
Eii 1R
=∩ )
1 ( x y z x yv ,v ,v , ,ω ω ) ( x y z y zv ,v ,v , ,ω ω ) ( x y z z xv ,v ,v , ,ω ω ) ( x y zv ,v ,v ) 2 ( x y z y zv ,v ,v , ,ω ω ) ( x y z z xv ,v ,v , ,ω ω ) ( x y z x yv ,v ,v , ,ω ω ) ( x y zv ,v ,v ) 3 ( x y z z xv ,v ,v , ,ω ω ) ( x y z x yv ,v ,v , ,ω ω ) ( x y z y zv ,v ,v , ,ω ω ) ( x y zv ,v ,v )
192 3 Structural analysis
Fig. 3.49. Delta Parallel mechanism 3RUU-type: kinematic structure (a), associ-ated graph (b), complex chain associated with a limb (c)
fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb RUU-type is com-posed of nine revolute joints. Two closed loops with two revolute joints with the same axis exist in each universal joint.
The moving platform 6 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.49c) may realize three translation velocities x y zv ,v ,v and two angular velocities iω and i+1ω (i=1 for A-limb, i=3 for
B-limb and i=5 for C-limb). In the general case, each angular velocity has
3.4 Parallel mechanisms with complex limbs 193
three components x y z, ,ω ω ω but just one can be independent. Various bases of the operational velocity space of each complex limb are systema-tized in Table 3.2. They are set up by taking into account Eq. (2.206) and Remark 3 in chapter 2. The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is D
6 / 1S = E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational ve-locities xv , yv and zv exist between the mobile platform 6 and the refer-ence base 1≡0 in the parallel mechanism D ← E1-E2-E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =27.One
joint parameter loses its independence in each closed loop of the universal joints (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=15-3+12=24. The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=27-15+3-12=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mecha-nism. These parameters are the three angular displacements q1, q2 and q3 in the rotating actuators mounted on the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=27-15-12=0).
Equation (2.217) gives ND=84-15+3-12=60. Each closed loop in the universal joint introduces five overconstraints. Usually these overcon-straints in the universal joints are eliminated by a precise alignment of ro-tation axes and the technical solutions used for the revolute joints.
As we have shown in the previous example, to avoid these overcon-straints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the mecha-nism in Fig. 3.49a may be considered as a parallel mechanism C ← A1-A2-A3 with three simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-2C-3C-4C-5C). The following structural parameters are associated with mechanism C ← A1-A2-A3: m=14, p=15, q=2, k1=3, k2=0, SA1=SA2=SA3=5, C
6 / 1S = 3, dim(U)= pii 1
f=∑ =15, rl=0, rC=12, MC=3, TC=0
and NC=0. In this case, mechanism C ← A1-A2-A3 in Fig. 49a may be con-sidered non overconstrained.
Example 29. The mechanism in Fig. 3.50 represents a translational Tri-cept-type parallel robot invented by Neumann (1998). This translational parallel mechanism D ← E1-E2-E3-E4 is obtained by parallel concatenation of 4 complex limbs E1 (1A≡0-2A-3A-4A-5A),
194 3 Structural analysis
Fig. 3.50. Tricept parallel mechanism invented by K-E. Neumann and presented in US Patent 4732525 (Neumann 1988)
Fig. 3.51. Tricept parallel mechanism: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)
3.4 Parallel mechanisms with complex limbs 195
E2 (1B≡0-2B-3B-4B-5B), E3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D) – Fig. 3.51a. The mechanism has the following structural characteristics: m=13 links including the fixed base, p=20 joints, 8 independent closed-loops (q=20-13+1=8) and four complex limbs (k1=0, k2=4). The mobile platform is 5≡5A≡5B≡5C≡4D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). Five closed loops with two revolute joints with the same axis exist in the mechanism, one in each limb Ai (i=1,2,3) and two in the universal joint of limb A4.
The moving platform 5 in each limb A1, A2 and A3 isolated from the Tri-cept parallel mechanism may realize six independent velocities, x y z x yv ,v ,v , ,ω ω and zω . In limb A4 isolated from the Tricept parallel
mechanism, the moving platform 5 may realize just three independent mo-tions which may be three translations, or combinations of rotations and translations. Various bases may be set up for the operational velocity space of limb A4 isolated from the Tricept parallel mechanism: ( x y zv ,v ,v ), ( x y xv ,v ,ω ), ( x y yv ,v ,ω ), ( y z zv ,v ,ω ), ( x x zv , ,ω ω ), ( y x zv , ,ω ω ), ( z x zv , ,ω ω ), ( x y zv , ,ω ω ), ( y y zv , ,ω ω ) or ( z y zv , ,ω ω ).
The dimensions of the four operational spaces give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3-E4: SEi=6 (i=1,2,3) and SE4=3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3-E4, given by Eq. (2.198), is D
5 / 1S = E1 E 2 E3 E4dim( R R R R ) 3∩ ∩ ∩ = . Just three independent veloci-ties exist between the mobile platform 5 and the reference base 1≡0 in par-allel mechanism D ← E1-E2-E3-E4. The basis of the operational velocity space of the moving platform 5 in the Tricept parallel mechanism coin-cides with the basis of the operational velocity space of limb A4 isolated from the Tricept parallel mechanism above mentioned.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =25. One
joint parameter loses its independence in each closed loop, with just two revolute joints of same axis (rl=5). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=21-3+5=23. The mobility of the parallel mechanism D ← E1-E2-E3-E4 given by Eq. (2.202) is MD=26-21+3-5=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted in the telescopic limbs A1, A2 and A3. These actuators are not adjacent to the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=26-21-5=0). Equation (2.217) gives ND=48-21+3-5=25. These overconstraints are introduced in the five
196 3 Structural analysis
Fig. 3.52. Kinematic structure of Tricept robot considered as a parallel mechanism with four simple limbs
closed loops with two revolute joints of same axis. Each closed loop intro-duces 5 overconstraints. Usually these overconstraints are eliminated by a precise alignment of rotation axes, and by the technical solutions used for the revolute joints.
As we have shown in the previous example, to avoid these overcon-straints in static and dynamic analysis of such a mechanisms, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the Tricept parallel mecha-nism (Fig. 3.52) may be considered as a parallel mechanism C ← A1-A2-A3-A4 with four simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D). The following structural parameters are associated with mechanism C ← A1-A2-A3-A4: m=13, p=15, q=3, k1=0, k2=1, k=1, C
5 / 1S = 3, dim(U)= pii 1
f=∑ =21, rl=0, rC=18, MC=3,
TC=0 and NC=0. The parallel mechanism C ← A1-A2-A3 in Fig. 3.52 is non overconstrained.
Example 30. The mechanism in Fig. 3.53 represents the hybrid robot IRB 940 fabricated by ABB by concatenating a Tricept parallel mecha-nism and a serial wrist RRR-type. This hybrid mechanism may be consid-ered to be a complex open kinematic chain F with the following structural characteristics: m=16 links including the fixed base, p=23 joints, eight in-dependent loops (q=23-16+1=6) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parameters that lose their independence in the eight closed
3.4 Parallel mechanisms with complex limbs 197
Fig. 3.53. Hybrid mechanism obtained by concatenating a Tricept parallel mecha-nism and a serial wrist: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)
loops of this mechanism is r=rD=23, where rD was calculated above for the mechanism in Fig 3.51 in the previous example. Equations (2.223)-(2.225) give the following structural parameters for the mechanism in Fig. 3.53: M=6, N=25, T=0. As in the previous example, the 25 overconstraints are introduced by the five closed loops with just two revolute joints with the same axis. If we take into account just one revolute joint kinematically equivalent to the two revolute joints of same axis (Fig. 3.54) the following structural parameters are associated: m=13, p=15, q=3, k1=0, k2=1, k=1, SG1=SF=3, dim(U)= p
ii 1f
=∑ =21. The number of joint parameters that lose their independence in the eight closed loops is r=rC=18,
198 3 Structural analysis
Fig. 3.54. A hybrid mechanism obtained by concatenating a Tricept parallel mechanism with simple limbs and a serial wrist: kinematic structure (a), associ-ated graph (b)
where rC was calculated above for the mechanism in Fig 3.52 in the previ-ous example.
Equations (2.223)-(2.225) give the following structural parameters for the mechanism in Fig. 3.54: M=3, T=0 and N=0. The mechanism is non overconstrained.
Example 31. The parallel robot H4 (Fig. 3.55) was developed by Mont-pellier Laboratory of Computer Science, Robotics, and Microelectronics –LIRMM (Pierrot and Company 1999; Company and Pierrot 1999; Pierrot et al. 2003, Company et al. 2003, 2006). This mechanism F ← G1-G2 is ob-tained by parallel concatenation of 2 hybrid limbs G1 and G2 with the mov-ing platform 6 (Fig. 3.56a). Each limb Gi ← Di-Ri (i=1,2) is obtained by concatenating a complex kinematic chain Di with a revolute joint Ri. Com-plex chains D1 ← E1-E2 and D2 ← E3-E4 represent parallel mechanisms with two complex limbs E1-E2 (Fig. 3.56b) and E3-E4 (Fig. 3.56c) connecting the fixed base to two intermediary moving platforms 5≡5A≡5B and 5’≡5C≡5D. Each complex limb Ei combines a revolute joint and a spatial parallelogram loop with four spherical joints Pa4s: E1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D-5D).
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=22 joints, 7 independent closed loops (q=22-16+1=7) and two complex limbs (k1=0, k2=2, k=2).
3.4 Parallel mechanisms with complex limbs 199
Fig. 3.55. H4 parallel robot developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (cour-tesy LIRMM)
The four complex limbs Ei isolated from the parallel robot (Figs. 3.57 and 3.58) have the following bases and dimensions for their operational velocities spaces: (RE1)=( x y z 1 2v ,v ,v , ,ω ω ),(RE2)=( x y z 3 4v ,v ,v , ,ω ω ), (RE1)=( x y z 5 6v ,v ,v , ,ω ω ),(RE2)=( x y z 7 8v ,v ,v , ,ω ω ) and SEi=5 (i=1,...,4).
The connectivity of the intermediary platforms 5 in the parallel mecha-nisms D1 ← E1-E2, given by Eq. (2.198), is D1
5 / 1S = 3.
200 3 Structural analysis
Fig. 3.56. H4 parallel robot, kinematic structure (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel robot
Just three independent translational velocities xv , yv and zv exist between 5 and reference base 1A≡1B ≡0 in parallel mechanism D1 ← E1-E2 isolated from the parallel robot. Equation (2.209) indicates that 19 joint parameters lose their independence in the closed loops of parallel mechanism D1 ← E1-E2 (rD1=10-3+12=19). We have taken into account that 6 joint parameters
3.4 Parallel mechanisms with complex limbs 201
Fig. 3.57. Parallel mechanism D1 ← E1-E2 (a) and its complex limbs E1 (b) and E2 (c) isolated from the parallel mechanism D1
lose their independence in each parallelogram closed loop integrated in complex limbs E1 and E2. The same results are obtained for limb D2:
D25'/ 1S 3= and rD2=19. In each hybrid limb G1 and G2 isolated from the H4 parallel robot (Fig.
3.56b and c), the moving platform 6 may realize four independent veloci-ties, x y zv ,v ,v and zω . We have: (RG1)=(RG2)=( x y z zv ,v ,v ,ω ) and SG1=SG2=4.
202 3 Structural analysis
Fig. 3.58. Parallel mechanism D2 ← E3-E4 (a) and its complex limbs E3 (b) and E4 (c) isolated from the parallel mechanism D2
The connectivity of mobile platform 6 in the parallel robot H4 (Fig. 3.56a) is F
6 / 1S = G1 G2dim( R R ) 4∩ = . Four independent velocities exist be-tween mobile platform 6 and reference base 1≡0 in parallel mechanism F ← G1-G2 associated with H4 robot. The basis of the operational velocity space of the moving platform 6 in H4 parallel robot coincides with the ba-
3.4 Parallel mechanisms with complex limbs 203
sis of the operational velocity space of limbs G1 and G1 isolated from the parallel mechanism F ← G1-G2 .
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =54. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+38=42. We have taken into account in Eq. (2.228) that 19 joint parameters lose their inde-pendence in each hybrid limb Gi (i=1,2) as we have calculated above. The mobility of the parallel mechanism F ← G1-G2 given by Eq. (2.223) is M=54-42=12. Twelve joint parameters have to be known to define the po-sition and the orientation of each link of the parallel mechanism F ← G1-G2. Equation (2.225) gives eight degrees of redundancy (T=12-4=8) for H4 parallel robot. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight degrees of redundancy represent in-ternal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mo-bility are external mobilities that are transmitted from the input to the out-put of the parallel mechanism. They represent the independent parameters that are necessary to define the position and the orientation of the moving platform 6. These parameters are the angular displacements in the four ro-tating actuators mounted on the fixed base.
Equation (2.224) indicates that H4 parallel robot is not overconstrained N=42-42=0.
Example 32. Isoglide4-T3R1 in Fig. 3.59 is the translational parallel ro-bot with decoupled motions proposed in (Gogu 2002; 2005e). This mechanism F ← G1-G2 is obtained by parallel concatenation of a hybrid limb G1 and a simple limb G2 to the moving platform 6 (Fig. 3.59a,b). The hybrid limb G1 ← C-R (Fig. 3.59c) is obtained by serial concatenation of a complex kinematic chain C with a revolute joint R. The complex chain C ← A1-A2-A3 represents a translational parallel mechanisms with three sim-ple limbs Isoglide3-T3 of type 3-P||R||R||R analysed in example 15. The intermediary moving platform 5 of Isoglide3-T3 is connected to the mov-ing platform 6 by a revolute joint. The simple limb G2 (1D≡0-2D-3D-4D-5D-6) is of type P||R||R||R⊥R (Fig. 3.49d). The revolute joints connecting the limbs G1 and G2 to the moving platform have parallel axes.
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=18 joints, 3 independent closed loops (q=18-16+1=3), one simple limb and one complex limb (k1=1, k2=1, k=2).
204 3 Structural analysis
Fig. 3.59. Parallel robot Isoglide4-T3R1 with one simple and one complex limb (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel mechanism
3.4 Parallel mechanisms with complex limbs 205
Fig. 3.60. CAD model of parallel robot Isoglide4-T3R1 with one simple and one complex limb
The two limbs isolated from the parallel robot (Figs. 3.59c and d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z yv ,v ,v ,ω ), (RG2)=( x y z y zv ,v ,v , ,ω ω ), and SG1=4, SG2=5.
As we have seen in Example 15, nine joint parameters lose their independence in the closed loops of the translational parallel mechanism C ← A1-A2-A3 included in the hybrid limb G1 (rl= G1
lr =rC=9). The connec-tivity of the mobile platform 6 in Isoglide4-T3R1 (Fig. 3.59a) is
F6 / 1S = G1 G2dim( R R ) 4∩ = . Four independent velocities exist between the
mobile platform 6 and the reference base 1≡1A≡1B≡1C≡1D≡0 in the paral-lel mechanism F ← G1-G2 associated with this parallel robot of the Isoglide family.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=9-4+9=14. The mobility of the parallel mechanism F ← G1-G2 given by Eq. (2.223) is M=18-14=4. Four joint parameters have to be known to define the position and the ori-entation of each link of the parallel mechanism F ← G1-G2. These parame-ters are the linear displacements of the four linear actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that parallel robot Isoglide4-T3R1 in Fig. 3.59a has four degrees of overconstraints (N=18-14=4).
206 3 Structural analysis
Figure 3.60 presents a CAD model of the parallel robot Isoglide4-T3R1 with one hybrid and one simple limb.
Example 33. Isoglide3-T2R1 in Fig. 3.61 is a fully-isotropic planar par-allel robot proposed in (Gogu 2004c). This mechanism F ← G1-G2-G3 is ob-tained by parallel concatenation of two simple limbs G1 and G2 and a hy-brid limb G3. The moving platform is denoted by 8, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.61a,b). Simple limbs G1 (1A≡0-2A-3A-4A-5A-8), and G2 (1B≡0-2B-3B-4B-5B-8) are of type P||R||R||R⊥R (Figs. 3.61c and d). Hybrid limb G3 (Fig. 3.61e) is obtained by concatenating a revolute pair with two interconnected planar parallelogram loops (2C-3C-4C-6C) and (7C-6C-4C-5C -8). The revolute pairs connecting the three limbs to the moving platform have the same axis.
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=19-16+1=4), two simple limbs and one complex limb (k1=2, k2=1, k=3).
The three limbs isolated from the parallel robot (Figs. 3.61c,d,e) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z x zv ,v ,v , ,ω ω ), (RG2)=( x y z y zv ,v ,v , ,ω ω ), (RG3)=( x y zv ,v ,ω ), and SG1= SG2=5, SG3=3.
To calculate the number of joint parameters that lose their independence in the two closed loops integrated in hybrid limb G3 ← R-C, we consider the
Fig. 3.61. Fully-isotropic planar parallel robot Isoglide3-T2R1 with one hybrid and two simple limbs (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 207
Fig. 3.62. Limbs G1 (c), G2 (d) and G3 (e) isolated from the parallel mechanism of Isoglide3-T2R1 in Fig. 3.61
kinematic chain formed by these two closed loops as a parallel kinematic chain C ← A1-A2-A3 with three simple limbs connecting the ternary links 6C and 4C (Fig. 3. 63). The three simple limbs are: A1 (6C-4C) of type R||R (Fig. 3.63b), A2 (6C-2C-3C-4C) of type R||R||R (Fig. 3.63c) and A3 (6C-7C-8-5C-4C) of type R||R||R||R (Fig. 3.63d).
208 3 Structural analysis
Fig. 3.63. Parallel kinematic chain C ← A1-A2-A3 with distal links 4C and 6C and as-sociated with the closed loops in limb G3 ← R-C (a), limb A1 (b), limb A2 (c) and limb A3 (d)
The three limbs isolated from the parallel kinematic chain C ← A1-A2-A3 (Figs. 3.63b,c,d) have the following bases and dimensions for their opera-tional velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RA3)=( x y zv ,v ,ω ) and SA1=1, SA2= SA3=3. The connectivity of link 4C with respect to link 6C in parallel kinematic chain C ← A1-A2-A3 (Fig. 3.62a) is C
4C / 6 CS =
A1 A2 A3dim( R R R ) 1∩ ∩ = . Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their inde-pendence in the parallel kinematic chain C ← A1-A2-A3 is rC=7-1=6. This value is given by Eq. (2. 191) by taking by taking into account that in the
3.4 Parallel mechanisms with complex limbs 209
parallel kinematic chain C ← A1-A2-A3 we have considered 6C as the refer-ence link, and 4C as the distal link. We thus obtain G3
l l Cr r r 6= = = . The connectivity of the mobile platform 8 in Isoglide3-T2R1 with two
simple and one hybrid limb in Fig. 3.61a is F8 / 1S = G1 G2 G3dim( R R R )∩ ∩
=3. Three independent velocities ( x yv ,v and zω ) exist between mobile platform 8 and reference base 1≡1A≡1B≡1C≡0 in parallel mechanism F ← G1-G2-G3 associated with this parallel robot of Isoglide family.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =19. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2-G3 given by Eq. (2.226) is r=13-3+6=16. The mo-bility of the parallel mechanism F ← G1-G2-G3 given by Eq. (2.223) is M=19-16=3. Three joint parameters have to be known to define the posi-tion and the orientation of each link of the parallel mechanism F ← G1-G2-G3. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=3-3=0) and Eq. (2.224) indicates that parallel robot Isoglide3-T2R1 in Fig. 3.61a has eight degrees of overconstraint (N=24-16=8). Six of these overconstraints are in the two planar parallelogram loops.
Example 34. Isoglide3-T2R1 in Fig. 3.64 is a slightly modified version of the fully-isotropic planar parallel robot presented in Fig. 3.61 (Gogu, 2004c). This mechanism F ← G1-G2 is obtained by parallel concatenation of two hybrid limbs G1 and G2. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.64a,b). This solution is derived from the solution in Fig. 3.61 by superposing two revolute joints in just one joint. Limb G1 ← B-R is obtained by serially concatenating a closed loop B ← A1-A2 with a revolute joint. Limbs G2 in Fig. 3.64a and G3 in Fig. 3.61a are identical. The revolute pairs connecting the two limbs to the moving platform have the same axis. The mechanism has the following structural characteristics: m=15 links including the fixed base, p=18 joints, 4 inde-pendent closed loops (q=18-15+1=4) and two hybrid limbs (k=2).
The two limbs isolated from the parallel robot (Figs. 3.64c,d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z zv ,v ,v ,ω ), (RG2)=( x y zv ,v ,ω ) and SG1=4, SG2=3.
Five joint parameters lose their independence in the closed loop of limb G1 (see Example 5), G1
l Br r 5= = , and six in the closed loops of limb G2 (see Example 33), G2
lr 6= .
210 3 Structural analysis
Fig. 3.64. Fully-isotropic planar parallel robot Isoglide3-T2R1 with two hybrid limbs (a), associated graph (b), limb G1 (c) and limb G2 (d)
The connectivity of mobile platform 8 in Isoglide3-T2R1 with two hybrid limbs in (Fig. 3.64a) is F
8 / 1S = G1 G2 G3dim( R R R ) 3∩ ∩ = . Three inde-pendent velocities ( x yv ,v and zω ) exist between the mobile platform 8 and the reference base 1≡1A≡1B≡1C≡0 in the parallel mechanism F ← G1-G2 as-sociated with this parallel robot of the Isoglide family.
3.4 Parallel mechanisms with complex limbs 211
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=7-3+11=15. We have taken into account that Eq. (2.228) gives rl=11. The mobility of the paral-lel mechanism F ← G1-G2 given by Eq. (2.223) is M=18-15=3. Three joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1-G2. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=3-3=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 in Fig. 3.63a has nine degrees of overconstraint (N=24-15=9). Six of these overconstraints are in the two planar parallelogram loops.
Example 35. Isoglide3-T2R1 in Fig. 3.65 is a slightly modified version of the fully-isotropic planar parallel robot presented in Fig. 3.64 (Gogu, 2004c). The three limbs of this mechanism are interconnected in one com-plex kinematic chain F. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.65a,b). The mechanism has the fol-lowing structural characteristics: m=15 links including the fixed base, p=18 joints, 4 independent closed loops (q=18-15+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in Isoglide3-T2R1 with one complex limb in (Fig. 3.65a) is F
8 / 1S = G1S = 3. Three independent ve-locities ( x yv ,v and zω ) exist between mobile platform 8 and reference base
Fig. 3.65. Fully-isotropic planar parallel robot Isoglide3-T2R1 with three inter-connected limbs (a), associated graph (b)
212 3 Structural analysis
Fig. 3.66. Parallel kinematic chain F ← G1-G2 with distal link 7 associated with fully-isotropic parallel robot Isoglide3-T2R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)
3.4 Parallel mechanisms with complex limbs 213
Fig. 3.67. CAD model of Isoglide3-T2R1 with interconnected limbs
1≡1A≡1B≡1C≡0 in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1-G2 as a parallel mecha-nism with two hybrid limbs G1 and G2 connecting reference link 1 and dis-tal link 7 (Fig. 3.66a). As we have seen in chapter 2, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal links. The two hybrid limbs G1 and G2 in Figs. 3.66c,d have the same structural parameters with limbs G1 and G2 in Figs. 3.64c,d analysed in Example 34: G1
lr 5= and G2lr 6= . Eq.
(2.228) gives rl=11 as in the previous example. The number of joint pa-rameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=7-3+11=15.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18 and
the mobility of the mechanism F in Fig. 3.65a, given by Eq. (2.223) is M=18-15=3. Three joint parameters have to be known to define the posi-tion and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actua-tors mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=3-3=0), and Eq. (2.224) indicates that the parallel robot Isoglide3-T2R1 with intercon-nected limbs in Fig. 3.65a has nine degrees of overconstraint (N=24-15=9). Six of these overconstraints are in the two planar parallelogram loops.
The two versions of Isoglide4-T3R1 in Figs. 3.64a and 3.65a have the same structural parameters. A CAD model of Isoglide3-T2R1 with inter-connected limbs is presented in Fig. 3.67.
214 3 Structural analysis
Example 36. Isoglide4-T3R1 in Fig. 3.68 is a fully-isotropic parallel ro-bot with Schönflies motions (Gogu 2004a, 2007a). The three limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 9, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.68a,b). The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=19-16+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in the fully-isotropic Isoglide4-T3R1 in Fig. 3.68a is
F9 / 1S = G1S = 4. Four independent velocities ( x y zv ,v ,v and zω ) exist be-
tween the mobile platform 9 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1-G2 as a parallel mecha-nism with two hybrid limbs G1 and G2 connecting the reference link 1 with the distal link 8 (Fig. 3.69a). As we have seen in chapter 2 and in the pre-vious example, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal link. The hybrid limb G1 (Figs. 3.69b) has the same structural pa-rameters as limb G1 in Figs. 3.64c and 3.66b analysed in Examples 34 and 35: (RG1)=( x y z zv ,v ,v ,ω ), SG1=4, and G1
lr 5= . The hybrid limb G2 (Fig. 3.69c) is obtained by concatenating a prismatic joint with two planar paral-lelogram loops. We simply observe by inspection that (RG2)=( x y z zv ,v ,v ,ω ) and SG2=4.
Fig. 3.68. Fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 215
Fig. 3.69. The parallel kinematic chain F ← G1-G2 with distal link 8 associated with the fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)
216 3 Structural analysis
Fig. 3.70. The parallel kinematic chain C ← A1-A2-A3 with reference link 7C and distal link 5C associated with the closed loops in limb G2 ← P-C (a), limb A1 (b), limb A2 (c) and limb A3 (d)
To calculate the number of joint parameters that lose their independence in the two closed loops integrated in the hybrid limb G3 ← P-C, we consider the kinematic chain formed by these two closed loops as a parallel kine-matic chain C ← A1-A2-A3 with three simple limbs connecting the ternary links 5C and 7C (Fig. 3. 70a). The three simple limbs are: A1 (7C-5C) of type R||R (Fig. 3.70b), A2 (7C-8-9-6C-5C) of type R||R||R||R (Fig. 3.70c) and A3 (7C-2C-3C-4C-5C) of type R||R||R||R (Fig. 3.70d). The three limbs isolated from the parallel kinematic chain C ← A1-A2-A3 (Figs. 3.70b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RA3)=( x y zv ,v ,ω ) and SA1=1, SA2= SA3=3.
3.4 Parallel mechanisms with complex limbs 217
Fig. 3.71. CAD model of Isoglide4-T3R1 with interconnected limbs
The connectivity of link 5C with respect to link 7C in the parallel kine-matic chain C ← A1-A2-A3 is C
5C / 7CS = A1 A2 A3dim( R R R ) 1∩ ∩ = (Fig. 3.62a). Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain C ← A1-A2-A3 is rC=7-1=6. This value is given by Eq. (2. 191) by taking into account that in the parallel kinematic chain C ← A1-A2-A3 we have considered 7C as the reference link, and 5C as the distal link. We thus obtain G2
l Cr r 6= = . The number of joint parameters that lose their independence in the par-
allel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+11=15. We have taken into account that the number of joint parameters that lose their inde-pendence in the two limbs of the parallel mechanism F ← G1-G2, given by Eq. (2.228) is G1 G2
l l lr r r 11= + = . The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =19, and the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=19-15=4. Four joint parameters have to be known to define the posi-tion and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) of the three linear actuators mounted on the fixed base, and one rotating actuator mounted on link 2C in the prox-imity of the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 with three inter-connected limbs in Fig. 3.68a has nine degrees of overconstraint (N=24-15=9). Six of these overconstraints are in the two planar parallelogram loops. Figure 3.71 presents the CAD model of this solution.
218 3 Structural analysis
Example 37. Isoglide4-T3R1 in Fig. 3.72 represents the first solution of a fully-isotropic parallel robot with Schönflies motions presented in the lit-erature (Gogu 2004a). The limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 10, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.72a,b). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 joints, 5 independent closed loops (q=21-17+1=5) and one complex limb (k=1). The connectivity of the mobile platform 10 in the fully-isotropic Isoglide4-T3R1 in Fig. 3.72a is F
10 / 1S = G1S = 4. Four inde-pendent velocities ( x y zv ,v ,v and zω ) exist between mobile platform 10 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the five closed loops, we may consider F ← G1-G2 as a parallel mecha-nism with two complex limbs G1 and G2 connecting reference link 1 with distal link 9 (Fig. 3.73a). As we have seen in chapter 2 and in the previous examples, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the dis-tal link. The hybrid limb G1 (Figs. 3.73b) has the same structural parame-ters as the limb G1 in Figs. 3.64c, 3.66b and 3.69b analysed in Examples 34-36: (RG1)=( x y z zv ,v ,v ,ω ), SG1=4, and G1
lr 5= . Three closed loops are combined in complex limb G2 (Fig. 3.73c). We simply observe by inspec-tion that (RG2)=( x y z zv ,v ,v ,ω ) and SG2=4.
Fig. 3.72. The fully-isotropic parallel robot Isoglide4-T3R1 with three intercon-nected limbs and the actuators mounted on the fixed base (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 219
Fig. 3.73. The parallel kinematic chain F ← G1-G2 with distal link 8 associated with the fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs and actuators on the fixed base (a), limb G1 (b) and limb G2 (c)
220 3 Structural analysis
Fig. 3.74. Parallel kinematic chain D ← A1-A2-A3-E1 with reference link 8C and dis-tal link 6C associated with the closed loops in limb G2 (a), limb A1 (b), limb A2 (c) and limb E1 (d)
3.4 Parallel mechanisms with complex limbs 221
To calculate the number of joint parameters that lose their independence in the three closed loops integrated in the complex limb G3, we consider the kinematic chain formed by these three closed loops as a parallel kine-matic chain D ← A1-A2-E1 with two simple limbs (A1 and A2) and one com-plex limb E1 connecting the ternary links 6C and 8C (Fig. 3. 74a). The two simple limbs are: A1 (8C-6C) of type R||R (Fig. 3.74b) and A2 (8C-9-10-7C-6C) of type R||R||R||R (Fig. 3.74c). Link 6C is connected to link 8C by the complex limb E1 (Fig. 3.74d) including a closed loop B (1-2C-3C-4C-1). The three limbs isolated from the parallel kinematic chain D ← A1-A2-E1 (Figs. 3.74b,c,d) have the following bases and dimensions for their opera-tional velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RE1)=( x y zv ,v ,ω ) and SA1=1, SA2= SE1=3.
We observe that the operational velocity space of relative velocities be-tween links 6C and 8C in the complex limb E1 is not influenced by closed loop B. The three independent velocities in (RE1) are in fact given by the three revolute pairs with parallel axes outside the closed loop B. This closed loop B ←
B B1 2A A− (Fig. 3.75a) is obtained by putting together the
distal links of two simple chains B1A and B
2A of type P||R and R||P respec-tively (Fig. 3.75b,c). Chains B
1A and B2A have (RA1B)=(RA1B)=( z zv ,ω ) and
SAIB=SA2B= 2. Two joint parameters lose their independence in the closed loop B (rB=2+2-2=2).
The connectivity of link 6C with respect to link 8C in the parallel kine-matic chain D ← A1-A2-E1 is D
6C / 8CS = A1 A2 E1dim( R R R ) 1∩ ∩ = (Fig. 3.73a). Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1-A2-E1 is rD=7-1+2=8. This value is given by Eq. (2. 191) by taking by taking into account that in the parallel kinematic chain D ← A1-A2-E1 we have considered 8C as the reference link, and 6C as the distal link. We thus obtain G2
l Dr r 8= = . The number of joint parameters that lose their independence in the par-
allel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+13=17. We have taken into account that the number of joint parameters that lose their ind-pendence in the two limbs of parallel mechanism F ← G1-G2, given by Eq. (2.228) is G1 G2
l l lr r r 13= + = . The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =21 and the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=21-17=4. Four joint parameters have to be known to define the posi-tion and the orientation of each link of the mechanism F. These parameters
222 3 Structural analysis
Fig. 3.75. Closed loop B ←
B B1 2A A− integrated in limb G2 (a) and its associated
simple open kinematic chains B1A (b) and B
2A (c)
are the displacements (qi, i=1,2,3,4) in the three linear and one rotating ac-tuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 with three inter-connected limbs and actuators on the fixed base in Fig. 3.72a has nine de-grees of overconstraint (N=30-17=13). Six of these overconstraints are in the two planar parallelogram loops, and four in the closed loop B inte-grated in limb G2.
Example 38. I4R in Fig. 3.76 is an innovative solution for a parallel ro-bot with Schönflies motions for high-speed pick-and-place operations, de-veloped by LIRMM - Montpellier Laboratory of Computer Science, Ro-botics, and Microelectronics (Krut et al. 2004b, Company et al. 2006). The limbs of this mechanism are interconnected in one complex kinematic chain F ← G1. The robot uses a two-part moving platform 5 and 5’ con-nected by a prismatic joint (Fig. 3.77).
3.4 Parallel mechanisms with complex limbs 223
(a) (b)
(c) Fig. 3.76. The I4R parallel robot developed by Montpellier Laboratory of Com-puter Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (courtesy LIRMM)
224 3 Structural analysis
Fig. 3.77. I4R parallel robot: kinematic structure (a) and associated graph (b)
3.4 Parallel mechanisms with complex limbs 225
Fig. 3.78. Pulley-cable system used in I4R parallel robot as technological solution for rolling without sliding joint (courtesy LIRMM)
The relative translation between the two parts of the moving platform is transformed in rotational motion of an end-effector 6 by a rolling-without- sliding joint denoted by Rw. (Fig. 3.77). This solution gives a high tilting angle of the end-effector connected to haf-platform 5 by a revolute joint and to half-platform 5’ by the rolling-without-sliding joint. A pulley-cable system is used as a technological solution for the rolling without sliding joint (Fig. 3.78)
Each half-platform is connected to two complex limbs RPa4S-type (Fig. 3.77). The complex mechanism associated with the I4R robot may be con-sidered to be a parallel mechanism D ← A1-A2-E1 connecting a reference link 5 and a distal link 5’ (Fig. 3.79). This mechanism is obtained by ki-nematic inversion with respect to link 5. This inversion conserves the structural parameters of the mechanism. It is equivalent to positioning the “observer” on link 5 instead on the fixed base 1≡1A≡1B≡1C≡1D≡0. The parallel mechanism D ← A1-A2-E1 has one complex and two simple limbs. The two simple limbs are: A1 (5-6-5’) of type R||Rw (Fig. 3.80a) and A2 (5-5’) reduced to a prismatic joint (Fig. 3.80b). The complex limb E1 com-bines 6 closed loops (Fig. 3.80c). In Example 31, we have demonstrated that 38 joint parameters lose their independence in the closed loops com-bined in limbs G1 and G2 of the parallel robot H4. These closed loops are structurally identical with those of the complex limb E1 of I4R. Subse-quently, rl= E1
lr =38 for the I4R parallel robot. The three limbs isolated from the parallel mechanism D ← A1-A2-E1 (Figs. 3.80) have the following bases and dimensions for their operational velocities spaces: (RA1)=( x zv ,ω ), (RA2)= ( xv ), (RE1)=( x y zv ,v ,v ) and SA1=2, SA2= 1, SE3=3.
The connectivity of link 5’ with respect to link 5 in the parallel mecha-nism D ← A1-A2-E1 (Fig. 3.79) is D
5'/ 5S = A1 A2 E1dim( R R R ) 1∩ ∩ = . This
226 3 Structural analysis
Fig. 3.79. Parallel mechanism D ← A1-A2-E1 with reference link 5 and distal link 5’ associated with parallel robot I4R by kinematic inversion with respect to link 5 (a) and its associated graph (b)
3.4 Parallel mechanisms with complex limbs 227
Fig. 3.80. Limbs isolated from parallel mechanism D ← A1-A2-E1 associated with parallel robot I4R by kinematic inversion with respect to link 5: simple limbs A1 (a) and A2 (b) and complex limb E1(c)
228 3 Structural analysis
result holds because just one translation about the x-axis is possible be-tween the two half-platforms 5 and 5’. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1-A2-E1 given by Eq. (2. 200) is r=rD=6-1+38=43.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =55 and
the mobility of the mechanism associated with I4R parallel robot given by Eq. (2.223) is M=55-43=12. Twelve joint parameters have to be known to define the position and the orientation of each link of the parallel mecha-nism I4R. Equation (2.225) gives eight degrees of redundancy (T=12-4=8). As for the H4 robot analysed in Example 31, each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb about a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight de-grees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are trans-mitted from the input to the output of the parallel mechanism. They repre-sent the independent parameters that are necessary to define the position and the orientation of the end-effector 6. These parameters are the angular displacements of the four rotating actuators mounted on the fixed base.
Equation (2.224) indicates that the I4R parallel robot has five degrees of overconstraint (N=48-43=5).
3.5 Other multi-loop kinematic chains
The structural formulae demonstrated in chapter 2 for parallel robots may also be applied to various multi-loop mechanisms that do not obey to the classical CGK formulae. The three following examples illustrate the appli-cation of the new formulae presented in chapter 2 to complex mechanisms that may be considered to be parallel mechanisms with simple limbs.
Example 39. The mechanism proposed by De Roberval (1670) may be considered as a parallel mechanism F ← G1-G2-G3 (Fig. 3.81) obtained by the concatenation of two planar parallelogram loops. The six revolute joints have parallel axes. The CGK formulae give zero degrees of mobility for this mechanism, which is clearly wrong.
3.5 Other multi-loop kinematic chains 229
Fig. 3.81. The De Roberval mechanism (a), associated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mecha-nism
The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by three simple limbs R||R-type: G1(1A-2A-3A), G2(1B-2B-3B) and G3(1C-2C-3C). The mechanism has the following structural characteristics: m=5 links including the fixed base, p=6 revolute joints, 2 structurally independent closed loops (q=6-5+1=2) and three simple limbs (k=3). We may observe by inspection that the three simple limbs have the following bases for their operational spaces: (RG1)=( x yv ,v ) (RG2)=( x xv ,ω ) or (RG2)=( y xv ,ω ) and (RG3)=( x yv ,v ) – see Fig. 3.81c-e. The dimensions of these operational vector spaces give the connectivity of the kinematic chains associated with each limb Gi of the parallel mecha-nism: SGi=2, i=1,2,3. We have considered the characteristic point H be-longing to the moving platform, point O0 as the reference point, and the velocities of H are expressed in the coordinate system O0x0y0z0. In accor-dance with remark 2 and Eq. (2.197), the point H and the bases of RGi
(i=1,2,3) are selected such that the minimum value of F3 / 1S is obtained. By
this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
3 / 1S = G1 G2 G3dim( R R R )∩ ∩ =1. Only one relative independent velocity xv or yv exists between the mobile platform 3 and the reference link 1≡0. The dimension of the joint
230 3 Structural analysis
Fig. 3.82. Parallel mechanism equivalent to De Roberval mechanism by kinematic inversion with respect to link 1 (a), associated graph (b)
vector space U is dim(U)= pii 1
f=∑ =6. The number of joint parameters that
lose their independence in De Roberval’s mechanism, given by Eq.(2.226) is r=6-1=5. The mobility of the mechanism, given by Eq. (2.223), is M=6-5=1. Equation (2.224) gives N=7 and T=0. De Roberval mechanism in Fig. 3.81 is non redundant with seven degrees of overconstraint.
It is not necessary to treat this mechanism as a special mechanism with passive freedoms or mechanisms with over closed loops. No special equa-tions are necessary to calculate the mobility of this mechanism; it obeys the general formulae proposed in Chapter 2 for calculating the degree of mobility of parallel mechanisms with simple limbs.
The same results are obtained for the De Roberval mechanism in Fig. 3.82. This may be transformed into the mechanism in Fig. 3.81 by kine-matic inversion with respect to link 1. As we have seen in chapter 2 kine-matic inversion does not alter the mobility.
Example 40. The mechanism F ← G1-G2 in Fig. 3.83 is obtained by in-terconnecting two De Roberval mechanisms. The eight revolute joints have parallel axes. The CGK formula gives a negative mobility for this mechanism, which is wrong.
The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by one simple limb G1(1A-2A-3A) of type R||R, and a complex limb G2 analysed in Example 38. The mechanism has the following structural characteristics: m=6 links including the fixed base, p=8 revolute joints, 3 structurally independent closed loops (q=8-6+1=3) and two limbs (k=2). We may simply observe by inspection that the two limbs have the following bases and dimensions of their operational spaces: (RG1)=( x yv ,v ), (RG2)=( xv ) or (RG2)=( yv ) and SG1=2, SG2=1– see Fig. 3.83c, d. We have considered the characteristic point H belonging to the moving platform, point O0 as the reference point, and the velocities of H are
3.5 Other multi-loop kinematic chains 231
Fig. 3.83. Complex mechanism with one simple and one complex limb F ← G1-G2 (a), associated graph (b), kinematic chains associated with limbs G1 (c) and G2 (d) isolated from the mechanism
expressed in the coordinate system O0x0y0z0. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) are selected so that the minimum value of F
3 / 1S is obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
3 / 1S = G1 G2dim( R R )∩ =1. Only one relative independent velocity xv or yv exists between the mobile platform 3 and the reference link 1≡0. As we have seen in Example 38, five joint parameters lose their independence in the two closed loops in limb G2 (rl=5). The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =8. The number of joint parameters that lose their independence in the mechanism in Fig. 3.83 given by Eq.(2.226) is r=3-1+5=7. The mobility of the mechanism given by Eq. (2.223), is M=8-7=1. Equation (2.224) gives N=11 and T=0. The mechanism in Fig. 3.83 is non redundant with 11 degrees of overconstraint.
Example 41. The complex mechanism presented in Fig. 3.84a can also be considered a parallel mechanism F ← G1-G2-G3 with three simple limbs,
232 3 Structural analysis
in which ternary links 1 and 4 are the reference and the “distal” links. They are connected by three simple limbs (Fig. 3.84a,b): G1(1A-2A-3A-4A) of type R||R||R , G2(1B-2B) reduced to a single revolute joint and G3(1C-2C-3C-4C) of type C||C||R. All revolute (R) and cylindrical (C) joints have parallel axes.
The mechanism has the following structural characteristics: m=6 links including the fixed base, p=7 joints, 2 structurally independent closed loops (q=7-6+1=2) and three simple limbs (k=3). The CGK formulae give different results (M=1 or M=2) depending on the choice of the two struc-turally independent loops. If the closed loops (1-2A-3A-4-3C-2C-1) and (1-4-3C-2C-1) are selected, these formulae give the erroneous result M=1; the other two combination of two closed loops give M=2. Theoretically, any two loops of the graph presented in Fig. 3.84d can be considered as struc-turally independent. In fact, the two loops can be independent from a struc-tural point of view, in the sense of graph theory, but they are not always independent from a kinematic point of view, as we have shown in chapter 2. This ambiguity can be avoided by using the new formulae.
By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.84c-e) have the following bases and dimensions: (RG1)=( x y zv ,v ,ω ), (RG2)=( zω ), (RG3)=( x y z zv ,v ,v ,ω ) and SG1=3, SG2=1,
Fig. 3.84. The complex mechanism F ← G1-G2-G3 with three simple limbs (a), as-sociated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mechanism
3.5 Other multi-loop kinematic chains 233
SG3=4. We have considered the characteristic point H situated on link 4 on the axis of the revolute joint connecting this link to the fixed base.
The connectivity of link 4 in the parallel mechanism F ← G1-G2-G3 is F4 / 1 A1 A2 A3S dim( R R R ) 1= ∩ ∩ = . Only one relative independent velocity
( zω ) exists between links 4 and 1. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =9. The number of joint parameters that lose their in-dependence in this complex mechanism is r=8-1=7 and the correct value of the mobility is given by Eq. (2.223) is M=9-7=2. The mechanism has one degree of redundancy (T=2-1=1) and five degrees of overconstraint (N=12-7=5). We can see that this mechanism also obeys Eq. (2.223), and the mobility is determined without any ambiguity.
Example 42. The complex mechanism presented in Fig. 3.85a was pro-posed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a). It may be considered a parallel mechanism F ← G1-G2-G3 with three simple limbs, in which ternary links 1 and 4 are the reference and the distal links respectively. They are connected by three simple limbs (Fig. 3.85a): G1(1-2-4) of type C-S, G2(1-4) reduced to a planar joint Pn and G3(1-3-4) of
Fig. 3.85. Complex mechanism F ← G1-G2-G3 with three simple proposed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a) - (a), simple open chains associated with limbs G1 (b), G2 (c) and G3 (d) isolated from the mechanism
234 3 Structural analysis
type C-S. The mechanism has the following structural characteristics: m=4 links including the reference link, p=5 joints, 2 structurally independent closed loops (q=5-4+1=2) and three simple limbs (k=3).
By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.85b-d) have the following bases and dimensions: (RG1)=( y z x y zv ,v , , ,ω ω ω ), (RG2)=( x y zv ,v ,ω ), (RG3)=( y x y zv , , ,ω ω ω ) and SG1=5, SG2=3, SG3=4. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) have been selected such that the minimum value of F
4 / 1S to be obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
4 / 1S = G1 G2 G3dim( R R R )∩ ∩ =2. Two independent velocity yv and zω exist between the mobile platform 4 and the reference link 1. The dimen-sion of the joint vector space U is dim(U)= p
ii 1f
=∑ =13. The number of joint parameters that lose their independence in the
mechanism in Fig. 3.85 given by Eq.(2.226) is r=12-2=10. The mobility of the mechanism given by Eq. (2.223), is M=13-10=3. Equation (2.224) gives N=2 and T=1. The mechanism in Fig. 3.85 has two degrees of over-constraint and one degree of redundancy. One internal mobility exists in limb G3 – the rotation of the link 3 that cannot be transmitted to link 4. The remaining two degrees of mobility are external mobilities that have to be actuated to obtain the translation and the rotation of distal link 4. Transla-tion about the y-axis may be actuated in one of the three joints connected to the fixed base. Rotation about the z-axis may be actuated by a circular translation in the planar joint.
4 Kinematic analysis
Kinematics of parallel robots is very important for structural synthesis. We have seen in chapter 2 that the structural parameters of parallel robots are mainly kinematically defined. The duality between velocity and force al-lows us to give alternative definitions of structural parameters based on statics. The author has preferred the kinematic definition, considering that the kinematic nature of the structural parameters facilitates the understand-ing of their meaning. Apparently, we are more skilled in imagining the motion than imaging the static or dynamic forces.
In this chapter, we will introduce the main models and performance in-dices used in parallel robots. In general, to study robot motions, geometric and kinematic models are used. We will present the general approach used to set up these models and we emphasize the Jacobian matrix which is the main issue in defining robot kinematics, singularities and performance in-dices. We saw in chapter 2 that the Jacobian is the matrix of the linear transformation between the velocity joint space and the operational space. This matrix contains very important information concerning robot behav-ior from a kinematic and static (kinetostatic) point of view.
The Jacobian will be used to define and to characterize motion decoup-ling in parallel robots. In modern design theories, decoupling the func-tional requirements has become a basic design principle (Suh 1990). In ro-botics, this principle leads to significant advantages in design, modeling, motion planning and control (Vukobratovic and Kircanski 1983; Konstan-tinov et al. 1987; Patarinski et al. 1991; Patarinski and Uichiyama 1993; Youcef-Toumi 1992; Galabov et al. 1999).
In many robotic applications the possibility of decoupling end-effector motion becomes a cornerstone criterion in the choice of the robot mecha-nism. The problem is that in the early stage of parallel robot design the Jacobian matrix is not yet known. For structural synthesis of parallel ro-bots we use an alternative method to define motion decoupling and singu-larities, which does not need prior knowledge of the Jacobian matrix. This method is based on the main structural parameters introduced in chapter 2 via the theory of linear transformation.
236 4 Kinematic analysis
4.1. Decoupling in axiomatic design
Axiomatic design is a new design theory developed by Suh (1990) to estab-lish a scientific foundation for the design field, and provide a fundamental basis for the creation of products, processes, systems, software and organi-zations. This is a significant departure from the conventional design proc-esses largely dominated by empiricism and intuition.
Axiomatic design considers that the design objective is always stated in the functional domain, whereas the physical solution is always generated in the physical domain. These two domains are in permanent interconnec-tion at every hierarchical level of the design process. To satisfy the design objectives defined in terms of functional requirements (FRs), a physical embodiment defined in terms of design parameters (DPs) must be created. In this way, the design is formally defined in axiomatic design as the crea-tion of synthesized solutions in the form of products, processes or systems that satisfy the perceived needs through the mapping between the func-tional requirements in the functional domain and the design parameters of the physical domain, through the proper selection of DPs that satisfy FRs. This mapping process in not unique, and the actual outcome depends on the designer’s individual creative process. Therefore, there can be a large number of plausible design solutions and mapping techniques. The design axioms, theorems and corollaries formulated in axiomatic design provide the principles that the mapping techniques must satisfy to produce a good design, and offer a basis for comparing and selecting design solutions. Axiomatic design is based on two axioms that govern good design: the in-dependence axiom and the information axiom. The independence axiom states that “in an acceptable design, the DPs and the FRs are related in such a way that a specific DP can be adjusted to satisfy its corresponding FR without affecting other functional requirements”. The information axiom states that “the best design is a functionally uncoupled design that has minimum information content” – see page 48 of “Principles of Design” by Suh (1990). We can see that decoupling is the keyword of both axioms.
The independence axiom is mathematically represented by the design equation
{FR}=[D]{DP} (4.1)
where {FR} is the set of functional requirements, {DP} is the set of design parameters and [D] is a m×n matrix called the design matrix. The ele-ments of this matrix are denoted by aij (i=1,2,…n and j=1,2,…,m). The left-hand side of the design equation represents “what we want in terms of design goals” and the right-hand side represents “how we hope to satisfy
4.1. Decoupling in axiomatic design 237
these goals in terms of a design solution”. We mention that in (Suh 1990), {FR} and {DP} are called “vectors”. In this work we call them “sets” be-cause they may not always satisfy the mathematical definition of a vector.
A design that can be represented by Eq. (4.1) with [D] being a diagonal matrix, i.e. aij=0 unless i=j, satisfies the first axiom since the independ-ence of FRs is assured when each DP is changed. That is, FRi can be satis-fied independently by simply changing the corresponding DPi without af-fecting any other FR. Such a design is called an uncoupled design. If the design matrix is triangular the design can be decoupled and is called a de-coupled or quasi-coupled design. A truly coupled design is one in which the design matrix is neither diagonal nor triangular.
Two general quantitative measures are proposed in (Suh 1990) for func-tional independence: reangularity R and semangularity S . They are the following functions of the design matrix elements aij:
R
12 2n
ki kjk 1
n ni 1,n 1 2 2j 1 i ,n ki kj
k 1 k 1
a a1
a a
=
= −= +
= =
= −
∑∏
∑ ∑
(4.2)
and
S n
jj1
j 1 n 22kj
k 1
a
a=
=
=
∏∑
.
(4.3)
Reangularity measures the degree of orthogonality between the DPs; . the maximum value of the reangularity occurs when the DPs are orthogo-nal. As the degree of coupling increases, so the value of R decreases. The sums in the denominator of Eq. (4.2) are the squares of the magnitudes of the columns in the design matrix. These terms are used to normalize the columns of the matrix. When the columns are already normalized, the su-ms are equal to one. Normalization is necessary in order to make the measure insensitive to changes in the scale of any of the DPs. The sum in the numerator is the dot product of the ith and jth columns of the design matrix. This sum, when properly normalized by the denominator, can be thought of as the cosine of the angle between the vectors in the ith and jth columns of the design matrix. Since this cosine term is squared and sub-
238 4 Kinematic analysis
tracted from unity, the argument of the product operator is the square of the sine of the same angle.
Semangularity characterizes the alignment of DPs with their corre-sponding FRs. The denominator of Eq. (4.3) is a normalizing factor that is equal to unity if the columns of the design matrix are normalized. When the design matrix is normalized, semangularity is simply the product of the absolute values of the diagonal elements of the design matrix. If the design matrix is properly normalized, then the diagonals are unity when all off-diagonal elements are zero. When the semangularity is unity, the DPs are aligned with the FRs. An uncoupled design has R =1 and S =1.
By analogy with the three types of design, in the next section we will define parallel robots with coupled, decoupled and uncoupled motions by considering the Jacobian as the design matrix.
4.2. Geometric modeling
In general in robot analysis, geometric modeling is set up in terms of finite motions; it is also called geometric analysis or pose analysis.
Definition 4.1 (Geometric problem). The geometric problem in robot analysis involves setting up the implicit relation between the finite motions of the actuated joints and the moving platform.
Usually, the geometric model is expressed in terms of linear and/or an-gular finite displacements of the actuated joints; moving platform pose is defined by its spatial position and orientation with respect to the reference frame. This model can be direct or inverse. In the direct geometric model the inputs are the finite displacement of the actuated joints, and the output is the pose of the moving platform. In the inverse geometric model the in-put is the pose of the moving platform and the outputs are the finite dis-placements of the actuated joints.
Let us consider the general case of a parallel robot F ← G1-G2-…Gk in which the moving platform n≡nGi is connected to the reference link 1≡1Gi by k limbs Gi (1Gi-2Gi-…nGi), i=1,2,…k. The robot has mobility M, and the connectivity of the moving platform is SF. The kinematic chains associated with limbs Gi can be simple or complex, and each limb Gi could integrate gi linear and/or rotation actuators. We have seen in chapter 2 that the total number of actuators is equal to the mobility
k
ii 1
g M=
=∑ . (4.4)
4.2. Geometric modeling 239
The geometric constraints imposed by the limbs on the motion of the moving platform give, for the ith limb, the following set of gi general rela-tions
( )ij ij 1 2 6f q ,w ,w ,...,w 0= , i=1,2,…,k ; j=1,2,…,gi. (4.5)
where qij (i=1,2,…,k ; j=1,2,…,gi) represent the linear or angular displace-ments in the actuated joints and w1, w2, …, w6 represent the parameters used to describe the spatial position and orientation of the moving plat-form; these are called the general coordinates of the moving platform.
We note that various parameters can be used to represent the spatial pose of the moving platform: quaternions, biquaternions, homogeneous operators, rotation matrices, complex matrices, etc. More details on these parameters are presented in the book (Gogu and Coiffet 1996).
In general, relations (4.5) can be obtained by expressing the vector- loop equation of each limb in the fixed coordinate frame, and eliminating the variables of the passive joints. Setting up analytical closed-form relations (4.5) is possible when an analytical inverse geometric model is known for each limb. In the general case, setting up analytical closed-form relations (4.5) may not always be possible for all limb architectures. However, in parallel robots, chains Gi are usually very simple, and such relations can be easily obtained.
A very common way to set up closed-form relations (4.5) is to start from the equalities
( ) ( ) ( ) ( )2 2 2Ai Bi Ai Bi Ai Bi ij ij 1 2 6x x y y z z h q ,w ,w ,...,w− + − + − = (4.6)
written for each limb Gi (i=1,2,…,M). The distance constraints can be solved algebraically or geometrically
(Thomas et al. 2004). The left-hand of Eq. (4.6) represents the square of the distance AiBi be-
tween the extreme points of each limb (Fig. 4.1). Points Ai are situated on the fixed base and have coordinates xAi, yAi and zAi. Points Bi are situated on the moving platform and have coordinates xBi, yBi and zBi. The coordinates of point Ai are the components of position vector O0Ai expressed in the fixed reference frame 0 0 0 0O x y z . The coordinates of point Bi are the com-ponents of vector O0Bi also expressed in the fixed reference frame by
O0Bi=O0On+R0nOnBi (4.7)
where OnBi is the position vector of point Bi in the mobile frame Onxnynzn attached to the moving platform, O0On is the position vector of point On in the fixed reference frame 0 0 0 0O x y z , and R0n is the 3× 3 rotation matrix
240 4 Kinematic analysis
giving the relative orientation of the frame n n n nO x y z with respect to
0 0 0 0O x y z . Vectors O0Ai and OnBi are constant and known for a given ge-ometry of the fixed and moving platforms. In the general case, vector O0On depends on three generalized coordinates, these define the position of the characteristic point of a moving platform. The rotation matrix R0n depends on three generalized coordinates used to define the orientation of the mov-ing platform. In this way, the coordinates of point Bi in the fixed reference frame depend on six general coordinates.
The right-hand of Eq. (4.6) is a function depending on at least qij. Usu-ally, in the parallel robots with telescopic limbs this function depends only on qij, that is hij=hij(qij). For example in Gough-type parallel robots gi=1 (i=1,2,…,6) and hij= 2
i1q . In other cases, this function could also depend on up to six general coordinates.
Eqs. (4.5) could be used to solve the direct and inverse geometric prob-lem of parallel manipulators.
Definition 4.2 (Direct geometric problem). The direct geometric prob-lem involves the calculation of the general coordinates w1, w2, …, w6 when the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) are known in Eq. (4.5).
Definition 4.3 (Inverse geometric problem). The inverse geometric problem involves the calculation of the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) when the general coordinates w1, w2, …, w6 of the moving platform are known in Eq. (4.5).
Fig. 4.1. General case of a parallel robot with k limbs
241
We note that contrary to the situation in serial robots, the direct geomet-ric problem is generally more difficult than the inverse one. The inverse geometric problem can be easily solved analytically by using Eqs. (4.6). The direct geometric problem is usually solved numerically. Various methods can be used: elimination, homotopy, Gröbner basis, interval analysis or other ad-hoc methods (Merlet 2005). We will see in the exam-ples presented at the end of this chapter that parallel robots with decoupled motions can have analytical solutions for both inverse and direct problems obtained in a very straightforward manner. This is an important advantage both for kinematic and dynamic modelling and path planning and control.
4.3 Kinematic modeling
In general in robot analysis, kinematic modeling is set up in terms of in-finitesimal motions; it is also called differential motion analysis.
Definition 4.4 (Kinematic problem). The kinematic problem in robot analysis involves setting up the implicit relations between the infinitesimal motions of the actuated joints and the moving platform.
Usually, the kinematic models are expressed in term of robot velocity and eventually robot acceleration. The kinematic model can also be direct and inverse. In the direct kinematic model the input is the velocity of the actuated joints and the output the velocity of the moving platform. In the inverse kinematic model the input is the velocity of the moving platform and the outputs are the velocities of the actuated joints. In robot kinemat-ics, the linear mapping between the velocity of the actuated joints and moving platform is set up by using the Jacobian matrix. Various methods can be used to calculate the Jacobian of a parallel robot: velocity vector-loop equations (Mohamed et al. 1983, Khalil and Dombre 2002), theory of reciprocal screws (Mohamed and Duffy 1985), motor algebra (Sugimoto 1987), Grassmann geometry (Merlet 1989), Plücker line coordinates (Hunt et al. 1991), etc. Various types of Jacobian matrices are known: conven-tional Jacobian (Tsai 1999), screw-based Jacobian (Tsai 1998), and overall Jacobian (Joshi and Tsai 2002). The screw-based Jacobian is developed by using the concept of reciprocal screws introduced by Ball (1900). The screw-based Jacobian becomes the conventional Jacobian after interchang-ing the first three columns with the last three columns in the parallel Jaco-bian. The overall Jacobian or full Jacobian (Merlet 2005) was proposed by Joshi and Tsai (2002) to provide information about both architecture and constraint singularities of parallel robots with less than six degrees of free-dom. In this section, we introduce the design and conventional Jacobian and we emphasize the difference between these two Jacobian matrices.
4.3. Kinematic modeling
242 4 Kinematic analysis
The design Jacobian is useful for defining and understanding: (i) the me-chanical nature of motion decoupling in parallel robots, (ii) the nature of conventional Jacobian matrix, and (iii) the formulation nature of certain singularities in parallel manipulators.
4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation
By differentiating Eq. (4.5) with respect to time, we obtain
ij ij ij ijij 1 2 6
ij 1 2 6
f f f fq w w ... w 0
q w w w∂ ∂ ∂ ∂
+ + + + =∂ ∂ ∂ ∂
, (4.8)
or
ij ij ij ijij 1 2 6
ij 1 2 6
f f f fq w w ... w
q w w w∂ ∂ ∂ ∂
= − − − −∂ ∂ ∂ ∂
,
i=1,2,…,k; j=1,2,…,gi.
(4.9)
In matrix form, Eq. (4.9) can be written as follows
Js q =Jp w (4.10)
where q =1 2 k
T
11 1g 21 2 g k1 kgq ... q q ... q ... q ... q is the veloc-
ity vector of actuated joints, w = [ ]T1 2 6w w ... w is the velocity vector of the moving platform,
k
k
11
11
12
12s
kg
kg
f 0 0q
f0 0qJ
f0 0
q
∂ ∂ ∂ ∂=
∂ ∂
,
(4.11)
is the direct kinematic matrix and
4.3. Kinematic modeling 243
k k k
11 11 11
1 2 6
12 12 12
1 2 6p
kg kg kg
1 2 6
f f fw w wf f fw w wJ
f f fw w w
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= − ∂ ∂ ∂ ∂ ∂ ∂
(4.12)
is the parallel kinematic matrix. In the literature, Js and Jp are also named serial and parallel Jacobian matrices (Wanger and Chablat 2000) by anal-ogy with the corresponding matrix of Eq. (4.8) written in matrix form. The latter is called Jacobian, although this is not correct from a strictly mathe-matical point of view. Matrices Js and Jp were used by Gosselin and Ange-les (1990) for singularity analysis and were also called direct kinematics and inverse kinematics Jacobians in (Tsai, 1999). We use the names of di-rect and inverse kinematics matrices coined by Chablat and Wenger (1998).
Eq. (4.10) represents the general kinematic model of parallel robots in which Js is a M×M matrix and Jp is a 6×M matrix. If the mobility M is greater than connectivity SF, the parallel robot is structurally redundant. If SF<6 just SF generalized coordinates are independent and (6-SF) are de-pendent variables. If the number of limbs is equal to the mobility, and each limb is actuated (k=M), the robot is fully-parallel.
4.3.2 Design and conventional Jacobian matrices
Usually robot mobility satisfies 2≤M≤6 and M=SF. In this case, we can eliminate (6-SF) rows of JS corresponding to dependent velocities of the moving platform and JS becomes a M×M square matrix. As M=SF, the M×M parallel Jacobian is non-singular and Eq. (4.10) gives
w =J q (4.13)
where
J= 1pJ − sJ (4.14)
is the M×M Jacobian of the parallel robot. It represents the matrix of the linear transformation between the M-dimensional velocity vector q of the
244 4 Kinematic analysis
actuated joints and the M-dimensional operational velocity vector w of the moving platform. The element of this matrix situated in line i and column j indicate how an infinitesimal change in qj affects the infinitesimal change in wi.
Eq. (4.13) is the mathematical expression of the direct kinematic model and
q = 1J − w (4.15)
is the mathematical expression of the inverse kinematic model. We can see that
1J − = 1sJ − pJ (4.16)
Usually the inverse Jacobian matrix can be calculated easier than the Jaco-bian matrix by taking into account of the fact that sJ is easier to invert than pJ .
Definition 4.5 (Direct kinematic problem). The direct kinematic prob-lem involves the calculation of the Jacobian matrix in Eq. (4.13). This ma-trix represents the linear transformation from the velocity joint space of ac-tuated joints to the operational velocity space of the moving platform.
Definition 4.6 (Inverse kinematic problem). The inverse kinematic problem involves the calculation of the inverse Jacobian matrix in Eq. (4.15). This matrix represents the linear transformation from the opera-tional velocity space of the moving platform to the velocity joint space of the actuated joints.
Design Jacobian
Introducing the Jacobian matrices by differentiating Eq. (4.5) we implicitly consider that the velocities of the moving platform result from direct dif-ferentiation of general coordinates w1, w2, …, w6 used to define the pose of the moving platform. To respect this hypothesis, the orientation of the moving platform must be defined by three consecutive rotation angles
,α β and δ used in the rotation matrix R0n. They could be roll-pitch-yaw angles ( nz′ - ny′′ - nx′′′ -type), Euler angles ( nz′ - nx′′ - nz′′′ -type or nz′ - ny′′ - nz′′′ -type) or Bryant angles ( nx′ - ny′′ - nz′′′ -type) or any other combination of three ordered rotations (Gogu, Coiffet, 1996). The axes mentioned in the brack-ets indicate the rotation axes na′ - nb′′ - nc′′′ of the three rotations ,α β and δ . They are the axes of the three intermediary coordinate systems used to de-
245
fine the relative position and orientation of n n n nO x y z with respect to
0 0 0 0O x y z
0 0 0 0 n n n n n n n n n n n n n n n nO x y z O x y z O x y z O x y z O x y z′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′′ ′′′ ′′′ ′′′→ → → → (4.17)
in the following order of transformations a b c
0n' 0 n n n n n n nT ( O O ) R ( ) R ( ) R ( )α β δ′ ′′ ′′′′ ′′ ′ ′′ ′ ′′− − − . (4.18)
The notation 0n' 0 nT ( O O ) indicates that the reference coordinate system
0 0 0 0O x y z is translated by the vector 0 nO O into the coordinate system
n n n nO x y z′ ′ ′ ′ . The notation an nR ( )α′′ ′′ indicates that n n n nO x y z′ ′ ′ ′ is superposed
on the coordinate system n n n nO x y z′′ ′′ ′′ ′′ by a rotation through an angle α , and so on. For roll-pitch-yaw and Euler angles the first rotation axis na′ is
nz′ ; for Bryant angles na′ is nx′ . The second rotation axis nb′′ is ny′′ except for the Euler angles nz′ - nx′′ - nz′′′ -type when nb′′ is nx′′ . The third rotation axis
nc′′′ is nz′′′ except for roll-pitch-yaw angles, when nc′′′ is nx′′′ . The rotation matrix R0n giving the orientation of the moving frame
n n n nO x y z with respect to the fixed frame 0 0 0 0O x y z is obtained by multi-plying the rotation matrices associated with the transformations defined in (4.18)
a b c0n n n n n n nR R ( )R ( )R ( )α β δ′ ′′ ′′′
′ ′′ ′ ′′ ′ ′′= . (4.19)
In the general case of a parallel robot with six degrees of freedoms, we denote the translational velocities of the characteristic point of the moving platform by iw (i=1,2,3), and the angular velocities of the moving plat-form by iw (i=4,5,6). The reference frame origin On is usually chosen as the characteristic point of the moving platform.
Usually in robot kinematics, iw (i=1,2,3) are associated with the trans-lational velocities x yv ,v and zv of the characteristic point in the directions
parallel to the x0-, y0 and z0-axis, that is 0v = [ ]T1 2 3w ,w ,w = T
x y zv ,v ,v .
The other three velocities iw (i=4,5,6) in Eq. (4.13) are associated with
the angular velocity of the moving platform ω = [ ]T4 5 6w ,w ,w =T
, ,α β δ
=T
, ,α β δω ω ω . In this case, the Jacobian matrices in Eqs. (4.13)-(4.16) are called design Jacobian matrices.
4.3. Kinematic modeling
246 4 Kinematic analysis
Definition 4.7 (Design Jacobian). The design Jacobian of a parallel ro-bot is a matrix used in setting up the relation between joint and operational velocities when the angular velocities of the moving platform are defined as the time derivatives of the rotation angles of the moving platform.
The direct kinematic model of a parallel robot with SF=6 founded on the design Jacobian is
0vω
= J q (4.20)
with [ 0v ,ω ]T=[ x y zv ,v ,v , , ,α β δω ω ω ]T=[ x, y,z, , ,α β δ ]T and
q =[ 1 2 3 4 5 6q ,q ,q ,q ,q ,q ]T. The direct kinematic model of a lower-connectivity parallel robot with
1<SF<6 founded on the design Jacobian is defined by analogy with Eq. (4.27) and taking into account the fact that the base of the operational ve-locity space of such a robot can be represented by any combination of two or more independent translational or angular velocities of the moving plat-form.
The design Jacobian J is always a square matrix SF× SF with 1<SF<≤ 6. We have derived its name from the design matrix used in axiomatic de-sign. The design Jacobian is used mainly in the structural synthesis ap-proach presented in this book.
Conventional Jacobian
The conventional Jacobian is used when the angular velocity of the mov-ing platform is expressed by its components in the reference frame. We denote this velocity by 0ω =
T
x y z, ,ω ω ω . Definition 4.8 (Conventional Jacobian). The conventional Jacobian of
a parallel robot is a matrix used in setting up the relation between joint and operational velocities when the angular velocity of the moving platform is expressed in the reference frame.
In this case, the angular velocity components cannot be directly associ-ated with the time derivatives of the rotation angles of the moving plat-form. For this reason, the conventional Jacobian does not represent a Jaco-bian matrix in the strict mathematical sense. This might be the meaning of “conventional” in the original name of this matrix.
Vector ω = [ ]T4 5 6w ,w ,w =T
, ,α β δ =T
, ,α β δω ω ω can be used to cal-
culate the angular velocity vector of the moving platform with respect to the fixed reference frame, by using the following mapping
247
0ω = rJ ω , (4.21)
where rJ is the Jacobian of angular velocity mapping. Angular velocity 0ω and moving platform velocities [ 0v , 0ω ]T=
[ x y z x y zv ,v ,v , , ,ω ω ω ]T can also be calculated directly without calculating ω . Eqs. (4.10) and (4.13) become
sJ q = pJ 0
0
vω
, (4.22)
and
0
0
vω
= J q (4.23)
In this case, matrices Js and Jp and J represent conventional Jacobian ma-trices and are calculated without time derivatives of rotation angles ,α β and δ by formulating a velocity vector-loop equation for each limb. This equation is obtained by differentiating, with respect to time, the geometri-cal loop-closure equation of each limb (Fig. 4.1) written as
O0Ai+ AiBi =O0On+OnBi . (4.24)
When we differentiate the geometric loop-closure equation (4.24) we must take into account the fact that the time derivative of a vector is the sum of its “apparent” time derivative in the rotating frame and the vector cross product of the angular vector of the frame and the vector itself.
For example, for a parallel robot with telescopic limbs, the time deriva-tive of Eq. (4.24) gives the following velocity vector-loop equation
iq ai + qi i iaω × = 0v + 0 ibω × (4.25)
where qi=AiBi and ai denote the variable length of ith limb, and a unit vec-tor along AiBi, respectively, ib = OnBi, and iω denotes the angular velocity of the ith limb with respect to the fixed reference frame.
To eliminate iω , we dot-multiply both sides of Eq. (4.25) by ai
iq = i 0a v⋅ + ( )×i i 0b a ω . (4.26)
For Gough-type parallel robot with six telescopic limbs, Eq. (4.26) written six times (i=1,2,…6) yields six scalar equations which can be assembled in matrix form (4.22) or (4.23) with
4.3. Kinematic modeling
248 4 Kinematic analysis
pJ =
( )( )
( )
× × ×
TT1 1 1
TT2 2 2
TT6 6 6
a b a
a b a
a b a
(4.27)
sJ = 6 6I × and 0J = 1pJ − , where 6 6I × denotes the 6× 6 identity matrix.
Property 4.1. The design and conventional Jacobian matrices are iden-tical for all parallel robots with up to three translational and one rotational degrees of freedom of the moving platform. Differences appear between the two matrices, just when the moving platform has two or three rota-tional degrees of freedom.
4.4 Types of workspaces and singularities
Workspace, singularity and performance indices are three connected sub-jects in parallel robot analysis. Various performance indices characterize robot behaviour and its workspace in close relation with its singular con-figurations.
4.4.1 Types of workspaces
An important parameter in robot design is the dextrous workspace, defined as all possible locations of an operating point for which all orientations are possible. In general, a major difficulty in the workspace and dexterity analysis of parallel robots is the strong coupling of positioning and orienta-tion parameters (Dasgupta and Mruthyunjaya 2000; Merlet 2006b). To overcome this difficulty, various types of partial representations of the workspace of parallel robots are used: (i) translation workspace with con-stant orientation of the moving platform, (ii) orientation workspace with a fixed location of an operation point of the moving platform, (iii) reachable workspace with a range of defined orientations.
In addition, dexterity workspace analysis should preferably be done in association with singularities, accuracy and rigidity. Near singularities, the accuracy and the rigidity change drastically, especially in high dynamics conditions. The accuracy of parallel robots is also strongly dependent on the static and dynamic rigidities of the limbs. Dimensions, shape, toler-ances, mechanical properties and technological solutions for links and
4.4 Types of workspaces and singularities 249
joints play a crucial role in robot accuracy and dexterity. In the design process, they represent the main parameters for robot dimensioning and optimization along with other factors, including reliability, safety, cost, ease of operation, and settling time. In the early stages of structural synthe-sis not all of these parameters can be integrated. Great efforts are made to integrate all these parameters into workspace analysis. More details on this topic may be found in the textbooks specialising in parallel robot analysis (Merlet 2006b; Tsai 1999).
4.4.2 Types of singularities
In a general mathematical sense, a singularity is a point at which a given mathematical model is not defined or fails to be well-behaved in some par-ticular way. In engineering, a singularity is a configuration of a system in which the subsequent behaviour cannot be predicted. In a singular configu-ration, the underlying equations of the mechanical model exhibit a mathe-matical singularity. The corresponding motions, forces, or other physical parameters modelled by these equations become undetermined.
Numerous investigations have been conducted on singular configura-tions of mechanisms, with recent emphasis on parallel robotic manipula-tors. The identification of singular configurations in parallel robots is of great importance. Near singular configuration, infinitesimal changes in in-put/output velocities or forces can produce huge variations of out-puts/inputs. In singular configurations, the mechanism loses the ability to transmit resolutely motion and force, and becomes uncontrollable; the ki-nematic and static behaviour of the mechanism change dramatically; the mechanism gains extra degrees of freedom and loses its stiffness. Further-more, the actuator forces may become very intense; this might lead to ro-bot failure or permanent damage of the robot and surrounding equipment, including a breakdown of the mechanism. Therefore, much effort has been devoted to avoiding singularities by a proper design. When they do exist in the workspace, they must be circumvented by proper path planning (Das-gupta and Mruthyunjaya 1998a). Singularity avoidance and a singularity-free workspace represent important design objectives (Carricato and Par-enti-Castelli 2002; Carricato 2005; Gogu 2005g; Wolf and Shohan 2003; Ben-Horin and Shoham 2006a,b).
Various tools have been used in singularity analysis of parallel robots: screw theory (Hunt 1978; Kumar 1992; Huang et al. 1999; Joshi and Tsai 2002; Fang and Tsai 2003; Zhao 2005) Grassmann geometry or line ge-ometry (Merlet 1989, 2006; Collins and Long 1995; Hao and McCarty, 1998, Notash 1998; Simaan and Shoham 2001; Gosselin and Angeles
250 4 Kinematic analysis
1990; Pottman et al 1999; Ebert-Uphoff et al. 2000), Study’s parameteriza-tion of Euclidean displacements (Husty and Karger 2000, 2002; Karger and Husty 1996; Karger 1998, 2003) Clifford algebra (Colins and McCarty 1998), Grassmann-Cayley algebra (Ben Horin ans Shoham 2006a,b,c), Morse function theory and differential forms associated with the constraint functions (Liu at al. 2003), Rodrigues parameters (Di Gregorio 2001a), mixed product of vectors (Di Gregorio 2002), linear decomposition of the determinant (Li et al. 2006), differential-geometric approaches (Park and Kim 1999; Ghosal and Ravani 1998; Kong and Gosselin 2001a; Takeda 2005) geometric approach based on the concept of common tangent (Basu and Ghosal 1997), etc. These tools have been applied to investigate the singular configurations based on the reciprocal screws, the linear transformation between joint and operational velocity spaces and Jacobian matrix calculation, the equilibrium equations governing the forces exerted on the output link by the environment and connecting chains, velocity vector-loop equation, simple geometric rules, characteristic tetrahedron associated with the wrench, the pressure angle in the mechanism, con-straint equations, self-motions of the moving platform, local mobilities in the connecting chains, etc.
Beginning with the pioneering work of Hunt (1978) and Merlet (1989), line geometry has been successfully applied by several researchers to the kinematics and statics of parallel manipulators with extendable limbs. Line geometry is used because the rows of the Jacobian matrix in a linearly ac-tuated fully-parallel manipulator are the Plücker line coordinates of the axes of its extendable links (Hunt 1991). Hence, linear dependency of these lines determines the conditions for instability and singularity of a parallel manipulator.
Much effort was made to understand the nature of singularities in paral-lel robots and to categorize them.
Singularities of parallel robots fall into two basic classes (Kong Gosselin 2001a): the first one, called stationary, kinematic, twist, or in-verse kinematic singularity, in which the moving platform loses one or more degrees of freedom; and, a second one, called uncertainty, static, wrench, or forward kinematic singularity, in which the moving platform may undergo infinitesimal or finite motion when all inputs are locked. These basic classes have been analysed in depth and more complex classi-fications have been proposed.
Gosselin and Angeles (1990) have suggested a classification of singu-larities in closed kinematic chains by considering the matrices Jp and Js in-troduced in the previous section. They have identified three types of singu-larities: (i) singularity of type I, when matrix Js is singular, (ii) singularity of type II, when matrix Jp is singular, and (iii) singularity of type III when
4.4 Types of workspaces and singularities 251
matrices Js and Jp are both singular. Tsai (1999) terms the three types of singularities: inverse, forward and combined singularities. In accordance with the name of Js and Jp, Wanger and Chablat (2000) use serial and par-allel singularities to define the first two types. Serial singularities occur when the determinant of Js becomes zero at the boundary of the opera-tional space. Parallel singularities occur when the determinant of Jp is zero. They are hard to pin down by simple geometric analysis. In a serial singu-larity, the velocity of the moving platform can be zero for a non-zero ac-tuator velocity vector. In a parallel singularity, the velocity of the moving platform can be non-zero when the actuator velocity vector is zero.
Singularities of type III have been redefined many times in the litera-ture. This singularity-type was considered in (Ma and Angeles 1991) as a subset of architecture singularities. In (Sefrioui and Gosselin 1995) it is considered that this type of singularity occurs when the nonlinear kine-matic constraints degenerate. It is mentioned in (Daniali at al. 1995a,b), that this type of singularity arises merely from the way in which the kine-matic relations are formulated, representing in fact a formulation singular-ity that is not necessarily architecture-dependent, contrary to the earlier claim (Ma and Angeles 1991). Later it was observed that configurations in which both matrices JS and Jp are singular exists for general designs of most 3-DOF planar parallel manipulators (Bonev et al. 2003). Generally, there is not uncontrollable finite motion in these configurations. These sin-gularities do not occur just for particular designs and do not necessarily correspond to self motions (Bonev et al. 2003), although, type III singular-ity continue to be mentioned in the literature in its initial definition.
Zlatanov et al. (1994a,b, 1995a,b) have further refined the previous clas-sification by introducing six singularity types; each singularity configura-tion belongs to at least two types. The six singularities are: (i) redundant input (RI-type) which occurs when a non-zero input is possible with zero output, (ii) redundant output (RO-type) which occurs when a non-zero output is possible with zero input, (iii) impossible input (II-type) when a certain input is not feasible for any output, (iv) impossible output (IO-type) when certain output is not feasible for any input, (v) redundant passive mo-tion (RPM-type) which occurs when a non-zero instantaneous motion is possible with both the input and output being equal to zero, and (vi) in-creased instantaneous mobility (IIM-type) which occurs when the transi-tory or instantaneous mobility is higher than the full-cycle mobility of the kinematic chain. This classification is based on a slightly modified defini-tion of singular configurations by taking into consideration the forward and inverse relations between the input and the output of the mechanism, and by considering mobility and singularity as input/output-dependent pa-rameters. This classification is applied to non-redundant mechanisms; the
252 4 Kinematic analysis
input defines the actuated joint velocities and the output defines the instan-taneous velocities of the output link. We can see that type I and type II singularities are in fact RI- and RO-type singularities. The definition of ki-nematic singularity adopted in (Merlet 2006a,b) corresponds to a IIM-type singularity and is called an uncertain configuration by Hunt (1978).
Later on, Park and his co-workers (Han et al. 2002) identified a new singularity in 3-UPU parallel mechanisms that did not obeye the previous classifications. Zlatanov et al. (2001) called it a constraint singularity.
Many other classifications have been proposed to systematize the singu-lar configurations of parallel robots: formulation singularities, configura-tion singularities and architecture singularities (Ma and Angeles 1991), ro-tational and translational singularities (Di Gregorio and Parenti-Castelli, 2002), parameterization singularities with end-effector and actuator singu-larities as two special cases (Liu et al. 2003).
Ma and Angeles (1991) proposed three groups to make a distinction be-tween the singularities which arise merely from the way in which the ki-nematic equations are formulated, the instantaneous singularities or the permanent singularities. A platform which is always in a singular position within the realm of specific displacement is called by Wohlhart (2000) an architecturally shaky platform with regard to these displacements. Com-plete lists of architecture singularities have been recently given by (Karger and Husty 1996), Karger (1998, 2003) and Husty and Karger (2000, 2002).
Sen and Mruthyunjaya (1998) have presented a centro-based characteri-zation of singularities in planar closed-loop manipulators. The singularities have been classified into five types: cusp, ordinary, fixed, force and inter-mediate. The first two are grouped as point singularities, and the last three as link singularities. Singularities of planar parallel robots have also been extensively studied in (Daniali et al. 1995a,b; Sefrioui and Gosselin 1995; Bonev et al. 2003).
Fang and Tsai (2002) divided the singularity cases into limb singularity, manipulator singularity and actuation singularity.
Ider (2005) studied drive singularities; these occur when the actuators lose control of the end-effector in certain directions. If certain conditions related to the consistency of the dynamic equations are satisfied by a proper design of the trajectory, the manipulator can pass through singular positions while the actuator forces and the system motion remain stable.
Three general solutions exist for coping with singular configurations in-side the workspace: (i) developing special control strategies (Bhattacharya et al 1998; Dash et al. 2003; Dasgupata and Mruthyunjaya 1998a, 2000), (ii) using redundantly actuated robots (Wang and Gosselin 2004; Kurtz and Hayward 1992; Merlet 1997; Firmani and Podhorodeski 2004; Notash and Podhorodeski 1994; Kim et al. 2001) or (iii) designing robots with
4.5. Kinetostatic performance indices 253
free-of-singularity workspace (Chablat and Wenger 2003), (Calimbari and Tarantini 2003) Ottaviano and Cecarelli 2002; Gallant and Boudreau 2002; Carricato and Parenti-Castelli 2002; Kong and Gosselin 2002a, Kim and Tsai 2002, Gogu 2004a-d, 2005f-i, 2006a-e, 2007a-c). The last two solu-tions will be integrated in the structural synthesis approach proposed in this book.
4.5. Kinetostatic performance indices
Various performance characteristics are used in robot analysis (Chedmail et al. 1998, Dombre 2001, Merlet 2006). These characteristics are com-monly called performance indices, and are usually calculated for a given robot architecture by using the kinematic, static or dynamic models. They are defined as the robot’s capability to achieve some design criteria related to motion and force transmission, repeatability, singularity avoidance, ac-curacy, workspace, etc. In this section, we discuss the kinematic and static (kinetostatic) instantaneous performance indices of parallel robots based on Jacobian matrix. We emphasize their correlation with motion decoup-ling. These indices are related to motion and force decoupling, manipula-bility, transmission factor, minimal singular value, condition number, con-ditioning ellipsoid, minimum singular value, and isotropy. These indices can be defined on the basis of the conventional or the design Jacobian. We will integrate them in the structural synthesis approach via motion decoup-ling. In the early engineering design stage of structural synthesis, it is es-pecially important that the designer understand the formal nature of these indices, their interconnections and their effect on robot behaviour.
Eqs. (4.20) and (4.23) can be generalized to express the direct kinemat-ics problem
p= J q (4.28)
where p represents the vector of operational velocities, q the vector of joint velocities, and J could be the conventional or the design Jacobian.
The corresponding inverse kinematics problem can be written in the form
q = 1J − p. (4.29)
The Jacobian matrix can be used to describe various kinetostatic rela-tionships related to direct and inverse problems between the following:
a) joint velocity vector and operational velocity vector, as in Eqs. (4.28) and (4.29);
254 4 Kinematic analysis
b) infinitesimal motion of joints ∆q and infinitesimal motion of the end-effector ∆w
∆w =J∆q (4.30)
or
∆q = 1J − ∆w (4.31)
c) the vector of the joint forces τ , and the vector µ of the forces de-veloped on the end-effector (vector of operational forces)
µ = TJ − τ (4.32)
or
τ = TJ µ (4.33)
Eqs. (4.28), (4.30) and (4.32) are associated with the direct problems in which the input is defined in the joint space, and the output in the opera-tional space. Eqs. (4.29), (4.31) and (4.33) are associated with the inverse problems in which the input is defined in the operational space, and the output in the joint space. We note Eq. (4.33) expressing the inverse static problem follows from the principle of power conservation in robot me-chanical system:
⋅Tµ p = ⋅Tτ q (4.34)
and by taking into account Eq. (2.28). Eqs. (4.28), (4.29) and (4.30)-(4.33) show that the Jacobian matrix con-
tains important information concerning the kinetostatic characteristics of the robot.
The various kinetostatic indices used to qualify robot behaviour are de-fined in a Euclidean vector space in which vector norms play an essential role. We have seen that Eq. (4.28) represents a linear transformation be-tween joint and operational velocity spaces. In defining this linear trans-formation there is no need to consider the two vector spaces to be Euclid-ean.
When joint velocity vector q or operational velocity vector p contain components with different physical units, the norms of these two vectors become physically inconsistent, however Eq. (4.28) remains physically consistent. Physical consistency refers to the physical validity of equations combining quantities of different units in a meaningful way. The simplest
4.5. Kinetostatic performance indices 255
example of physically inconsistency is the attempt to add quantities with different units.
The translation and angular velocities of the moving platform given by Eq. (4.28) will always have the same units (length/time and 1/time) as the velocities of the actuators. The linear transformation between actuated and operational vector spaces expressed by this equation does not alter velocity units. For this reason Eq. (4.28) is always physically consistent, although the norms of the velocity vectors of the moving platform and actuated joint could be physically inconsistent. The two vector norms are physically con-sistent just for four types of parallel robots: (i-ii) translational parallel ro-bots with two or three degrees of mobility actuated by linear motors and (iii-iv) parallel wrists with two or three degrees of mobility actuated by ro-tation motors. All other parallel robots have inconsistent norms for the two velocity vectors, because these vectors include terms with dimension length/time and 1/time. The vector velocity of the moving platform is not Euclidian since its elements do not share the same physical unit. For a ma-nipulator with mixed revolute and prismatic actuated joints, the joint-rate vector is not Euclidian either. As a consequence, operational velocity space and joint velocity space are Euclidian vector spaces just for the four types of above mentioned parallel robots. The Jacobian matrix contains only dimensionless numbers when the norms of the two velocity vectors are physically consistent and a mix of dimensionless numbers and terms with length and/or time units when the norms of the two velocity vectors are physically inconsistent.
This warning is also applies to the forces and moments of the actuated joints, and the operational forces in Eq. (4.32) and (4.33).
A general study of incommensurate systems and physical consistency in robotics have been presented by Schwartz at al. (2002, 2003) and we will follow their lead with respect to physical consistency. An incommensurate system involves physical quantities with different units that are nonethe-less described and controlled by physically consistent equations.
The kinetostatic performance indices are based on the norm of velocity vectors, the determinant and eigenvalues of the Jacobian matrix J, or other matrices such as TJ , 1J − , TJ − , TJJ , TJ J , T 1( JJ )− , T 1( J J )− . We have denoted T 1( J )− = 1 T( J )− by TJ − . The mathematical equations used to calculate these parameters often obscure the interaction between units, and lead to results that may be misinterpreted, erroneous, or simply arbitrary. We must always check the physical consistency of these mathematical en-tities. The general definitions and formulae presented in this section for ki-netostatic indices are introduced under the hypothesis of respecting physi-cal consistency.
256 4 Kinematic analysis
In the structural synthesis approach, we will integrate the notion of mo-tion decoupling, and in this section we will emphasize its connection with the kinetostatic performance indices.
4.5.1 Cross-coupling indices
Cross-coupling indices represent measures of decoupling between the col-umn vectors of the matrices used in formalizing the direct and inverse problems in robot kinematics and statics.
These indices take values between zero and one. A unit value of one of these indices indicates that the two column vectors are orthogonal. When all cross-coupling indices have maximum values, the robot kinemat-ics/statics are completely decoupled.
We have seen that Eqs. (4.28)-(4.33) represent linear transformations between vectors of the joint space and operational space. In the general case, these equations can be written in the form
y=Ax. (4.35)
Vector x could be q or p and vector y could also be q or p in any combi-nation with the condition that x≠ y. Matrix A could be the Jacobian J, its transpose JT or an inverse matrix 1J − or TJ − .
For n× n square matrix A and a unit vector x, Eq. (4.35) becomes
y=n
ii i=∑A
(4.36)
where Ai is the vector associated with of the ith column of matrix A. The angular relationship ijθ between two columns Ai and Aj of matrix A can be obtained by taking the dot product
cos ijθ = i j
i j
⋅A AA A
. (4.37)
If we denote by aij the element of matrix A situated in the ith row and jth column, Eq. (4.37) becomes
4.5. Kinetostatic performance indices 257
cos ijθ
n
ki kjk 1
1n n 2
2 2ki kj
k 1 k 1
a a
a a
=
= =
=
∑
∑ ∑.
(4.38)
Definition 4.9. (Local cross-coupling index). The local cross-coupling index ijκ is a measure of decoupling between two column vectors Ai and Aj of the matrix A and is given by the sine of intersection angle ijθ be-tween Ai and Aj
ijκ =sin ijθ . (4.39)
By taking into account Eqs. (4.37) and (4.38), the cross-coupling index can be expressed by
ijκ =
12 2
i j
i j
1 ⋅ −
A AA A
(4.40)
or
ijκ
12 2n
ki kjk 1
n n2 2ki kj
k 1 k 1
a a1
a a
=
= =
= −
∑
∑ ∑.
(4.41)
Since the angle of interaction between column vectors is in the range of (0,π ) we can see that 0≤ ijκ ≤ 1. For any non-singular matrix A, all n col-umns of A are linearly independent. When two column vectors iA and jA are linearly dependent, i.e. iA =λ jA , matrix A is singular and ijκ =0. In this case we say that matrix A is rank deficient; the two column vectors are aligned, that is ijθ =0. When ijκ =1, the two column vectors iA and jA of matrix A are mutually orthogonal and the robot joint motions are com-pletely decoupled. The name of this index is derived from the fact that its maximum value corresponds to a decoupled motion.
A more general cross-coupling index can be defined by extending the partial index to the entire matrix A.
258 4 Kinematic analysis
Definition 4.10. (Cross-coupling index). The cross-coupling index is a measure of decoupling between all the column vectors of matrix A, and is given by the product of the local cross-coupling indices
iji 1,n 1j 1 i ,n
κ κ= −= +
= ∏ (4.42)
or 1
2 2n
ki kjk 1
n ni 1,n 1 2 2j 1 i ,n ki kj
k 1 k 1
a a1
a aκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.43)
We can see that the cross-coupling index defined by Eq. (4.43) is equivalent to the reangularity index in axiomatic design - see Eq. (4.2).
We derive various cross-coupling indices from the two indices defined above by associating the matrix A with the transformation matrices of the kinematics and statics problems defined in Eqs. (4.28), (4.29), (4.32), (4.33). These Jacobian matrices are n× n square matrices with n=SF=M.
Kinematics cross-coupling indices
Cross-coupling indices in direct and inverse kinematics problems are de-rived from the local and general indices defined above by associating the matrix A with the Jacobian J and its inverse 1J − . We denote two column vectors of the Jacobian J by Ji and Jj, and two column vectors of the in-verse Jacobian 1J − by 1
i−J and 1
j−J .
Definition 4.11. (Local cross-coupling index of the direct kinematics problem). The local cross-coupling index of the direct kinematic problem is a measure of decoupling between two column vectors Ji and Jj of the Jacobian matrix J given by the sine of the intersection angle between Ji and Jj in an instantaneous pose of the moving platform.
This index is given by Eqs. (4.40) or (4.41) in which A=J and Ai=Ji (i=1,2,…,n).
This instantaneous index is named the local kinematic cross-coupling index in (Lee et al. 1993) and is used to measure the degree of linear de-pendency between column vectors of the conventional Jacobian matrix.
For any non-singular pose of the moving platform, all n=SF columns of the Jacobian matrix are linearly independent. These column vectors be-
4.5. Kinetostatic performance indices 259
come linearly dependent only in singular poses as we have mentioned above.
Definition 4.12. (Local cross-coupling index of the inverse kinemat-ics problem). The local cross-coupling index of the inverse kinematic problem is a measure of decoupling between two column vectors 1
i−J and
1j−J of inverse Jacobian matrix 1J − , and is given by the sine of intersec-
tion angle between 1i−J and 1
j−J in an instantaneous pose of the moving
platform. This index is given by Eqs. (4.40) or (4.41) in which A= 1J − and
iA = 1i−J (i=1,2,…,n).
Definition 4.13. (Cross-coupling index of the direct kinematics prob-lem). The cross-coupling index of the direct kinematic problem is a meas-ure of decoupling between the column vectors of Jacobian matrix J, and is given by the product of the corresponding local cross-coupling indices.
This index is given by Eqs. (4.42) or (4.43) in which A=J and Ai=Ji (i=1,2,…,n).
Definition 4.14. (Cross-coupling index of the inverse kinematics problem). The cross-coupling index of the inverse kinematic problem is a measure of decoupling between the column vectors of inverse Jacobian matrix J-1 and is given by the product of the corresponding local cross-coupling indices.
It is given by Eqs. (4.42) or (4.43) in which A= 1J − and iA = 1i−J
(i=1,2,…,n).
Statics cross-coupling indices
The cross-coupling indices in direct and inverse statics problems are de-rived from the local and general indices defined above by associating ma-trix A with Jacobian TJ and its inverse TJ − . We denote two column vec-tors of the transpose Jacobian TJ by T
iJ and TjJ , and two column vectors
of the inverse transpose Jacobian TJ − by Ti−J and T
j−J .
Definition 4.15A. (Local cross-coupling index of the inverse statics problem). The local cross-coupling index of the inverse statics problem is a measure of decoupling between two column vectors T
iJ and TjJ of
transpose Jacobian TJ , and is given by the sine of intersection angle be-tween T
iJ and TjJ in an instantaneous pose of the moving platform.
260 4 Kinematic analysis
This index is given by Eqs. (4.40) or (4.41) in which A= TJ and
iA = TiJ (i=1,2,…,n).
The local cross-coupling index of the inverse statics problem can also be defined and calculated by using Jacobian matrix J and taking into account that the columns of the transpose Jacobian TJ are the rows of J.
Definition 4.15B. (Local cross-coupling index of the inverse statics problem). The local cross-coupling index of the inverse statics problem is a measure of decoupling between two row vectors i
*J and j*J of the
Jacobian matrix J and is given by the sine of intersection angle between
i*J and j
*J in an instantaneous pose of the moving platform
ijκ =
12 2* *
i j* *
i j
1 ⋅ −
J JJ J
(4.44)
or
ijκ
12 2n
ik jkk 1
n n2 2ik jk
k 1 k 1
a a1
a a
=
= =
= −
∑
∑ ∑.
(4.45)
Definition 4.16A. (Local cross-coupling index of the direct statics problem). The local cross-coupling index of the direct statics problem is a measure of decoupling between two column vectors T
i−J and T
j−J of the
inverse transpose Jacobian matrix TJ − given by the sine of the intersection angle between -T
iJ and -TjJ in an instantaneous pose of the moving plat-
form. This index can be calculated by Eqs. (4.40) or (4.41) in which A= TJ −
and iA = Ti−J (i=1,2,…,n).
The local cross-coupling index of the direct statics problem can also be defined and calculated by using the inverse Jacobian matrix by taking into account that the columns of TJ − are the rows of 1J − .
Definition 4.16B. (Local cross-coupling index of the direct statics problem). The local cross-coupling index of the direct statics problem is a
4.5. Kinetostatic performance indices 261
measure of decoupling between two row vectors 1i−*J and 1
j−*J of the in-
verse Jacobian 1J − , and is given by the sine of intersection angle between 1
i−*J and 1
j−*J in an instantaneous pose of the moving platform
ijκ =
12 2-1 * -1
i j-1 * -1i j
1 ⋅ −
*
*
J JJ J
(4.46)
or
ijκ
12 2n
ik jkk 1
n n2 2ik jk
k 1 k 1
b b1
b b
=
= =
= −
∑
∑ ∑
(4.47)
where bij is the element situated on ith row and jth column of matrix 1J − . Definition 4.17A. (Cross-coupling index of the inverse statics prob-
lem). The cross-coupling index of the inverse kinematic problem is a measure of decoupling between the column vectors of the transpose Jaco-bian matrix TJ given by the product of the corresponding local cross-coupling indices.
It can be calculated by Eqs. (4.42) or (4.43) in which A= TJ and
iA = TiJ (i=1,2,…,n).
Definition 4.17B. (Cross-coupling index of the inverse statics prob-lem). The cross-coupling index of the inverse kinematic problem is a measure of decoupling between the row vectors of the transpose Jacobian matrix TJ given by the product of the corresponding local cross-coupling indices
12 2n
ik jkk 1
n ni 1,n 1 2 2j 1 i ,n ik jk
k 1 k 1
a a1
a aκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.48)
Definition 4.18A. (Cross-coupling index of the direct statics prob-lem). The cross-coupling index of the direct statics problem is a measure
262 4 Kinematic analysis
of decoupling between the column vectors of the inverse transpose Jaco-bian matrix TJ − given by the product of the corresponding local cross-coupling indices.
It can be calculated by Eqs. (4.42) or (4.43) in which A= TJ − and
iA = Ti−J (i=1,2,…,n).
Definition 4.18B. (Cross-coupling index of the direct statics prob-lem). The cross-coupling index of the direct static problem is a measure of decoupling between the row vectors of the inverse Jacobian matrix 1J − given by the product of the corresponding local cross-coupling indices
12 2n
ik jkk 1
n ni 1,n 1 2 2j 1 i ,n ik jk
k 1 k 1
b b1
b bκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.49)
where bij is the element situated in the ith row and jth column of matrix 1J − .
4.5.2 Indices of input-output propensity
The indices of input-output propensity represent a measure of the predilec-tion of input components for their corresponding output components de-fined by the equations of the direct and inverse problems in robot kinemat-ics and statics. The word propensity is used here in the sense of predilection of a “personalized” input for its corresponding output.
These indices also take values between zero and one. The unit value of these indices indicates that the correspondence between the input-and out-put has a maximum intensity. When all indices of input-output propensity have unit value robot kinematic/statics are completely decoupled.
We first define these indices for the general case of the linear transfor-mation set up by Eq. (4.34) in which we consider vector x as input and y as output. A pair of corresponding input-output components xi, yi is formed by the ith components of both x and y vectors.
Definition 4.19. (Local index of input-output propensity). The local index of input-output propensity is a measure of the predilection of the ith component of input vector for the ith component of output vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector iA of matrix A
4.5. Kinetostatic performance indices 263
iη = ii1
n 22ki
k 1
a
a=
∑
.
(4.50)
We can see that iη is a measure of the relative value of aii in the column vector iA of matrix A.
A more general index of input-output propensity can be defined by ex-tending the partial index to n=SF=M input-output pairs.
Definition 4.20. (Index of input-output propensity). The index of in-put-output propensity is a measure of the of the predilection of n=SF=M input components for their corresponding output components, and is given by the product of the local corresponding indices
n nii
i 1i 1 i 1 n 2
2ki
k 1
a
a
η η= =
=
= =
∏ ∏∑
.
(4.51)
We can see that the index of input-output propensity defined by Eq. (4.51) is equivalent to the semangularity in axiomatic design - see Eq. (4.3).
We derive various indices of input-output propensity from the two indi-ces defined above by associating matrix A with the transformation matrices of the kinematics and statics problems defined in Eqs. (4.28), (4.29), (4.32), (4.33).
Kinematics indices of input-output propensity
Indices of input-output propensity in direct and inverse kinematics prob-lems are derived from the local and general indices defined above by asso-ciating matrix A with Jacobian J and its inverse 1J − .
Definition 4.21. (Local index of input-output propensity in direct kinematics). The local index of input-output propensity in direct kinemat-ics is a measure of the predilection of the ith component of the joint veloc-ity vector for the ith component of operational velocity vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector iJ of the Jacobian matrix J.
This index is given by Eq. (4.50) in which A=J and iA = iJ Definition 4.22. (Local index of input-output propensity in inverse
kinematics). The local index of input-output propensity in inverse kine-
264 4 Kinematic analysis
matics is a measure of the predilection of the ith component of operational velocity vector for the ith component of joint velocity vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector 1
i−J of Jacobian matrix J.
It is given by Eq. (4.50) in which A= 1J − and iA = 1i−J .
Definition 4.23. (Index of input-output propensity in direct kinemat-ics). The index of input-output propensity in direct kinematics is a measure of the of the predilection of the components of joint velocity vector for their corresponding components of operational velocity vector, and is given by the product of the local corresponding indices.
It is given by Eq. (4.51) in which A=J and iA = iJ . Definition 4.24. (Index of input-output propensity in inverse kine-
matics). The index of input-output propensity in inverse kinematics is a measure of the predilection of components of operational velocity vector for their corresponding components of joint velocity vector, and is given by the product of the local corresponding indices.
It is given by Eq. (4.51) in which A= 1J − and iA = 1i−J .
Statics indices of input-output propensity
Indices of input-output propensity in direct and inverse statics problems are derived from the local and general indices defined above by associating matrix A with transpose Jacobian TJ and its inverse TJ − .
Definition 4.25A. (Local index of input-output propensity in inverse statics). The local index of input-output propensity in inverse statics is a measure of the predilection of the ith component of operational force vec-tor for the ith component of joint force vector and is given by the ratio be-tween the absolute value of the element aii and the norm of the column vector T
iJ of transpose Jacobian matrix TJ . It is given by Eq. (4.50) in which A= TJ and iA = T
iJ . Definition 4.25B. (Local index of input-output propensity in inverse
statics). The local index of input-output propensity in inverse statics is a measure of the predilection of the ith component of operational force vec-tor for the ith component of joint force vector, and is given by the ratio be-tween the absolute value of the element aii and the norm of the row vector
i*J of Jacobian matrix J
4.5. Kinetostatic performance indices 265
iη = ii1
n 22ik
k 1
a
a=
∑
.
(4.52)
Definition 4.26A. (Local index of input-output propensity in direct statics). The local index of input-output propensity in direct statics is a measure of the predilection of the ith component of joint force vector for the ith component of operational force vector, and is given by the ratio be-tween the absolute value of the element aii and the norm of the column vector T
i−J of inverse transpose Jacobian TJ − .
It is given by Eq. (4.50) in which A= TJ − and iA = Ti−J .
Definition 4.26B. (Local index of input-output propensity in direct statics). The local index of input-output propensity in direct statics is a measure of the predilection of the ith component of joint force vector for the ith component of operational force vector, and is given by the ratio be-tween the absolute value of the element bii and the norm of the row vector
-1i
*J of Jacobian matrix 1J −
iη = ii1
n 22ik
k 1
b
b=
∑
.
(4.53)
where bii is the element situated on ith row and jth column of matrix 1J − . Definition 4.27. (Index of input-output propensity in inverse statics).
The index of input-output propensity in inverse statics is a measure of the predilection of components of operational force vector for their corre-sponding components of joint force vector, and is given by the product of the local corresponding indices.
This index can be calculated by Eq. (4.51) in which A= TJ and iA = TiJ
or by n n
iii 1
i 1 i 1 n 22ik
k 1
a
a
η η= =
=
= =
∏ ∏∑
.
(4.54)
Definition 4.28. (Index of input-output propensity in direct statics). The index of input-output propensity in direct statics is a measure of the predilection of components of joint force vector for their corresponding
266 4 Kinematic analysis
components of operational force vector, and is given by the product of the local corresponding indices.
It can be calculated by Eq. (4.51) in which A= TJ − and iA = Ti−J or by
n nii
i 1i 1 i 1 n 2
2ik
k 1
b
b
η η= =
=
= =
∏ ∏∑
.
(4.55)
where bii is the element situated in the ith row and jth column of matrix 1J − .
4.5.3 Kinetostatic indices defined in connection with manipulability ellipsoids and polytops
Manipulability ellipsoids and polytops of the Jacobian matrix of robotic manipulators were introduced by Salisbury and Craig (1982) and Yoshi-kawa (1985a,b, 1990) to characterize robot behaviour. They can be defined from the viewpoints of kinematics, statics and dynamics. The shape and volume of this ellipsoid/polytop is often used to characterize the manipula-tor dexterity and motion/force amplification. In this section, we simply present the kinetostatic indices defined in connection with manipulability ellipsoids and polytops and we emphasise their connection with motion decoupling that we integrate in the structural synthesis approach.
The manipulability ellipsoid/polytop is associated with the mapping be-tween the joint and operational velocity/force spaces. We mention that n-dimensional ellipsoids are the images of spheres under invertible linear transformations and the polytop represents the generalization to any di-mension of a polygon in two dimensions, and a polyhedron in three dimen-sions. The difference between manipulability ellipsoid and manipulability polytop resides in the norm used for the vectors of the joint and operational spaces (Yoshikawa 1985a,b, 1990; Merlet 2006a,b). Numerous types of norms can be used to represent the manipulability ellipsoid and polytop in an n-dimensional space. For a non-redundant robot the dimension of this space is equal to robot mobility and end-effector connectivity (n=M=SF). The most common are L1, L2 and L∞ norms. The L1-norm is the sum of absolute magnitudes of each element. The L2-norm, also called Euclidean norm is the square root of the sum of the squares of the magnitudes of each element, and the infinity norm is the maximum of the magnitudes of the elements.
4.5. Kinetostatic performance indices 267
The eigenvalues and the singular values of Jacobian matrix are used to define several kinetostatic indices in connection with the n-dimensional el-lipsoid. In this section, we present the force and velocity manipulability el-lipsoids and their associated indices.
Velocity and force ellipsoids
Definition 4.29. (Velocity ellipsoid). The velocity ellipsoid represents the image in robot operational velocity space of an n-dimensional sphere of unit radius defined in the joint velocity space.
This map between joint and operational velocity spaces is expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.28).
Let us consider the set of all operational velocities corresponding to all joint velocities with Euclidean norm bounded by unity:
1≤q . (4.56)
We can write 2 2 2 2
1 2 nq q ... q 1= = + + + ≤Tq q q . (4.57)
In the n-dimensional joint velocity space, the upper bound of Eq. (4.57) can be associated with an n-dimensional sphere of unit radius.
Eq. (4.28) maps this hypersphere, defined in the joint velocity space, to an n-dimensional ellipsoid in the operational space
( ) 1T TJJ 1−
≤p p . (4.58)
The end-effector velocity is amplified in the direction of the major axes and reduced in the direction of the minor axis of this hyperellipsoid for a unit actuated joint velocity. If the ellipsoid is a sphere, the end-effector moves uniformly in all directions. Yoshikawa have called this the manipu-lability ellipsoid (Yshikawa 1985a). This ellipsoid was later called the ve-locity eigen-ellipsoid (Lee et al. 1993).
The eigenvectors of matrix T 1( JJ )− define the directions of the principal axes of this manipulability ellipsoid. The magnitudes of singular values of J represent the lengths of the principal axes. We recall that the singular values of J are the square roots of the eigenvalues of TJJ .
By taking into account the reciprocity force-velocity we can also define a force ellipsoid.
268 4 Kinematic analysis
Definition 4.30. (Force ellipsoid). The force ellipsoid represents the image in robot operational force space of an n-dimensional sphere of unit radius defined in the actuated joint force space.
This map between joint and operational force spaces is expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.33).
Let us consider the set of all operational forces statically exerted by the end-effector on an object corresponding to all actuated joint forces with Euclidean norm bounded by unity
1≤τ . (4.59)
We can write 2 T 2 2 2
1 2 n... 1τ τ τ= = + + + ≤τ τ τ . (4.60)
In the n-dimensional joint velocity space, the upper bound of Eq. (4.60) can be associated with an n-dimensional sphere of unit radius.
Eq. (4.33) maps this force hypersphere, defined in the joint space of ac-tuator forces, to an n-dimensional force ellipsoid in the operational force space described by
T TJJ 1≤τ τ . (4.61)
The end-effector exerts high forces in the direction of the major axes and low forces in the direction of the minor axes of this hyperellipsoid. If the ellipsoid is a sphere, the end-effector exerts forces in all directions uni-formly. Yoshikawa (1990) have called it manipulating-force ellipsoid. This ellipsoid was also called later force-eigen-ellipsoid (Lee et al. 1993).
The eigenvectors of matrix JJT define the directions of the principal axes of this force ellipsoid. The magnitudes of inverse singular values of J rep-resent the lengths of the principal axes.
Velocity and force polytops
Similar representations of manipulability can be obtained if we use the in-finity norm instead of the Euclidean norm when bounding the joint veloci-ties and forces.
Definition 4.31. (Velocity polytop). The velocity polytop represents the image in robot operational velocity space of an n-dimensional cube of unit edge-length defined in the joint velocity space.
This map between joint and operational velocity spaces is also ex-pressed by the linear transformation with the Jacobian matrix defined in Eq. (4.28).
For the manipulability polytop, each joint velocity is bounded by unity
4.5. Kinetostatic performance indices 269
iq 1≤ , i=1,2,…,n. (4.62)
In the n-dimensional joint velocity space, the upper bound of Eq. (4.62) can be associated with an n-dimensional cube of unit edge-length.
Eq. (4.28) maps this hypercube, defined in the joint velocity space, to an n-dimensional polytop in the operational velocity space. This polytop was introduced by Yoshikawa (1990) and called the manipulability parallelepi-ped in n-dimensional space, and later the manipulability polyhedron (Mer-let 2006a,b) or simply the operational polytop that can include the largest manipulability ellipsoid (Krut et 2004). A possible advantage of using the polytop instead of the ellipsoid in manipulability analysis resides in the fa-cility of building and intersecting polytops.
Definition 4.32. (Force polytop). The force polytop represents the im-age in robot operational force space of an n-dimensional cube of unit edge-length in the actuated joint force space.
This map between joint and operational force spaces is also expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.33).
For the manipulating-force polytop, each joint force is bounded to unity
i 1≤τ , i=1,2,…,n. (4.63)
In the n-dimensional force space of the actuated joints, Eq. (4.63) can also be associated with an n-dimensional cube of unit edge-length.
Eq. (4.33) maps this hypercube, defined in the force space of the actu-ated joints, to an n-dimensional polytop in the operational space of forces exerted by the end-effector.
Several indices are defined by using the manipulability ellipsoid or polytop. We denote by iυ (i=1,2,…5) and iψ (i=1,2, …,n) these various indices. These indices are applicable for both serial and parallel robots when the physical consistency is respected, as mentioned at the beginning of this section. They are defined by using the volume of the manipulability ellipsoid and the length of its principal axes, and are mainly used in di-mensional synthesis (Bouzgarrou et al. 2000; Chablat et al. 2000).
Isotropy and conditioning indices
A robotic manipulator is called isotropic if its Jacobian matrix is isotropic in at least one configuration (Salisbury and Craig 1982; Angeles and Rojas 1987; Angeles and López-Cajún 1988; Angeles 1992). A given matrix B is called isotropic if its singular values are all equal and nonzero. Isotropy in robotic manipulators is closely related to condition number of the Jacobian
270 4 Kinematic analysis
matrix. The condition number of the matrix B is defined as the ratio of the greatest and the smallest singular values of J (Golub and Van Loan 1983)
max
min
scs
= . (4.64)
If B is a square matrix than c is the product of the norms of B and B-1: 1c B B−= . (4.65)
The magnitude of the condition number depends on the choice of the matrix norm. Eqs. (4.64) and (4.65) give similar result if 2-norm is used in Eq. (4.65). When 1-norm, Frobenius norm or infinity norm are used the two equations give different values of the condition number. We recall that: (i) the 1-norm of matrix B is the largest column sum, where the col-umn sum is the sum of the magnitudes of the elements in a given column, (ii) the 2-norm of matrix B is the square root of the largest eigenvalue of BBT, (iii) the Frobenius norm of matrix B is the square root of the sum of the squares of the magnitudes of each element and (iv) the infinity norm of matrix B is the maximum row sum, where the row sum is the sum of the magnitudes of the elements in a given row.
Isotropic matrices have a condition number of unity. Furthermore, the condition number of either square singular matrices or rank-deficient ma-trices is infinity, and hence isotropy and singularity are apparently at the opposite side of the spectrum with regard to matrix invertibility. We recall that for a m×n matrix B with m≤n, its singular values are the eigenvalues of the m×m symmetric matrix BBT. Hence matrix B is isotropic if all its singularvalues are identical and nonzero. This is equivalent to saying that, if B is isotropic, than the product BBT is proportional to the identity matrix; i.e., a positive scalar exists in this case for which
BBT=λ I (4.66)
where I is m×m identity matrix. The condition number of matrix B is configuration dependent and is
used by Angeles (1992) to measure the distance of the given pose to a sin-gular configuration.
A parallel robot is fully-isotropic if the condition number is unity in the entire workspace.
Definition 4.33. (Conditioning index 1υ ). The conditioning index of a parallel manipulator is the inverse of the condition number in a given pose, and is given by the ratio of the smallest and the greatest singular values of its Jacobian matrix
4.5. Kinetostatic performance indices 271
1υ = min
max
ss
(4.67)
Definition 4.34. (Minimum conditioning index 2υ ). The minimum conditioning index of a parallel manipulator is the minimum value of the conditioning index corresponding to the entire robot workspace.
2υ =min( 1υ ) (4.68)
As we have already mentioned, the dimensional in homogeneity of the elements of Jacobian matrix leads to physical inconsistency when evaluat-ing the manipulator conditioning index. There are various ways of coping with this problem:
a) Redefine the manipulator Jacobian as the matrix mapping joint rates into velocities of a set of points on the end-effector, thereby eliminating the end-effector angular velocity and hence producing a dimensionally homogeneous Jacobian (Gosselin 1990; Kim and Ryu 2003). Although this inverse Jacobian matrix is homogeneous in terms of units, the condition number will no longer describe the intrinsic behaviour of the robot as its value depends on the choice of points on the end-effector (Merlet 2006a,b).
b) Write the point-velocity manipulator equations in dimensionless form by suitably dividing both sides of the velocity equation by a natural length of the robotic manipulator (Ma and Angeles 1991; Angeles 1992; Fattah and Hasan Ghasemi 2002). This length, called a characteristic length, is defined as that minimizing the condition number of the arising dimensionless Jacobian so ob-tained.
c) Calculate two condition numbers by splitting the Jacobian matrix into two parts corresponding to linear mapping of translational and angular velocities, or forces and moments (Kosuge et al 1993; Ahn and Hong 1996; Hong and Kim 2000).
Manipulability indices
Definition 4.35. (Manipulability index 3υ ). The manipulability index 3υ represents the absolute value of the determinant of the parallel robot Jaco-bian matrix in a current pose
3υ = det J . (4.69)
272 4 Kinematic analysis
The manipulability index 3υ is closely related to the singular values si (i=1,2,…,n) of Jacobian matrix J and to the volume Ve of the manipulabil-ity ellipsoid
3υ =n
ii 1
s=∏
(4.70)
and
3υ = e
m
Ve
(7.71)
where
( )( )
( )( )
n2
m n 12
2if n is even
2 4 6 n 2 ne
2 2if n is odd
1 3 5 n 2 n
−
⋅ ⋅ ⋅ ⋅ ⋅ − ⋅=
⋅ ⋅ ⋅ ⋅ ⋅ − ⋅
π
π.
Definition 4.36. (Manipulability index 4υ ). The manipulability index
4υ represents the ratio between the absolute value of the Jacobian deter-minant in a current pose and its maximum absolute value corresponding to the overall workspace
34
3maxmax
det Jdet J
υυυ
= = . (4.72)
This normalized index was proposed in (Lee et al. 1998) and extended in (Zhang et al. 2000) to redundant parallel robots.
Definition 4.37. (Performance index 5υ ). The performance index 5υ represents the minimum singular value of the Jacobian matrix
5 minsυ = (4.73)
This performance index was proposed in (Klein and Blaho 1987) to characterize the distance to singular configuration.
Transmission factors
Definition 4.38. (Velocity transmission factors iψ , i=1,2,…,n) The ve-locity transmission factors in the directions of the principal axes, iψ
4.5. Kinetostatic performance indices 273
(i=1,2,…,n) are the singular values of 1J − ; that is, the inverses of the square roots of the eigenvalues of TJJ
ii
1s
ψ = , i=1,2,…,n. (4.73)
We recall that the singular values of J are the square roots of the eigen-values of TJJ and the singular values of 1J − are the inverses of the singu-lar values of J.
The velocity transmission factors are used in (Chablat and Wenger, 2003) to determine the link length and the linear joint limits of the transla-tional parallel robot Orthoglide.
We note that Eqs. (4.67), (4.68), (4.70), (4.71) and (4.73) are applicable to non-redundant and redundant robotic manipulators. For redundant ro-botic manipulators, Eqs. (4.69) and (4.72) become
( )T3 det JJυ = (4.74)
and
( )( )
T3
4 T3maxmax
det JJ
det JJυ
υυ
= = . (4.75)
We note that the manipulability index 3υ also called the manipulability measure was introduced by Yoshikawa (1985a,b, 1990) to calculate a kind of distance of the robot’s current configuration from singular ones, starting from the observation that in a singular position the determinant of the Jacobian matrix and one or more eigenvalues become zero, leading to
3υ =0. Yoshikawa considered that when the manipulability index 3υ reaches its maximum, the robot is furthest away from the singular configu-ration. The use of manipulability indices is critically reviewed by Angeles (1992) pointing out that the determinant of a square matrix, or that of the product of a rectangular matrix and its transpose does not measure how far from singularity a matrix is (Forsythe and Moler 1967). In fact, the non zero determinant of a square matrix merely indicates that a matrix is in-vertible, but it does not measure the invertibility of the matrix.
The determinant of the Jacobian matrix has the advantage of being in-variant with respect to changes of reference frame and, by normalization, it can be made invariant with respect to changes of scale and physical units. In this case, the manipulability index could exhibit consistent values with
274 4 Kinematic analysis
respect to positioning errors, as in the example presented in (Merlet, 2006). The use of condition number along with manipulability as kinetostatic per-formance indices is critically reviewed by Merlet (2006) when accuracy of a parallel robot is analysed. This analysis emphasizes the sensitivity of the condition number of a normalized Jacobian with respect to various types of norms used for its calculation.
Since there are no natural norms in the spaces used to represent general-ized velocities and forces (Park 1995), kinetostatic performance indices are not invariant under changes of the norms used. This non-invariance prop-erty introduces certain applicability limitations of these indices as meas-ures of distance from singularity, as presented in (Staffetti et al. 2002).
4.6 Design Jacobian and motion decoupling
Motion decoupling in parallel robots can be defined by analogy with de-sign decoupling in axiomatic design. We consider the design Jacobian as a design matrix and we use it to introduce various decoupling measures and types of parallel robots.
We have seen in 4.3.2 that the angular velocity 0ω =T
x y z, ,ω ω ω can-not be calculated by direct differentiation of the rotation angles of the moving platform. The corresponding rotation angles do not have a physical meaning when using 0ω . Moreover, Eqs. (4.20)-(4.22) used to calculate 0ω with or without time derivatives of the rotation angles ,α β and δ , intro-duce some hypothetical complementary coupling between ω and 0ω and implicitly between the velocity of the actuated joints and the velocity of the moving platform. This coupling has no physical correspondence in the mechanical architecture of the kinematic chain associated with the robot. It is due merely to the theoretical approach used in modelling robot kinemat-ics by using 0ω . This hypothetical coupling is unavoidable in calculating
0ω for any robot with two or three rotational degrees of freedom of the end-effector. Even the simplest orientation mechanism with just two or-thogonal revolute pairs is affected by this hypothetical coupling. For this reason, in motion coupling analysis we use ω =
T, ,α β δ =
T, ,α β δω ω ω
and [ 0v ,ω ]T= [ x y zv ,v ,v , , ,α β δω ω ω ]T to define the velocity of the moving platform. This choice allows us to characterise the real nature of motion coupling due to robot mechanical architecture.
4.6 Design Jacobian and motion decoupling 275
Parallel robots with coupled, decoupled, or uncoupled motions, and maximally regular robots can be defined by using the design Jacobian de-fined in connection with the direct kinematics problem of non-redundant parallel robots
J=w q (4.76)
where [ ]T1 nw ... w=w is the vector of operational velocities of the mov-
ing platform, [ ]T1 nq ... q=q are the velocity vectors of the actuated joints, and J is the design Jacobian. We recall that n is equal to robot mo-bility and the connectivity of the moving platform n=M=SF.
The degree of motion coupling in a non redundant parallel robot defines the number of interconnections existing between the joint and operational motions. We define an instantaneous and a global degree of motion cou-pling and we denote them by Ci and C .
Definition 4.39 (Degree of instantaneous motion coupling). The de-gree of instantaneous motion coupling of a non redundant parallel robot is given by the number of non zero off-diagonal terms of the design Jacobian in a given non-singular pose of the moving platform.
We have seen that the design Jacobian is a square n× n matrix. The maximum value of the absolute motion coupling is n(n-1) and indicates that all n=M actuators participate in n=SF independent motions of the mov-ing platform (n=M=SF). In non redundant parallel robots the degree of mo-tion coupling could vary from 2 to 30; the maximum value corresponds to parallel robots with n=M=SF=6, and the minimum to robots with n=M=SF=2.
Definition 4.40 (Global degree of motion coupling). The global degree of motion coupling of a non redundant parallel robot is given by the maximum value of the instantaneous degree of motion coupling
C =min(Ci). (4.77)
The degree of instantaneous motion coupling is defined numerically for the design Jacobian. The global degree can be easily determined on the Jacobian defined symbolically with no need to use Eq. (4.77). In this case, the global degree of motion coupling of a non redundant parallel robot is given by the number of non zero off-diagonal terms in the symbolical de-sign Jacobian.
In defining various types of parallel robots we use the global degree that will simply be called degree of motion coupling.
276 4 Kinematic analysis
Definition 4.41 (Parallel robot with fully-coupled motions). A non redundant parallel robot has fully-coupled motions if the design Jacobian symbolically defined contains no elements with zero values.
4.6.1. Parallel robots with coupled motions
A parallel robot with coupled motions represents the archetype of in-parallel actuated robot arms beginning with the first industrial parallel ro-bots developed in the first half of the twelfth century and the first compre-hensive study of their structural synthesis made by Hunt (1983). The ma-jority of parallel robots used today have coupled motions.
Definition 4.42A (Parallel robot with coupled motions). A parallel robot has coupled motions if the design Jacobian cannot be set up, in a non-singular pose, as a diagonal or triangular matrix under alternative re-ordering of lines and columns, as long as ordering preserves the associa-tion of each actuated joint velocity with its corresponding operational ve-locity.
In the general case of direct kinematics problems, the component iw (i=1,2,…,M) of the operational velocity vector of a parallel robot with
coupled motions is a function of M components of the actuated velocity space iq (i=1,2,…,M): i i 1 Mw w ( q ,...,q )= , i=1,2,…,M, where M is the de-gree of mobility of the parallel robot.
To preserve the association of each actuated joint velocity with its corre-sponding operational velocity, interchanging between two lines of the de-sign Jacobian must be accompanied by a corresponding interchange in the sequence of components in the operational velocity vector. In the same way, interchanging two columns must be accompanied by a corresponding interchange in the sequence of components in the actuated velocity vector.
Reordering lines and columns by preserving the association between each actuated joint velocity and its corresponding operational velocity does not affect the nature of the linear mapping set up by the Jacobian matrix. For example, such a reordering is the usual way to switch from the opera-tional velocity vector [ 0v , 0ω ]T= [ x y z x y zv ,v ,v , , ,ω ω ω ]T, calculated by the conventional Jacobian, and the corresponding twist $=[ x y z x y z, , ,v ,v ,v ,ω ω ω ], calculated by the screw-based Jacobian.
In general for a parallel robot, the inverse Jacobian is easier to calculate than the direct Jacobian. Motion coupling could be also defined with re-spect to inverse design Jacobian in connection with the inverse kinematics problem of non-redundant parallel robots
4.6 Design Jacobian and motion decoupling 277
1J −=q w . (4.78)
The property of being diagonal or triangular is conserved by matrix in-version in non-singular poses. In a singular pose, some elements of the Jacobian matrix may become zero and the instantaneous degree of motion coupling could change. For this reason, the degrees of motion coupled are defined in non-singular poses.
Definition 4.42B (Parallel robot with coupled motions). A parallel robot has coupled motions if the inverse design Jacobian cannot be set up, in a non-singular pose, as a diagonal or triangular matrix under alternative reordering of lines and columns, as long as ordering preserves the associa-tion of each actuated joint velocity with its corresponding operational ve-locity.
In the general case of the inverse kinematics problem, the component iq (i=1,2,…,M) of the actuated velocity space of a parallel robot with cou-
pled motions is a function of M components iw (i=1,2,…,M) of the opera-tional velocity vector: i i 1 Mq q ( w ,...,w )= , i=1,2,…,M.
Orthoglide parallel robot
We investigate the motion coupling of the parallel robot Orthoglide (Wanger and Chablat 2000; Chablat and Wanger 2003) developed by IRCCyN and analysed in example 24 in chapter 3. For convenience, the kinematic structure of the robot is completed with the necessary notations and presented in Fig. 4.2. We develop design and conventional Jacobian matrices by using the procedures presented in section 4.3 and we show that the two Jacobian matrices are the same for this robot.
The moving platform of Orthoglide moves in the Cartesian space with fixed orientation. We denote by x, y and z the three spatial coordinates of the characteristic point H, that is w1=x, w2=y, w3=z. The operational veloc-ity of the moving platform is w = 0v = [ ]Tx, y,z =
T
x y yv ,v ,v . The robot is actuated by three linear motors which are mounted or-
thogonally. The velocity vector of the actuated joints is q = [ ]T1 2 3q ,q ,q . The distance di=CiBi (i=1,2,3) is maintained constant by the parallelogram loops used in the three limbs. For each limb we can write:
( ) ( ) ( )2 2 22i Ci Bi Ci Bi Ci Bid x x y y z z= − + − + − , i=1,2,3. (4.79)
The coordinates of points Ci and Bi in the fixed reference frame 0 0 0 0O x y z are: C1 (q1+a, 0, 0), C2 (0, q2+b, 0), C3 (0, 0, q1+a) and B1 (x+a,
278 4 Kinematic analysis
Fig. 4.2. Orthoglide and the notations used in kinematic modelling
y, z), B2 (x, y+b, z), B3 (x, y, z+c) with a, b and c as constant lengths. Intro-ducing these coordinates in Eq. (4.79) and taking its derivatives with re-spect to time three times, once for each i=1,2 and 3, yields three scalar equations, which can be arranged in matrix form
sJ q = pJ v0 (4.80)
or
q = 1sJ −
pJ v0= 1J − v0 (4.81)
with the following matrices
sJ1
2
3
x q 0 00 y q 00 0 z q
− = − −
,
(4.82)
pJ1
2
3
x q y zx y q zx y z q
− = − −
(4.83)
4.6 Design Jacobian and motion decoupling 279
and
1J −
1 1
2 2
3 3
y z1x q x q
x z1y q y q
x y 1z q z q
− −
= − −
− −
(4.84)
The same result can be obtained by using a loop-closure equation writ-ten for each limb
O0Ci+CiBi=O0H+HBi , i=1,2,3. (4.85)
A velocity vector-loop equation is obtained by taking the derivative of Eq. (4.85) with respect to time
iq ni+ di i iaω × = 0v (4.86)
where ia and in denotes the unit vectors along CiBi and O0Ci, respec-tively, and iω denotes the angular velocity of a virtual link CiBi with re-spect to fixed reference frame. This virtual link is associated with the par-allelogram loop in ith limb.
Eq. (4.86) written three times, once for each i=1,2 and 3, yields three scalar equations, which can be arranged in matrix form (4.80) with
sJ =0 0
0 00 0
⋅ ⋅ ⋅
1 1
2 2
3 3
n a n a
n a,
(4.87)
and
pJ =
T1T2T3
aaa
,
(4.88)
To eliminate iω , we dot-multiply both sides of Eq. (4.86) by ia :
iq ni ia⋅ = 0v ia⋅ , i=1,2,3. (4.88)
280 4 Kinematic analysis
1J − =
T1
1 1
T2
2 2
T3
3 3
⋅
⋅
⋅
an a
an a
an a
(4.90)
Taking into account the fact that the unit vectors ni point in the positive directions of the axes of the fixed reference frame 0 0 0 0O x y z , that is n1=i0, n2=j0 and n3=k0, we can see that the expressions of the matrices defined by Eqs. (4.88)-(4.90) and (4-82)-(4-84) are the same.
The inverse geometric solution of Orthoglide can be obtained by intro-ducing the coordinates of points Ci and Bi in Eq. (4.79)
2 2 21 1
2 2 22 1
2 2 23 1
q x d y z
q y d x z
q z d x y
= + − −
= + − −
= + − −
.
(4.91)
By taking into account Eqs. (4.91), the Jacobian matrix (4.84) becomes
1J −
2 2 2 2 2 21 1
2 2 2 2 2 22 2
2 2 2 2 2 23 3
y z1d y z d y z
x z1d x z d x z
x y 1d x y d x y
− − − − − − = − − − − − − − − − − − −
.
(4.92)
In a general position of the workspace the Jacobian matrix J-1 is a full ma-trix indicating that Orthoglide is a parallel robot with fully-coupled mo-tions (C=6). There is just one point in the workspace of Orthoglide, de-fined by x=y=z=0, where the inverse Jacobians (4.83) or (4.89) become instantaneously the identity matrix. This point is reached when the charac-teristic point H of the moving platform is situated in the origin of the fixed reference frame, as in Fig. 4.3, and ia and in point in opposite directions. In this pose instantaneous degree of motion coupling is Ci =0.
4.6 Design Jacobian and motion decoupling 281
Fig. 4.3. Paricular position of Orthoglide with instantaneous degree of motion coupling Ci =0
Structural solutions derived from parallel robots with fully-coupled motions
Various solutions of parallel robots can be derived from parallel robots with fully-coupled motions aiming to decrease the degree of motion cou-pling. In this section, we illustrate two solutions (Figs. 4.4 and 4.5) derived from Orthoglide (Gogu 2004b,d). The moving platform of each solution also moves in the Cartesian space, 0v = [ ]Tx, y,z =
T
x y yv ,v ,v , with fixed orientation, and the actuators are mounted on the fixed base.
The first solution 2PR1PaR1-1PR1R1R1-type is derived by replacing one limb (PR1PaR1-type) with a limb of PR1R1R1-type used in the CPM robot in Fig. 2.1. We recall that the revolute pairs with the same index have par-allel axes. As with Orthoglide, the first solution is actuated by three linear motors which are mounted orthogonally on the fixed base with q = [ ]T1 2 3q ,q ,q - see Fig. 4.4.
Eqs. (4.79) are applicable just for the limbs of PRPaR-type for i=2 and 3. Referring to Fig. 4.4a, the following simple geometric relation can be written for the limb of PRRR-type:
282 4 Kinematic analysis
Fig. 4.4. Translational parallel robot of 2PR1PaR1-1PR1R1R1-type with decoupled motions in a general position with C =4 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 283
x = q1-a. (4.93)
Eq. (4.93) replaces Eq. (4.79) for i=1. Matrices sJ , pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ 2
3
1 0 00 y q 00 0 z q
= − −
(4.94)
pJ 2
3
1 0 0x y q zx y z q
= − −
,
(4.95)
1J −
2 2
3 3
1 0 0x z1
y q y qx y 1
z q z q
= − −
− −
.
(4.96)
In a general position (Fig. 4.4a) in the workspace the global degree of motion coupling is C =4. When characteristic point H of the moving plat-form is situated in the origin of the fixed reference frame (x=y=z=0), as in Fig. 4.4b, the instantaneous degree of motion coupling is zero.
The second solution 2PR1PaR1-1R2R2R1R1R1-type is derived by replac-ing one limb PR1PaR1-type with a limb 1R2R2R1R1R1-type. This solution is actuated by a rotation motor and two linear motors (Fig. 4.5).
Eqs. (4.79) are also applicable just for the limbs of PRPaR-type for i=2 and 3. Referring to Fig. 4.5a, the following simple geometric relation can be written for the limb 1R2R2R1R1R1-type
x=d1sinq1. (4.97)
Eq. (4.97) replaces Eq. (4.79) for i=1. Matrices Js, Jp and J in Eqs. (4.80) and (4.81) have the following expressions:
284 4 Kinematic analysis
Fig. 4.5. Translational parallel robot of 2PR1PaR1-1R2R2R1R1R1-type with decoup-led motions in a general position with C =4 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 285
sJ1 1
2
3
d cos q 0 00 y q 00 0 z q
= − −
(4.98)
pJ 2
3
1 0 0x y q zx y z q
= − −
(4.99)
and
1J −
1 1
2 2
3 3
1 0 0d cos q
x z1y q y q
x y 1z q z q
= − −
− −
.
(4.100)
This robot has also Ci=0 when characteristic point H of the moving plat-form is situated in the origin of the fixed reference frame (x=y=z=0), as in Fig. 4.5b.
4.6.2. Parallel robots with decoupled motions
The first solutions of parallel robots with decoupled motions have been proposed recently (Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2004a-d, 2005f-i, 2006a-e, 2007a-c).
Definition 4.43 (Parallel robot with decoupled motions). A parallel robot has decoupled motions if the design Jacobian (or its inverse) can be set up as a triangular matrix under alternative reordering of lines and col-umns, as long as ordering preserves the association of each actuated joint velocity with its corresponding operational velocity.
In the direct kinematics problem, the component iw (i=1,2,…,M) of op-erational velocity vector of a parallel robot with decoupled motions is a function of just the first i components iq of the actuated velocity space:
i i 1 iw w ( q ,...,q )= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
286 4 Kinematic analysis
In the inverse kinematics problem, the component iq (i=1,2,…,M) of the actuated velocity space of a parallel robot with decoupled motions is a function of just the first i components iw of operational velocity vector:
i i 1 Mq q ( w ,...,w )= , i=1,2,…,M. Definition 4.44 (Parallel robot with half-decoupled motions). A par-
allel robot has half-decoupled motions if: (i) the design Jacobian (or its in-verse) cannot be set up as a diagonal or triangular matrix under alternative reordering of lines and columns and (ii) the translational motions are de-coupled with respect to rotational motions of the moving platform.
In this case, the design Jacobian can be partitioned in four blocks. The two blocks situated on the main diagonal are translation- and rotation-related (sub)matrices. The other two blocks contains only zero elements.
Various solutions of parallel robots with decoupled motions can be de-rived from their counterparts with coupled motions. In this section, we il-lustrate two solutions derived from previous solutions with coupled mo-tions (Gogu 2004b,d). These solutions are also translational parallel robots with 0v = [ ]Tx, y,z =
T
x y yv ,v ,v , and the actuators are mounted on the fixed base.
The first solution 1PR1PaR1-2PR1R1R1-type is also derived from Or-thoglide by replacing a second limb PR1PaR1-type with a limb PR1R1R1-type. Orthoglide, this solution is also actuated by three linear motors which are mounted orthogonally on the fixed base with q = [ ]T1 2 3q ,q ,q - see Fig. 4.6.
Eqs. (4.79) are applicable just for the limb PRPaR-type for i=3. Refer-ring to Fig. 4.6a, the following simple geometric relation can be written for the two limbs PRRR-type:
x = q1-a (4.101)
and
y = q2-b. (4.102)
Eqs. (4.101) and (4.102) replace Eq. (4.79) for i=1 and 3. Matrices sJ ,
pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ
3
1 0 00 1 00 0 z q
= −
(4.103)
4.6 Design Jacobian and motion decoupling 287
Fig. 4.6. Translational parallel robot of 1PR1PaR1-2PR1R1R1-type with decoupled motions in a general position with C =2 (a) and a particular position when Ci =0 (b)
288 4 Kinematic analysis
pJ
3
1 0 00 1 0x y z q
= −
, (4.104)
1J −
3 3
1 0 00 1 0x y 1
z q z q
= − −
. (4.105)
By taking into account the inverse geometric solution, Eq. (4.105) be-comes
1J −
2 2 2 2 2 23 3
1 0 00 1 0x y 1
d x y d x y
= − − − − − −
.
(4.106)
The various kinetostatic indices of translational parallel robot 1PR1PaR1-2PR1R1R1-type with decoupled motions are presented in Figs. 4.7-4.11.
The values of the conditioning index are 0.101≤ 1υ ≤1 and velocity transmission factors satisfy 0.318≤ iψ ≤3.146 (Fig. 4.7).
Local cross-coupling indices 12κ , 13κ , 23κ and cross-coupling index κ of the direct kinematics problem for this parallel robot are the same corre-sponding indices of the direct kinematics. For this reason, we name both groups as the kinematics cross-coupling indices. The diagrams presented in Fig. 4.8 indicate a strong coupling at the extremities of the working space and a complete uncoupled kinematics in the central point of the working space (x=y=0) where 12 13 23 1κ κ κ κ= = = = . The length d3 must be correlated with the dimensions of the working space such as
2 23d x y> + . When ( )2 2 2
3x y d+ → , the parallel robot is in the vicinity of workspace limit singularity and the cross-coupling indices tend to zero. We have considered d3=B3C3 =3m.
4.6 Design Jacobian and motion decoupling 289
Fig.4 .7. Conditioning index 1υ (a) and velocity transmission factors iψ (b)-(d) of parallel robot 2PRRR-1PRPaR-type
The corresponding indices of the direct and inverse statics problem for this parallel robot are also the same and we call both groups as the statics cross-coupling indices. The diagrams presented in Fig. 4.9 show that, for this robot, cross-coupling in the statics problems is more important than in kinematics problems.
For this parallel robot, the indices of input-output propensity 1η , 2η , 3η and η of the direct and inverse kinematics problems are also the same, and we called both groups as the indices of kinematics input-output propensity. The diagrams presented in Fig. 4.10 indicate a constant input-output kine-matics propensity in the case of 3η and the strongest propensity in the cen-tral point of the working space (x=y=0) where 1η = 2η = 3η =η . The input-output propensity is more important in statics than in kinematics problems.
The second solution 1PR1PaR1-2R2R2R1R1R1-type is also derived by Or-tyhoglide by replacing a second limb PR1PaR1-type with a limb R2R2R1R1R1-type. This solution is actuated by two rotation motors and one linear motor (Fig. 4.12).
Eqs. (4.79) are applicable just for the limb PRPaR-type for i=3. Refer-ring to Fig. 4.7a, the following simple geometric relation can be written for the two limbs of 1R2R2R1R1R1-type
290 4 Kinematic analysis
Fig. 4.8. Kinematics cross-coupling indices of parallel robot 2PRRR-1PRPaR-type
Fig. 4.9. Statics cross-coupling indices of parallel robot 2PRRR-1PRPaR-type
4.6 Design Jacobian and motion decoupling 291
Fig. 4.10. Indices of kinematics input-output propensity of parallel robot 2PRRR-1PRPaR-type
Fig. 4.11. Indices of statics input-output propensity of parallel robot 2PRRR-1PRPaR-type
292 4 Kinematic analysis
Fig.4.12. Translational parallel robot 1PR1PaR1-2R2R2R1R1R1-type with decoupled motions in a general position with C =2 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 293
x=d1cosq1 (4.107)
and
y=d2cosq2. (4.108)
Eqs. (4.107) and (4.108) replace Eq. (4.79) for i=1 and 3. Matrices sJ ,
pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ1 1
2 2
3
d cos q 0 00 d cos q 00 0 z q
= −
(4.109)
pJ
3
1 0 00 1 0x y z q
= −
(4.110)
and
1 1
1
2 2
3 3
1 0 0d cos q
1J 0 0d cos q
x y 1z q z q
−
=
− −
.
(4.111)
Both solutions presented in Figs. 4.6 and 4.12 have a global degree of mo-tion coupling C =2. When the characteristic point H of the moving plat-form is situated in the origin of the fixed reference frame (x=y=z=0), as in Figs. 4.6b and 4.12b, the instantaneous degree of motion coupling be-comes Ci=0.
4.6.3. Parallel robots with uncoupled motions
Along with parallel robot with decoupled motions, parallel robots with un-coupled motions have been proposed recently (Kong and Gosselin 2002b, Gogu 2004a-d, 2005f-i, 2006a-e, 2007a-c).
294 4 Kinematic analysis
Fig. 4.13. Translational parallel robot 3R2R2R1R1R1-type with uncoupled motions in a general position (a) and the initial position when qi=0, i=1,2,3 (b)
4.6 Design Jacobian and motion decoupling 295
Definition 4.45 (Parallel robot with uncoupled motions). A parallel robot has uncoupled motions if the design Jacobian (or its inverse) can be set up as a diagonal matrix.
In the kinematics problem, the component iw (i=1,2,…,M) of opera-tional velocity vector of a parallel robot with uncoupled motions is merely a function of ith component iq of the actuated velocity space: i i iw w ( q )= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
In a parallel robot with uncoupled motions, pJ becomes the identity matrix and the design and direct kinematics matrix have the same expres-sion
J= sJ . (4.112)
The parallel robots with uncoupled motions have unity value for the cross-coupling indices and for the indices of input-output propensity. Various solutions of parallel robots with uncoupled motions can be derived from their counterparts with decoupled motions. In this section, we illus-trate a solution derived from the parallel robot with decoupled motions 1PR1PaR1-2R2R2R1R1R1-type analysed in the previous section (Gogu 2004d). This solution, 3R2R2R1R1R1-type, is a translational parallel robot with 0v = [ ]Tx, y,z =
T
x y yv ,v ,v and has three rotation actuators mounted on the fixed base (Fig. 4.13). It is derived from its counterpart with de-coupled motions in Fig. 4.12 by replacing the remaining limb of type 1PR1PaR1 by a limb of type 2R2R2R1R1R1. In this way, the parallel robot with uncoupled motions 3R2R2R1R1R1-type has three structurally identical limbs.
Referring to Fig. 4.13a, the following simple geometric relation can be written for the three limbs of 1R2R2R1R1R1-type
x=d1cosq1 (4.113)
y=d2cosq2 (4.114)
and
z=d3cosq3. (4.115)
Matrices sJ and J in Eqs. (4.80) and (4.81) have the same simple ex-pression
296 4 Kinematic analysis
sJ1 1
2 2
3 3
d cos q 0 0J 0 d cos q 0
0 0 d cos q
= =
(4.116)
The Jacobian is a diagonal matrix and C =Ci=0 in any non-singular po-sition with / 2iq π≠ ± including the initial pose when characteristic point H of the moving platform is situated at the origin of the fixed reference frame (x=y=z=0) and qi=0 (i=1,2,3) - Fig. 4.13b.
4.6.4. Maximally regular parallel robots
Maximally regular parallel robots represent a particular case within the category of parallel robots with uncoupled motions; the design Jacobian is the identity matrix. Along with parallel robots with decoupled and uncou-pled motions, maximally regular parallel robots have quite recently been developed as fully-isotropic parallel robots. The term maximally regular parallel robot was recently proposed by Merlet (2006) to define isotropic robots. We extend this term, and we use it to define parallel robots with unity value for all kinetostatic performance indices presented in the previ-ous section.
Definition 4.46 (Maximally regular parallel robot). A parallel robot with connectivity SF of the moving platform is maximally regular if the de-sign Jacobian is the SF× SF unit matrix.
In the kinematics problem, the component iw (i=1,2,…,M) of the opera-tional velocity vector of a maximally regular parallel robot is equal to its corresponding component iq of the actuated velocity space: i iw q= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
We recall that connectivity indicates the number of independent motions of the moving platform and is also known as the degree of freedom.
Maximally regular parallel robots have unit values for all kinetostatic performance indices presented in the previous sections: cross-coupling in-dices, indices of input-output propensity, conditioning indices, manipula-bility indices, and transmission factors. The manipulability ellipsoid of maximally regular parallel robots is a hypersphere of unit radius.
The first solution of such a robot was the translational parallel robot pre-sented in Fig. 2.1. This type of robot was developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author at the French Institute of Advanced Mechanics. In
4.6 Design Jacobian and motion decoupling 297
2002, the four groups published the first results of their works (Carricato and Parenti-Castelli 2002; Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2002b). Each of the last three groups has built a prototype of this ro-bot in their research laboratories and has called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3-T3 (Gogu 2004b). The first practical implementation of this robot was the CPM developed at University of California.
The author has also published the first solutions of the following fully-isotropic parallel robots: parallel wrists with two (Gogu 2005f) and three (Gogu 2007c) degrees of mobility, planar parallel robots (Gogu 2004c), parallel robots of T2R1-type (Gogu 2005i), parallel robots with Schönflies motions (Gogu 2004a, 2005g, 2006c, 2007a), parallel robots of T3R2 type (Gogu 2006a,b,d), hexapodes with six degrees of freedom (Gogu 2006e). These solutions belong to a modular family called IsoglideN-TxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The moving platform of the robots belonging to the Isogliden-TaRb family (Gogu 2007b) can have any combination of n independent translations (T) and rotations (R). This fam-ily is composed of three groups marked by A, B and C:
a) parallel robots with decoupled motions: Isogliden-TaRb-Ax, b) parallel robots with uncoupled motions: Isogliden-TaRb-Bx, c) maximally regular parallel robots: Isogliden-TaRb-Cx,
where x=1,2,… is used to make a distinction between the different solu-tions in each group.
The Isogliden-TaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics in Cler-mont-Ferrand.
The presentation of the systematic approach of the structural synthesis of the innovative Isogliden-TaRb family with simple or complex limbs, whether fully-parallel or not, whether redundant or not redundant, whether overconstrained or not, is included in the second part of this book.
5 Structural synthesis
The method proposed in this book for structural synthesis of parallel mechanisms is founded on the theory of linear transformations and evolu-tionary morphology. The structural parameters defined via the theory of linear transformations in chapter 2 are integrated in an original approach to innovative design proposed in (Gogu 2005a, 2007a) and called evolution-ary morphology. This method allows us to obtain the structural solutions of decoupled, uncoupled, fully-isotropic and maximally regular parallel robots in a systematic way. Overconstrained/isostatic solutions with ele-mentary/complex limbs actuated by linear/rotary actuators with/without idle mobilities could be obtained.
The first part of this chapter presents the general paradigms of structural synthesis and evolutionary morphology as structured approaches to inven-tive engineering design. In the second part, the general bases are set up in applying evolutionary morphology to structural synthesis of parallel robots with coupled, decoupled, uncoupled and maximally regular parallel robots.
5.1 Structural synthesis: a systematic approach in mechanism design
Mechanism design has always fascinated mankind. The fascination arises from the fact that understanding how a mechanism functions is fairly easy, but comprehending how it originated and why it was designed in a particu-lar form in which it exists is more difficult (Kota 1992).
Leonardo da Vinci in his works The Madrid Codex and Atlantic Codex was the first to advocate the necessity of a science of mechanisms, and the wide application of mathematical methods to the construction of machines. He had the vision that any machine can be built with a set of mechanisms, and he even listed 22 elements from which machines can be constructed.
Willis (1841) called for a science “that provides us with methods by which all forms and arrangements that are applicable to the desired pur-poses can be obtained”. Instead of selecting mechanisms based on intuition and experience the designer could choose the most suitable solutions from
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this list of possibilities. Willis developed a classification scheme of mechanisms according to type of motion transmission.
Reuleaux (1870) laid the foundation for a systematic study of mecha-nisms by proposing a formal classification of mechanisms and their sym-bolic representation. In this classification, he recognized the preeminence of joints and identified them as kinematic pairs. While Reuleaux’s sys-tematization of kinematic pairs into lower and higher pairs, and his classi-fication of mechanisms based on their types of joints, are still used today, his symbolic notation is regarded as too unwieldy for use in structural syn-thesis procedures (Erdman 1992). Davies and Crossley (1966) remarked that “although Reuleaux’s work has done much to clarify thinking about mechanisms, his notation is rarely used, probably because in making it both comprehensive and concise, the graphic quality of a drawing is en-tirely lost.” This drawback is still valid for modern-day systematization of mechanisms based on symbolic notations without graphical counterparts, or with a poor graphical representation in which the types of joints and the relative position of their axes are represented ambiguously.
Researches in design formalization have found that the fundamental task of conceptualizing devices is a mixture of art and science. Researchers in mechanism theory could have conceded that mechanism design is an art and then left its pursuit to the gifted few. Instead, they continue to search for a scientific basis for creative design, attempting to offer a gift of crea-tivity to the engineering community. (Kota 1992).
Creative design, in the context of mechanisms, is what is more widely known as systematics, and consists in generating the entire set of solutions defined by certain structural parameters without regard to the dimensions of the components. Sometimes systematics is associated with systematiza-tion, classification end simple enumeration of the known solutions without giving emphasis to the generating process (Artobolevskii 1976). The sys-tematic generating process of mechanisms is generally called structural synthesis. Various other terms are also used to define this systematic syn-thesis approach as shown in Table 5.1.
Type synthesis and number synthesis were identified by Hain (1961) and Crosley (1964) as two main approaches in the systematics of mechanisms. Type synthesis is based on the selection of a particular type of mechanism (linkage, cam, gear, etc.). Number synthesis is the process of finding the arrangements of a given number of bodies and joints which result in kine-matic chains with the desired mobility (Crosley 1964). Crossley (1980) also identified type synthesis, that is the “determination of the simplest type(s) of mechanisms that are capable of producing the desired motion” as the first stage in mechanism synthesis. He introduced the synthesis
5.1 Structural synthesis: a systematic approach in mechanism design 301
Table 5.1. Terminology used to define the systematic synthesis of mechanisms
Terminology
References
Structural synthesis
Manolescu and Tempea 1967a,b; Manolescu 1968; Mruthyunjaya 1979, 1984a,b,c 2003; Yang 1985; Dudiţă and Diaconescu 1986; Dudiţă et al. 1989; 2001a,b; Hervé and Sparacino 1991; Hwang and Hwang 1992; Wenhui et al. 1995; Rao 1997; Rao and Deshmukh 2001;Rao et al. 2001; Alizade and Bayram 2004; Alizade et al. 200 Gogu 2004b, 2007a; Sunkari and Schmidt 2006.
Structure synthesis Linda et al. 2000; Schmitd et al. 2000; Jin and Yang 2001, 2002; Fang and Tsai 2002, 2004b.
Structural and kinematic synthesis
Manolescu and Tempea 1967
Structural kinematics
Hunt 1983
Kinematic synthesis
Denavit and Hartenberg 1964; Vucina and Freudenstein 1991; Sandor and Erdman 1984; Erdman and Sandor 1996
Topological synthesis
Sardain 1997; Murphy et al. 1996; Dar-Zen and Kang-Li 2000; Carricato and Parenti-Castelli 2004.
Topology synthesis
Jin and Yang 2004.
Type synthesis Hain 1961;Crosley 1964; Alizade et al. 1985; Olson et al. 1985; Dijksman and Timmermans 1994; Huang and Li 2002a, 2003; Huang 2004; Kong and Gosselin 2002b, 2004a-e, 2005, 2006; Lu and Leinonen 2005.
Qualitative synthesis
Angeles 2004.
Number synthesis Hain 1961; Crosley 1964; Yang 1983; Raghavan 1996.
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of permutations based on the study of the permutations of given combina-tions of links and joints as a second stage. Both stages are founded on the systematic generation of all possible mechanisms that can give rise to the desired kinematic performance. Freudenstein (1967) also associated type synthesis with “the determination of the structure of the mechanism from desired kinematic performance”. Type and number synthesis does not nec-essarily mean identifying a single mechanism type from a multitude of possibilities (Kota 1992).
Erdman (1987) pointed out that an improper choice of kinematic con-figuration may demand sophisticated control algorithms to undo poor type selection. Kota (1992) emphasized that a rational approach to mechanism design requires an exhaustive survey of all possibilities before the various solutions are compared and analyzed.
Eckhardt (1998) pointed out that the design of a machine or mechanism or any moving mechanical system always starts with kinematic synthesis defined by (i) the functional relationship between the parts, (ii) how those parts are interconnected, (iii) how these parts move relative to each other. Only after choices have been made regarding these factors, can matters such as materials, strengths, manufacturing, and costs be seriously ad-dressed. Failure to devote the proper attention to kinematics “up front” can, and often does, lead to the design of a system with substandard or sub-optimal performance and/or unsatisfactory reliability.
The main issues in systematic synthesis are the following: proper classi-fication of mechanisms, systematic representation of mechanisms (sym-bolically or otherwise) and means for generating different possibilities in a systematic way (Kota 1992). The process of generating all possible distinct kinematic chains is dificult and time consuming (Rao and Deshmukh 2001). Nevertheless, it is desirable to enumerate the kinematic chains sys-tematically and also to know the inherent characteristics of a chain related to its structure so that all the acceptable chains can be evaluated in depth before making the final selection for a specified task.
The common point of the various methods developed for generating sys-tematically the possible solutions is the use of a mobility criterion as the main structural parameter for the synthesis approach of mechanisms. The main difference between these methods resides in the tool used to define the generating process.
Various approaches are reported in the literature for the synthesis of mechanisms from intuitive and empirical knowledge to formalized meth-ods, as shown in Table 5.2. The large part of these methods was applied to synthesis of planar mechanism.
5.1 Structural synthesis: a systematic approach in mechanism design 303
Table 5.2. Approaches used for the synthesis of mechanisms
Approach
References
Visual inspection and intuition
Grübler 1883, 1885; Alt 1921; Hain 1955; Crossley 1964.
Assur groups
Assur 1913, 1914; Manolescu et al. 1964; Manolescu 1964, 1965, 1968; Manolescu and Tempea 1967a,b.
Franke’s notation
Franke 1958. Davies and Crossley 1966; Haas and Crossley 1969.
Structural graphs
Paul 1960; Freudenstein and Dobrjanskyj 1964; Manolescu 1964; Crossley 1965; Woo 1967; Dobrjanskyj and Freudenstein 1967;Freudenstein 1967; Huang and Soni; Kiper and Schian 1975; Turtle 1987. Dudiţă and Diaconescu 1986; Dudiţă et al. 1989; 2001a,b;
Adjacency matrices
Hunag and Soni 1972; Hwang and Hwnag 1992; Wenhui et al. 1995; Murphy et al. 1996, Lu and Leinonen 2005.
Baranov trusses
Manolescu 1973.
Joint substitution
Jensen 1991.
Associated linkage
Johnson 1978.
Transformation of binary chains
Mruthyunjaya 1979, 1984a,b,c.
Finite groups
Tuttle and Peterson 1987; Tuttle et al. 1987, 1988, 1989.
Prime chain
Dijksman and Timmermans 1994.
Melbourne method
Tischler et al. 1995a,b.
Structural groups
Mitrouchev, 2001; Alizade and Bayram 2004, Alizade et al. 2006.
Evolutionary morphology
Gogu 2005a, 2005a, 2007a.
Virtual chain Kong and Gosselin 2006.
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As it was shown in Table 1.6, the systematic synthesis of spatial mecha-nisms used in parallel robots is founded on: four main methods: (i) group theory, (ii) screw theory, (iii) velocity-loop equations, and (iv) theory of linear transformations. The four methods determine, by using different mathematical tools, the set of kinematic chains defined by a given number and type of independent motions of a distal link with respect to a reference link. The various limb topologies and solutions for their connection in par-allel mechanisms are generated in two sequential stages.
This chapter focuses on the presentation of a new systematic approach to structural synthesis of parallel mechanisms, integrating evolutionary morphology and the complete set of structural parameters (mobility, con-nectivity, overconstraint and redundancy) demonstrated via the theory of linear transformations.
5.2. Morphological and evolutionary approaches
In general, any abstract task can be thought of as solving a problem, which, in turn, can be perceived as a search through a space of potential solutions. Generation of this space is an inventive activity inherent to any innovation-oriented synthesis approach. Too much innovation is guided by mystery. An inventive structured approach eliminates mystery, errors and time bombs, and reveals new solutions while remaining adaptable to future change. In an inventive structured approach, the inventive effort becomes an inherent judgment process, traceable and reproducible. An inventive structured approach is ideally suited to engineering design and for inven-tive problem solving (Ritchey 1998; Gogu 2000). Evolutionary morphol-ogy is such a structured approach to inventive design, combining features of morphological research and evolutionary algorithms. The morphologi-cal approach is a methodology, first proposed by Zwicky (1969) to en-hance the systems engineering process. The morphological approach de-termines the requirements and conceptual design solutions of a system by considering the "totality" of all solutions included in a morphological box. This is an initial phase before the optimization process looks for “the best solution”. The evolutionary algorithms are stochastic techniques whose search methods for an optimal solution model some natural phenomena: genetic inheritance and Darwinian striving for survival.
We use “morphosis” as the morphological terminology for change, re-placing one system by the next. This concept is based on the fundamental scientific method of alternating between analysis and synthesis. It can be trusted as a useful, qualitative method for investigating complex problems
5.2. Morphological and evolutionary approaches 305
which cannot be treated by direct mathematical formalization (equations), causal modelling or simulation (Ritchey 1998). We differentiate between static, dynamic and evolutionary morphosis. In static morphosis, a system is disassembled into very tiny components, and reassembled into a com-pletely different system that has no obvious similarity with the previous system. In dynamic morphosis, an existing system changes by enhancing, adding or reducing existing properties (parameters). Evolutionary morpho-sis has the characteristics of both static and dynamic morphosis.
5.2.1 Morphological approaches
The term morphology comes from the ancient Greek (morphe) and means shape or form. The general definition of morphology is "the study of form or pattern", i.e. the shape and arrangement of parts of an object, and how these "conform" to create a “whole” or Gestalt. The "objects" in question can be physical objects (e.g. an organism, an anatomy, an ecosystem) or mental objects (e.g. linguistic forms, concepts or systems of ideas).
The term morphology, as an explicitly defined scientific method, was first mentioned by J. W. Goethe (1749-1832). In addition to poetry, dra-mas, and novels, Goethe continued his work in science, studying structures and shapes in mineralogy, geology, and osteology (the study of bones). In 1784, he had made the discovery, by methods which foreshadowed the sci-ence of comparative morphology, that the human jawbone contained traces of a structure similar to the intermaxillary bone in other mammals. In 1790, he wrote Versuch, die Metamorphose der Pflanzen zu erklären (Es-say on the Metamorphosis of Plants), which further developed his ideas on comparative morphology and to some extent foreshadowed Darwin's ideas on organic evolution. His "comparative morphology" of botany describing how animals and plants changed shape and evolved into new species was a breakthrough in thinking.
Today, morphology is associated with a number of scientific disciplines. In linguistics, morphology is the study of word formation and how the parts of words relate to each other. Morphology is used in disciplines where formal structure and structural interrelations, and not necessarily quantity, are the central issues, e.g. geology, anatomy, botany, zoology etc.
Zwicky (1969) proposed a generalised form of morphological research. He proposed to generalize and systematize the concept of morphological research and include not only the study of the shapes of geometrical, geo-logical, biological, and generally material structures, but also to study the more abstract structural interrelations among phenomena, concepts, and ideas, whatever their character might be.
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Essentially, general morphological analysis is a method for identifying and investigating the total set of possible relationships or “configurations” contained in a given complex problem. In this sense, it is closely related to typology construction (Bailey 1994), although it is more generalised in form and conceptual range.
Although Zwicky coined the term morphological analysis, the technique predates him and can be traced back to the “universal method of thinking” presented by the Catalan philosopher Ramon Lull (1233-1315) in his work Ars Magna (The Great Art) published in 1275. Zwicky was the first to use the technique in modern-day applications. He used morphological research in new product and process ideation such as jet engines (Zwicky 1947), propulsive power generation (Zwicky 1962) and astronomical investiga-tion (Zwicky 1957). After his main work on the subject (Zwicky 1969) originally published in 1966 by Droemersche Verlagsanstalt Th. Knaur Nachf., München/Zürich, under the title Entdecken, Erfinden, Forschen im morphologischen Weltblid, morphological research was extensively ap-plied to engineering design (Norris 1963), manufacturing and process de-sign (Royston 1969; Willis and Hawthone 1969), systems engineering (Hall 1969), technological forecasting and planning (Ayres 1969; Gerardin 1973).
Zwicky presented three morphological methods based on systematic field coverage, morphological box, and negation and construction (Zwicky 1969). The method of complete field coverage searches all solutions of a well-defined problem. Starting from known pegs of knowledge, infiltrating the surrounding fields, and using a sufficient number of “principles of thought” we strive to discover new facts, formulate new problems, and if necessary invent and produce needed new materials, instruments and pro-cedures. Some of the major fundamental principles of thought that we need are based on our ability to count and recognize identities, equalities, differ-ences and coincidences in time and space. The method is applicable to three types of cases; these involve interrelations among certain objects, in-terrelations among certain natural phenomena, and the interplay between certain abstract concepts.
Among the methods and procedures that have been conceived and per-fected through the morphological approach, the construction of the mor-phological box of all possible aspects and solutions of a given problem, as well as the unbiased evaluation of the solutions found, are two of the most effective. A morphological box – also fittingly known as a ”Zwicky box” – is constructed by setting the parameters against each other in an n-dimensional morphological box. Each cell of the n-dimensional box con-tains one particular ”value” or condition from each of the parameters, and thus marks out a particular state or configuration of the problem. Zwicky
5.2. Morphological and evolutionary approaches 307
(1969) summarises the five iterative steps of the morphological box method:
1. The problem to be solved must be concisely formulated. 2. All of the parameters that might be of importance for the solution of
the given problem must be localized and analysed. 3. The morphological box or multidimensional matrix, which contains
all of the potential solutions of the given problem, is constructed. 4. All the solutions contained in the morphological box are closely scru-
tinized and evaluated with respect to the purposes that are to be achieved. 5. The optimally suitable solutions are selected and applied, provided
the necessary means are available. Of course, the matrixing of parameters, in order to uncover the multi-
plicity of relationships associated with a complex problem, is nothing new. The virtually universal use of “four-fold tables” and the study of typology construction as a classification technique attest to this fact (Bayley 1994). The morphological box represents a static morphosis based on a simple combinatorial approach. However, Zwicky’s highly systematic approach to this field should not be underestimated. Used properly – and on the right types of problems – the method is both deceptively complex and rich.
The method of negation and construction, the third morphological method presented by Zwicky, is inspired by pure mathematics. The insight gained as a result of any negation is immediately used for the purpose of future construction. Following up the well-reasoning negation of apparent truths and of the so called absolute facts with the constructive use of the vistas that thus open themselves, we succeed not only in making sporadic discoveries and inventions but whole groups and entire classes of them.
The three morphological methods proposed by Zwicky have little theo-retical base. They combine analysis and synthesis with some general crea-tivity paradigms, called “principles of thought” such as combination, in-version and hypothesis smashing.
5.2.2 Evolutionary algorithms
The beginnings of the evolutionary algorithms can be traced back to the early 1950s when several biologists used computers to simulate biological systems (Goldberg 1989). There are several variants of evolutionary algo-rithms (genetic algorithms, evolution strategies, evolutionary program-ming, genetic programming) and there are also many hybrid systems which incorporate various features of these paradigms. However, the struc-tures of evolutionary methods are very much the same (Dasgupta and Michalewicz 1997).
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An evolutionary algorithm maintains a population of individuals, { }1( ) , . . . ,t t
nP t x x= for iteration t. Each individual represents a
potential solution to the problem at hand, and is implemented as a data structure S. Each solution 1
tx is evaluated to give some measure of its fit-ness. Then, a new population (iteration t+1) is formed by selecting the fit-test individuals. Some members of the new population undergo transfor-mations by means of genetic operators to form new solutions. There are unary transformations im (mutation type), which create new individuals by small changes in a single individual ( :im S S→ ), and higher order transformations jc (crossover type), which create new individuals by com-bining parts from several (two or more) individuals ( : ...jc S S S× × → ). Af-ter a number of generations, the algorithm converges to a “best individ-ual”. It is hoped that the best individual represents a near-optimum (reasonable) solution.
Despite powerful similarities between various evolutionary algorithms, there are also many differences between them, often hidden on a lower level of abstraction. They use different data structures for their chromoso-mal representation; consequently, the genetic operators are also different. They may or may not incorporate some complementary information to control the search process. There are many methods for selecting individu-als for survival and reproduction. These methods include: a) proportional selection, where the probability of selection is proportional to the individ-ual’s fitness, b) ranking methods, where all individuals in a population are sorted from the best to the worst and probabilities of their selection are un-changed for the whole evolution process, and c) tournament selection, where some number of individuals (usually two) compete for selection to the next generation. This competition (tournament) step is repeated throughout the population a number of times. The generational policy is also an important strategy. It is possible to replace the whole population by a population of offspring, or it is possible to select the best individuals from two populations, a population of parents and a population of off-spring. This selection can be done in a deterministic or nondeterministic way. It is also possible to produce a few (in particular a single) offspring which replace some individuals in a steady state generation policy. Also we can use an elitist model which keeps the best individual from one gen-eration to another.
The representations used for a particular problem together with the set of genetic operators constitute the essential components of any evolution-
5.2. Morphological and evolutionary approaches 309
ary algorithms. These are the key elements which allow us to distinguish between various paradigms of evolutionary methods.
Genetic algorithms, as they are known today, are closely related to the work done in the late 1960s and early 1970s at the University of Michigan under the direction of John Holland (1975). Genetic algorithms (GAs) were conceived to model adaptation processes, mainly operated on binary strings and used a recombination operator with mutation as a background operator. Mutation flips a bit in a chromosome, and a crossover exchanges genetic material between two parents. The fitness of an individual is as-signed proportionally to the value of the objective function for the individ-ual. Individuals are selected for the next generation on the basis of their fit-ness. The combined effect of selection, crossover and mutation gives a so-called reproductive schema growth equation. The crossover operator enables structured, yet random information exchange. The mutation opera-tor introduces greater variability into the population. To apply a genetic al-gorithm to a particular problem, it is necessary to design a mapping be-tween a space of potential solutions for the problem and a space of binary strings of some length. This is not a trivial task and, often, it has an impor-tant impact on the final solution (Michalewicz 1996), (Bäck 1996). Evolu-tionary morphology contributes to solving this task. Conceptual aspects in-spired by evolutionary theory are integrated into evolutionary morphology to generate qualitative diversification of the initial population.
Evolution strategies (ESs) were developed to solve parameter optimiza-tion problems (Schwefel 1994). The earliest evolution strategies were based on a population consisting of one individual only. There was also only one genetic operator used in the evolution process: a mutation. The interesting idea (not presented in GAs) was to use a chromosome that represents an individual as a pair of float-valued vectors i.e ( , )v v x σ= . The first vector x represents a point in the search space and the second vector is a vector of the standard deviations. Mutations are realized by replacing x by 1 (0, )t tx x N σ+ = + , where (0, )N σ is a vector of independent random Gaussian numbers with a mean of zero and standard deviation σ , which remains unchanged during the evolution process. This is in accordance with the biological observation that smaller changes occur more often than larger ones. The offspring, as a mutated individual, is accepted as a new member of the population, and it replaces its parents if it has better fitness and all imposed constraints are satisfied. Otherwise, the offspring is elimi-nated and the population remains unchanged (Schwefel 1995).
Evolutionary algorithms are methods for solving optimization-oriented problems which are not suited for solving conceptual design oriented prob-
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lems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions.
5.3 Evolutionary morphology
Evolutionary morphology (EM) is a new structured approach to inventive design proposed by the author (Gogu 2005a).
In this section we emphasize the structure of EM and its specific differ-ences with respect to morphological research and evolutionary algorithms. To help understanding of the proposed formalization, the main paradigms of EM are presented in a closed relation with the biological background of the synthetic theory of evolution.
EM is defined as a 6-tuple
t t t tEM ( ,E, , , , )Φ Ο Ψ Σ ι= (5.1)
applicable to each generation t of the morphological evolutionary process, in which :
1 n( ,..., )Φ φ φ= is the set of design objectives,
1( ,..., )nE ε ε= - the set of constituent elements,
t 1 n( ,..., )Ο ο ο= - the set of morphological operators applicable at each generation t,
1t n( ,..., )Τ τ τ= - the set of evolution criteria from generation t to genera-tion t+1
1( ,..., )t nσ σΣ = - the set of solutions at each generation t, called morpho-logical products or morphologies at generation t,
{ }: ,i true falseι ι → - a termination criterion for EM.
5.3.1 Design objectives
The set of design objectives 1 n( ,..., )Φ φ φ= represents the required charac-teristics of the final solution; these could be expressed by any kind of for-mal language: mathematical (equalities, inequalities), semantic (textual de-scription), graphical (parametric design), etc. The design objective is not necessarily expressed by an objective function as in genetic algorithms. EM is not an optimisation technique; it is an approach towards generating the total set of possible solutions respecting the complete set of design ob-jectives. To help generate the total set of possible solutions, the EM oper-
5.3 Evolutionary morphology 311
ates preferably with qualitative characteristics or with quantified character-istics limited to few discrete values.
5.3.2 Constituent elements
The set of constituent elements 1( ,..., )nE ε ε= represents qualitative “pro-toelements” (from the Greek “proteios” or primary and element), used to build the solutions during the morphological evolutionary process.
The constituent elements belong to homogeneous groups, called primary groups, which could be one of the following:
a) material parts (subsystems) of the final system, b) geometrical, physical or chemical characteristics, c) geometrical, physical, chemical or organisational principles, d) natural phenomena, e) human behaviour, f) interrelations between certain parts, characteristics, principles, natural
phenomena and human behaviour, …etc. This list of primary groups is not exhaustive; it could be completed with
problem-specific groups. The protoelements are qualitative elements. For instance distance, as a geometrical characteristic, could be a qualitative constituent element when it is important as a qualitative dimension and not as a specific value with a specific unit of measurement. Distance, as a con-stituent element, is equivalent to a length in parametric design. In concep-tual design the most important thing is to consider or ignore the existence of this distance as a design parameter. The absolute value and the unit of measurement are not important at this design stage.
The constituent elements represent the evolving units in EM. They are equivalent to nucleic acids and proteins of a cell structure. EM is equiva-lent to chromosome generation, a common gene pool included in the geno-type of the individuals. The genetic algorithms manipulate the genotype of the initial population to find the “best” individual. EM generates the initial population genotype.
The identification and study of genes are of great interest to biologists, and are also of medical importance when a particular gene is involved in disease. All the genes of an organism, distributed over several chromo-somes, are summarized under the term genome. Biological research has revealed that the human genome contains approximately 100,000 genes, of which about 4,000 may be associated with disease. A globally coordinated effort called the Human Genome Project was started in 1990 to character-ize the entire human genome. The primary goal of the Human Genome
312 5 Structural synthesis
Project is the generation of various genome maps, including the entire nu-cleotide sequence of the human genome. The Human Genome Project has been greatly assisted by the ability to clone large fragments of DNA into artificial yeast chromosome vectors for further analysis, and the automa-tion of many techniques such as DNA sequencing. The identification and study of constituent elements in EM is equivalent to the generation of the “genome of the technical system” in engineering conceptual design. This generation could help both in understanding and avoiding functional flaws as well as in diversifying and improving the technical systems designed by the engineer.
A constituent element has a monary or a binary structure. The monary structure ( iε ) indicates that the constituent element maintains the same feature. The binary structure indicates that the constituent element has two different features. In conceptual design we often use ( iε =0 or 0iε ≠ ) which correspond to the absence ( iε =0) or the presence ( 0iε ≠ ) of a fea-ture a in a solution jσ . Any other two distinct features could also be used; for instance the parallel and perpendicular geometric position ( iε ⇒ or iε ⇒⊥ ). The solutions built up exclusively from constituent elements having monary structure are equivalent to haploid organisms in evolution-ary theory. The solutions built up exclusively from constituent elements having binary structure are equivalent to diploid organisms. Haploid or-ganisms are modelled by one set of genetic information per individual which is unicellular from a biological point of view. In the diploid organ-isms (e.g., in the case of humans) each body cell includes two sets of chromosomes, such that for each chromosome two homologous forms ex-ist. The structure of both forms of chromosomes is identical, but their ge-netic information content may be different. In contrast, haploid organisms possess only one set of chromosomes per body cell. For diploid organisms, corresponding genes in homologous chromosomes are called alleles. The fact that alleles of diploid organisms may be identical (homozygotic) or different (heterozygotic) is worthy of mention here. The constituent ele-ment with a monary structure in EM is homozygotic for a particular gene in evolutionary theory. A constituent element with a binary structure is heterozygotic for the particular gene. Both alleles are carried in the feature of the constituent element, as in the genetic material of the individual, but if one is dominant, only that one will be expressed. The other allele is called recessive.
5.3 Evolutionary morphology 313
5.3.3 Morphological operators
The morphological operators t 1 n( ,..., )Ο ο ο= are (re)combination, muta-tion, migration and selection. These operators are deterministic and can be applied at each generation t of the EM.
Morphological combination
Morphological combination is applied directly to the constituent elements to obtain the complete set of solutions at the first generation. Morphologi-cal recombination is used to combine solutions of generation t with solu-tions of the previous generations, including the constituent elements, to ob-tain the complete set of solutions at generation t+1. We use the morphological box to generate the total set of possible solutions at each generation. In EM, (re)combination is a deterministic operator and not a probabilistic one as in genetic algorithms. The mechanism of morphologi-cal (re)combination is combinatorial. The evolution is made by adding parts/characteristics to the previous morphology, and not by interchanging (crossing-over) characteristics as in intrachromosomous recombination (crossover) specific to genetic algorithms. The (re)combination in EM is equivalent to chromosome generation in ontogenesis.
We know that chromosomes contain the genetic blueprints for a specific organism. Several thousand genes are arranged in a single line on a chro-mosome, a threadlike structure of nucleic acids and protein. Different groups of organisms have different numbers of chromosomes; for exam-ple, human beings have 23 pairs (46 total) of chromosomes, divided into 8 different sets according to their size and shape. In humans 23 of the chro-mosomes come from the mother and 23 from the father. The variation pre-sent in individuals is a reflection of the genetic recombination of these sets of chromosomes from generation to generation in phylogeny.
Morphological mutation
Morphological mutation modifies variability in the process of solution generation, as in genetic algorithms; the difference is that morphological mutation is a deterministic operator (all mutations are predefined) and not a probabilistic one. Morphological mutations act in two ways:
a) increasing variability by interchanging a dominant allele by a reces-sive one,
b) diminishing variability by changing a binary constituent element into a monary one (by cutting up the dominant or the recessive allele).
314 5 Structural synthesis
Morphological migration
Morphological migration acts by replacing dominant constituent element ( iε ) by a new dominant constituent element ( jε ). The constituent elements
iε and jε must belong to the same group of homogeneous elements. Mor-phological migration, like mutation in genetic algorithms, introduces more variability in the result of the generation process.
Morphological selection and evolution criteria
Morphological selection acts to eliminate the non-evolving intermediary solutions and the incompatible final solutions. Non-evolving intermediary solutions are the solutions in a generation t that did not evolve with respect to the previous generation. Incompatible final solutions are solutions which do not conform to the set of final objectives. The set of evolution criteria from generation t to t+1 allow us to eliminate the non-evolving so-lutions at generation t, and to stop them spreading to the following genera-tions.
5.3.4 Set of solutions
The set of solutions 1( ,..., )t nσ σΣ = at each generation t is also the called morphology at generation t. The set of solutions in the final generation represents the result of EM which we call the morphological product. It corresponds to the first generation that satisfies all the final objectives. When the morphological product is obtained, the termination criterion stops the evolution algorithm.
5.3.5 General structure of the evolutionary morphology
The general structure of evolutionary morphology is presented in Fig. 5.1. EM evolves from the constituent elements to the final generation t=n in the following steps:
• Define the set of final objectives. • Define the set of constituent elements and identify their struc-
ture (monary or binary constituent element) and the dominant allele for the binary elements.
• Define the set of morphological operators and identify the con-stituent elements and the morphologies on which they act.
5.3 Evolutionary morphology 315
Fig. 5.1. General structure of the evolutionary morphology
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the morphological combination of constituent elements to obtain the total set of morphologies of the first generation.
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the set of evolution criteria from generation 1 to genera-tion 2 and eliminate the non-evolving solutions.
• Apply the morphological recombination of the constituent ele-ments and the morphologies issued in generation 1 to generate the total set of morphologies of generation 2.
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the set of evolution criteria from generation 2 to genera-tion 3 and eliminate the non-evolving solutions from generation 2.
… • Apply the morphological recombination of the constituent ele-
ments and the morphologies issued in generation t=n-1 to gen-erate the total set of morphologies of generation t=n.
• Verify the termination criterion, { }: i falseι ϕ → , that means t=n is the last generation.
316 5 Structural synthesis
Procedure 1 while (not termination-criterion) do while (migration operator defined) do apply migration operator by interchanging a domi-
nant constituent element ( iε ) by a new dominant constituent element ( jε )
apply the selection criterion to eliminate incompatible solutions
end while (mutation operator defined) do apply mutation operator by interchanging a domi-
nant constituent element by the recessive one apply the selection criterion to eliminate
incompatible solutions end end The inventive structured approach is a key ingredient in conceptual en-
gineering design. It allows the designer to examine all possible options at minimum cost by an inventive approach. EM is based on the fundamental scientific method of alternating between analysis and synthesis. For this reason, it can be trusted as a useful, qualitative method for investigating inventive problems which cannot be treated by formal mathematical meth-ods, causal modelling or simulation. Furthermore, EM gives a fairly clear “audit trail”, which makes the judgmental processes relatively traceable, reproducible and inventive. The “solution space” of EM consists of the subset of configurations which satisfy some criteria (design objectives).
Of course, as is the case with everything else, the output of an EM is no better than the quality of its input. The final solutions depend a lot on the choice of constituent elements. This choice is a problem-dependent task. The main advantage of this approach consists in finding the complete set of possible solutions in an evolutionary sequence based on a set of proto-elements. EM may help us to invent new relationships or configurations, which may not be so evident, or which we might have overlooked by other less structured methods. Most importantly, it encourages the identification and investigation of boundary conditions, i.e. the limits and extremes of different contexts and factors. The evolutionary sequence limits the gen-eration and the proliferation of incompatible solutions.
The main paradigms of EM are inspired from the synthetic theory of evolution developed by modern biology. EM integrates some features of morphological research and evolutionary algorithms but it represents a
5.4 General approach to structural synthesis of parallel robots 317
radically different approach. Evolutionary algorithms start from a given initial population to obtain an optimum solution with respect to a fitness function. EM creates this initial population to enhance the chance to obtain a “more global optimum” by qualitative diversification of the initial popu-lation. EM and evolutionary algorithms are complementary methods. Evo-lutionary algorithms are optimized oriented methods; EM is a conceptual design oriented method.
5.4 General approach to structural synthesis of parallel robots
We consider the general case of a parallel robot F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the fixed base 1≡1Gi≡0 by k≥2 sim-ple or complex kinematic chains Gi (1Gi-2Gi-…nGi) i=1,2,…,k. The parallel robot has p joints. The limbs can be identical or different. The first link 1Gi of each limb is fixed to the base 1Gi≡0, and the distal link is the moving platform nAi≡n. Link 1≡1Gi≡0 is also called the fixed platform. We denote by iq (i=1,…,MF) the velocities of the actuated joints, and by iw (i=1,…, SF) the translational and angular velocities of the moving platform. MF represents the mobility of the parallel robot F, and F
n / 1S the connectivity between mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in the parallel robot F. The parallel robot F ← G1-G2-…Gk is TxRy-type, with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3. The mobile platform of the parallel ro-bot TxRy-type can make x independent translations and y independent rota-tions.
5.4.1 General conditions for structural synthesis of parallel robots via theory of linear transformations
In chapter 2 we saw that a mechanism F corresponds to a linear transfor-mation F from joint velocity vector space U into external velocity vector space W. Velocity spaces U and W are finite dimensional vector spaces. The Jacobian matrix J is the matrix of this linear mapping. To each vector q , in the joint vector space U, the linear transformation F assigns a uniquely determined vector ( )F q in the vector space W. In parallel ro-bots, the vector q represents the actuated joint velocity vector, and it is composed of the relative velocities of the actuated joints. The vector ( )F q in W is the image of the vector q in U under the transformation F. The
318 5 Structural synthesis
image of U in W represents the range RF of F, and is called the operational velocity vector space. The connectivity /
Fa bS between two links a and b in
the mechanism F represents the number of independent relative velocities allowed by the mechanism between the two links. It is given by the dimen-sion of the vector space /
Fa bR of relative velocities between links a and b in
mechanism F. The range /Fa bR is a sub-space of RF, and implicitly a sub-
space of W. The connectivity characterizes the degree of freedom of links a and b in the mechanism F. The number of overconstraints of F is de-noted by NF, and its degree of structural redundancy by TF. All these pa-rameters are defined in chapter 2.
The existence of F ← G1-G2-…Gk involves the following set of general conditions for any position of the mechanism when 1q ≠0 (i=1,2,…,MF):
F G1 G 2 Gk( R ) ( R R ... R )= ∩ ∩ ∩ (5.2)
SF≤MF≤ MGi (i=1,…,k), (5.3)
MGi≥ SGi≤6 (i=1,…,k), (5.4)
MF≥p
ii 1
f 6q=
−∑ , (5.5)
where: RGi is the operational velocity vector space of limb Gi isolated from the parallel robot F, MGi is the mobility of limb Gi isolated from F, SGi= Gi
nGi / 1S =dim(RGi) is the connectivity between the mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in limb Gi isolated from F, SF=dim(RF) is the connectivity between the mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in the parallel robot F, q is the number of independent closed loops of F, fi is the mobility of the ith joint. Conditions (5.3)-(5.5) follow from Eqs. (2.211), (2.220) and (2.222).
Conditions (5.3)-(5.4) become
SF=MF≤ MGi (i=1,…,k), (5.6)
MGi=SGi≤6 (i=1,…,k), (5.7)
for non redundant parallel robots (TF=0); and
SF<MF≤ MGi (i=1,…,k), (5.8)
5.4 General approach to structural synthesis of parallel robots 319
MGi>SGi≤6 (i=1,…,k), (5.9)
for redundant parallel robots with TF>0. Condition (5.5) becomes
MF=p
ii 1
f 6q=
−∑ , (5.10)
for non overconstrained parallel robots (NF=0); and
MF>p
ii 1
f 6q=
−∑ , (5.11)
for overconstrained parallel robots with NF>0. We recall that MF, NF, TF are given by Eqs. (2.211), (2.220), (2.222) and
Gip
Gi i Gii 1
M f r=
= −∑ (5.12)
where pGi represent the number of joints of the limb Gi and rGi the number of joint parameters that lose their independence in the closed loops of limb Gi. For simple (elementary) limbs, rGi=0; and for complex limbs rGi is given by Eq. (2.209) – see also section 2.3.8.
Eqs. (5.2)-(5.11) are applicable to all parallel robots with any degree of motion coupling.
Parallel robots with uncoupled motions obey some additional condi-tions. In this case, the operational velocity vector space RF has (i) a unique basis (RF) of dimension SF for any position of the mechanism when iq ≠0 (i=1,2,…,MF) and (ii) to each actuator locked, i.e. iq =0, a specific unique basis of dimension SF-1 is associated with the parallel robot. These condi-tions express the possibility of direct actuation of each independent motion of the mobile platform by just one specific actuator.
5.4.2 General approach to structural synthesis of parallel robots via evolutionary morphology
Evolutionary morphology is used for structural synthesis of each limb Gi (i=1,…,k) by taking into account the general conditions defined in the pre-vious chapter via the theory of linear transformations.
320 5 Structural synthesis
Set of design objectives
In structural synthesis of TxRy-type PMs with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3 we adopt the general set of design objectives
1 8 i( ,..., )Φ φ φ += i=1,2,…,MF presented in Table 5.3. Design objectives 1 8,...,φ φ are applicable to the structural design of all
parallel robots, and 9 8 i,...,φ φ + (i=1,2,…,MF) represent additional objectives for PMs with uncoupled motions. Design objectives 1 2,φ φ and 3φ are represented by qualitative characteris-tics. In the first stage of structural synthesis, PMs with orthogonal non re-dundant limbs are generated. In such a limb, a link is connected by revo-lute (R) and prismatic pairs (P) with parallel (║) or orthogonal (⊥ ) axes/directions. The revolute and prismatic pairs with orthogonal position are adopted for ease of manufacture. We know that any kind of motion of the mobile platform can be generated by concatenating revolute and pris-matic pairs. Various multi-degrees-of freedom joints can be obtained by combining revolute and prismatic joints; for example a cylindrical joint (C) is equivalent to a combination R║P or P║R, a spherical joint (S) is equivalent to three revolute pairs with intersecting axes (R⊥R⊥R), a combination of a revolute and a helical joint (H) with the same axis (R║H
Table 5.3. Set of design objectives
No. Notation Objective Related equations
1 1φ k≥2 elementary limbs or complex limbs integrating c≥1 closed loops in a limb
-
2 2φ Generic limbs with revolute (R) and prismatic (P) pairs
-
3 3φ Orthogonal limbs - 4 4φ Connectivity of the moving platform SF Eq. (2.207) 5 5φ Unique basis of the operational velocity
space (RF) Eq. (5.2)
6 6φ Mobility MF>2 Eq. (2.211) 7 7φ Number of overconstraints QN 0≥ Eqs. (2.220),
8 8φ Degree of structural redundancy FT 0≥ Eq. (2.222) 9 8 iφ + Direct actuation of unidirectional mo-
tion iw by iq (i=1,2,…,MF) in PMs with
decoupled motions
Eqs. (2.207), (5.2)
5.4 General approach to structural synthesis of parallel robots 321
or H║R) is equivalent to a cylindrical joint and implicitly to a combination R║P or P║R. The same is true for a planar pair or any other pair with more than one degree of freedom.
Set of constituent elements
The following set of constituent elements 1i 4iE ( ,..., )ε ε= is used: a) 1iε - kinematic pair (KPi) between the links i -1 and i, if i iKP P⇒ (prismatic pair) than 1i 1ε = , and if i iKP R⇒ (revolute pair) then 1iε =0, b) 2iε - axial offset (ai) of the link i, if ia 0≠ than 2i 1ε = , and if ai=0 than 2iε =0, c) 3iε - length (bi) of the link i, if 0ib ≠ than 3i 1ε = , and if bi=0 than 3iε =0, d) 4iε - relative position pi of the axes/directions of the two joints adja-
cent to the same link i, if ip ⇒⊥ than 4i 1ε = , and if ip ⇒ than 4iε =0. We consider ji 1ε = as dominant and ji 0ε = as recessive alleles. In accordance with the TCS method for geometric modeling of robot
manipulators (Gogu et al. 1997), a link i could have two dimensions in an orthogonal kinematic chain: axial offset ai, measured on the axis of the pair (i-1,i), and link length bi, defined by the normal to the axes of the two pairs adjacent to element i (Fig. 5.2, a-e).
In general, the previous constituent elements have binary structure, ex-cept for the following elements, which have a monary structure:
a) 21ε =0, we make a1=0 by a proper choice of reference frame used in geometric modelling and Jacobian matrix calculation,
b) 2iε =0 and 3iε =0 if KPi between the links i -1 and i is a prismatic pair, that is the prismatic pair has only a direction and not a defined axis (Fig. 5.2, g-j).
The constituent elements with monary structure are not affected by the morphological mutation operators.
Fig. 5.2 illustrates the geometrical meaning of the constituent elements included in the “genetic code” of link i. Kinematic pair KPi belongs to a constituent group representing material subsystems of the parallel
322 5 Structural synthesis
Fig. 5.2. Geometrical meaning of the constituent elements included in the “genetic code” of link i
manipulator (the joints). Dimensions ai and bi belong to a constituent group representing geometrical characteristics of the links, and relative po-sition pi belongs to a constituent group representing interrelations among the axes of the revolute joints. We note that the constituent elements are qualitative. In this case, distance is important as a qualitative dimension and not as a specific value with a specific unit of measurement. Distance, as a constituent element, is equivalent to length in parametric design. In conceptual design the most important thing is to consider or ignore the ex-istence of distance as a design parameter. The absolute value and the unit of measurement are not important at this design stage.
Other types of constituent elements could be introduced in the evolu-tionary process by the mutation operator, as we will see in the next sec-tions, to obtain the diversification of the initial solutions.
5.4 General approach to structural synthesis of parallel robots 323
Set of evolution criteria
The evolutionary process from generation t to generation t+1 is analyzed by the set of evolution criteria t 1 2T ( , )τ τ= where 1τ and 2τ are connec-tivity criteria defined by
St<St+1 for Mt+1<MGi (5.13)
and
St=St+1= GiS for Mt+1=MGi. (5.14)
Evolution criterion (5.13) is applicable to the generations of non redun-dant limbs with Mt+1<MAi. The evolutionary process from generation t to generation t+1 with Mt+1=MGi is analyzed by the connectivity criterion 2τ defined by (5.14). This criterion indicates that the process is stationary. The solutions of the generation t+1 that do not obey Eqs. (5.13) and (5.14) are eliminated from the evolutionary process. This allows us to avoid the proliferation of non-evolving solutions in the next generations. These non-evolving solutions represent kinematic chains with structural singularities. For example, open kinematic chains with two parallel prismatic pairs, three prismatic pairs with directions parallel to the same plane, four paral-lel revolute pairs, etc., are non-evolving solutions with structural singulari-ties that are not allowed to proliferate in the evolutionary formation of non-redundant limbs.
Set of morphological operators
The set of morphological operators t 1 2 3 4( , , , )Ο ο ο ο ο= is composed of 1ο - (re)combination, 2ο - selection, 3ο - mutation and 4ο - migration.
The mechanism of morphological (re)combination is combinatorial. Evolution is achieved by adding structural components (kinematic pairs-KPi) and geometric characteristics (ai, bi and pi) to the previous morphol-ogy and not by interchanging (crossing-over) characteristics as in cross-over specific to genetic algorithms.
At least MF=SF generations are necessary to evolve by successive com-binations from the first generation of protoelements with MGi=SGi=1 to a first solution of elementary limb with MF=SF satisfying the set of design objectives.
Morphological migration could introduce new constituent elements formed by new joints or combinations of joints into the evolutionary proc-ess. Some examples are presented in Table 5.4. The new and the replaced joints/combination of joints have the same mobility. We note that the num-
324 5 Structural synthesis
number of joint parameters that lose their independence in a planar closed loop integrated in a complex limb is r=3.
The notations ⊥ and ║ used in Table 5.4 indicate the perpendicular or parallel positions of the joint axes. For example the notation R P⊥ ⊥ ║R
Table 5.4. Examples of new joints or combinations of joints that could be intro-duced in the evolutionary process by morphological migration
No. New joint or combination of joints
Replaced joint or combination of joints
1 Helical joint (H) Revolute joint (R) 2 Cylindrical joint (C) Two consecutive joints:
(i) a revolute and a prismatic joint with parallel axes/directions, (ii) two helical joints with parallel axes, (iii) a helical and a prismatic joint with parallel axes/directions, (iv) a helical and a revolute joint with parallel axes
3 Universal joint (U) Two orthogonal intersecting revolute joints ( R R⊥ ) adjacent to the link i when
4i 1ε = ( ip ⇒⊥ ), 2iε =0 (ai=0) and
3iε =0 (bi=0), 4 Spherical joint (S) Three orthogonal revolute joints with inter-
secting axes ( R R R⊥ ⊥ ) 5 Planar joint (Pl) Open planar kinematic chain with three de-
grees of freedom 6 Planar parallelogram
closed loop (Pa) with one degree of freedom
Prismatic joint (P)
7 R║Rb║R-type or R║Rb1║…║Rbn║R-type planar kinematic chains in-tegrating one or more rhombus loops (Rb)
R P⊥ ⊥ ║R-type planr kinematic chain
8 Pn2║R-type planar kine-matic chain integrating a planar loop with two de-grees of freedom (Pn2)
P P R⊥⊥ ⊥ -type planar kinematic chain
9 3Pn planar loop with three degrees of freedom
Open planar kinematic chain with three de-grees of freedom
5.4 General approach to structural synthesis of parallel robots 325
indicates that the axis of the first pair is perpendicular to the direction of the second pair, the axis of the third pair is also perpendicular to the direc-tion of the second pair, and at the same time parallel to the direction of the first pair. The notation R║Rb║R indicates that the axes of the first and last revolute joints are parallel to the revolute axes of the rhombus loop.
At each stage, morphological selection acts to eliminate the non-evolving intermediate solutions and the incompatible solutions. The non-evolving intermediate solutions are the solutions for a generation t that did not evolve with respect to the previous generation. The incompatible solu-tions are the solutions which do not satisfy the set of design objectives.
The set of evolution criteria from generation t to t+1 allows us to elimi-nate the non-evolving solutions at generation t, and to stop them spreading to the following generations.
Set of final solutions
The set of solutions 1( ,..., )t nσ σΣ = at each generation t is also called the morphology at generation t. The set of solutions that satisfies all design ob-jectives represents the result of the EM; we call this the morphological product. When the morphological product is obtained, the termination cri-terion stops the evolution algorithm.
The various types of limbs generated by the general approach to struc-tural synthesis can be systematized by taking into account their structural parameters: the type (simple or complex), mobility MG, connectivity SG, basis of operational velocity space (RG), degree of overconstraint (NG) and degree of structural redundancy (TG). If SG≥SF than a limb with connec-tivity SG could be used in a parallel robots F with connectivity SF of the moving platform. This allows the same limb to be used in different types of parallel robots. For this reason, limb synthesis is a key issue in the struc-tural synthesis of parallel robots.
The various types of limbs generated by this systematic synthesis ap-proach could combine simple or complex kinematic chains with prismatic or revolute pairs connected to the fixed base. The prismatic pair could ac-commodate a linear actuator, and the revolute pair a rotary actuator mounted on the fixed base. Usually the complex limbs integrate planar closed loops with one, two or three degrees of mobility. The simple limbs are not overconstrained (NG=0). Between three and six joint parameters could lose their independence in a planar closed loop. Usually the limbs are not redundant (TG=MG-SG=0). Redundant limbs with MG>SG could be used in order to obtain simple solutions for parallel robots which have un-coupled motions, and are maximally regular.
326 5 Structural synthesis
The following chapters of this book point out on the general systemati-zation of limbs which satisfy the set of design objectives
1 8( ,..., )Φ φ φ= defined in section 5.4.2. Limb connectivity is the main cri-terion for this systematization.
We focus on solutions, with a unique basis of the operational velocity space, that are useful for generating various topologies of parallel robots with decoupled, uncoupled, fully-isotropic and maximally regular parallel robots. Limbs with multiple bases of the operational velocity space and re-dundant limbs are also presented in these chapters. These limb solutions are systematized with respect to various combinations of independent mo-tions of the distal link.
Other solutions with non-orthogonal intersecting axes may be also used in parallel robots. As we have already mentioned, only solutions with or-thogonal and/or parallel axes of the kinematic joints are presented in this work. Such links and joints are easy to make.
Limbs with two degrees of connectivity are systematized in chapter 6. They are defined by the following independent motions of the distal link: two translational motions (section 6.1), two rotational motions (section 6.2) and one translational and one rotational motion (section 6.3). Exam-ples of limbs with two degrees of connectivity and multiple bases of the operational velocity space are presented in section 6.4, and redundant limbs in section 6.5.
Limbs with three degrees of connectivity are systematized in chapter 7. They are defined by the following independent motions of the distal link: three translational motions (section 7.1), one rotational and two transla-tional motions (sections 7.2 and 7.3), one translational and two rotational motions (section 7.4), and three rotational motions (section 7.5). Examples of limbs with three degrees of connectivity and multiple bases of the op-erational velocity space are presented in section 7.6, and redundant limbs in section 7.7.
Limbs with four degrees of connectivity are systematized in chapter 8. They are defined by the following independent motions of the distal link: one rotational and three translational motions (section 8.1), two transla-tional and two rotational motions (section 8.2) and one translational and three rotational motions (section 8.3). Examples of limbs with four degrees of connectivity and multiple bases of the operational velocity space are presented in section 8.4, and redundant limbs in section 8.5.
Limbs with five degrees of connectivity are systematized in chapter 9. They are defined by three translational motions and two rotational motions (section 9.1) and two translational and three rotational motions (section 9.2). Examples of limbs with five degrees of connectivity and multiple
5.4 General approach to structural synthesis of parallel robots 327
bases of the operational velocity space are also presented in section 9.3 and redundant limbs in section 9.4.
The last chapter of this book presents some examples of non-redundant and redundant limbs with six degrees of connectivity. The complete set of solutions of non-redundant limbs has been generated by the systematic synthesis approach presented in this chapter. Due to space limitation the entire set of solutions cannot be presented. In fact, these limbs do not in-troduce restrictions in the motion of the distal link. For this reason, just the simplest solutions of limbs with six degrees of connectivity are used in parallel robots. They are presented in chapter 10.
The limbs presented in chapters 6-10 are defined by symbolic notations and illustrated in about 250 figures containing more than 1500 structural diagrams. Special attention was paid to graphic quality of structural dia-grams to ensure a clear correspondence between the symbolic and graphic notation of joints and the relative position of axes.
The kinematic chains presented in chapters 6-10 are useful as innovative solutions of limbs in parallel, serial or hybrid robots. In fact, serial and hy-brid robots may be considered as particular case of parallel robots, with just one limb which can be a simple, complex or hybrid kinematic chain, as we have seen in chapter 3 of this book.
The work is organized in two parts published in two distinct books. Part 1 presents the methodology proposed for structural synthesis and Part 2 the various topologies of parallel robots systematically generated by the struc-tural synthesis approach. The limbs presented in chapters 6-10, will be used in Part 2 of this work for structural synthesis of TxRy-type parallel robots with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3 . Part 2 of this work will present the following topologies: translational PMs T2-type (chapter 1), T1R1-type PMs with screw motion (chapter 2), other T1R1-type PMs (chapter 3), R2-type spherical parallel wrists (chapter 4), transla-tional PMs T3-type (chapter 5), T2R1-type PMs with planar motion (chap-ter 6), other T2R1-type PMs (chapter 7), T1R2-type PMs (chapter 8), R3-type spherical parallel wrists (chapter 9), T3R1-type PMs with Schönflies motion (chapter 10), other T3R1-type PMs (chapter 11), T2R2-type PMs (chapter 12), T1R3-type PMs (chapter 13), T3R2-type PMs (chapter 14), T2R3-type PMs (chapter 15) and PMs with six degrees of freedom (chap-ter 16).
Parallel robots with coupled, decoupled and uncoupled motions, along with fully-isotropic and maximally regular solutions, are systematically presented in each chapter of Part 2. Innovative solutions of overcon-strained or non-overconstrained, non-redundant or redundantly actuated parallel robots with simple or complex limbs actuated by linear or rotating
328 5 Structural synthesis
motors are set up in Part 2 by applying the structural synthesis methodol-ogy presented in this chapter.
As we have already mentioned, many solutions for parallel robots ob-tained through this systematic approach of structural synthesis are pre-sented here for the first time in the literature. The author had to make a dif-ficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. By publishing them in this work, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products such as hap-tic devices, parallel kinematic machines, guiding, testing, control and tracking devices.
6 Limbs with two degrees of connectivity
Limbs with two degrees of connectivity (SG=2) are used in parallel robots F with the connectivity of the moving platform SF=2. The basis of the op-erational velocity space of these limbs contains just two independent ve-locities that can be two orthogonal translations, two orthogonal rotations or combination of one translation and one rotation. In this chapter and the fol-lowing chapters as well, we denote by iv and iω (i=1,2,3) the independ-ent translational and angular velocities of a characteristic point H situated on the distal link of a kinematic chain.
Non-redundant solutions generated by the evolutionary morphology ap-proach are presented in the first four sections of this chapter. Some exam-ples of redundant solutions are also presented in the last section. The first four sections present solutions of limbs defined by the following: (i) two translational motions, (ii) two rotational motions, (iii) one translational and one rotational motion, and (iv) multiple bases of the operational velocity space.
6.1 Limbs with two translational motions
These limbs give two translational velocities v1 and v2 in the basis of op-erational velocity space (RG)=( 1 2,v v ) for any position of the characteristic point H on the distal link. They give rise to two independent translations along with a constant orientation of the moving platform.
The simplest structural solutions for both simply and complex non re-dundant limbs with only revolute R and prismatic P joints are presented in Figs. 6.1 and 6.2. Prismatic or revolute joints may be connected to the fixed base. The following structural parameters MG=SG=2, (RG)=( 1 2,v v ) and TG=0 characterize these solutions.
The complex limbs integrate one or two planar parallelogram loops. As presented in chapter 3, a closed parallelogram loop Pa could combine up to three idle mobilities that may be two orthogonal rotations and one trans-lation. The direction of the translation is perpendicular to the two idle
330 6 Limbs with two degrees of connectivity
Fig. 6.1. Simple limb PP-type (a) and complex limbs of types PPa (b) and PaPat (c) defined by MG=SG=2, (RG)=( 1 2,v v )
Fig. 6.2. Complex limbs of types PaP (a)-(b) and PaPa (c) defined by MG=SG=2, (RG)=( 1 2,v v )
6.1 Limbs with two translational motions 331
Fig. 6.3. Parallelogram loops of types Pa (a), Pac (b), Pau (c), Pas (d), Pauu (e), Pacu (f), Pa* (g), Pass (h)
332 6 Limbs with two degrees of connectivity
Table 6.1. Structural parameters NG and rG of various types of limbs defined by (RG)=( 1 2,v v )
No. Limb-type NG rG 1 P ⊥ P 0 0 2 P ⊥ Pa and Pa ⊥ P 3 3 3 P ⊥ Pac, Pac ⊥ P, P ⊥ Pau and Pau ⊥ P 2 4 4 P ⊥ Pas, Pas ⊥ P, P ⊥ Pauu, Pauu ⊥ P, P ⊥ Pacu and Pacu ⊥ P 1 5 5 P ⊥ Pa* and Pa* ⊥ P 0 6 6 Pa ⊥ Pa 6 6 7 Pa ⊥ Pac, Pac ⊥ Pa, Pa ⊥ Pau and Pau ⊥ Pa 5 7 8 Pa ⊥ Pas, Pas ⊥ Pa, Pa ⊥ Pauu, Pauu ⊥ Pa, Pa ⊥ Pacu , Pacu ⊥ Pa,
Pac ⊥ Pac, Pau ⊥ Pau, Pac ⊥ Pau and Pau ⊥ Pac 4 8
9 Pa ⊥ Pa*, Pa* ⊥ Pa, Pac ⊥ Pas, Pas ⊥ Pac, Pac ⊥ Pauu, Pauu ⊥ Pac, Pac ⊥ Pacu, Pacu ⊥ Pac, Pau ⊥ Pas, Pas ⊥ Pau, Pau ⊥ Pauu, Pauu ⊥ Pau, Pau ⊥ Pacu and Pacu ⊥ Pau
3 9
10 Pa* ⊥ Pac, Pac ⊥ Pa*, Pa* ⊥ Pau, Pau ⊥ Pa*, Pas ⊥ Pas, Pas ⊥ Pauu, Pauu ⊥ Pas, Pas ⊥ Pacu, Pacu ⊥ Pas, Pauu ⊥ Pas, Pas ⊥ Pauu, Pauu ⊥ Pauu, Pauu ⊥ Pacu, Pacu ⊥ Pauu, Pacu ⊥ Pas, Pas ⊥ Pacu, Pacu ⊥ Pauu, Pauu ⊥ Pacu and Pacu ⊥ Pacu
2 10
11 Pa* ⊥ Pas, Pas ⊥ Pa*, Pa* ⊥ Pauu, Pauu ⊥ Pa*, Pa* ⊥ Pacu and Pacu ⊥ Pa*
1 11
10 Pa* ⊥ Pa* 0 12 NG degree of overconstraint, rG number of joint parameters that lose their inde-pendence in the closed loops that might be integrated in G-limb, (RG) basis of the operational velocity space
rotations and parallel to the rotation axes of the revolute pairs of the paral-lelogram loop. The idle mobilities are usually introduced in the parallelo-gram loop by replacing a revolute pair with: (i) a cylindrical pair with the same axis, (ii) a universal pair that has one revolute axis superposed on the axis of the replaced revolute joint or (iii) a spherical pair.
A closed parallelogram loop integrating idle mobilities is denoted by: (i) Pac, when an idle mobility of translation is integrated in a cylindrical joint (Fig. 6.3b), (ii) Pau, when an idle mobility of rotation is integrated in a uni-versal joint (Fig.6.3c), (iii) Pas, when two idle mobilities of rotation are integrated in a spherical joint (Fig. 6.3d), (iv) Pauu, when two idle mobili-ties of rotation are integrated in two universal joints (Fig. 6.3e), (v) Pacu, when one idle mobility of translation is integrated in a cylindrical joint and one idle mobility of rotation is integrated in a universal joint (Fig. 6.3f), (vi) Pacs (also denoted by Pa*), when the three idle mobilities are inte-grated in a cylindrical and a spherical pair (Fig. 6.3g) and (vii) Pass (also denoted by Pa+R), when the three idle mobilities are integrated in two
6.2 Limbs with two rotational motions 333
Table 6.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=2, (RG)=( 1 2,v v ) and TG=0 presented in Figs. 6.1 and 6.2
Limb-type Joint Structural parameters No. arrangement m p q rG NG
Examples
1 PP P⊥ P 3 2 - - - Fig. 6.1a 2 PPa P⊥ Pa 5 5 1 3 3 Fig. 6.1b 3 PaPa Pa||Pat 7 8 2 6 6 Fig. 6.1c 4 PaP Pa⊥ P 5 5 1 3 3 Fig. 6.2a,b 5 PaPa Pa||Pa 7 8 2 6 6 Fig. 6.2c MG degree of mobility, SG connectivity, (RG) basis of the operational velocity space, TG degree of structural redundancy, m number of links including the fixed base, p number of joints, q number of independent closed loops, NG degree of overconstraint, rG number of joint parameters that lose their independence in the closed loops integrated in the limb. a Structural parameters rG, MG, NG and TG are given by Eqs. 2.209, 2.211, 2.220 and 2.222 taking into account the fact that for an isolated limb r=rG=rl, M=MG, N=NG, T=TG, k=1 and SG=SF.
spherical joints adjacent to the same link (Fig. 6.3g). In the latter, an addi-tional mobility of rotation is introduced by the internal mobility of the link adjacent to the two spherical joints. The solution in Fig. 6.1c combines a telescopic parallelogram loop denoted by Pat with a constant length paral-lelogram loop Pa. A revolute and a prismatic joint are connected to the fixed base.
Three joint parameters lose their independence in a planar parallelogram loop without idle mobilities, that is r=3 in Pa- and Pat-type loops.
In a parallelogram loop with idle mobilities 4≤ r≤ 6: r=4 for Pac and Pau, r=5 for Pas, Pauu and Pacu, r=6 for Pa* and Pass. The degree of over-constraint and the number of joint parameters that lose their independence in the closed loops that might be integrated in the various types of limbs with MG=SG=2, (RG)=( 1 2,v v ) and TG=0 are systematized in Table 6.1.
The joint arrangement and structural parameters of the solutions pre-sented in Figs. 6.1 and 6.2 are systematized in Table 6.2.
6.2 Limbs with two rotational motions
of non redundant limb contains just two revolute joints. If the characteris-tic point H of the distal link is situated at the intersection of the axes of the
These limbs give two rotation velocities 1ω and 2ω in the basis of the operational velocity space (RG)=( 1 2,ω ω ). This simplest structural solution
334 6 Limbs with two degrees of connectivity
Fig. 6.4. Simple limb RR-type defined by MG=SG=2 and (RG)=( 1 2,ω ω )
two revolute pairs, this limb gives two rotations along with a constant posi-tion of the characteristic point (Fig. 6.4). The limb is U-type and has MG=SG=2, (RG)=( 1 2,ω ω ), NG=0, TG=0, rG=0.
Other solutions with non-orthogonal intersecting axes may also be used. As we have mentioned in chapter 5, just solutions with orthogonal and/or parallel axes of the kinematic joints are presented in this work. They are the easiest to make.
If the characteristic point H is not situated at the intersection of the axes of the revolute joints, the basis of the operational velocity space may con-tain (i) two angular velocities, (ii) two translational velocities or (iii) one rotational and two translational velocities.
6.3 Limbs with one translational and one rotational motion
These limbs give the translational and angular velocities 1v and 1ω in the basis of the operational velocity space (RG)=( 1 1,v ω ). The simplest struc-tural solutions of non redundant limbs contain just one revolute and one prismatic joint. These limbs give a rotation of the moving platform and a translation of the characteristic point H situated on the rotation axes (Figs. 6.5 and 6.6). They have the following structural parameters: MG=SG=2, (RG)=( 1 1,v ω ), NG=0, TG=0, rG=0.
The direction of the translation could be either parallel (Fig. 6.5) or per-pendicular (Fig. 6.6) to the rotation axis. Both translation and rotation have fixed directions. The order of the rotation and the translation is not impor-tant when the direction of the translation is parallel to the rotation axis (Fig. 6.5).
6.4 Other limbs with two degrees of connectivity 335
Fig. 6.5. Simple limbs PR-type and RP-type defined by MG=SG=2, (RG)=( 1 1,v ω ) and the direction of the translation parallel to the rotation axis
Fig. 6.6. Simple limb PR-type defined by MG=SG=2, (RG)=( 1 1,v ω ) and the direc-tion of the translation perpendicular to the rotation axis
If the characteristic point H is not situated on the rotation axis, the basis of the operational velocity space may contain (i) two translational veloci-ties or (ii) one translational and one rotational velocity.
6.4 Other limbs with two degrees of connectivity
The limbs presented in this section may be defined by multiple bases of the operational velocity space for any position of the characteristic point on the distal link. These bases combine translational and angular velocities. For example, the basis of the operational velocity space of the limb in Fig. 6.7 may be defined by two translational velocities ( 1 2,v v ) or by one trans-lational and one rotational velocity ( i 1,v ω ), i=1,2.
Examples of simple and complex limbs with two degrees of connec-tivity and multiple bases of the operational velocity space are presented in
336 6 Limbs with two degrees of connectivity
Figs. 6.7-6.9. The complex limbs may contain planar or spatial closed loops. The planar loops are of type parallelogram Pa (Fig. 6.9a-f) rhombus Rb (Fig. 6.8a,b) or other planar loops Pn2 (Fig. 6.9g,h).
Various solutions of planar bimobile closed loops Pn2 containing five revolute or prismatic joints may be used, for example: R||R||R||R||R, P⊥R||R||R||R, R⊥P⊥ ||R||R||R, R||R⊥P⊥ ||R||R, P ⊥ P ⊥⊥ R||R||R, R⊥P ⊥⊥ P ⊥⊥ R||R, P⊥R ⊥⊥ P⊥ ||R||R, R⊥P⊥ R⊥P⊥ ||R, P⊥R||R⊥P⊥ ||R, P⊥R||R||R⊥P, P⊥P ⊥⊥ R||R⊥P.
Fig. 6.7. Simple limbs RP-type (a) and RR-type (b) and (c) defined by MG=SG=2 and multiple bases of the operational velocity space
Fig. 6.8. Complex limbs RRb-type (a) and RRbRb-type (b) defined by MG=SG=2 and multiple bases of the operational velocity space
6.4 Other limbs with two degrees of connectivity 337
Fig. 6.9. Complex limbs of types RPa (a)-(c), PaR (d)-(f), and Pn2 (g), (h) de-fined by MG=SG=2 and multiple bases of the operational velocity space
338 6 Limbs with two degrees of connectivity
Table 6.3. Structural parametersa of the limbs with MG=SG=2 and TG=0 presented in Figs. 6.7-6.9
Structural parameters
No. Limb-type Joint arrangement
m p q rG NG
Examples
1 RP R⊥ P 3 2 - - - Fig. 6.7a 2 RR R||R 3 2 - - - Fig. 6.7b 3 RR R⊥ R 3 2 - - - Fig. 6.7c 4 RRb R||Rb 5 5 1 3 3 Fig. 6.8a 5 RRbRb R||Rb||Rb 7 8 2 6 6 Fig. 6.8b 6 RPa R⊥ Pa 5 5 1 3 3 Fig. 6.9a,b 7 RPa R||Pa 5 5 1 3 3 Fig. 6.9c 8 PaR Pa⊥ R 5 5 1 3 3 Fig. 6.9d-e 9 PaR Pa||R 5 5 1 3 3 Fig. 6.9f 9 Pa2 Pa2 5 5 1 3 3 Figs. 6.9g,h aSee footnote of Table 6.2 for the nomenclature of structural parameters.
The solutions of trimobile closed loops Pn3 contain six revolute and prismatic joints, for example: R||R||R||R||R||R, P⊥R||R|| R||R||R, R⊥P⊥ ||R||R||R||R, R||R⊥P⊥ ||R||R||R, P⊥P ⊥⊥ R||R||R||R, R⊥P ⊥⊥ P ⊥⊥ R||R||R, P⊥R ⊥⊥ P⊥ ||R||R||R, R⊥P⊥ ||R⊥P⊥ ||R||R, P⊥R||R⊥P⊥ ||R||R, P⊥R||R||R⊥P⊥ ||R, R||R⊥P ⊥⊥ P ⊥⊥ R||R, P⊥R||R||R||R⊥P, R⊥P⊥ ||R||R⊥P⊥ ||R, P⊥P ⊥⊥ R||R||R⊥P.
The revolute joints integrated in the closed loops Pn2 and Pn3 have par-allel axes and the directions of the prismatic joints are contained in a plane perpendicular to the revolute axes.
A complex limb may integrate one, two or more rhombus (Rb) loops. In Fig. 8.8 just two examples with one and two rhombus loops are presented.
Three joint parameterslose their independence in each planar closed loop Rb-, Pn2- or Pn3-type.
The structural parameters of the limbs presented in Figs.6.7-6.9 are sys-tematized in Table 6.3.
6.5 Redundant limbs with two degrees of connectivity
A large variety of redundant limbs with SG=2 and MG>2 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just some exam-ples are presented in Fig. 6.10. These examples are obtained from the non-
6.5 Redundant limbs with two degrees of connectivity 339
Fig. 6.10. Example of translational limbs of types PPP (a) PPaP (b) and PaPatP (c) with one degree of redundancy defined by MG=3, SG=2 and (RG)=( 1 2,v v )
redundant solutions in Fig. 6.1 by adding a prismatic joint in the plane of motion.
The structural parameters are defined as: (i) m=4, p=3 q=0, rG=0, SG=2, MG=3, (RG)=( 1 2,v v ), NG=0 and TG=1, for the example in Fig. 6.10a, (ii) m=6, p=6 q=1, rG=3, SG=2, MG=3, (RG)=( 1 2,v v ), NG=3 and TG=1, for the example in Fig. 6.10b, and (iii) m=8, p=9 q=2, rG=6, SG=2, MG=3, (RG)=( 1 2,v v ), NG=6 and TG=1, for the example in Fig. 6.10c. Three joint parameters lose their independence in each parallelogram loop Pa- and Pat-type. Each solution has one degree of redundancy which may be used to increase the work space or to increase the precision by setting up a two-stage mechanism with a macro and a micro level with different precision. For example the redundant translation may be used for the micro level stage to achieve a high precision or a compliance motion.
7 Limbs with three degrees of connectivity
Limbs with three degrees of connectivity (SG=3) are used in parallel robots F with the connectivity of the moving platform SF≤ 3. The base of the op-erational space (RG) of these limbs contains three independent velocities. Non-redundant solutions generated by the evolutionary morphology ap-proach are presented in the first six sections of this chapter. Some exam-ples of redundant solutions are presented in the last section. The first six sections present solutions of limbs defined by the following: (i) three trans-lational motions, (ii)-(iii) one rotational and two translational motions, (iv) one translational and two rotational motions, (v) three rotational motions, and (vi) multiple bases of the operational velocity space.
7.1 Limbs with three translational motions
These limbs give three translational velocities v1, v2 and v3 in the basis of the operational velocity space (RG)=( 1 2 3, ,v v v ) along with a constant orien-tation of the distal link. The characteristic point H may have any position on the distal link. The simplest structural solutions for simple and complex non-redundant limbs with only prismatic and revolute pairs are presented in Figs. 7.1-7.6.
The complex limbs combine up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could combine up to three idle mobilities that may be two orthogonal rotations and one trans-lation.
The degree of overconstraint and the number of joint parameters that lose their independence in the closed loops combined in the various types of limbs defined by MG=SG=3, (RG)=( 1 2 3, ,v v v ) and TG=0 are systematized in Table 7.1.
In the solutions presented in Figs. 7.2b,d, 7.4b and 7.5a,d,f, the number of joints may be reduced by replacing the sequence of joints P||Pa or Pa||P by a parallelogram loop Pacc-type with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop (Fig. 7.7). Parallelogram loop Pacc-type has the following structural
342 7 Limbs with three degrees of connectivity
Table 7.1. Structural parameters NG and rG of the complex limbs with parallelo-gram loops
No. Complex limbs containing: NG rG 1 one loop Pa-type 3 3 2 one loop Pac- or Pau-type 2 4 3 one loop Pas-, Pauu- or Pacu-type 1 5 4 one loop Pa*-type 0 6 5 one loop Pa4u-type 0 6 6 two loops Pa-type 6 6 7 two loops of type: (i) Pa and Pac, (ii) Pa and Pau 5 7 8 a) two loops Pac- or Pau-type
b) two loops of different types: (i) Pa and Pas, (ii) Pa and Pauu, (iii) Pa and Pacu , (iv) Pac and Pau
4 8
9 two loops of different types: (i) Pa and Pa*, (ii) Pas and Pac, (iii) Pas and Pau, (iv) Pauu and Pac, (v) Pauu and Pau, (vi) Pacu and Pac, (vii) Pacu and Pau
3 9
10 a) two loops Pas-, Pacu- or Pauu-type b) two loops of different types: (i) Pas and Pauu, (ii) Pas and Pacu, (iii) Pauu and Pacu, (vii) Pa* and Pac, (viii) Pa* and Pau,
2 10
11 two loops of different types: (i) Pa* and Pauu, (ii) Pa* and Pacu , (iii) Pa* and Pas
1 11
12 two loops Pa*-type 0 12 13 three loops Pa-type 9 9 14 two loops Pa-type and one loop Pac- or Pau-type 8 10 15 a) two loops Pac- or Pau-type and one loop Pa-type
b) two loops Pa-type and one loop Pas-, Pauu- or Pacu-type 7 11
16 a) three loops Pac- or Pau-type b) two loops Pa-type and one loop Pa*-type c) one loop Pac- or Pau-type, one loop Pas-, Pauu- or Pacu-type and one loop Pa-type
6 12
17 a) one loop Pas-, Pauu- or Pacu-type and the other two loops of types Pac or Pau b) one loop Pa-type and the other two loops of types Pas, Pauu or Pacu c) one loop Pac- or Pau-type, one loop Pa*-type and one loop Pa-type
5 13
18 a) one loop Pa*-type and the other two loops of types Pac or Pau b) one loop Pac- or Pau-type and the other two loops of types Pas, Pauu or Pacu c) one loop Pa*-type, one loop Pa-type and one loop Pas-, Pauu- or Pacu-type
4 14
19 a) one loop Pa*-type, one loop Pac- or Pau-type and one loop Pas-, Pauu- or Pacu-type b) three loops of types Pas, Pauu or Pacu c) one loop Pa*-type, one loop Pa-type and one loop Pas-, Pauu- or Pacu-type
3 15
7.1 Limbs with three translational motions 343
Table 7.1. (cont.)
20 a) one loop Pa*-type and the other two loops of types Pas, Pauu or Pacu b) two loops Pa*-type and one loop Pac- or Pau-type
2 16
21 two loops Pa*-type and one loop Pas-, Pauu- or Pacu-type 1 17 22 three loops of Pa*-type 0 18 NG degree of overconstraint, rG number of joint parameters that lose their inde-pendence in the closed loops that may be integrated in G-limb
parameters: MG=2, NG=2 and rG=4. Limbs defined by MG=SG=3 and (RG)= ( 1 2 3, ,v v v ) may be obtained by using parallelogram loops Pacc-type (Figs. 7.8 and 7.11).
The various solutions illustrated in Figs. 7.1-7.11 may be systematized in by taking into account various criteria: a) limb type a1) simple limbs composed by a simple open kinematic chain (Fig. 7.1) a2) complex limbs containing at least one closed loop (Figs. 7.2-7.6, 7.8-7.11) b) type of kinematic joint connected to the fixed base b1) prismatic joint (Figs. 7.1, 7.2, 7.3a, 7.8) b2) revolute joint (Figs. 7.4-7.6, 7.9-7.11) b3) prismatic and revolute joints (Figs. 7.3b) c) type of joints c1) with only prismatic and/or revolute joints (Figs. 7.1-7.6) c2) with prismatic, revolute and cylindrical joints (Figs. 7.8-7.11) d) number of closed loops integrated in the limb (q≥ 1) d1) with one closed loop (Figs. 7.2, 7.4, 7.8, 7.9) d2) with two closed loops (Figs. 7.3, 7.5, 7.10) d3) with three closed loops (Fig. 7.6) e) number of joint parameters that lose their independence in the closed loops integrated in the limb (3≤ rG≤ 12) e1) rG=3 (Figs. 7.2a-d, 7.4) e2) rG=4 (Figs. 7.8, 7.9) e3) rG=6 (Figs. 7.3, 7.5) e4) rG=7 (Figs. 7.10, 7.11)
e5) rG=9 (Fig. 7.6). The joint arrangement and structural parameters of the solutions pre-
sented in Figs. 7.1-7.11 are systematized in Table 7.2.
344 7 Limbs with three degrees of connectivity
Table 7.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 3, ,v v v ) and TG=0 presented in Figs. 7.1-7.11
Structural parameters No. Limb-type Joint arrangement m p q rG NG
Examples
1 PPP P⊥ P⊥ P 4 3 - - - Fig. 7.1 2 PPPa P⊥ P⊥ ||Pa 6 6 1 3 3 Fig. 7.2a 3 PPPa P⊥ P||Pa 6 6 1 3 3 Fig. 7.2b 4 PPaP P⊥ Pa||P 6 6 1 3 3 Fig. 7.2c 5 PPaP P||Pa⊥ P 6 6 1 3 3 Fig. 7.2d,e 6 PPaPa P||Pa||Pa 8 9 2 6 6 Fig. 7.3a 7 PPaPa Pa||Pat|| P 8 9 2 6 6 Fig. 7.3b 8 PaPP Pa⊥ P⊥ ||P 6 6 1 3 3 Fig. 7.4a,b 9 PaPP Pa||P⊥ P 6 6 1 3 3 Fig. 7.4c,d 10 PaPaP Pa⊥ Pa||P 8 9 2 6 6 Fig. 7.5a 11 PaPaP Pa⊥ Pa⊥ || P 8 9 2 6 6 Fig. 7.5b 12 PaPaP Pa⊥ Pa ⊥⊥ P 8 9 2 6 6 Fig. 7.5c 13 PaPaP Pa||Pa||P 8 9 2 6 6 Fig. 7.5d 14 PaPPa Pa||P⊥ Pa 8 9 2 6 6 Fig. 7.5e 15 PaPPa Pa⊥ P ⊥⊥ Pa 8 9 2 6 6 Fig. 7.5f 16 PaPPa Pa⊥ P||Pa 8 9 2 6 6 Fig. 7.5g 17 PaPPa Pa||P||Pa 8 9 2 6 6 Fig. 7.5h 18 PaPaPa Pa⊥ Pa||Pa 10 12 3 9 9 Fig. 7.6a 19 PaPaPa Pa||Pa⊥ Pa 10 12 3 9 9 Fig. 7.6b 20 PaPaPa Pa⊥ Pa⊥ Pa 10 12 3 9 9 Fig. 7.6c 21 PPacc P⊥ Pacc 5 5 1 4 3 Fig. 7.8a-c 22 PaccP Pacc ⊥ P 5 5 1 4 2 Fig. 7.9a-d 23 PaccPa Pacc||Pa 7 8 2 7 5 Fig. 7.10a,b 24 PaccPa Pacc ⊥ Pa 7 8 2 7 5 Fig. 7.11a,b 25 PaPacc Pa||Pacc 7 8 2 7 5 Fig. 7.10c-e 26 PaPacc Pa⊥ Pacc 7 8 2 7 5 Fig. 7.11c-e aSee footnote of Table 6.2 for the nomenclature of structural parameters
7.1 Limbs with three translational motions 345
Fig. 7.1. Simple limb PPP-type defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.2. Complex limbs of types PPPa (a-c), PPaP (d-e) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
346 7 Limbs with three degrees of connectivity
Fig. 7.3. Complex limbs of type PPaPa (a) and PaPatP (b) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.4. Complex limbs PaPP-type defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
7.1 Limbs with three translational motions 347
Fig. 7.5. Complex limbs of types PaPaP (a)-(d) and PaPPa (e)-(h) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
348 7 Limbs with three degrees of connectivity
Fig. 7.6. Complex limbs PaPaPa-type defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.7. Parallelogram loop with two cylindrical joints Pacc-type used to replace a kinematic chain P||Pa- or Pa||P-type
7.1 Limbs with three translational motions 349
Fig. 7.8. Complex limbs PPacc-type defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.9. Complex limbs PaccP-type defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
350 7 Limbs with three degrees of connectivity
Fig. 7.10. Complex limbs of types PaccP (a, b) and PPacc (c-e) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
7.1 Limbs with three translational motions 351
Fig. 7.11. Complex limbs of types PaccPa (a)-(c) and PaPacc (d, e) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
352 7 Limbs with three degrees of connectivity
7.2 Planar limbs with one rotational and two translational motions
These limbs give one rotational and two translational velocities in the basis of the operational velocity space (RG)=( 1 2 1, ,v v ω ). The axis of angular ve-locity 1ω is perpendicular to the plane of translational velocities v1 and v2.
Structural solutions for non-redundant limbs with only revolute and pris-matic pairs defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω ) are presented in Figs. 7.12-7.18.
These solutions have no idle mobilities and the plane of translational ve-locities has a constant orientation. Solutions for simple limbs are illustrated in Figs. 7.12-7.14. They have prismatic (Fig. 7.12 and 7.13) or revolute (Fig. 7.14) pairs adjacent to the fixed base. The complex limbs may con-tain one (Figs. 7.13, 7.15 and 7.16) or two (Fig. 7.17) parallelogram loops and have prismatic (Fig. 7.13) or revolute (Fig. 7.15-7.18) pairs adjacent to the fixed base. The solution in Fig. 7.17e integrates a parallelogram loop Pa-type connected to an extensible parallelogram loop denoted by Pat.
In the complex limbs, the revolute joints connected to the fixed base may be integrated in a parallelogram loop (Figs. 7.16 and 7.17a and b) or outside the closed loop (Figs. 7.15, 7.17c-e and 7.18).
Each planar parallelogram loop could integrate up to three idle mobili-ties as we have seen in the previous sections. The degree of overconstraint and the number of joint parameters that lose their independence in the par-allelogram loops integrated in the various types of limbs are given in Table 7.1. The complex limb in Fig. 7.17e are defined by NG=6 and rG=6.
Fig. 7.12. Simple limbs of types PPR (a), PRR (b) and PRP (c) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 353
Table 7.3. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 1, ,v v ω ) and TG=0 presented in Figs. 7.12-7.18
Structural parameters No. Limb-type Joint arrangement m p q rG NG
Examples
1 PPR P⊥ P ⊥⊥ R 4 3 - - - Fig. 7.12a 2 PRR P⊥ R||R 4 3 - - - Fig. 7.12b 3 PRP P⊥ R⊥ P 4 3 - - - Fig. 7.12c 4 PPaR P⊥ Pa||R 6 6 1 3 3 Fig. 7.13a 5 PRPa P⊥ R||Pa 6 6 1 3 3 Fig. 7.13b 6 RRR R||R||R 4 3 - - - Fig. 7.14a 7 RPR R⊥ P⊥ ||R 4 3 - - - Fig. 7.14b 8 RRP R||R⊥ P 4 3 - - - Fig. 7.14c 9 RPP R⊥ P ⊥⊥ P 4 3 - - - Fig. 7.14d 10 RPPa R⊥ P⊥ || Pa 6 6 1 3 3 Fig. 7.15a 11 RPaP R||Pa⊥ P 6 6 1 3 3 Fig. 7.15b,c 12 RRPa R||R||Pa 6 6 1 3 3 Fig. 7.15d 13 PaPR Pa⊥ P⊥ ||R 6 6 1 3 3 Fig. 7.16a,b 14 PaRP Pa||R⊥ P 6 6 1 3 3 Fig. 7.16c 15 PaRR Pa||R||R 6 6 1 3 3 Fig. 7.16d 16 PaPaR Pa||Pa||R 8 9 2 6 6 Fig. 7.17a 17 PaRPa Pa||R||Pa 8 9 2 6 6 Fig. 7.17b 18 RPaPa R||Pa||Pa 8 9 2 6 6 Fig. 7.17c,d 19 RPaPat R||Pa||Pat 8 9 2 6 6 Fig. 7.17e 20 RPaPat R||Pa||Pat 8 9 2 6 6 Fig. 7.17e 21 RRbR R|| Rb|| R 6 6 1 3 3 Fig. 7.18a 22 RRbRbR R|| Rb|| Rb|| R 8 9 2 6 6 Fig. 7.18b 23 Pn2 R Pn2|| R 6 6 1 3 3 Fig. 7.18c,d 24 Pn3 Pn3 6 6 1 3 3 Fig. 7.18e,f aSee footnote of Table 6.2 for the nomenclature of structural parameters
The joint arrangement and structural parameters of the solutions pre-
sented in Figs. 7.12-7.18 are systematized in Table 7.3.
354 7 Limbs with three degrees of connectivity
Fig. 7.13. Complex limbs of types PPaR (a) and PRPa (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.14. Simple limbs of types RRR (a), RPR (b), RRP (c) and RPP (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 355
Fig. 7.15. Complex limbs of types PRPa (a), RPaP (b), (c) and RRPa (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
356 7 Limbs with three degrees of connectivity
Fig. 7.16. Complex limbs of types PaPR (a), (b), PaRP (c) and PaRR (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 357
Fig. 7.17. Complex limbs of types PaPaR (a), PaRPa (b), RPaPa (c, d) and RPaPat (e) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
358 7 Limbs with three degrees of connectivity
Fig. 7.18. Complex limbs of types RRbR (a), RRbRbR (b) Pn2R (c, d) and Pn3 (e, f) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.3 Non planar limbs with one rotational and two translational motions 359
7.3 Non planar limbs with one rotational and two translational motions
In this case, the axis of angular velocity 1ω lies in the plane of transla-tional velocities v1 and v2. If the characteristic point H of the distal link is situated on the rotation axis and the plane of translations has a constant orientation, the basis of the operational velocities space is (RG)= ( 1 2 1, ,v v ω ).
Structural solutions fort non-redundant limbs with only revolute and prismatic pairs are presented in Figs. 7.19-7.22. These solutions have no idle mobilities and the plane of translational velocities has a constant ori-entation. The rotation axis also has a constant direction in the plane of translations. Solutions for simple limbs with prismatic pairs adjacent to the fixed base are illustrated in Figs. 7.19. The complex limbs may contain one (Figs. 7.20 and 7.21) or two (Fig. 7.22) parallelogram loops and have pris-matic (Fig. 7.20) or revolute (Fig. 7.21 and 7.22a, b) pairs adjacent to the fixed base. The solutions in Fig. 7.22c and have prismatic and revolute joints connected to the fixed base.
Each planar parallelogram loop could integrate up to three idle mobili-ties as we have seen in the previous sections. The degree of overconstraint and the number of joint parameters that lose their independence in the par-allelogram loops integrated in the various types of limbs presented in Figs. 7.19-7.22 are given in Table 7.1.
In the solutions presented in Figs. 7.19-7.22, the number of joints may be reduced by replacing:
a) two consecutive revolute and prismatic joints with the same axes/direction (R||P or P||R) by a cylindrical joint such as in Fig. 7.23,
b) a sequence of joints R⊥Pa- or Pa⊥R-type by a parallelogram loop Pass-type with two spherical joints adjacent to the distal link such as in Fig. 7.24.
Solutions with a cylindrical joint and a parallelogram loop Pass-type are presented in Figs. 7.25 and 7.26. The solution presented in Fig. 7.26a has NG=0, rG=6 and the solution in Fig. 7.26b are defined by NG=3 and rG=9.
The joint arrangement and structural parameters of the solutions pre-sented in Figs. 7.19-7.26 are systematized in Table 7.4.
If the characteristic pint of the distal link is not on situated on the rota-tion axis the basis of the operational velocities space can be ( 1 2 1, ,v v ω ) or ( 1 2 3, ,v v v ).
360 7 Limbs with three degrees of connectivity
Table 7.4. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 1, ,v v ω ) and TG=0 presented in Figs. 7.19-7.26
Structural parameters No. Limb-type Joint arrangement m p q rG NG
Examples
1 PPR P⊥ P⊥ || R 4 3 - - - Fig. 7.19a 2 PPR P⊥ P||R 4 3 - - - Fig. 7.19b 3 PPaR P⊥ Pa⊥ || R 6 6 1 3 3 Fig. 7.20a 4 PPaR P⊥ Pa ⊥⊥ R 6 6 1 3 3 Fig. 7.20b 5 PaPR Pa⊥ P ⊥⊥ R 6 6 1 3 3 Fig. 7.21a,b 6 PaPR Pa⊥ P||R 6 6 1 3 3 Fig. 7.21c,d 7 PaRP Pa⊥ R||P 6 6 1 3 3 Fig. 7.21e,f 8 PaPaR Pa||Pa⊥ R 8 9 2 6 6 Fig. 7.22a,b 9 PaPaR Pa||Pat ⊥ R 8 9 2 6 6 Fig. 7.22c,d 10 PC P⊥ C 3 2 - - - Fig. 7.25a 11 PaC Pa⊥ C 5 5 1 3 3 Fig. 7.25b 12 PPass P⊥ Pass 5 5 1 6 0 Fig. 7.26a 13 PPass P||Pass 7 8 2 9 3 Fig. 7.26b aSee footnote of Table 6.2 for the nomenclature of structural parameters
Fig. 7.19. Non planar simple limbs of types PPR (a, b) and PRP (c) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.3 Non planar limbs with one rotational and two translational motions 361
Fig. 7.20. Non planar complex limbs PPaR-type defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.21. Non planar complex limbs of types PaPR (a-d) and PaRP (e, f) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
362 7 Limbs with three degrees of connectivity
Fig. 7.22. Non planar complex limbs of types PaPaR (a, b) and PaPatR (c, d) de-fined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.23. Cylindrical joint used to replace a kinematic chain R||P- or P||R-type
7.3 Non planar limbs with one rotational and two translational motions 363
Fig. 7.24. Parallelogram loop Pass-type used to replace a sequence of joints Pa⊥R- or R⊥ Pa-type
Fig. 7.25. Non planar simple limb PC-type (a) and complex limbs PaC-type (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.26. Non planar complex limbs of type PPass (a) and PaPass (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
364 7 Limbs with three degrees of connectivity
Fig. 7.27. Simple limbs of types PRR (a, b) and RPR (c) defined by MG=SG=3 and (RG)=( 1 1 2, ,v ω ω )
7.4 Limbs with one translational and two rotational motions
These limbs have one translational 1v and two rotation velocities 1ω and
2ω in the basis of the operational velocity space (RG)=( 1 1 2, ,v ω ω ). They allow two rotations along with a variable position of characteristic point H of the distal link. Point H is situated at the intersection of the rotation axes on a line with a fixed direction (Fig. 7.27). The simplest structural solu-tions for non-redundant limbs with only prismatic and revolute pairs have the following structural parameters: MG=SG=3, (RG)=( 1 1 2, ,v ω ω ), NG=0, TG=0, rG=0.
If the characteristic point H is not situated at the intersection of the axes of the revolute joints, the basis of the operational velocity space may con-tain (i) one translational and two angular velocities, (ii) one rotational and two translational velocities or (iii) three translational velocities.
7.5 Limbs with three rotational motions
These limbs has three angular velocities 1ω , 2ω and 3ω in the basis of the operational velocity space (RG)=( 1 2 3, ,ω ω ω ). The simplest structural solu-tion for non-redundant limbs has three revolute pairs with intersecting axes and are defined by MG=SG=3, (RG)=( 1 2 3, ,ω ω ω ), NG=0, TG=0, rG=0.
7.6 Other limbs with three degrees of connectivity 365
Fig. 7.28. Simple limb RRR-type defined by MG=SG=3 and (RG)=( 1 2 3, ,ω ω ω )
The solution in Fig. 7.28 gives three rotations of the distal link along with a constant position of the characteristic point H situated at the intersecting axes. Other solutions with non-orthogonal intersecting axes may also be used. As we have mentioned in the previous chapters, just solutions with orthogonal and/or parallel axes of the kinematic joints are presented in this work. They are the easiest to make.
If the characteristic point of the distal link is not situated at the intersect-ing axes of the revolute joints, the basis of the operational velocity space-may contain : (i) three angular velocities, (ii) one translational and two an-gular velocities, (iii) one rotational and two translational velocities or (iv) three translational velocities.
7.6 Other limbs with three degrees of connectivity
The limbs presented in this section have multiple bases for the operational velocity space for any position of the characteristic point on the distal link. These bases combine translational and angular velocities. For example, the basis of the operational velocity space of the limb in Fig. 7.29a may be de-fined by three translational velocities ( 1 2 3, ,v v v ) or by two translational and one angular velocities ( 1 i 1, ,v v ω ), and the basis of the operational velocity space of the limb in Fig. 7.29b by two translational and one angular veloci-ties ( 1 i i, ,v v ω ) or by one translational and two angular velocities ( 1 1 i, ,v ω ω ), i=2,3. These bases are valid for any position of the characteristic point H on distal link 4.
Examples of simple and complex limbs with three degrees of connec-tivity and multiple bases of the operational velocity space are presented in Figs. 7.29-7.35. The complex limbs may contain planar or spatial loops.
366 7 Limbs with three degrees of connectivity
Fig. 7.29. Simple limbs of types PRP (a), PRR (b-d), RPP (e, f) and RRR (g-i) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 367
Fig. 7.30. Simple limbs of types RPR (a-d) and RRP (e-g) defined by MG=SG=3 and multiple bases of the operational velocity space
368 7 Limbs with three degrees of connectivity
Fig. 7.31. Complex limbs of types PPaR (a-c) and PRPa (d-g) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 369
Fig. 7.32. Complex limbs of types RRPa (a-c) and RPaR (d-i) defined by MG=SG=3 and multiple bases of the operational velocity space
370 7 Limbs with three degrees of connectivity
Fig. 7.33. Complex limbs of types PaRR (a-c) and PaRP (d, e) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 371
Fig. 7.34. Complex limbs of types RPaP (a-h) and RPPa (i-l) defined by MG=SG=3 and multiple bases of the operational velocity space
372 7 Limbs with three degrees of connectivity
Fig. 7.35. Complex limbs of types PRRb (a), PRRbRb (b), and PPa2 (c-d) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 373
Fig. 7.36. Complex limbs of types RRRb (a), RRRbRb (b), and RPa2 (c-f) defined by MG=SG=3 and multiple bases of the operational velocity space
374 7 Limbs with three degrees of connectivity
Table 7.5. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3 and TG=0 presented in Figs. 7.29-7.36
Structural parameters No. Limb-type Joint arrangement m p q rG NG
Examples
1 PRP P||R⊥ P 4 3 - - - Fig. 7.29a 2 PRR P|| R⊥ P 4 3 - - - Fig. 7.29b 3 PRR P||R||P 4 3 - - - Fig. 7.29c 4 PRR P⊥ R⊥ R 4 3 - - - Fig. 7.29d 5 RPP R⊥ P⊥ || P 4 3 - - - Fig. 7.29e 6 RPP R||P⊥ P 4 3 - - - Fig. 7.29f 7 RRR R⊥ R||R 4 3 - - - Fig. 7.29g 8 RRR R||R⊥ R 4 3 - - - Fig. 7.29h 9 RRR R⊥ R⊥ R 4 3 - - - Fig. 7.29i 10 RPR R⊥ P ⊥⊥ R 4 3 - - - Fig. 7.30a 11 RPR R⊥ P||R 4 3 - - - Fig. 7.30b 12 RPR R||P||R 4 3 - - - Fig. 7.30c 13 RPR R||P⊥ R 4 3 - - - Fig. 7.30d 14 RRP R⊥ R||P 4 3 - - - Fig. 7.30e 15 RRP R⊥ R⊥ P 4 3 - - - Fig. 7.30f 16 RRP R||R||P 4 3 - - - Fig. 7.30g 17 PPaR P||Pa⊥ R 6 6 1 3 3 Fig. 7.31a,b 18 PPaR P||Pa||R 6 6 1 3 3 Fig. 7.31c 19 PRPa P⊥ R⊥ Pa 6 6 1 3 3 Fig. 7.31d,e 20 PRPa P||R⊥ Pa 6 6 1 3 3 Fig. 7.31f 21 PRPa P||R||Pa 6 6 1 3 3 Fig. 7.31g 22 RRPa R⊥ R||Pa 6 6 1 3 3 Fig. 7.32a 23 RRPa R⊥ R⊥ Pa 6 6 1 3 3 Fig. 7.32b,c 24 RPaR R⊥ Pa||R 6 6 1 3 3 Fig. 7.32d,e 25 RPaR R⊥ Pa⊥ R 6 6 1 3 3 Fig. 7.32f,g 26 RPaR R||Pa||R 6 6 1 3 3 Fig. 7.32h 27 RPaR R||Pa⊥ R 6 6 1 3 3 Fig. 7.32i 28 PaRR Pa⊥ R⊥ R 6 6 1 3 3 Fig. 7.33a,b 29 PaRR Pa||R⊥ R 6 6 1 3 3 Fig. 7.33c 30 PaRP Pa⊥ R⊥ P 6 6 1 3 3 Fig. 7.33d,e 31 RPaP R||Pa||P 6 6 1 3 3 Fig. 7.34a,g 32 RPaP R⊥ Pa⊥ || P 6 6 1 3 3 Fig. 7.34b,h 33 RPaP R⊥ Pa||P 6 6 1 3 3 Fig. 7.34c 34 RPaP R⊥ Pa ⊥⊥ P 6 6 1 3 3 Fig. 7.34d,f 35 RPPa R⊥ P||Pa 6 6 1 3 3 Fig. 7.34i 36 RPPa R||P||Pa 6 6 1 3 3 Fig. 7.34j 37 RPPa R||P⊥ Pa 6 6 1 3 3 Fig. 7.34k 38 RPPa R⊥ P⊥ Pa 6 6 1 3 3 Fig. 7.34l 39 PRRb P||R||Rb 6 6 1 3 3 Fig. 7.35a 40 PRRbRb P||R||Rb||Rb 8 9 2 6 6 Fig. 7.35b 42 PPn2 P||Pn2 6 6 1 3 3 Fig. 7.35c,d
7.7 Redundant limbs with three degrees of connectivity 375
Table 7.5. (cont.)
43 RRRb P⊥ R||Rb 6 6 1 3 3 Fig. 7.36a 44 RRRbRb P⊥ R||Rb||Rb 8 9 2 6 6 Fig. 7.36b 45 PPn2 R⊥ Pn2 6 6 1 3 3 Fig. 7.36c-f aSee footnote of Table 6.2 for the nomenclature of structural parameters
The planar loops are of type parallelogram Pa (Fig. 6.9a-f) rhombus Rb (Fig. 6.8a,b) or other planar loops Pn2 (Fig. 6.9h,i). The structural parame-ters of these limbs are systematized in Table 7.5.
7.7 Redundant limbs with three degrees of connectivity
A large variety of redundant limbs with SG=3 and MG>3 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just two examples are presented in Figs. 7.37 and 7.38.
Figure 7.37 represents a particular case of example 2 analyzed in chap-ter 3 with p=4 revolute joints with parallel axes. This solution is defined by the following structural parameters: m=5, q=0, rG=0, SG=3, MG=4 (RG)=( 1 2 1, ,v v ω ), NG=0 and TG=1. The kinematic chain has four degrees of mobility but distal link 5 can realize just three independent planar motions.
The complex limb in Fig. 7.38 combines a parallelogram loop Pa, a telescopic parallelogram loop Pat and a serial chain R║R -type. Three joint parameters lose their independence in each parallelogram loop Pa- and Pat-type. This solution is defined by the following structural parameters: m=9, p=10, q=2, rG=6, SG=3, MG=4, (RG)=( 1 2 1, ,v v ω ), NG=6 and TG=1.
As in the previous example, this kinematic chain has four degrees of mobility but distal link 9 can realize just three independent planar motions.
This degree of redundancy gives more dexterity in avoiding singularities and obstacles. Structural redundancy can also contribute to increasing the working space of the robot as we will present in Part 2 of this work.
376 7 Limbs with three degrees of connectivity
Fig. 7.37. Example of simple planar limb RRRR-type defined by MG=4, SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.38. Example of complex planar limb PaPatRR-type defined by MG=4, SG=3 and (RG)=( 1 2 1, ,v v ω )
8 Limbs with four degrees of connectivity
Limbs with four degrees of connectivity (SG=4) are used in parallel robots F where the connectivity of moving platform SF≤ 4. The basis of the op-erational space of these limbs contains four independent velocities. Non-redundant solutions generated by the evolutionary morphology approach are presented in the first four sections of this chapter. Some examples of redundant solutions are also presented in the last section. The first four sections present solutions of limbs defined by the following: (i) one rota-tional and three translational motions (Schönflies motion), (ii) two transla-tional and two rotational motions, (iii) one translational and three rota-tional motions, and (iv) multiple bases of the operational velocity space.
8.1 Limbs with Schönflies motion
These limbs give three translational velocities, v1, v2, v3, and one angular velocity 1ω about an axis of fixed direction. The basis of the operational velocity space of these limbs is (RG)=( 1 2 3 1, , ,v v v ω ). The characteristic point H may have any position on the distal link.
This motion was studied by the German mathematician Arthur Moritz Schönflies (Schoenflies) and is usually called Schönflies motion. The ref-erenced book of Schönflies (1893) is the translation from German into French of “Geometrie der Bewegung in synthetischer Darstellung”, Teub-ner, Leipzig, 1886 where the author’s name is spelt Schoenflies. This is why, probably for the first time, Bottema and Roth (1979) called this spe-cial motion type Schönflies motion. In Schönflies’ book, as well as in the Bottema and Roth book, a so-called Schoenflies motion is described as be-ing a rigid-body motion having axodes (surfaces generated by the instanta-neous screw axes in any time-dependent motion) that are cylinders (or prismatic surfaces) rather than being a four-degrees-of-freedom (4-dof) motion. Schönflies is better known for his important contribution to the theory of 230 discrete (non-continuous) subgroups of displacements as a basic tool in crystallography developed independently by Schönflies and
378 8 Limbs with four degrees of connectivity
Fedorov. An initial enumeration of 4-dof serial kinematic chains generat-ing Schönflies motions composed of 1-dof kinematic pairs and plane-hinged parallelogram is presented by Hervé and Sparacino (1991).
The simplest structural solutions of non redundant simple and complex limbs with only revolute and prismatic pairs defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) are presented in Figs. 8.1-8.50. These solutions have no idle mobilities.
In the solutions presented in Figs. 8.1-8.50 the number of joints may be reduced by replacing as follows:
a) two consecutive revolute and prismatic joints with the same axes/direction (R||P or P||R) by a cylindrical joint such as in Fig. 7.23,
b) a sequence of joints R⊥Pa or Pa⊥R by a parallelogram loop Pass-type with two spherical joints adjacent to the distal link such as in Fig. 7.24,
c) a sequence of joints P||Pa or Pa||P by a parallelogram loop Pacc-type with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop such as in Fig. 7.7.
The complex limbs may combine up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could contain up to three idle mobilities that may be two orthogonal rotations and one translation.
The degree of overconstraint and the number of joint parameters that lose their independence in the parallelogram closed loops that may be combined in the various types of limbs with MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and TG=0 are systematized in Table 7.1.
The various solutions illustrated in Figs. 8.1-8.50 may be systematized in by taking into account various criteria: a) limb type a1) simple limbs composed by a simple open kinematic chain (Figs. 8.1-8.7)
a2) complex limbs containing at least one closed loop (Figs. 8.8- 8.55) b) position of the rotation axis called rotation axis of the Schönflies motion
b1) fixed position and orientation (Figs. 8.1, 8.5, 8.8, 8.16, 8.21- 8.23, 8.26-8.28, 8.43)
b2) lying in a plane of fixed position and orientation (Figs. 8.2, 8.10, 8.11, 8.18, 8.33, 8.41, 8.42, 8.45)
b3) lying on a cylinder of fixed position, orientation and radius (Figs. 8.4, 8.6, 8.9, 8.17, 8.24, 8.34, 8.38, 8.39, 8.47)
8.1 Limbs with Schönflies motion 379
b4) passing by the characteristic point H of the distal link (Figs. 8.3, 8.7, 8.12-8.14, 8.20, 8.25, 8.29-8.31, 8.35-8.37, 8.44, 8.48-8.50)
b5) unchanging orientation (Figs. 8.15, 8.19, 8.32, 8.40, 8.46) c) type of kinematic joint connected to the fixed base c1) prismatic joint (Figs. 8.1-8.4, 8.8-8.20, 8.48-8.50) c2) revolute joint (Figs. 8.5-8.7, 8.21-8.47) d) type of closed loops
d1) parallelogram loops (Figs. 8.8-8.47) d2) extensible parallelogram loop (Fig. 8.25b) d3) rhombus loops (Fig. 8.48) d4) other loops (Figs. 8.49, 8.50)
e) number of closed loops combined in the limb (q≥ 1) e1) with one closed loop (Figs. 8.8-8.15, 8.21-8.23, 8.24, 8.29- 8.34, 8.48a, 8.49, 8.50) e2) with two closed loops (Figs. 8.16-8.19, 8.25-8.28, 8.35- 8.42, 8.48b) e3) with three closed loops (Figs. 8.43-8.47) f) number of parameters that lose their independence in the closed loops combined in the limb (3≤ rG≤ 12) f1) rG=3 (Figs. 8.8-8.15, 8.21-8.24, 8.29-8.34, 8.48a, 8.49, 8.50) f3) rG=6 (Figs. 8.16-8.20, 8.25, 8.26-8.28, 8.35-8.42, 8.48b) f5) rG=9 (Figs. 8.43-8.47). g) degree of mobility of a planar loops combined in the complex limbs g1) one degree of mobility (Figs. 8.8-8.47) g2) two degrees of mobility (Figs. 8.48, 8.49) g3) three degrees of mobility (Fig.8.50).
We note that in all cases, the rotation axis of the Schönflies keep a con-stant orientation. For this reason, we consider case b5 mentioned above as the general case. We denote it by S. The particular positions corresponding to cases b1-b4 are denoted by S1- S4. Rotation axis of the Schönflies motion may be: (i) in a fixed position - denoted by S1, (ii) lying in a plane of fixed position and orientation - denoted by S2, (iii) lying on a cylinder of fixed position-orientation and radius - denoted by S3, and (iv) passing by the characteristic point of the distal link – denoted by S4.
Various solutions of planar bimobile closed loops Pn2-type and Pn3-type may be used as presented in chapter 6. The revolute joints combined in the closed loops Pn2 and Pn3 have parallel axes and the directions of the prismatic joints are contained in a plane perpendicular to the revolute axes.
380 8 Limbs with four degrees of connectivity
Table 8.1. Joint arrangement and structural parametersa of the limbs defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and TG=0 presented in Figs. 8.1-8.55
Structural parameters No. Limb-type Rotation axis of the Schönflies motion b m p q rG NG
Figure
1 PPaRP S 7 7 1 3 3 Fig. 8.15a,b 2 PPRPa S 7 7 1 3 3 Fig. 8.15c 3 PPaRPa S 9 10 2 6 6 Fig. 8.19a,b 4 PaPaRPa S 11 13 3 9 9 Fig. 8.46a-f 5 PRPP S1 5 4 - - - Fig. 8.1 6 RPPP S1 5 4 - - - Fig. 8.5a-c 7 PRPPa S1 7 7 1 3 3 Fig. 8.8a,b 8 PRPaP S1 7 7 1 3 3 Fig. 8.8c-e 9 PRPaPa S1 9 10 2 6 6 Fig. 8.16 10 RPPaP S1 7 7 1 3 3 Fig. 8.21a-i 11 RPaPP S1 7 7 1 3 3 Fig. 8.22a-k 12 RPPPa S1 7 7 1 3 3 Fig. 8.23a-f 13 RPaPaP S1 9 10 2 6 6 Fig. 8.26a-l 14 RPaPPa S1 9 10 2 6 6 Fig. 8.27a-l 15 RPPaPa S1 9 10 2 6 6 Fig. 8.28a-c 16 RPaPaPa S1 11 13 3 9 9 Fig. 8.43a-i 17 PPRP S2 5 4 - - - Fig. 8.2a,b 18 PRPPa S2 5 4 - - - Fig. 8.2c,d 19 PPRPa S2 7 7 1 3 3 Fig. 8.10a-d 20 PRPPa S2 7 7 1 3 3 Fig. 8.10e-h 21 PRPaP S2 7 7 1 3 3 Fig. 8.11a-f 22 PPaRP S2 7 7 1 3 3 Fig. 8.11g,h 23 PRPaPa S2 9 10 2 6 6 Fig. 8.18a,b 24 PaPRP S2 7 7 1 3 3 Fig. 8.33a-c 25 PaRPP S2 7 7 1 3 3 Fig. 8.33d-f 26 PaRPaP S2 9 10 2 6 6 Fig. 8.41a-f 27 PaPRPa S2 9 10 2 6 6 Fig. 8.41g-i 28 PaRPPa S2 9 10 2 6 6 Fig. 8.42a-e 29 PaRPaP S2 9 10 2 6 6 Fig. 8.42f 30 PaPaRP S2 9 10 2 6 6 Fig. 8.42g 31 PaPRPa S2 9 10 2 6 6 Fig. 8.42h 32 PaRPPa S2 9 10 2 6 6 Fig. 8.42i 33 PaPaRPa S2 11 13 3 9 9 Fig. 8.45a,b 34 PaRPaPa S2 11 13 3 9 9 Fig. 8.45c-g
8.1 Limbs with Schönflies motion 381
Table 8.1. (cont.)
35 PRRP S3 5 4 - - - Fig. 8.4 36 RPRP S3 5 4 - - - Fig. 8.6a,b 37 RRPP S3 5 4 - - - Fig. 8.6c 38 PPaRP S3 7 7 1 3 3 Fig. 8.9a 39 PRRPa S3 7 7 1 3 3 Fig. 8.9b 40 PPaRPa S3 9 10 2 6 6 Fig. 8.17 41 RPRPa S3 7 7 1 3 3 Fig. 8.24a,b 42 RRPPa S3 7 7 1 3 3 Fig. 8.24c 43 RRPaP S3 7 7 1 3 3 Fig. 8.24d 44 PaPRP S3 7 7 1 3 3 Fig. 8.34a 45 PaRPP S3 7 7 1 3 3 Fig. 8.34b,c 46 PaRPaP S3 9 10 2 6 6 Fig. 8.38a-d 47 PaPRPa S3 9 10 2 6 6 Fig. 8.38e,f 48 PaRPPa S3 9 10 2 6 6 Fig. 8.39a-d 49 PaRPaPa S3 11 13 3 9 9 Fig. 8.47a-c 50 PPPR S4 5 4 - - - Fig. 8.3a-c 51 PPRP S4 5 4 - - - Fig. 8.3d 52 PPRR S4 5 4 - - - Fig. 8.3e,f 53 PRPR S4 5 4 - - - Fig. 8.3h 54 PRRR S4 5 4 - - - Fig. 8.3g 55 RPPR S4 5 4 - - - Fig. 8.7a,b 56 RPRP S4 5 4 - - - Fig. 8.7c 57 RPRR S4 5 4 - - - Fig. 8.7d,e 58 RRPR S4 5 4 - - - Fig. 8.7f,g 59 RRRP S4 5 4 - - - Fig. 8.7h 60 PPPaR S4 7 7 1 3 3 Fig. 8.12a-f 61 PPaRP S4 7 7 1 3 3 Fig. 8.12g-i 62 PPaPR S4 7 7 1 3 3 Fig. 8.13a-i 63 PPaRR S4 7 7 1 3 3 Fig. 8.14a 64 PRPaR S4 7 7 1 3 3 Fig. 8.14b 65 PPaPaR S4 9 10 2 6 6 Fig. 8.20a-c 66 RPPaPa S4 9 10 2 6 6 Fig. 8.25a 67 RPPaPat S4 9 10 2 6 6 Fig. 8.25b 68 PaPPR S4 7 7 1 3 3 Fig. 8.29a-l 69 PaPRP S4 7 7 1 3 3 Fig. 8.30a-d 70 PaRRP S4 7 7 1 3 3 Fig. 8.30e 71 PaRPR S4 7 7 1 3 3 Fig. 8.31a-c
382 8 Limbs with four degrees of connectivity
Table 8.1. (cont.)
72 PaRRR S4 7 7 1 3 3 Fig. 8.31d 73 PaPRR S4 7 7 1 3 3 Fig. 8.31e 74 PaPRP S4 7 7 1 3 3 Fig. 8.32a,b 75 PaPaPR S4 9 10 2 6 6 Fig. 8.35a-k 76 PaPPaR S4 9 10 2 6 6 Fig. 8.36a-l 77 PaPaRP S4 9 10 2 6 6 Fig. 8.37a-d 78 PaPaRP S4 9 10 2 6 6 Fig. 8.40a-c 79 PaPRPa S4 9 10 2 6 6 Fig. 8.40d-h 80 PaPaPaR S4 11 13 3 9 9 Fig. 8.44a-i 81 PRRbR S4 7 7 1 3 3 Fig. 8.48a 82 PRRbRbR S4 9 10 2 6 6 Fig. 8.48b 83 PPn2R S4 7 7 1 3 3 Fig. 8.49a,b 84 PPn3 S4 7 7 1 3 3 Fig. 8.50a,b aSee footnote of Table 6.2 for the nomenclature of structural parameters bS1 fixed position and orientation, S2 lying in a plane of fixed position and orienta-tion, S3 lying on a cylinder of fixed position, orientation and radius, S4 passing by the characteristic point of the distal link, S fixed orientation.
A complex limb may contain one, two or more rhombus (Rb) loops. In
Fig. 8.48 just two examples with one and two rhombus loops are pre-sented.
Three joint parameterslose their independence in each planar closed loop Rb-, Pn2 or Pn3-type.
The structural parameters of the solutions presented in Figs. 8.1-8.55 are systematized in Table 8.1.
8.1 Limbs with Schönflies motion 383
Fig. 8.1. Simple limb PRPP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.2. Simple limbs of types PPRP (a, b) and PRPP (c, d) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing in a plane
384 8 Limbs with four degrees of connectivity
Fig. 8.3. Simple limbs of types PPPR (a-c), PPRP (d), PPRR (e, f), PRRR (g) and PRPR (h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 385
Fig. 8.4. Simple limb PRRP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.5. Simple limbs RPPP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.6. Simple limbs of types RPRP (a, b) and RRPP (c) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cyl-inder
386 8 Limbs with four degrees of connectivity
Fig. 8.7. Simple limbs of types RPPR (a, b), RPRP (c), RPRR (d, e), RRPR (f, g) and RRRP (h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 387
Fig. 8.8. Complex limbs of types PRPPa (a, b) and PRPaP (c-e) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.9. Complex limbs of types PPaRP (a) and PRRPa (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing on a cylinder
388 8 Limbs with four degrees of connectivity
Fig. 8.10. Complex limbs of types PPRPa (a-d) and PRPPa (e-h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing in a plane
8.1 Limbs with Schönflies motion 389
Fig. 8.11. Complex limbs of types PRPaP (a-f) and PPaRP (g, h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing in a plane
390 8 Limbs with four degrees of connectivity
Fig. 8.12. Complex limbs of types PPPaR (a-f) and PPaRP (g-i) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 391
Fig. 8.13. Complex limbs PPaPR-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
392 8 Limbs with four degrees of connectivity
Fig. 8.14. Complex limbs of types PPaRR (a) and PRPaR (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
Fig. 8.15. Complex limbs of types PPaRP (a, b) and PPRPa (c) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schön-flies motion
8.1 Limbs with Schönflies motion 393
Fig. 8.16. Complex limb PRPaPa-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.17. Complex limbs PPaRPa-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cyl-inder
Fig. 8.18. Complex limbs PRPaPa-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
394 8 Limbs with four degrees of connectivity
Fig. 8.19. Complex limbs PPaRPa-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
Fig. 8.20. Complex limbs PPaPaR-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the char-acteristic point of the distal link
8.1 Limbs with Schönflies motion 395
Fig. 8.21. Complex limbs RPPaP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
396 8 Limbs with four degrees of connectivity
Fig. 8.22. Complex limbs RPaPP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
8.1 Limbs with Schönflies motion 397
Fig. 8.22 (cont.)
Fig. 8.23. Complex limbs RPPPa-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
398 8 Limbs with four degrees of connectivity
Fig. 8.24. Complex limbs of types RPRPa (a, b), RRPPa (c), and RRPaP (d) de-fined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.25. Complex limbs of types RPPaPa (a) and RPPaPat (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 399
Fig. 8.26. Complex limbs RPaPaP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed posi-tion
400 8 Limbs with four degrees of connectivity
Fig. 8.26 (cont.)
8.1 Limbs with Schönflies motion 401
Fig. 8.27. Complex limbs RPaPPa-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
402 8 Limbs with four degrees of connectivity
Fig. 8.27 (cont.)
8.1 Limbs with Schönflies motion 403
Fig. 8.28. Complex limbs RPPaPa-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
404 8 Limbs with four degrees of connectivity
Fig. 8.29. Complex limbs PaPPR-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 405
Fig. 8.29 (cont.)
Fig. 8.30. Complex limbs of types PaPRP (a-d) and PaRRP (e) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
406 8 Limbs with four degrees of connectivity
Fig. 8.31. Complex limbs of types PaRPR (a-c), PaRRR (d) and PaPRR (e) de-fined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
Fig. 8.32. Complex limbs PaPRP-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
8.1 Limbs with Schönflies motion 407
Fig. 8.33. Complex limbs of types PaPRP (a-c) and PRPP (d-f) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing in a plane
408 8 Limbs with four degrees of connectivity
Fig. 8.34. Complex limbs of types PaPRP (a) and PaRPP (b, c) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing on a cylinder
8.1 Limbs with Schönflies motion 409
Fig. 8.35. Complex limbs PaPaPR-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the char-acteristic point of the distal link
410 8 Limbs with four degrees of connectivity
Fig. 8.35 (cont.)
8.1 Limbs with Schönflies motion 411
Fig. 8.36. Complex limbs PaPPaR-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the char-acteristic point of the distal link
412 8 Limbs with four degrees of connectivity
Fig. 8.36 (cont.)
8.1 Limbs with Schönflies motion 413
Fig. 8.37. Complex limbs PaPaRP-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the char-acteristic point of the distal link
414 8 Limbs with four degrees of connectivity
Fig. 8.38. Complex limbs of types PaRPaP (a-d), and PaPRPa (e), (f) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing on a cylinder
8.1 Limbs with Schönflies motion 415
Fig. 8.39. Complex limbs PaRPPa-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
416 8 Limbs with four degrees of connectivity
Fig. 8.40. Complex limbs of types PaPaRP (a-c) and PaPRPa (d-h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schön-flies motion
8.1 Limbs with Schönflies motion 417
Fig. 8.41. Complex limbs of types PaRPaP (a-f) and PaPRPa (g-i) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion ly-ing in a plane
418 8 Limbs with four degrees of connectivity
Fig. 8.42. Complex limbs of types PaRPPa (a-e), PaRPaP (f), PaPaRP (g), PaPRPa (h) and PaRPPa (i) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
8.1 Limbs with Schönflies motion 419
Fig. 8.43. Complex limbs RPaPaPa-type defined by defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed posi-tion
420 8 Limbs with four degrees of connectivity
Fig. 8.44. Complex limbs PaPaPaR-type defined by defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 421
Fig. 8.45. Complex limbs of types PaPaRPa (a, b), and PaRPaPa (c-g) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
422 8 Limbs with four degrees of connectivity
Fig. 8.46. Complex limbs PaPaRPa-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
8.1 Limbs with Schönflies motion 423
Fig. 8.47. Complex limbs PaRPaPa-type defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.48. Examples of complex limbs of types PRRbR (a) and PRRbRbR (b) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
424 8 Limbs with four degrees of connectivity
Fig. 8.49. Examples of complex limbs PPn2R-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
Fig. 8.50. Examples of complex limbs PPn3-type defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.2 Limbs with two translational and two rotational motions 425
8.2 Limbs with two translational and two rotational motions
The basis of the operational velocities space of these limbs contains only two planar translational velocities v1, v2 and two angular velocities 1ω , 2ω if characteristic point H is situated on the second rotation axis. The plane of translational velocities has a constant orientation. The rotation axis of
1ω has a fixed direction which may be either parallel or perpendicular to the plane of translations. In all cases, the two rotation axes are orthogonal. If the rotation axes are concurrent, the characteristic point is situated at the intersecting axes. The rotation axis of 2ω lies in the plane of translations when the rotation axis of 1ω is perpendicular to this plane.
Structural solutions for non-redundant limbs with only revolute and prismatic pairs and MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) are presented in Figs. 8.51-8.58.
Solutions for simple limbs are illustrated in Figs. 8.51-8.52 and 8.55. They have prismatic (Fig. 8.51) or revolute (Figs. 8.52 and 8.55) pairs ad-jacent to the fixed base. The complex limbs may contain one (Figs. 8.53, 8.54, 8.56a-d, 8.57a-d and 8.58a-f) or two (Fig. 8.56e-f, 8.57e-f and 8.58g-h) parallelogram loops and have prismatic (Fig. 8.53 and 8.54) or revolute (Fig. 8.56-8.58) pairs adjacent to the fixed base. Solution in Fig. 8.56g contains a parallelogram loop Pa-type connected with an extensible paral-lelogram loop Pat-type.
In the complex limbs, the revolute joints connected to the fixed base may be combined in a parallelogram loop (Figs. 8.57 and 8.58) or remain outside the closed loop (Fig. 8.56).
Each planar parallelogram loop could combine up to three idle mobili-ties as we have seen in the previous sections. The degree of overconstraint and the number of joint parameters that lose their independence in the par-allelogram loops combined in the various types of limbs presented in Figs. 8.51-8.58 are given in Table 7.1. The complex limb in Fig. 8.56 has NG=6 and rG=6.
In the solutions presented in Figs. 8.51b, 8.55d, 8.56c, 8.58b,c,e,f, the number of joints may be reduced by replacing two consecutive revolute and prismatic joints with the same axes/direction (R║P or P║R) by a cylin-drical joint (Figs. 8.59 and 8.60). In the solutions presented in Figs. 8.53b, 8.54b, 8.56a,d,f, 8.57f, 8.58h the number of joints may be reduced by re-placing a sequence of joints Pa⊥R by a parallelogram loop Pass-type with two spherical joints adjacent to the distal link (Figs. 8.61 and 8.62).
426 8 Limbs with four degrees of connectivity
Fig. 8.51. Simple limbs of types PPRR (a, b), and PRPR (c) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction lying in the plane of translations
Fig. 8.52. Simple limbs of types PPRR (a), PRRR (b) and PRPR (c) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction perpendicu-lar to the plane of translations
8.2 Limbs with two translational and two rotational motions 427
Fig. 8.53. Complex limbs PPaRR-type defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction lying in the plane of translations
Fig. 8.54. Complex limbs of types PPaRR (a) and PRPaR (b) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction perpendicu-lar to the plane of translations
428 8 Limbs with four degrees of connectivity
Fig. 8.55. Simple limbs of types RRPR (a), RPRR (b), RRPR (c) and RPPR (d) de-fined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction per-pendicular to the plane of translations
8.2 Limbs with two translational and two rotational motions 429
Fig. 8.56. Complex limbs of types RPPaR (a), RPaPR (b, c), RRPaR (d), RPaPaR (e, f) and RPaPatR (g) defined by MG=SG=4 ,(RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction perpendicular to the plane of translations
430 8 Limbs with four degrees of connectivity
Fig. 8.57. Complex limbs of types PaPRR (a, b), PaRPR (c), PaRRR (d), PaPaRR (e) and PaRPaR (f) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction perpendicular to the plane of translations
8.2 Limbs with two translational and two rotational motions 431
Fig. 8.58. Complex limbs of type PaPRR (a-d), PaRPR (e), (f), and PaPaRR (g, h) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction lying in the plane of translations
432 8 Limbs with four degrees of connectivity
Fig. 8.59. Limbs of types PCR (a) and PaCR (b, c) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction lying in a plane
Fig. 8.60. Limbs of types RPC (a) and RPaC (b) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a fixed direction rotation axis perpendicular to the plane of translations
Fig. 8.61. Complex limbs of type PPassR (a) and PaPassR (b) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction lying in the plane of translations
8.2 Limbs with two translational and two rotational motions 433
Fig. 8.62. Complex limbs of types PRPass (a), RPPass (b), RRPass (c), RPaPass (d) and PaRPass (e) defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and a rotation axis of fixed direction perpendicular to the plane of translations
434 8 Limbs with four degrees of connectivity
Table 8.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=4, (RG)=( 1 2 1 2, , ,v v ω ω ) and TG=0 presented in Figs. 8.51-8.62
Structural parameters No. Limb-type Configuration of the rotation axes m p q rG NG
Figure
1 PPRR R║ 5 4 - - - Fig. 8.51a,b 2 PRPR R║ 5 4 - - - Fig. 8.51c 3 PPRR R ⊥ 5 4 - - - Fig. 8.52a 4 PRRR R ⊥ 5 4 - - - Fig. 8.52b 5 PRPR R ⊥ 5 4 - - - Fig. 8.52c 6 PPaRR R║ 7 7 1 3 3 Fig. 8.53a,b 7 PPaRR R ⊥ 7 7 1 3 3 Fig. 8.54a 8 PRPaR R ⊥ 7 7 1 3 3 Fig. 8.54b 9 RRRR R ⊥ 7 7 1 3 3 Fig. 8.55a 10 RPRR R ⊥ 5 4 - - - Fig. 8.55b 11 RRPR R ⊥ 5 4 - - - Fig. 8.55c 12 RPPR R ⊥ 5 4 - - - Fig. 8.55d 13 RPPaR R ⊥ 7 7 1 3 3 Fig. 8.56a 14 RPaPR R ⊥ 7 7 1 3 3 Fig. 8.56b,c 15 RRPaR R ⊥ 7 7 1 3 3 Fig. 8.56d 16 RPaPaR R ⊥ 9 10 2 6 6 Fig. 8.56e,f 17 RPaPatR R ⊥ 9 10 2 6 6 Fig. 8.56g 18 PaPRR R ⊥ 7 7 1 3 3 Fig. 8.57a,b 19 PaRPR R ⊥ 7 7 1 3 3 Fig. 8.57c 20 PaRRR R ⊥ 7 7 1 3 3 Fig. 8.57d 21 PaPaRR R ⊥ 9 10 2 6 6 Fig. 8.57e 22 PaRPaR R ⊥ 9 10 2 6 6 Fig. 8.57f 23 PaPRR R║ 7 7 1 3 3 Fig. 8.58a-d 24 PaRPR R║ 7 7 1 3 3 Fig. 8.58e,f 25 PaPaRR R║ 9 10 2 6 6 Fig. 8.58g,h 26 PCR R║ 4 3 - - - Fig. 8.59a 27 PaCR R║ 7 7 1 3 3 Fig. 8.59b,c 28 RPC R ⊥ 4 3 - - - Fig. 8.60a 29 RPaC R ⊥ 7 7 1 3 3 Fig. 8.60b 30 PPassR R║ 6 6 1 6 0 Fig. 8.61a
8.3 Limbs with one translational and three rotational motions 435
Table 8.2. (cont.)
31 PaPassR R║ 6 6 1 6 0 Fig. 8.61b 32 PRPass R ⊥ 6 6 1 6 0 Fig. 8.62a 33 RPPass R ⊥ 6 6 1 6 0 Fig. 8.62b 34 RRPass R ⊥ 6 6 1 6 0 Fig. 8.62c 35 RPaPass R ⊥ 8 9 2 9 3 Fig. 8.62d 36 PaRPass R ⊥ 8 9 2 9 3 Fig. 8.62e aSee footnote of Table 6.2 for the nomenclature of structural parameters b R║ the rotation axis of fixed direction lies in the plane of translations, R ⊥ the ro-tation axis of fixed direction is perpendicular to the plane of translations.
The rotation axis of fixed direction is perpendicular to the plane of
translations in the solutions in Figs. 8.51, 8.53, 8.58, 8.59, 8.61 and paral-lel to this plane in Figs. 8.52, 8.54-8.57, 8.60 and 8.62. The structural pa-rameters of the limbs presented in Figs. 8.51-8.62 are systematized in Ta-ble 8.2. In this table, the configuration of the rotation axes is denoted by R║ if the rotation axis of fixed direction lies in the plane of translations and by R ⊥ if this rotation axis is perpendicular to the plane of translations.
We note that the basis of the operational velocity space is defined as (RG)=( 1 2 1 2, , ,v v ω ω ) if the characteristic point H is situated on the last rota-tion axis or at the intersecting axes if the axes are concurrent. If these con-ditions are not fulfilled, the basis of the operational velocity space may contain: (i) two translational and two angular velocities or (ii) one rota-tional and three translational velocities.
8.3 Limbs with one translational and three rotational motions
The basis of the operational space of these limbs contains a unidirectional translational velocity v1 with a fixed direction and three angular velocities
1ω , 2ω and 3ω , when the characteristic point of the distal link is situated in the intersecting axes of the three angular velocities.
Structural solutions for non redundant limbs with only revolute and pris-matic joints and MG=SG=4, (RG)=( 1 1 2 3, , ,v ω ω ω ) are presented in Fig. 8.63. The direction of the translation may be perpendicular (Fig. 8.63a) or paral-lel (Fig. 8.63b,c) to the axis of the first revolute joint. In the solutions presented in Fig. 8.90, the number of joints may be reduced by replacing
436 8 Limbs with four degrees of connectivity
Fig. 8.63. Simple limbs of types PRRR (a) and RPRR (b) defined by MG=SG=4 and (RG)=( 1 1 2 3, , ,v ω ω ω )
Fig. 8.64. Simple limb PS-type defined by MG=SG=4 and (RG)=( 1 1 2 3, , ,v ω ω ω )
three revolute joints with intersecting axes by a spherical joint (Fig. 8.64).The solutions for non-redundant limbs in Figs. 8.63 and 8.64 are de-fined by the following structural parameters: MG=SG=3, (RG)=( 1 1 2, ,v ω ω ), NG=0, TG=0, rG=0. If the characteristic point of the distal link is not situ-ated at the intersecting axes of the revolute joints, the basis of the opera-tional velocity space may contain: (i) one translational and three angular velocities, (ii) two translational and two angular velocities or (iii) one rota-tional and three translational velocities.
8.4 Other limbs with four degrees of connectivity
The limbs presented in this section have multiple bases for the operational velocity space for any position of the characteristic point on the distal link.
8.4 Other limbs with four degrees of connectivity 437
Fig. 8.65. Simple limbs of types PPRR (a, b), PRRP (c, d), RPP (e, f), PRPR (e, f) and PRRR (g-i) defined by MG=SG=4 and multiple bases of the operational veloc-ity space
438 8 Limbs with four degrees of connectivity
Fig. 8.66. Simple limbs of types RPPR (a-c) and RPRR (d-i) defined by MG=SG=4 and multiple bases of the operational velocity space
8.4 Other limbs with four degrees of connectivity 439
Fig. 8.67. Simple limbs of types RRPR (a, b), RRPR (b-e) and RRRR (f-i) defined by MG=SG=4 and multiple bases of the operational velocity space
440 8 Limbs with four degrees of connectivity
Fig. 8.68. Complex limbs of types RRRbR (a), RRRbRbR (b), and RPa2R (c-f) de-fined by MG=SG=4 and multiple bases of the operational velocity space
8.4 Other limbs with four degrees of connectivity 441
Fig. 8.69. Complex limbs of types PRRbR (a), PRRbRbR (b), and PPa2R (c-d) de-fined by MG=SG=4 and multiple bases of the operational velocity space
These bases combine translational and angular velocities. For example, the basis of the operational velocity space of the limb in Fig. 8.65a may be defined by one rotational and three translational velocities ( 1 2 3 i, , ,v v v ω ) or by two translational and two angular velocities ( 1 2 1 i, , ,v v ω ω ), i=2,3. These bases are valid for any position of the characteristic point H on the distal link 5.
A large number of limbs with four degrees of connectivity and multiple bases of the operational velocity space are generated by the evolutionary morphology approach. Due to space limitations, just some examples are presented in Figs. 8.65-8.69. The simple limbs in Figs. 8.65-8.67 are de-fined by the following structural parameters: MG=SG=4, NG=0, TG=0 and rG=0. The complex limbs with one planar loop in Figs. 8.68a,c-f and 8.69a,c,d are defined by MG=SG=4, NG=3, TG=0 and rG=3, and the com-
442 8 Limbs with four degrees of connectivity
plex limbs with two rhombus loops in Figs. 8.68b and 8.69b by MG=SG=4, NG=6, TG=0 and rG=6.
8.5 Redundant limbs with four degrees of connectivity
A large variety of redundant limbs with SG=4 and MG>4 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just two examples are presented in Figs. 8.70 and 8.71.
Figure 8.70 represents a spatial pseudo-parallelogram closed loop with four spherical joints. Six joint parameters lose their independence in this Pa4s-type closed loop. This solution is defined by the following structural parameters: m=4, p=4 q=1, rG=6, SG=4, MG=6 (RG)=( 1 2 1 2, , ,v v ω ω ), NG=0 and TG=2. The parallelogram loop Pa4S-type contains two internal mobili-ties introduced by the rotation of links 2 and 3 along a rotation axis passing through the centre of the two spherical joints adjacent to these links. These two internal mobilities are associated with the two degrees of redundancy that cannot be transmitted to the distal link. The remaining four degrees of mobility are external mobilities associated with the independent motions of distal link 4. This solution has the advantage of reducing limb complexity by using a compact spherical joint as an alternative to the universal joint.
The complex limb in Fig. 8.71 combines a parallelogram loop, a tele-scopic parallelogram loop and a serial chain R║R║R-type. Six joint pa-rameters lose their independence in the two parallelogram loops.
Fig. 8.70. Spatial loop SSSS-type defined by MG=6, SG=4 and (RG)=( 1 2 1 2, , ,v v ω ω )
8.5 Redundant limbs with four degrees of connectivity 443
Fig. 8.71. Example of redundant limb defined by MG=5, SG=4 and (RG)= ( 1 2 3 1, , ,v v v ω )
Solution in Fig. 8.71 is defined by the following structural parameters: m=10, p=11, q=2, rG=6, SG=4, MG=5, (RG)=( 1 2 3 1, , ,v v v ω ), NG=6 and TG=1. This degree of redundancy gives an internal mobility in the limb as-sociated with the translation in the telescopic parallelogram loop. The re-maining three degrees of mobility are external mobilities associated with the independent motions of distal link 10. The internal mobility in this limb is useful for motion decoupling in parallel robots with Schönflies mo-tions as we will present in Part 2 of this work.
9 Limbs with five degrees of connectivity
Limbs with five degrees of connectivity (SG=5) are used in parallel robots F with the connectivity of moving platform SF≤ 5. The base of the opera-tional space of these limbs contains five independent velocities. Non-redundant solutions generated by the evolutionary morphology approach are presented in the first three sections of this chapter. Some examples of redundant solutions are presented in the last section. The first three sec-tions present solutions of limbs defined by the following: three transla-tional and two rotational motions and (ii) two translational and three rota-tional motions, (iii) multiple bases of the operational velocity space.
9.1 Limbs with two rotational and three translational motions
These limbs give three translational velocities v1, v2, v3 and two angular ve-locities 1ω and 2ω in the basis of the operational velocities space (RG)=( 1 2 3 1 2, , , ,v v v ω ω ). We denote by 1∆ and 2∆ the axes of the rotations. The rotation axis 1∆ has a fixed direction and the rotation axis 2∆ lies in a plane perpendicular to 1∆ . The characteristic point H may have any posi-tion on the distal link.
The simplest structural solutions of non redundant simple and complex limbs with only revolute and prismatic pairs and MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) are presented in Figs. 9.1-9.130. These solutions have not idle mobilities.
As in the previous solutions with less than five degrees of connectivity, the number of joints may be reduced by replacing as follows:
a) two consecutive revolute and prismatic joints with the same axes/direction (R P or P R) by a cylindrical joint such as in Fig. 6.29,
b) a sequence of joints R⊥Pa or Pa⊥R by a parallelogram loop Pass-type with two spherical joints adjacent to the distal link such as in Fig. 6.30,
446 9 Limbs with five degrees of connectivity
c) a sequence of joints P Pa or Pa P by a parallelogram loop Pacc-type with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop such as in Fig. 6.13.
The complex limbs may contain up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could contain up to three idle mobilities that may be two orthogonal rotations and one translation.
The degree of overconstraint and the number of joint parameters that lose their independence in the parallelogram loops that may be combined in the various types of limbs defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and TG=0 are systematized in Table 6.2. We recall that a parallelogram loop Pacc-type has MG=2, NG=2 and rG=4.
We note that, in all cases, the rotation axis of 1ω has an unchanging ori-entation. This rotation axis is denoted by 1∆ . The rotation axis of 2ω is perpendicular on the rotation axis of 1ω . This configuration of the rotation axes is valid for all solutions presented in Figs. 9.1-9.130. This general configuration is denoted by D. The rotation axis 1∆ may have the follow-ing particular positions: (i) fixed position; the configuration of the rotation axes is denoted by A, (ii) lying in a plane with unchanging orientation - configuration denoted by B, and (ii) lying on a cylinder with unchanging position, orientation and radius – configuration denoted by C. The rotation axis of 2ω may have the following particular positions: (i) passing by the characteristic point of the distal link; in this case index 1 is added to the letters A, B, C or D defining the configuration of the rotation axes, (ii) in-tersecting 1∆ ; in this case index 2 is used to define the configuration of the rotation axes, (iii) intersecting 1∆ and passing by the characteristic point of the distal link - index 12 is used to define the configuration of the rotation axes and (iii) intersecting 1∆ and both rotation axes passing by the charac-teristic point of the distal link – index 123 is used to define the configura-tion of the rotation axes.
The various solutions illustrated in Figs. 9.1-9.130 may be systematized by taking into account various criteria: a) limb type a1) simple limbs composed by a simple open kinematic chain (Figs. 9.1-9.11)
a2) complex limbs containing at least one closed loop (Figs. 9.12- 9.130) b) configuration of the rotation axes
9.1 Limbs with two rotational and three translational motions 447
b1) configuration A (Figs. 9.7a,f,g; 9.12a-d; 9.32a,b; 9.44a-k; 9.45a-l; 9.46a-i; 9.47a-i; 9.48a-d; 9.65a-p; 9.66a-p; 9.67a- q; 9.68a-r; 9.69a-f; 9.114a-p; 9.115a-l; 9.130a,b)
b2) configuration A1 (Figs.9.1a-c, 9.7e; 9.8a-f; 9.13a-f; 9.14a-f; 9.33a,b; 9.49a-l; 9.50a-p; 9.51a-r; 9.52a-e; 9.53a-f; 9.70a- x; 9.71a-x; 9.72a-f; 9.73a-f; 9.116a-r; 9.128a,b; 9.129a,b) b3) configuration A2 (Figs.9.1d, 9.7b-d,h,i; 9.8g-i; 9.15a-f; 9.16a-f; 9.34a,b; 9.54a-l; 9.55a-d; 9.56a-j; 9.57a-j; 9.58a-k; 9.59a- f; 9.60a-d; 9.74a-l; 9.75a-l; 9.76a-f; 9.77a-f; 9.78a-f; 9.79a-d; 9.117a-i; )
b4) configuration A12 (Figs.) b5) configuration A123 (Figs.) b6) configuration B (Figs.9.2a, 9.17a-h; 9.35a-d; 9.80a; 9.93a-f;
9.118a-k;) b7) configuration B1 (Figs. 9.2e,f,i; 9.3b-f; 9.18a-f; 9.19a-q; 9.35e-
h; 9.38a,b; 9.40a,b; 9.42a,b; 9.80b-f; 9.81a-e; 9.94a-i; 9.95a-i; 9.96a-f; 9.97a-e; 9.119a-n; )
b8) configuration B2 (Figs.9.2b,c; 9.3a; 9.22a-l; 9.23a-f; 9.24a-m; 9.35i-l; 9.82a-e; 9.83a-e; 9.98a-l; 9.99a-l; 9.100a-q; 9.120a-k; )
b9) configuration B12 (Figs.9.2d,g,h; 9.20a-f; 9.21a-f; 9.84a-d; 9.85a-d; 9.101a-e; )
b10) configuration B123 (Figs.) b11) configuration C (Figs. 9.61a,b; 9.86a; 9.102a-e; 9.121a-e; ) b12) configuration C1 (Figs. 9.9a,b; 9.10a; 9.25a-c; 9.36a,b; 9.61c-f;
9.62a-d; 9.86b-f; 9.103a-f; 9.104a-f; 9.105a-c; 9.122a-f; ) b13) configuration C2 (Figs. 9.9e,f,i; 9.10d; 9.26a,c,d; 9.36c,d;
9.63a-j; 9.87a-f; 9.106a-l; 9.107a-e; 9.18a-d; 9.123a-f; ) b14) configuration C12 (Figs. 9.9c,d,g,h; 9.10b,c; 9.26b,e; 9.88a-e;
9.109a-e; ) b15) configuration C123 (Figs. 9.6a,b; ) b16) configuration D (Figs. 9.4a-g; 9.124a,b; ) b17) configuration D1 (Figs.9.5a,b; 9.27a-c; 9.37a-d; 9.43a,b;
9.89a,b; 9.110a-i; 9.124c-k; ) b18) configuration D2 (Figs. 9.29a-d; 9.37e-h; 9.89c-k; 9.111a-k;
9.125a-j; ) b19) configuration D12 (Figs. 9.28a-c; 9.90a-d; 9.112a-f; 9.113a-c;
9.126a-f; )
448 9 Limbs with five degrees of connectivity
b20) configuration D123 (Figs. 9.11a-h; 9.30a-i; 9.31a-k; 9.37i-k; 9.39a,b; 9.41a,b; 9.64a,b; 9.91a-i; 9.92a-f; 9.127a-f; ) c) type of kinematic joint connected to the fixed base c1) prismatic joint (Figs. 9.1-9.6; 9.12-9.43; ) c2) revolute joint (Figs. 9.7-9.11; 9.44-9.130) d) type of closed loops combined in the limb
d1) parallelogram loops (Figs. 9.12-9.37; 9.44-9.127) d2) extensible parallelogram loop (Fig. 9.64b) d3) rhombus loops (Fig. 9.128) d4) other planar loops (Figs. 9.40-9.43, 9.129, 9.130)
e) number of closed loops combined in the limb (q≥ 1) e1) one closed loop (Figs. 9.12-9.31, 9.38a, 9.39a, 9.40-9.63, 9.89- 9.92, 9.128a, 9.129, 9.130) e2) two closed loops (Figs. 9.32-9.37, 9.38b, 9.39b, 9.64-9.79, 9.93-9.113, 9.128b) e3) three closed loops (Figs. 9.114-9.127) f) number of parameters that lose their independence in the closed loops combined in the limb (3≤ rG≤ 12) f1) rG=3 (Figs. 9.12-9.31, 9.38a, 9.39a, 9.40-9.63, 9.80-9.92, 9.128a, 9.129, 9.130) f2) rG=6 (Figs. 9.32-9.37, 9.38b, 9.39b, 9.64-9.79, 9.93-9.113, 9.128b) f3) rG=9 (Figs. 9.114-9.127) g) degree of mobility of a planar loops combined in the complex limbs g1) one degree of mobility (Figs. 9.12-9.39, 9.44-9.128) g2) two degrees of mobility (Figs. 9.40, 9.41, 9.129) g3) three degrees of mobility (Figs. 9.42, 9.43, 9.64b, 9.130).
Various solutions of planar bimobile closed loops Pn2-type and Pn3-type may be used as presented in chapter 6. The revolute joints combined in the closed loops Pn2 and Pn3 have parallel axes and the directions of the prismatic joints are contained in a plane perpendicular to the revolute axes. Three joint parameters lose their independence in each planar closed loop Rb-, Pn2- or Pn3-type. A complex limb may contain one, two or more rhombus (Rb) loops. In Figs. 9.38 and 9.128 just two examples with one and two rhombus loops are presented.
Three joint parameters lose their independence in each planar closed loop Rb-, Pn2 or Pn3-type.
The structural parameters of the solutions presented in Figs. 9.1-9.130 are systematized in Tables 9.1-9.4.
9.1 Limbs with two rotational and three translational motions 449
Table 9.1. Joint arrangement and structural parametersa of the limbs defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), TG=0 and the rotation axes in the configura-tions of type A
Structural parameters No. Limb-type Configuration of the rotation axes m p q rG NG
Examples
1 RPRPP A 6 5 - - - Fig. 9.7a,f,g 2 PRPaRP A 8 8 1 3 3 Fig. 9.12a-c 3 PRPRPa A 8 8 1 3 3 Fig. 9.12d 4 PRPaRPa A 10 11 2 6 6 Fig. 9.32a-b 5 RPPRPa A 8 8 1 3 3 Fig. 9.44a-h 6 RPRPPa A 8 8 1 3 3 Fig. 9.44i-k 7 RPPaRP A 8 8 1 3 3 Fig. 9.45a-l 8 RPaRPP A 8 8 1 3 3 Fig. 9.46a-i 9 RPaPRP A 8 8 1 3 3 Fig. 9.47a-i 10 RPRPaP A 8 8 1 3 3 Fig. 9.48a-d 11 RPaRPaP A 10 11 2 6 6 Fig. 9.65a-p 12 RPaPaRP A 10 11 2 6 6 Fig. 9.66a-p 13 RPaRPPa A 10 11 2 6 6 Fig. 9.67a-q 14 RPaPRPa A 10 11 2 6 6 Fig. 9.68a-r 15 RPPaRPa A 10 11 2 6 6 Fig. 9.69a-f 16 RPaPaRPa A 12 14 3 9 9 Fig. 9.114a-p 17 RPaRPaPa A 12 14 3 9 9 Fig. 9.115a-l 18 RPPn3 A 8 8 1 3 3 Fig. 9.130a,b 19 PRPRP A1 6 5 - - - Fig. 9.1a,b 20 PRPPR A1 6 5 - - - Fig. 9.1c 21 RPPRP A1 6 5 - - - Fig. 9.7e 22 RPPPR A1 6 5 - - - Fig. 9.8a-f 23 PRPPaR A1 8 8 1 3 3 Fig. 9.13a-d 24 PRPaRP A1 8 8 1 3 3 Fig. 9.13e,f 25 PRPaPR A1 8 8 1 3 3 Fig. 9.14a-f 26 PRPaPaR A1 10 11 2 6 6 Fig. 9.33a,b 27 RPPPaR A1 8 8 1 3 3 Fig. 9.49a-l 28 RPPaPR A1 8 8 1 3 3 Fig. 9.50a-p 29 RPaPPR A1 8 8 1 3 3 Fig. 9.51a-r 30 RPaPRP A1 8 8 1 3 3 Fig. 9.52a-e 31 RPPaRP A1 8 8 1 3 3 Fig. 9.53a,b 32 RPaRRR A1 8 8 1 3 3 Fig. 9.53c,d 33 RPaRPR A1 8 8 1 3 3 Fig. 9.53e,f
450 9 Limbs with five degrees of connectivity
Table 9.1. (cont.)
34 RPaPaPR A1 10 11 2 6 6 Fig. 9.70a-x 35 RPaPPaR A1 10 11 2 6 6 Fig. 9.71a-x 36 RPaPaRP A1 10 11 2 6 6 Fig. 9.72a-f 37 RPPaPaR A1 10 11 2 6 6 Fig. 9.73a-f 38 RPaPaPaR A1 12 14 3 9 9 Fig. 9.116a-r 39 RPRRbR A1 8 8 1 3 3 Fig. 9.128a 40 RPRRbRbR A1 10 11 2 6 6 Fig. 9.128b 41 RPPn2R A1 8 8 1 3 3 Fig. 9.129a,b 42 PRPPR A2 6 5 - - - Fig. 9.1d 43 RPRPP A2 6 5 - - - Fig. 9.7b-d 44 RPPRP A2 6 5 - - - Fig. 9.7h,i 45 RRPPP A2 6 5 - - - Fig. 9.8g-i 46 PRRPPa A2 8 8 1 3 3 Fig. 9.15a-d 47 PRPRPa A2 8 8 1 3 3 Fig. 9.15e 48 PRPaRP A2 8 8 1 3 3 Fig. 9.15f 49 PRRPaP A2 8 8 1 3 3 Fig. 9.16a-f 50 PRRPaPa A2 10 11 2 6 6 Fig. 9.34a,b 51 RRPPPa A2 8 8 1 3 3 Fig. 9.54a-g 52 RPRPPa A2 8 8 1 3 3 Fig. 9.54h-l 53 RPPRPa A2 8 8 1 3 3 Fig. 9.55a-d 54 RRPPaP A2 8 8 1 3 3 Fig. 9.56a-j 55 RPRPaP A2 8 8 1 3 3 Fig. 9.57a-j 56 RRPaPP A2 8 8 1 3 3 Fig. 9.58a-k 57 RPaRPP A2 8 8 1 3 3 Fig. 9.59a-f 58 RPaPRP A2 8 8 1 3 3 Fig. 9.60a-c 59 RPPaRP A2 8 8 1 3 3 Fig. 9.60d 60 RRPaPaP A2 10 11 2 6 6 Fig. 9.74a-l 61 RRPaPPa A2 10 11 2 6 6 Fig. 9.75a-l 62 RRPPaPa A2 10 11 2 6 6 Fig. 9.76a-c 63 RPRPaPa A2 10 11 2 6 6 Fig. 9.76d-f 64 RPaRPaP A2 10 11 2 6 6 Fig. 9.77a-f 65 RPaRPPa A2 10 11 2 6 6 Fig. 9.78a-f 66 RPaPRPa A2 10 11 2 6 6 Fig. 9.79a-d 67 RRPaPaPa A2 12 14 3 9 9 Fig. 9.117a-h 68 RPaRPaPa A2 12 14 3 9 9 Fig. 9.117i-l aSee footnote of Table 6.2 for the nomenclature of structural parameters
9.1 Limbs with two rotational and three translational motions 451
Table 9.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), TG=0 and the rotation axes in the configura-tions of type B
Structural parameters No. Limb-type Configuration of the rotation axes m p q rG NG
Examples
1 PRPRP B 6 5 - - - Fig. 9.2a 2 PRPaRP B 8 8 1 3 3 Fig. 9.17a-f 3 PRPRPa B 8 8 1 3 3 Fig. 9.17g,h 4 PRPaRPa B 10 11 2 6 6 Fig. 9.35a-d 5 PaRPRP B 8 8 1 3 3 Fig. 9.80a,b 6 PaRPaRP B 10 11 2 6 6 Fig. 9.93a,b 7 PaRPRPa B 10 11 2 6 6 Fig. 9.93c-e 8 PaRPaRP B 10 11 2 6 6 Fig. 9.93f 9 PaPaRRPa B 12 14 3 9 9 Fig. 9.118a 10 PaRRPaPa B 12 14 3 9 9 Fig. 9.118b,c 11 PaRPaRPa B 12 14 3 9 9 Fig. 9.118d-k 12 PPRPR B1 6 5 - - - Fig. 9.2e,f 13 PRPRR B1 6 5 - - - Fig. 9.2i 14 PRPPR B1 6 5 - - - Fig. 9.3b-d 15 PRRPR B1 6 5 - - - Fig. 9.3e 16 PRRRR B1 6 5 - - - Fig. 9.3f 17 PPRPaR B1 8 8 1 3 3 Fig. 9.18a-f 18 PRPPaR B1 8 8 1 3 3 Fig. 9.19a-g 19 PRPaPR B1 8 8 1 3 3 Fig. 9.19h-n 20 PPaRPR B1 8 8 1 3 3 Fig. 9.19o,p 21 PRPaRP B1 8 8 1 3 3 Fig. 9.19q 22 PRPaPaR B1 10 11 2 6 6 Fig. 9.35e-h 23 PRRRbR B1 8 8 1 3 3 Fig. 9.38a 24 PRRRbRbR B1 10 11 2 6 6 Fig. 9.38b 25 PRPn2R B1 8 8 1 3 3 Fig. 9.40a,b 26 PRPn3 B1 8 8 1 3 3 Fig. 9.42a,b 27 PaRPPR B1 8 8 1 3 3 Fig. 9.80c-f 28 PaPRPR B1 8 8 1 3 3 Fig. 9.81a-c 29 PaRRPR B1 8 8 1 3 3 Fig. 9.81d 30 PaRRRR B1 8 8 1 3 3 Fig. 9.81e 31 PaRPaPR B1 10 11 2 6 6 Fig. 9.94a-l 32 PaRPPaR B1 10 11 2 6 6 Fig. 9.95a-i 33 PaRPaRP B1 10 11 2 6 6 Fig. 9.95j-l
452 9 Limbs with five degrees of connectivity
Table 9.2. (cont.)
34 PaPaRRP B1 10 11 2 6 6 Fig. 9.96a,b 35 PaPaRPR B1 10 11 2 6 6 Fig. 9.96c-f 36 PaPRPaR B1 10 11 2 6 6 Fig. 9.97a-e 37 PaPaRPaR B1 12 14 3 9 9 Fig. 9.119a-d 38 PaRPaPaR B1 12 14 3 9 9 Fig. 9.119e-n 39 PRPRP B2 6 5 - - - Fig. 9.2b,c 40 PPRRPa B2 8 8 1 3 3 Fig. 9.22a-g 41 PRRPPa B2 8 8 1 3 3 Fig. 9.22h-l 42 PRPRPa B2 8 8 1 3 3 Fig. 9.23a-f 43 PRRPaP B2 8 8 1 3 3 Fig. 9.24a-j 44 PRPaRP B2 8 8 1 3 3 Fig. 9.24k-m 45 PaRRPaP B2 10 11 2 6 6 Fig. 9.98a-l 46 PaRRPPa B2 10 11 2 6 6 Fig. 9.99a-h 47 PaRPaRP B2 10 11 2 6 6 Fig. 9.99i-l 48 PaRPRPa B2 10 11 2 6 6 Fig. 9.100a-h 49 PaPRRPa B2 10 11 2 6 6 Fig. 9.100i-p 50 PaPaRRP B2 10 11 2 6 6 Fig. 9.100q 51 PaPaRRPa B2 12 14 3 9 9 Fig. 9.120a-c 52 PaRRPaPa B2 12 14 3 9 9 Fig. 9.120d-i 53 PaRPaRPa B2 12 14 3 9 9 Fig. 9.120j,k 54 PRPRP B12 6 5 - - - Fig. 9.2d,g,h 55 PRPaPR B12 8 8 1 3 3 Fig. 9.20a-d 56 PPRPaR B12 8 8 1 3 3 Fig. 9.20e,f 57 PRPaRP B12 8 8 1 3 3 Fig. 9.21a 58 PRPPaR B12 8 8 1 3 3 Fig. 9.21b 59 PPaRRP B12 8 8 1 3 3 Fig. 9.21c,d 60 PPaRPR B12 8 8 1 3 3 Fig. 9.21e,f 61 PaPRPR B12 8 8 1 3 3 Fig. 9.84a-c 62 PaRPRP B12 8 8 1 3 3 Fig. 9.84d 63 PaRPPR B12 8 8 1 3 3 Fig. 9.85a,b 64 PaPRRP B12 8 8 1 3 3 Fig. 9.85c,d 65 PaRPaPR B12 10 11 2 6 6 Fig. 9.101a 66 PaRPPaR B12 10 11 2 6 6 Fig. 9.101b,c 67 PaPRPaR B12 10 11 2 6 6 Fig. 9.101d,e aSee footnote of Table 6.2 for the nomenclature of structural parameters
9.1 Limbs with two rotational and three translational motions 453
Table 9.3. Joint arrangement and structural parametersa of the limbs defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), TG=0 and the rotation axes in the configura-tions of type C
Structural parameters No. Limb-type Configuration of the rotation axes m p q rG NG
Examples
1 RRPaRP C 8 8 1 3 3 Fig. 9.61a,b 2 PaRPRP C 8 8 1 3 3 Fig. 9.86a 3 PaRPaRP C 10 11 2 6 6 Fig. 9.102a-c 4 PaRPRPa C 10 11 2 6 6 Fig. 9.102d,e 5 PaRPaRPa C 12 14 3 9 9 Fig. 9.121a-e 6 RPRPR C1 6 5 - - - Fig. 9.9a,b 7 RRPPR C1 6 5 - - Fig. 9.10a 8 PRRPaR C1 8 8 1 3 3 Fig. 9.25a,b 9 PPaRPR C1 8 8 1 3 3 Fig. 9.25c 10 PPaRPaR C1 10 11 2 6 6 Fig. 9.36a,b 11 RPRPaR C1 8 8 1 3 3 Fig. 9.61c-f 12 RRPPaR C1 8 8 1 3 3 Fig. 9.62a,b 13 RRPaPPR C1 8 8 1 3 3 Fig. 9.62c,d 14 PaRPPR C1 8 8 1 3 3 Fig. 9.86b,c 15 PaPRPR C1 8 8 1 3 3 Fig. 9.86d 16 PaRRPR C1 8 8 1 3 3 Fig. 9.86e 17 PaRRRR C1 8 8 1 3 3 Fig. 9.86f 18 PaRPaPR C1 10 11 2 6 6 Fig. 9.103a-f 19 PaRPPaR C1 10 11 2 6 6 Fig. 9.104a-e 20 PaRPaRP C1 10 11 2 6 6 Fig. 9.104f 21 PaPRPaR C1 10 11 2 6 6 Fig. 9.105a-c 22 PaRPaPaR C1 12 14 3 9 9 Fig. 9.122a-f 23 RPRRP C2 6 5 - - - Fig. 9.9e,f 24 RRRPP C2 6 5 - - - Fig. 9.9i 25 RRPRP C2 6 5 - - Fig. 9.10d 26 PPaRRP C2 8 8 1 3 3 Fig. 9.26a 27 PRRRPa C2 8 8 1 3 3 Fig. 9.26c,d 28 PPaRRPa C2 10 11 2 6 6 Fig. 9.36c,d 29 RPRRPa C2 8 8 1 3 3 Fig. 9.63a-d 30 RRRPPa C2 8 8 1 3 3 Fig. 9.63e,f 31 RRPRPa C2 8 8 1 3 3 Fig. 9.63g,h 32 RRRPaP C2 8 8 1 3 3 Fig. 9.63i,j 33 PaPRRP C2 8 8 1 3 3 Fig. 9.87a
454 9 Limbs with five degrees of connectivity
Table 9.3. (cont.)
34 PaRRPP C2 8 8 1 3 3 Fig. 9.87b-d 35 PaRPRP C2 8 8 1 3 3 Fig. 9.87e,f 36 PaRRPaP C2 10 11 2 6 6 Fig. 9.106a-g 37 PaRRPPa C2 10 11 2 6 6 Fig. 9.106h-l 38 PaPRRPa C2 10 11 2 6 6 Fig. 9.107a-c 39 PaRPaRP C2 10 11 2 6 6 Fig. 9.107d,e 40 PaRPRPa C2 10 11 2 6 6 Fig. 9.108a-d 41 PaRRPaPa C2 12 14 3 9 9 Fig. 9.123a-f 42 RPRPR C12 6 5 - - - Fig. 9.9c,d 43 RPRRP C12 6 5 - - - Fig. 9.9g,h 44 RRPPR C12 6 5 - - Fig. 9.10b 45 RRPRP C12 6 5 - - Fig. 9.10c 46 PPaRRP C12 8 8 1 3 3 Fig. 9.26b 47 PPaRPR C12 8 8 1 3 3 Fig. 9.26e 48 PaPRRP C12 8 8 1 3 3 Fig. 9.88a 49 PaPRPR C12 8 8 1 3 3 Fig. 9.88b 50 PaRPPR C12 8 8 1 3 3 Fig. 9.88c,d 51 PaRPRP C12 8 8 1 3 3 Fig. 9.88e 52 PaRPaPR C12 10 11 2 6 6 Fig. 9.109a,b 53 PaRPaRP C12 10 11 2 6 6 Fig. 9.109c 54 PaPRPaR C12 10 11 2 6 6 Fig. 9.109d 55 PaRPPaR C12 10 11 2 6 6 Fig. 9.109e 56 PRRRP C123 6 5 - - - Fig. 9.6a 57 PRRPR C123 6 5 - - - Fig. 9.6b aSee footnote of Table 6.2 for the nomenclature of structural parameters
9.1 Limbs with two rotational and three translational motions 455
Table 9.4. Joint arrangement and structural parametersa of the limbs defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), TG=0 and the rotation axes in the configura-tions of type D
Structural parameters No. Limb-type Configuration of the rotation axes m p q rG NG
Examples
1 PaPaRRPa D 12 14 3 9 9 Fig. 9.124a,b 2 PPRRR D1 6 5 - - - Fig. 9.5a 3 PRRRR D1 6 5 - - - Fig. 9.5b 4 PPRPaR D1 8 8 1 3 3 Fig. 9.27a 5 PPaRPR D1 8 8 1 3 3 Fig. 9.27b,c 6 PPaRPaR D1 10 11 2 6 6 Fig. 9.37a-d 7 PPn3R D1 8 8 1 3 3 Fig. 9.43a,b 8 PaPRPR D1 8 8 1 3 3 Fig. 9.89a,b 9 PaPaRPR D1 10 11 2 6 6 Fig. 9.110a,b 10 PaPRPaR D1 10 11 2 6 6 Fig. 9.110c-i 11 PaPaRPaR D1 12 14 3 9 9 Fig. 9.124c-k 12 PPaRRP D2 8 8 1 3 3 Fig. 9.29a-c 13 PPRRPa D2 8 8 1 3 3 Fig. 9.29d 14 PPaRRPa D2 10 11 2 6 6 Fig. 9.37e-h 15 PaRRRP D2 8 8 1 3 3 Fig. 9.89c 16 PaPRRP D2 8 8 1 3 3 Fig. 9.89d-k 17 PaPaRRP D2 10 11 2 6 6 Fig. 9.111a-c 18 PaPRRPa D2 10 11 2 6 6 Fig. 9.111d-k 19 PaPaRRPa D2 12 14 3 9 9 Fig. 9.125a-j 20 PPaRRP D12 8 8 1 3 3 Fig. 9.28a 21 PPaRPR D12 8 8 1 3 3 Fig. 9.28b 22 PPRPaR D12 8 8 1 3 3 Fig. 9.28c 23 PaPRRP D12 8 8 1 3 3 Fig. 9.90a,b 24 PaPRPR D12 8 8 1 3 3 Fig. 9.90c,d 25 PaPaRRP D12 10 11 2 6 6 Fig. 9.112a-c 26 PaPRPaR D12 10 11 2 6 6 Fig.9. 112d-f 27 PaPaRPR D12 10 11 2 6 6 Fig.9 113a-c 28 PaPaRPaR D12 12 14 3 9 9 Fig. 9.126a-c 29 PaPaPaRR D12 12 14 3 9 9 Fig. 9.126d-f 30 PPPRR D123 6 5 - - - Fig. 9.4a-c 31 PPRPR D123 6 5 - - - Fig. 9.4d 32 PPRRR D123 6 5 - - - Fig. 9.4e 33 PRPRR D123 6 5 - - - Fig. 9.4f
456 9 Limbs with five degrees of connectivity
Table 9.4. (cont.)
34 PRRRR D123 6 5 - - - Fig. 9.4g 35 RPPRR D123 6 5 - - Fig. 9.11a,b 36 RPRPR D123 6 5 - - Fig. 9.11c 37 RPRRR D123 6 5 - - Fig. 9.11d,e 38 RRPRR D123 6 5 - - Fig. 9.11f,g 39 RRRPR D123 6 5 - - Fig. 9.11h 40 PPaPRR D123 8 8 1 3 3 Fig. 9.30a-i 41 PPPaRR D123 8 8 1 3 3 Fig. 9.31a-f 42 PRPaRR D123 8 8 1 3 3 Fig. 9.31g 43 PPaRPR D123 8 8 1 3 3 Fig. 9.31h-j 44 PPaRRR D123 10 11 2 6 6 Fig. 9.31k 45 PPaPaRR D123 10 11 2 6 6 Fig. 9.37i-k 46 PRRbRR D123 8 8 1 3 3 Fig. 9.39a 47 PRRbRbRR D123 10 11 2 6 6 Fig. 9.39b 48 PPn2RR D123 8 8 1 3 3 Fig. 9.41a,b 49 RPPaPaR D123 10 11 2 6 6 Fig. 9.64a 50 RPPaPatR D123 10 11 2 6 6 Fig. 9.64b 51 PaPPRR D123 8 8 1 3 3 Fig. 9.91a-i 52 PaRPRR D123 8 8 1 3 3 Fig. 9.91j,k 53 PaRRPR D123 8 8 1 3 3 Fig. 9.91l 54 PaPRPR D123 8 8 1 3 3 Fig. 9.92a-d 55 PaPRRR D123 8 8 1 3 3 Fig. 9.92e 56 PaRRRR D123 8 8 1 3 3 Fig. 9.92f 57 PaPaPaRR D123 12 14 3 9 9 Fig. 9.127a-f aSee footnote of Table 6.2 for the nomenclature of structural parameters
9.1 Limbs with two rotational and three translational motions 457
Fig. 9.1. Simple limbs of types PRPRP (a, b), and PRPPR (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types A1 (a-c) and A2 (d)
458 9 Limbs with five degrees of connectivity
Fig. 9.2. Simple limbs of types PRPRP (a-d), PPRPR (e-h) and PRPRR (i), de-fined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types B (a), B1 (e, f, i), B2 (b, c) and B12 (d, g, h)
9.1 Limbs with two rotational and three translational motions 459
Fig. 9.3. Simple limbs of types PRPPR (a-d), PRRPR (e) and PRRRR (f), defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types B1 (b-f) and B2 (a)
460 9 Limbs with five degrees of connectivity
Fig. 9.4. Simple limbs of types PPPRR (a-c), PPRPR (d), PPRRR (e), PRPRR (f) and PRRRR (g) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in D123-type configuration
9.1 Limbs with two rotational and three translational motions 461
Fig. 9.5. Simple limbs of types PPRRR (a) and PRRRR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in D1-type configuration
Fig. 9.6. Simple limbs of types PRRRP (a) and PRRPR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axis in C123-type configuration
462 9 Limbs with five degrees of connectivity
Fig. 9.7. Simple limbs of types RPRPP (a-d) and RPPRP (e-i) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types A (a, f, g), A1 (e), and A2 (b-d, h, i)
9.1 Limbs with two rotational and three translational motions 463
Fig. 9.8. Simple limbs of types RPPPR (a-f) and RRPPP (g-i) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types A1 (a-f), and A2 (g-i)
464 9 Limbs with five degrees of connectivity
Fig. 9.9. Simple limbs of types RPRPR (a-d), RPRRP (e-h) and RRRPP (i) de-fined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types C1(a, b), C2 (e, f, i), and C12 (c, d, g, h)
9.1 Limbs with two rotational and three translational motions 465
Fig. 9.10. Simple limbs of types RRPPR (a, b) and RRPRP (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types C1(a), C2 (d), and C12 (b, c)
466 9 Limbs with five degrees of connectivity
Fig. 9.11. Simple limbs of types RPPRR (a, b), RPRPR (c), RPRRR (d, e), RRPRR (f, g), RRRPR (h) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
9.1 Limbs with two rotational and three translational motions 467
Fig. 9.12. Complex limbs of types PRPaRP (a-c) and PRPRPa (d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
468 9 Limbs with five degrees of connectivity
Fig. 9.13. Complex limbs of types PRPPaR (a-d) and PRPaRP (e, f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configura-tion
9.1 Limbs with two rotational and three translational motions 469
Fig. 9.14. Complex limbs PRPaPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
470 9 Limbs with five degrees of connectivity
Fig. 9.15. Complex limbs of types PRRPPa (a-d), PRPRPa (e) and PRPaRP (f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2
configuration
9.1 Limbs with two rotational and three translational motions 471
Fig. 9.16. Complex limbs PRRPaP defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
472 9 Limbs with five degrees of connectivity
Fig. 9.17. Complex limbs of types PRPaRP (a-f) and PRPRPa (g, h) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B configuration
9.1 Limbs with two rotational and three translational motions 473
Fig. 9.18. Complex limbs PPRPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
474 9 Limbs with five degrees of connectivity
Fig. 9.19. Complex limbs of types PRPPaR (a-g), PRPaPR (h-n), PPaRPR (o, p) and PRPaRP (q) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
9.1 Limbs with two rotational and three translational motions 475
Fig. 9.19 (cont.)
476 9 Limbs with five degrees of connectivity
Fig. 9.19 (cont.)
9.1 Limbs with two rotational and three translational motions 477
Fig. 9.20. Complex limbs of types PRPaPR (a-d) and PPRPaR (e, f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B12 configura-tion
478 9 Limbs with five degrees of connectivity
Fig. 9.21. Complex limbs of types PRPaRP (a), PRPPaR (b), PPaRRP (c, d) and PPaRPR (e, f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B12 configuration
9.1 Limbs with two rotational and three translational motions 479
Fig. 9.22. Complex limbs of types PPRRPa (a-g) and PRRPPa (h-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
480 9 Limbs with five degrees of connectivity
Fig. 9.22 (cont.)
9.1 Limbs with two rotational and three translational motions 481
Fig. 9.23. Complex limbs PRPRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
482 9 Limbs with five degrees of connectivity
Fig. 9.24. Complex limbs of types PRRPaP (a-j) and PRPaRP (k-m) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
9.1 Limbs with two rotational and three translational motions 483
Fig. 9.24 (cont.)
484 9 Limbs with five degrees of connectivity
Fig. 9.24 (cont.)
Fig. 9.25. Complex limbs of types PRRPaR (a, b) and PPaRPR (c) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
9.1 Limbs with two rotational and three translational motions 485
Fig. 9.26. Complex limbs of types PPaRRP (a, b), PRRRPa (c ,d) and PPaRPR (e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configu-rations of types C2 (a, c, d) and C12 (b, e)
486 9 Limbs with five degrees of connectivity
Fig. 9.27. Complex limbs of types PPRPaR (a) and PPaRPR (b ,c) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D1 configuration
Fig. 9.28. Complex limbs of types PPaRRP (a), PPaRPR (b) and PPRPaR (c) de-fined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D12 configuration
9.1 Limbs with two rotational and three translational motions 487
Fig. 9.29. Complex limbs of types PPaRRP (a-c) and PPRRPa (d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D2 configura-tion
488 9 Limbs with five degrees of connectivity
Fig. 9.30. Complex limbs PPaPRR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
9.1 Limbs with two rotational and three translational motions 489
Fig. 9.31. Complex limbs of types PPPaRR (a-f), PRPaRR (g), PPaRPR (h-j) and PPaRRR (k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
490 9 Limbs with five degrees of connectivity
Fig. 9.31 (cont.)
Fig. 9.32. Complex limbs PRPaRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 491
Fig. 9.33. Complex limbs PRPaPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
Fig. 9.34. Complex limbs PRRPaPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
492 9 Limbs with five degrees of connectivity
Fig. 9.35. Complex limbs of types PRPaRPa (a-d), PRPaPaR (e-h) and PRRPaPa (i-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types B (a-d), B1 (e-h) and B2 (i-l)
9.1 Limbs with two rotational and three translational motions 493
Fig. 9.35 (cont.)
494 9 Limbs with five degrees of connectivity
Fig. 9.36. Complex limbs of types PPaRPaR (a, b) and PPaRRPa (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types C1 (a, b) and C2 (c-d)
9.1 Limbs with two rotational and three translational motions 495
Fig. 9.37. Complex limbs of types PPaRPaR (a-d), PPaRRPa (e-h) and PPaPaRR (i-k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in con-figurations of types D1 (a-d), D2 (e-h) and D123 (i-k)
496 9 Limbs with five degrees of connectivity
Fig. 9.37 (cont.)
9.1 Limbs with two rotational and three translational motions 497
Fig. 9.38. Examples of complex limbs of types PRRRbR (a) and PRRRbRbR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
Fig. 9.39. Examples of complex limbs of types PRRbRR (a) and PRRbRbRR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
498 9 Limbs with five degrees of connectivity
Fig. 9.40. Examples of complex limbs PRPn2R-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
Fig. 9.41. Examples of complex limbs PPn2RR-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
9.1 Limbs with two rotational and three translational motions 499
Fig. 9.42. Examples of complex limbs PRPn3-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
Fig. 9.43. Examples of complex limbs PPn3R-type (a) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D1 configuration
500 9 Limbs with five degrees of connectivity
Fig. 9.44. Complex limbs of types RPPRPa (a-h) and RPRPPa (i-k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 501
Fig. 9.44 (cont.)
502 9 Limbs with five degrees of connectivity
Fig. 9.45. Complex limbs RPPaRP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 503
Fig. 9.45 (cont.)
504 9 Limbs with five degrees of connectivity
Fig. 9.46. Complex limbs RPaRPP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 505
Fig. 9.46 (cont.)
506 9 Limbs with five degrees of connectivity
Fig. 9.47. Complex limbs RPaPRP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 507
Fig. 9.47 (cont.)
508 9 Limbs with five degrees of connectivity
Fig. 9.48. Complex limbs RPRPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 509
Fig. 9.49. Complex limbs RPPPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
510 9 Limbs with five degrees of connectivity
Fig. 9.49 (cont.)
9.1 Limbs with two rotational and three translational motions 511
Fig. 9.50. Complex limbs RPPaPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
512 9 Limbs with five degrees of connectivity
Fig. 9.50 (cont.)
9.1 Limbs with two rotational and three translational motions 513
Fig. 9.50 (cont.)
514 9 Limbs with five degrees of connectivity
Fig. 9.51. Complex limbs RPaPPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
9.1 Limbs with two rotational and three translational motions 515
Fig. 9.51 (cont.)
516 9 Limbs with five degrees of connectivity
Fig. 9.51 (cont.)
9.1 Limbs with two rotational and three translational motions 517
Fig. 9.52. Complex limbs RPaPRP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
518 9 Limbs with five degrees of connectivity
Fig. 9.53. Complex limbs of types RPPaRP (a,b), RPaRRR (c,d) and RPaRPR (e, f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
9.1 Limbs with two rotational and three translational motions 519
Fig. 9.54. Complex limbs of types RRPPPa (a-g) and RPRPPa (h-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configura-tion
520 9 Limbs with five degrees of connectivity
Fig. 9.54 (cont.)
9.1 Limbs with two rotational and three translational motions 521
Fig. 9.55. Complex limbs RPPRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
522 9 Limbs with five degrees of connectivity
Fig. 9.56. Complex limbs RRPPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 523
Fig. 9.56 (cont.)
524 9 Limbs with five degrees of connectivity
Fig. 9.57. Complex limbs RPRPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 525
Fig. 9.57 (cont.)
526 9 Limbs with five degrees of connectivity
Fig. 9.58. Complex limbs RRPaPP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 527
Fig. 9.58 (cont.)
528 9 Limbs with five degrees of connectivity
Fig. 9.59. Complex limbs RPaRPP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 529
Fig. 9.60. Complex limbs of types RPaPRP (a-c) and RPPaRP (d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configura-tion
530 9 Limbs with five degrees of connectivity
Fig. 9.61. Complex limbs of types RRPaRP (a, b) and RPRPaR (c-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types C (a, b) and C1 (c-f)
9.1 Limbs with two rotational and three translational motions 531
Fig. 9.62. Complex limbs of types RRPPaR (a, b) and RRPaPR (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
532 9 Limbs with five degrees of connectivity
Fig. 9.63. Complex limbs of types RPRRPa (a-d), RRRPPa (e, f), RRPRPa (g, h) and RRRPaP (i, j) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configuration
9.1 Limbs with two rotational and three translational motions 533
Fig. 9.63 (cont.)
Fig. 9.64. Complex limbs of types RPPaPaR (a) and RPPaPatR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configura-tion
534 9 Limbs with five degrees of connectivity
Fig. 9.65. Complex limbs RPaRPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 535
Fig. 9.65 (cont.)
536 9 Limbs with five degrees of connectivity
Fig. 9.65 (cont.)
9.1 Limbs with two rotational and three translational motions 537
Fig. 9.66. Complex limbs RPaPaRP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
538 9 Limbs with five degrees of connectivity
Fig. 9.66 (cont.)
9.1 Limbs with two rotational and three translational motions 539
Fig. 9.66 (cont.)
540 9 Limbs with five degrees of connectivity
Fig. 9.67. Complex limbs RPaRPPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 541
Fig. 9.67 (cont.)
542 9 Limbs with five degrees of connectivity
Fig. 9.67 (cont.)
9.1 Limbs with two rotational and three translational motions 543
Fig. 9.68. Complex limbs RPaPRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
544 9 Limbs with five degrees of connectivity
Fig. 9.68 (cont.)
9.1 Limbs with two rotational and three translational motions 545
Fig. 9.68 (cont.)
546 9 Limbs with five degrees of connectivity
Fig. 9.69. Complex limbs RPPaRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 547
Fig. 9.70. Complex limbs RPaPaPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
548 9 Limbs with five degrees of connectivity
Fig. 9.70 (cont.)
9.1 Limbs with two rotational and three translational motions 549
Fig. 9.70 (cont.)
550 9 Limbs with five degrees of connectivity
Fig. 9.70 (cont.)
9.1 Limbs with two rotational and three translational motions 551
Fig. 9.71. Complex limbs RPaPPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
552 9 Limbs with five degrees of connectivity
Fig. 9.71 (cont.)
9.1 Limbs with two rotational and three translational motions 553
Fig. 9.71 (cont.)
554 9 Limbs with five degrees of connectivity
Fig. 9.71 (cont.)
9.1 Limbs with two rotational and three translational motions 555
Fig. 9.72. Complex limbs RPaPaRP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
556 9 Limbs with five degrees of connectivity
Fig. 9.73. Complex limbs RPPaPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
9.1 Limbs with two rotational and three translational motions 557
Fig. 9.74. Complex limbs RRPaPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
558 9 Limbs with five degrees of connectivity
Fig. 9.74 (cont.)
9.1 Limbs with two rotational and three translational motions 559
Fig. 9.75. Complex limbs RRPaPPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
560 9 Limbs with five degrees of connectivity
Fig. 9.75 (cont.)
9.1 Limbs with two rotational and three translational motions 561
Fig. 9.76. Complex limbs of types RRPPaPa (a-c) and RPRPaPa (d-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configura-tion
562 9 Limbs with five degrees of connectivity
Fig. 9.77. Complex limbs RPaRPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 563
Fig. 9.78. Complex limbs RPaRPPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
564 9 Limbs with five degrees of connectivity
Fig. 9.79. Complex limbs RPaPRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configuration
9.1 Limbs with two rotational and three translational motions 565
Fig. 9.80. Complex limbs of types PaRPRP (a, b) and PaRPPR (c-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types B (a) and B1 (b-f)
566 9 Limbs with five degrees of connectivity
Fig. 9.81. Complex limbs of types PaPRPR (a-c), PaRRPR (d) and PaRRRR (e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
9.1 Limbs with two rotational and three translational motions 567
Fig. 9.82. Complex limbs PaRRPP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
568 9 Limbs with five degrees of connectivity
Fig. 9.83. Complex limbs of types PaPRRP (a, b) and PaRPRP (c-e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
9.1 Limbs with two rotational and three translational motions 569
Fig. 9.84. Complex limbs of types PaPRPR (a-c) and PaRPRP (d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B12 configura-tion
570 9 Limbs with five degrees of connectivity
Fig. 9.85. Complex limbs of types PaRPPR (a, b) and PaPRRP (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B12 configura-tion
9.1 Limbs with two rotational and three translational motions 571
Fig. 9.86. Complex limbs of types PaRPRP (a), PaRPPR (b, c), PaPRPR (d), PaRRPR (e) and PaRRRR (f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types C (a) and C1 (b-f)
572 9 Limbs with five degrees of connectivity
Fig. 9.87. Complex limbs of types PaPRRP (a), PaRRPP (b-d) and PaRPRP (e-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configuration
9.1 Limbs with two rotational and three translational motions 573
Fig. 9.88. Complex limbs of types PaPRRP (a), PaPRPR (b), PaRPPR (c, d) and PaRPRP (e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C12 configuration
574 9 Limbs with five degrees of connectivity
Fig. 9.89. Complex limbs of types PaPRPR (a, b), PaRRRP (c) and PaPRRP (d-k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types D1 (a, b) and D2 (c-k)
9.1 Limbs with two rotational and three translational motions 575
Fig. 9.89 (cont.)
576 9 Limbs with five degrees of connectivity
Fig. 9.90. Complex limbs of types PaPRRP (a, b) and PaPRPR (c, d) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D12 configura-tion
9.1 Limbs with two rotational and three translational motions 577
Fig. 9.91. Complex limbs of types PaPPRR (a-i), PaRPRR (j, k) and PaRRPR (l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
578 9 Limbs with five degrees of connectivity
Fig. 9.91 (cont.)
9.1 Limbs with two rotational and three translational motions 579
Fig. 9.92. Complex limbs of types PaPRPR (a-d), PaPRRR (e) and PaRRRR (f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
580 9 Limbs with five degrees of connectivity
Fig. 9.93. Complex limbs of types PaRPaRP (a, b), PaRPRPa (c-e) and PaR-PaRP (f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and defined by two paral-lelogram loops and the rotation axes in a type B configuration
9.1 Limbs with two rotational and three translational motions 581
Fig. 9.94. Complex limbs PaRPaPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
582 9 Limbs with five degrees of connectivity
Fig. 9.94 (cont.)
9.1 Limbs with two rotational and three translational motions 583
Fig. 9.95. Complex limbs of types PaRPPaR (a-i) and PaRPaRP (j-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
584 9 Limbs with five degrees of connectivity
Fig. 9.95 (cont.)
9.1 Limbs with two rotational and three translational motions 585
Fig. 9.96. Complex limbs of types PaPaRRP (a, b) and PaPaRPR (c-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configura-tion
586 9 Limbs with five degrees of connectivity
Fig. 9.97. Complex limbs PaPRPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 configuration
9.1 Limbs with two rotational and three translational motions 587
Fig. 9.98. Complex limbs PaRRPaP-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
588 9 Limbs with five degrees of connectivity
Fig. 9.98 (cont.)
9.1 Limbs with two rotational and three translational motions 589
Fig. 9.99. Complex limbs of types PaRRPPa (a-h) and PaRPaRP (i-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
590 9 Limbs with five degrees of connectivity
Fig. 9.99 (cont.)
9.1 Limbs with two rotational and three translational motions 591
Fig. 9.100. Complex limbs of types PaRPRPa (a-g), PaPRRPa (h-o), PaRPRPa (p) and PaPaRRP (q) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rota-tion axes in a type B2 configuration
592 9 Limbs with five degrees of connectivity
Fig. 9.100 (cont.)
9.1 Limbs with two rotational and three translational motions 593
Fig. 9.100 (cont.)
594 9 Limbs with five degrees of connectivity
Fig. 9.101. Complex limbs of types PaRPaPR (a), PaRPPaR (b, c) and PaPRPaR (d, e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B12 configuration
9.1 Limbs with two rotational and three translational motions 595
Fig. 9.102. Complex limbs of types PaRPaRP (a-c) and PaRPRPa (d, e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C configura-tion
596 9 Limbs with five degrees of connectivity
Fig. 9.103. Complex limbs PaRPaPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
9.1 Limbs with two rotational and three translational motions 597
Fig. 9.104. Complex limbs of types PaRPPaR (a-e) and PaRPaRP (f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
598 9 Limbs with five degrees of connectivity
Fig. 9.105. Complex limbs PaPRPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
9.1 Limbs with two rotational and three translational motions 599
Fig. 9.106. Complex limbs of types PaRRPaP (a-g) and PaRRPPa (h-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configura-tion
600 9 Limbs with five degrees of connectivity
Fig. 9.106 (cont.)
9.1 Limbs with two rotational and three translational motions 601
Fig. 9.107. Complex limbs of types PaPRRPa (a-c) and PaRPaRP (d, e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configura-tion
602 9 Limbs with five degrees of connectivity
Fig. 9.108. Complex limbs PaRPRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configuration
9.1 Limbs with two rotational and three translational motions 603
Fig. 9.109. Complex limbs of types PaRPaPR (a, b), PaRPaRP (c), PaPRPaR (d) and PaRPPaR (e) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C12 configuration
604 9 Limbs with five degrees of connectivity
Fig. 9.110. Complex limbs of types PaPaRPR (a, b) and PaPRPaR (c-i) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D1 configu-ration
9.1 Limbs with two rotational and three translational motions 605
Fig. 9.110 (cont.)
606 9 Limbs with five degrees of connectivity
Fig. 9.111. Complex limbs of types PaPaRRP (a-c) and PaPRRPa (d-k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D2 configu-ration
9.1 Limbs with two rotational and three translational motions 607
Fig. 9.111 (cont.)
608 9 Limbs with five degrees of connectivity
Fig. 9.112. Complex limbs of types PaPaRRP (a-c) and PaPRPaR (d-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D12 configu-ration
9.1 Limbs with two rotational and three translational motions 609
Fig. 9.113. Complex limbs PaPaRPR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D12 configuration
610 9 Limbs with five degrees of connectivity
Fig. 9.114. Complex limbs RPaPaRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.1 Limbs with two rotational and three translational motions 611
Fig. 9.114 (cont.)
612 9 Limbs with five degrees of connectivity
Fig. 9.114 (cont.)
9.1 Limbs with two rotational and three translational motions 613
Fig. 9.115. Complex limbs RPaPaRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
614 9 Limbs with five degrees of connectivity
Fig. 9.115 (cont.)
9.1 Limbs with two rotational and three translational motions 615
Fig. 9.116. Complex limbs RPaPaPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
616 9 Limbs with five degrees of connectivity
Fig. 9.116 (cont.)
9.1 Limbs with two rotational and three translational motions 617
Fig. 9.116 (cont.)
618 9 Limbs with five degrees of connectivity
Fig. 9.117. Complex limbs of types RRPaPaPa (a-h) and RPaRPaPa (i-l) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A2 configu-ration
9.1 Limbs with two rotational and three translational motions 619
Fig. 9.117 (cont.)
620 9 Limbs with five degrees of connectivity
Fig. 9.118. Complex limbs of types PaPaRRPa (a), PaRRPaPa (b, c) and PaR-PaRPa (d-k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B configuration
9.1 Limbs with two rotational and three translational motions 621
Fig. 9.118 (cont.)
622 9 Limbs with five degrees of connectivity
Fig. 9.119. Complex limbs of types PaPaRPaR (a-d) and PaRPaPaR (e-n) de-fined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B1 con-figuration
9.1 Limbs with two rotational and three translational motions 623
Fig. 9.119 (cont.)
624 9 Limbs with five degrees of connectivity
Fig. 9.119 (cont.)
9.1 Limbs with two rotational and three translational motions 625
Fig. 9.120. Complex limbs of types PaPaRRPa (a-c), PaRRPaPa (d-i) and PaRPaRPa (j, k) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type B2 configuration
626 9 Limbs with five degrees of connectivity
Fig. 9.120 (cont.)
9.1 Limbs with two rotational and three translational motions 627
Fig. 9.121. Complex limbs PaRPaRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C configuration
628 9 Limbs with five degrees of connectivity
Fig. 9.122. Complex limbs PaRPaPaR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C1 configuration
9.1 Limbs with two rotational and three translational motions 629
Fig. 9.123. Complex limbs PaRRPaPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type C2 configuration
630 9 Limbs with five degrees of connectivity
Fig. 9.124. Complex limbs of type PaPaRRPa (a, b) and PaPaRPaR (c-k) defined by RPaPaRPa-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in configurations of types D (a, b) and D1 (c-k)
9.1 Limbs with two rotational and three translational motions 631
Fig. 9.124 (cont.)
632 9 Limbs with five degrees of connectivity
Fig. 9.125. Complex limbs PaPaRRPa-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D2 configuration
9.1 Limbs with two rotational and three translational motions 633
Fig. 9.125 (cont.)
634 9 Limbs with five degrees of connectivity
Fig. 9.126. Complex limbs of type PaPaRPaR (a-c) and PaPaPaRR (d-f) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D12 configu-ration
9.1 Limbs with two rotational and three translational motions 635
Fig. 9.127. Complex limbs PaPaPaRR-type defined by MG=SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type D123 configuration
636 9 Limbs with five degrees of connectivity
Fig. 9.128. Examples of complex limbs of types RPRRbR (a) and RPRRbRbR (b) defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
Fig. 9.129. Examples of complex limbs RPPn2R-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A1 configuration
9.2 Limbs with two translational and three rotational motions 637
Fig. 9.130. Examples of complex limbs RPPn3-type defined by MG=SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ) and the rotation axes in a type A configuration
9.2 Limbs with two translational and three rotational motions
These limbs give two translational velocities v1, v2, and three angular ve-locities 1ω , 2ω , 3ω in the basis of the operational velocities space (RG)=( 1 2 1 2 3, , , ,v v ω ω ω ). They contain two coplanar prismatic joints and three revolute joints with intersecting axes. These solutions give three rota-tions of the distal link along with a planar motion of characteristic point H situated at the intersecting axes of the three revolute joints. They are de-fined by MG=SG=5.
The simplest structural solutions of non redundant simple and complex limbs with only revolute and prismatic pairs and are presented in Figs. 9.131a-c, 9.132a-c, 9.133a-f and 9.134a-c. A spherical joint replaces the three revolute joints with intersecting axes in the solutions presented in Figs. 9.131d, 9.132d, 9.133g, h, and 9.134d.
The solutions in Figs. 9.131-9.134 have not idle mobilities. Three joint parameters lose their independence in each planar parallelogram closed loop combined in Figs. 8.132-9.134.
The structural parameters of the solutions presented in Figs. 9.131-9.134 are systematized in Table 9.5.
If the characteristic pint of the distal link is not on situated at the inter-secting axes the three revolute joints the basis of the operational velocities space can be ( 1 2 1 2 3, , , ,v v ω ω ω ) or ( 1 2 3 1 2, , , ,v v v ω ω ).
638 9 Limbs with five degrees of connectivity
Table 9.5. Joint arrangement and structural parametersa of the limbs defined by MG=SG=5, (RG)=( 1 2 1 2 3, , , ,v v ω ω ω ) and TG=0 presented in Figs. 9.131-9.134.
Structural parameters No. Limb-type m p q rG NG
Figure
1 PPRRR 6 5 - - - Fig. 9.131a-c 2 PPS 4 3 - - - Fig. 9.131d 3 PPaRRR 8 8 1 3 3 Fig. 9.132a-c 4 PPaS 6 6 1 3 3 Fig. 9.132d 5 PaPRRR 8 8 1 3 3 Fig. 9.133a-f 6 PPaS 6 6 1 3 3 Fig. 9.133g,h 7 PaPaRRR 10 11 2 6 6 Fig. 9.134a-c 8 PaPaS 8 9 2 6 6 Fig. 9.134d aSee footnote of Table 6.2 for the nomenclature of structural parameters
Other solutions with non-orthogonal intersecting axes may also be used.
As we have mentioned in the previous chapters, just solutions with or-thogonal and/or parallel axes of the kinematic joints are presented in this work. They are the easiest to make.
9.2 Limbs with two translational and three rotational motions 639
Fig. 9.131. Simple limbs of types PPRRR (a-c) and PPS (d) defined by MG=SG=5 and (RG)=( 1 2 1 2 3, , , ,v v ω ω ω )
640 9 Limbs with five degrees of connectivity
Fig. 9.132. Examples of complex limbs of types PPaRRR (a-c) and PPaS (d) de-fined by MG=SG=5 and (RG)=( 1 2 1 2 3, , , ,v v ω ω ω )
9.2 Limbs with two translational and three rotational motions 641
Fig. 9.133. Examples of complex limbs of types PaPRRR (a-f) and PPaS (g, h) defined by MG=SG=5 and (RG)=( 1 2 1 2 3, , , ,v v ω ω ω )
642 9 Limbs with five degrees of connectivity
Fig. 9.133 (cont.)
9.2 Limbs with two translational and three rotational motions 643
Fig. 9.134. Examples of complex of types PaPaRRR (a-c) and PaPaS (d) defined by MG=SG=5 and (RG)=( 1 2 1 2 3, , , ,v v ω ω ω )
644 9 Limbs with five degrees of connectivity
9.3 Other limbs with five degrees of connectivity
The limbs presented in this section have multiple bases for the operational velocity space for any position of the characteristic point on the distal link. These bases are defined by various combinations of translational and angu-lar velocities. For example, the basis of the operational velocity space of the limb in Fig. 9.135a may be defined by three translational and two an-gular velocities ( 1 2 3 1 2, , , ,v v v ω ω ) or by two translational and three angular velocities ( 1 2 1 i, , ,v v ω ω ), i=2,3. These bases are valid for any position of the characteristic point H on the distal link 5.
Fig. 9.135. Simple limbs of types PPRRR (a, b), PRRPR (c) and PRRPR (d-f) de-fined by MG=SG=5 and multiple bases of the operational velocity space
9.3 Other limbs with five degrees of connectivity 645
Fig. 9.136. Simple limbs of types PRPRR (a), PRRRR (b-d), RPPRR (e-g) and RRPRR (h, i) defined by MG=SG=5 and multiple bases of the operational velocity space
646 9 Limbs with five degrees of connectivity
Fig. 9.137. Simple limbs of types RRRPR (a) and RPRRR (b-i) defined by MG=SG=5 and multiple bases of the operational velocity space
9.3 Other limbs with five degrees of connectivity 647
Fig. 9.138. Simple limbs RRRRR-type defined by MG=SG=5 and multiple bases of the operational velocity space
648 9 Limbs with five degrees of connectivity
Fig. 9.139. Complex limbs of types RRRbRR (a), RRRbRbRR (b), and RPa2RR (c-f) defined by MG=SG=5 and multiple bases of the operational velocity space
9.3 Other limbs with five degrees of connectivity 649
Fig. 9.140. Complex limbs of types PRRbRR (a), PRRbRbRR (b), and PPa2RR (c-d) defined by MG=SG=5 and multiple bases of the operational velocity space
Many limbs with five degrees of connectivity and multiple bases of the operational velocity space are generated by the evolutionary morphology approach. Due to space limitations, just some examples are presented in Figs. 9.135-9.140. The simple limbs in Figs. 9.135-9.138 are defined by the following structural parameters: MG=SG=5, NG=0, TG=0 and rG=0. The complex limbs with one planar loop in Figs. 9.139a,c-f and 9.140a,c,d are defined by MG=SG=4, NG=3, TG=0 and rG=3, and the complex limbs with two rhombus loops in Figs. 9.139b and 9.140b by MG=SG=4, NG=6, TG=0 and rG=6.
650 9 Limbs with five degrees of connectivity
9.4 Redundant limbs with five degrees of connectivity
A large variety of redundant limbs with SG=5 and MG>5 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just two examples are presented in Figs. 9.141 and 9.142.
The two solutions in Fig. 9.141 contain a spatial pseudo-parallelogram closed loop with four spherical joints. Six joint parameters lose their inde-pendence in this Pa4s-type closed loop. These two solutions are defined by the following structural parameters: m=5, p=5, q=1, rG=6, SG=5, MG=7 (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), NG=0 and TG=2. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of links 3 and 4 along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These two internal mobilities are associated with the two degrees of redundancy that cannot be transmitted to the distal link. The remaining five degrees of mobility are external mobilities associated with the independent motions of distal link 5. These solutions have the ad-vantage of reducing limb complexity by using a compact spherical joint as an alternative to the universal joint.
The complex limb in Fig. 9.142 combines a parallelogram loop Pa-type, a telescopic parallelogram loop Pat-type and a serial chain R║R║R⊥R-type. Three joint parameters lose their independence in each parallelogram loop. This solution is defined by the following structural parameters: m=11, p=12, q=2, rG=6, SG=5, MG=6, (RG)=( 1 2 3 1 2, , , ,v v v ω ω ), NG=6 and TG=1.
Fig. 9.141. Examples of complex limbs with internal mobilities of types PSSSS (a) and RSSSS (b) defined by MG=7, SG=5, (RG)=( 1 2 3 1 2, , , ,v v v ω ω )
9.4 Redundant limbs with five degrees of connectivity 651
Fig. 9.142. Example of redundant limb defined by MG=6, SG=5, (RG)= ( 1 2 3 1 2, , , ,v v v ω ω )
The degree of redundancy gives an internal mobility in the limb associ-ated with the translation in the telescopic parallelogram loop. The remain-ing five degrees of mobility are external mobilities associated with the in-dependent motions of distal link 11. The internal mobility in this limb is useful for motion decoupling in parallel robots with five degrees of mobil-ity as we will present in Part 2 of this work.
10 Limbs with six degrees of connectivity
Limbs with six degrees of connectivity (SG=6) may be used in any parallel robot F with the connectivity of the moving platform 2≤SF≤6. The basis of the operational velocity space of these limbs contains the six independent translational and angular velocities (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω ) for any posi-tion of the characteristic point on the distal link. Non-redundant solutions generated by the evolutionary morphology approach are presented in the first section of this chapter. Some examples of redundant solutions are pre-sented in the second section.
10.1 Limbs with three translational and three rotational motions
Many non-redundant solutions with SG=MG=6 are obtained by the evolu-tionary morphology approach presented in chapter 5 and used in chapters 6-9 for generating the solutions of the previous generations with 2≤SG<6. The solutions with SG=MG=6 can be obtained as follows: (i) adding a third revolute joint in the non-redundant limbs with five degrees of connectivity presented in chapter 9 or (ii) adding a third prismatic joint in the non-redundant limbs defined by (RG)=( 1 2 1 2 3, , , ,v v ω ω ω ) and presented in sec-tion 9.2. Due to space limitation, just some examples are presented in Figs. 10.1-10.7. In these figures three revolute joints with concurrent axes are replaced by a spherical joint. For example, solution in Fig. 10.1 is obtained by adding a revolute joint to the solution with five degrees of connectivity in Fig. 9.1b.
The structural parameters of the solutions presented in Figs. 10.1-10.7 are systematized in Table 10.1. These solutions have not internal mobili-ties. We recall that three joint parameters lose their independence in each planar closed loop Pa- Rb-, Pn2 or Pn3-type.
As presented in chapter 9, the number of joints may be reduced in these solutions by replacing:
654 10 Limbs with six degrees of connectivity
a) two consecutive revolute and prismatic joints with the same axes/direction (R||P or P||R) by a cylindrical joint such as in Fig. 7.23,
b) a sequence of joints R⊥Pa or Pa⊥R by a parallelogram loop Pass-type with two spherical joints adjacent to the distal link such as in Fig. 7.24,
c) a sequence of joints P||Pa or Pa||P by a parallelogram loop Pacc-type with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop such as in Fig. 7.7.
The complex limbs may combine up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could contain up to three idle mobilities that may be two orthogonal rotations and one translation.
Table 10.1. Joint arrangement and structural parametersa of the limbs defined by MG=SG=6, (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω ) and TG=0 presented in Figs. 10.1-10.7
Structural parameters No. Limb-type m p q rG NG
Figure
1 PRPRPR 7 6 - - - Fig. 10.1a 2 PPPS 5 4 - - - Fig. 10.1b 3 PPRS 5 4 - - - Fig. 10.1c,d 4 PRPS 5 4 - - - Fig. 10.1e,f 5 PRRS 5 4 - - - Fig. 10.1g-i 6 RPPS 5 4 - - - Fig. 10.2a,b 7 RPRS 5 4 - - - Fig. 10.2c-f 8 RRPS 5 4 - - - Fig. 10.3a-c 9 RRRS 5 4 - - - Fig. 10.3d-f 10 PPaPS 7 7 1 3 3 Fig. 10.4a 11 PPaRS 7 7 1 3 3 Fig. 10.4b 12 PRPaS 7 7 1 3 3 Fig. 10.4c 13 PRRbRbS 9 10 2 6 6 Fig. 10.4d 14 PPn2S 7 7 1 3 3 Fig. 10.4e,f 15 RRRbS 7 7 1 3 3 Fig. 10.5a 16 RRRbRbS 9 10 2 6 6 Fig. 10.5b 17 RPn2S 7 7 1 3 3 Fig. 10.5c-f 18 PaPaPS 9 10 2 6 6 Fig. 10.6a,b 19 PPaPaS 9 10 2 6 6 Fig. 10.6c 20 PaPaPaS 9 10 2 6 6 Fig. 10.6d-f 21 RPPaPaRR 11 12 2 6 6 Fig. 10.7a 22 RPPaPatRR 11 12 2 6 6 Fig. 10.7b aSee footnote of Table 6.2 for the nomenclature of structural parameters
10.1 Limbs with three translational and three rotational motions 655
Fig. 10.1. Simple limb of types PRPRPR (a), PPPS (b), PPRS (c, d), PRPS (e, f) and PRRS (g-i) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
656 10 Limbs with six degrees of connectivity
Fig. 10.2. Simple limb of types RPPS (a, b) and RPRS (c-f) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
10.1 Limbs with three translational and three rotational motions 657
Fig. 10.3. Simple limb of types RRPS (a-c) and RRRS (d-f) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
658 10 Limbs with six degrees of connectivity
Fig. 10.4. Complex limb of types PPaPS (a), PPaRS (b), PRPaS (c), PRRbRbS (d) and PPn2S (e, f) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
10.1 Limbs with three translational and three rotational motions 659
Fig. 10.5. Complex limb of types RRRbS (a), RRRbRbS (b) and RPn2S (c-f) de-fined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
660 10 Limbs with six degrees of connectivity
Fig. 10.6. Complex limb of types PaPaPS (a, b), PPaPaS (c) and PaPaPaS (d-f) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
10.2 Redundant limbs with six degrees of connectivity 661
Fig. 10.7. Complex limb of types RPPaPaRR (a) and RPPaPatRR (b) defined by MG=SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
10.2 Redundant limbs with six degrees of connectivity
A large variety of redundant limbs with SG=6 and MG>6 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just two examples are presented in Figs. 10.8 and 10.9.
The solutions in Fig. 10.8 are defined by the following structural pa-rameters: m=4, p=3, q=0, rG=0, SG=6, MG=7 (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω ), NG=0 and TG=1. An internal mobility exists in the solutions in Fig. 10.8a and b introduced by the rotation of link 3 along the rotation axis passing through the centre of the two adjacent spherical joints. The internal mobil-ity in the solution in Fig. 10.8c is associated with the common rotation of the sub chain 3-4 along the rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. In both cases, this internal mobility cannot be transmitted to the distal link. The remaining six degrees of mobility are external mobilities associated with the independent motions of the distal link. These solutions have the advantage of reducing limb complexity by using a compact spherical joint as an alternative of univer-sal joint.
The complex limb in Fig. 10.9 combines a parallelogram loop Pa, a telescopic parallelogram loop Pat and a serial chain R║R║R⊥R⊥R-type. Three joint parameters lose their independence in each parallelogram loop.
662 10 Limbs with six degrees of connectivity
Fig. 10.8. Complex limb of types PSS (a), RSS (b) and SPS (c) defined by MG=7, SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
Fig. 10.9. Redundant complex limb PaPatRRRRR-type defined by MG=7, SG=6 and (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω )
10.2 Redundant limbs with six degrees of connectivity 663
Solution in Fig. 10.9 is defined by the following structural parameters: m=12, p=13, q=2, rG=6, SG=6, MG=7, (RG)=( 1 2 3 1 2 3, , , , ,v v v ω ω ω ), NG=6 and TG=1. The degree of redundancy gives an internal mobility in the limb associated with the translation in the telescopic parallelogram loop. The remaining six degrees of mobility are external mobilities associated with the independent motions of distal link 12. The internal mobility in this limb is useful for motion decoupling in parallel robots with five degrees of mobility as we will present in Part 2 of this work.
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Index
A
actuator, 16 adjacency matrices, 303 algorithm
evolutionary, 307 genetic, 309
angle Bryant, 245 Euler, 245 roll-pitch-yaw, 245
approach inventive, 299 structured, 299
associated linkage, 303 Assur groups, 303 axiomatic design, 236
B
bar, 12 Baranov trusses, 303 base, 12
fixed, 11
C
chain, 11 paradoxical, 33 virtual, 303
characteristic link, 127 point, 129
condition number, 270 conditioning index, 270 connectivity, 13, 59, 82
geometric definition, 82
kinematic definition, 82 proposition, 82, 84, 85, 86, 90,
91, 97, 98, 99, 100, 102, 107, 108, 110, 111, 113, 116, 117, 118, 119
constituent elements, 310 constraint, 16
geometric, 239 coupled motions, 276 CPM, 34 crossing-over, 313
D De Roberval, 120, 228 decoupled motions, 285 degree
of connectivity, 37 of constraint, 38 of freedom, 13 of mobility, 43 of motion coupling, 275
design, 24 conceptual, 304 creative, 300 matrix, 236 objectives, 310 parameter, 236 solution, 237
dimension joint space, 85 operational space, 85
direct task, 20
displacement, 18 angular, 18 finite, 79, 238
Index
694
infinitesimal, 79 drive, 16
E
eigenvalues, 267 element, 11
distal, 20 final, 20 mechanism, 15 pairing, 12
end-effector, 17 equation
design, 236 loop-closure, 247 vector-loop, 247
Euler’s formula, 66 evolution criterion, 323 evolutionary morphology, 310,
F
finite groups, 303 force
ellipsoid, 268 polytop, 269
frame, 11 moving, 17 reference, 279
Franke’s notation, 304 fully-coupled motions, 274 functional requirement, 236
G
geometric model, 238 problem, 238
group theory, 304 gripper, 17
H
H4, 198 half-decoupled motions, 286
I
I4R, 222 IFMA, 150 independent
motion, 73 parameter, 32, 73
index cross-coupling, 257 input-output propensity, 262
input link, 12
input-output propensity, 262 intuition, 303 inverse
task, 20 IRCCyN, 176 Isoglide family, 29, 297 Isoglide2-T2, 150 Isoglide3-T2R1, 206, 209, 211 Isoglide3-T3, 154 Isoglide4-T3R1, 155, 214, 218 Isoglide5-T3R2, 161 Isogliden-TaRb, 150 isomorphism, 16 isotropy, 26, 269
J
Jacobian conventional, 246 design, 242, 274 matrix, 26, 81, 83
joint, 13 Cardan, 14 force, 254 heterokinetic, 21 homokinetic, 21 Hooke, 14 passive, 125 space, 14 universal, 14 variable, 14 velocity space, 81, 110, 115,
Index
695
K
kinematic analysis, 20 chain, 15 constraint, 112 equation, 80 inversion, 15 joint, 15 problem, 241 singularity, 86
kinematic chain, 15 closed, 15 open, 15 parallel, 10 serial, 10 simple, 20, 82, 131 single-loop, 88, 137
kinematics, 20 kinetostatic, 235, 253
index, 256
L
leg, 10, 21 complex, 21 simple, 21
limb, 10, 21 linear
set, 68 transformation, 69
link, 11 binary, 11 distal, 17 monary, 11 polinary, 11 proximal, 17 reference, 11
linkage, 15 four-bar, 15
LIRMM, 198, 222 loop, 15
M
manipulability, 271 ellipsoid, 266
index, 271 manipulator, 17
parallel, 17 manoeuvrability, 20 matrix
transformation, 69 mechanism, 15, 127
Bennett, 145 Bricard, 148 complex, 20, 101, 127, 148 De Roberval, 229 element, 11 equivalent, 16 family, 60 four-bar, 16 guidance, 16 member, 11 multi-loop, 128 overconstrained, 120 paradoxical, 33 parallel, 10 planar, 15 serial, 127 simple, 127 single-loop, 128 spatial, 16 spherical, 15
Melbourne method, 303 mobility, 16
criterion, 77 full-cycle, 80 geometric definition, 79 general, 80 global, 33, 35 gross, 33 idle, 124 instantaneous, 33, 80 kinematic definition, 79 number, 59 redundant, 20 single-loop mechanism, 88 proposition, 80, 84, 90, 92, 93,
94, 107, 112, 117, 118 mobility formula
CGK, 58
Index
696
model direct, 238 geometric, 238 inverse, 238
modeling, 238 module, 17 morphological
analysis, 306 approach, 305 box, 306 combination, 313 method, 306 migration, 314 mutation, 313 operator, 313 research, 305 selection, 314
morphology, 305 morphosis, 305 motion, 17
absolute, 17 infinitesimal, 254 inverse, 17 planar, 19 relative, 17 spatial, 19 spherical, 19
N
nullity, 86 nutation, 18
O
operational space, 81 velocity space, 81, 101, 110, 115
orientation, 18, Orthoglide, 176, output
link, 12 overconstraint, 120
proposition, 122, 123
P
pair, 13 cylindrical, 14 helical, 14 higher, 14 kinematic, 13 lower, 13 passive, 23 planar, 14 revolute, 14 spherical, 14
parallel mechanism, 148 2-PPa-type, 168 3-RPS-type, 166 3-RRPRR-type, 166 3-UPU-type, 166 simple limbs, 148
parallel robot, 10 fully-isotropic, 27, 214, 218, 270,
279 Gough-Stewart, 23, 163 maximally regular, 27, 296 with complex limbs, 109, 168 with coupled motions, 27, 276,
275 with decoupled motions, 27, 285 with half-decoupled motions, 286 with simple limbs, 101, with uncoupled motions, 27, 295
pedipulator, 17 performance index, 272
kinematics, 258, 263 kinetostatic, 253 static, 259, 264
physical embodiment, 236 implementation, 23
platform, 20 moving, 20 reference, 20
point dead, 19 reference, 19
pose, 18
Index
697
position, 12 limit, 12, 19
precession, 18 problem
direct, 238, 244 geometric, 238 inverse, 238, 244
rank, 70 reangularity, 237 redundancy, 20, 86, 126
proposition, 86, 87, 126 rigid
body, 12 robot, 1, 17
antropomorphic, 17 fully parallel, 115 hexapod, 10 hybrid, 115 parallel, 115 redundant, 125 serial, 115
robotics, 7 rotation, 18
angle, 18 axis, 18 matrix, 245
S
screw axis, 18 instantaneous, 19 motion, 19
semangularity, 238 set of solutions, 314 singular
configuration, 20 singular values, 267 singularity, 249
instantaneous, 86 structural, 86
systematics, 300 spatiality, 59 structural
analysis, 131 diagram, 23
graph, 23, 303 independence, 66 parameters, 31, 127 redundancy, 87, 126 singularity, 86 synthesis, 300
structure, 16 synthesis, 24, 303
dimensional, 24 kinematic, 301 number, 300 qualitative, 303 structural, 24, 301 structure, 301 systematic, 301 topological, 301 topology, 301 type, 300
systematics, 302 systematization, 302
T
theory group, 28, 3004 linear transformation, 28, 3004 screw, 28, 304
trajectory, 18 transformation
linear, 81 translation, 18
rectilinear, 18 transmission factor, 272 tree, 15
U
uncoupled motions, 293
V
vector space Euclidean, 254
velocity, 19 absolute, 19 angular, 19, 245 ellipsoid, 267
Index
698
polytop, 269 relative, 19 translational, 245 vector space, 81
velocity space joint, 81,115 operational, 81, 115
W
working range, 19 space, 19
workspace, 248
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim ofthis series is to provide lucid accounts written by authoritative researchers giving vision and insightin answering these questions on the subject of mechanics as it relates to solids. The scope of theseries covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics;variational formulations; computational mechanics; statics, kinematics and dynamics of rigid andelastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories ofelasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes;structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.
1. R.T. Haftka, Z. Gurdal and M.P. Kamat: Elements of Structural Optimization. 2nd rev.ed., 1990ISBN 0-7923-0608-2
2. J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990 ISBN 0-7923-0712-73. P. Karasudhi: Foundations of Solid Mechanics. 1991 ISBN 0-7923-0772-04. Not published5. Not published.6. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and
Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-27. O.O. Ochoa and J.N. Reddy: Finite Element Analysis of Composite Laminates.
ISBN 0-7923-1125-68. M.H. Aliabadi and D.P. Rooke: Numerical Fracture Mechanics. ISBN 0-7923-1175-29. J. Angeles and C.S. Lopez-Cajun: Optimization of Cam Mechanisms. 1991
ISBN 0-7923-1355-010. D.E. Grierson, A. Franchi and P. Riva (eds.): Progress in Structural Engineering. 1991
ISBN 0-7923-1396-811. R.T. Haftka and Z. Gurdal: Elements of Structural Optimization. 3rd rev. and exp. ed. 1992
ISBN 0-7923-1504-9; Pb 0-7923-1505-712. J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-1610-X13. H.S. Tzou and G.L. Anderson (eds.): Intelligent Structural Systems. 1992
ISBN 0-7923-1920-614. E.E. Gdoutos: Fracture Mechanics. An Introduction. 1993 ISBN 0-7923-1932-X15. J.P. Ward: Solid Mechanics. An Introduction. 1992 ISBN 0-7923-1949-416. M. Farshad: Design and Analysis of Shell Structures. 1992 ISBN 0-7923-1950-817. H.S. Tzou and T. Fukuda (eds.): Precision Sensors, Actuators and Systems. 1992
ISBN 0-7923-2015-818. J.R. Vinson: The Behavior of Shells Composed of Isotropic and Composite Materials. 1993
ISBN 0-7923-2113-819. H.S. Tzou: Piezoelectric Shells. Distributed Sensing and Control of Continua. 1993
ISBN 0-7923-2186-320. W. Schiehlen (ed.): Advanced Multibody System Dynamics. Simulation and Software Tools.
1993 ISBN 0-7923-2192-821. C.-W. Lee: Vibration Analysis of Rotors. 1993 ISBN 0-7923-2300-922. D.R. Smith: An Introduction to Continuum Mechanics. 1993 ISBN 0-7923-2454-423. G.M.L. Gladwell: Inverse Problems in Scattering. An Introduction. 1993 ISBN 0-7923-2478-1
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
24. G. Prathap: The Finite Element Method in Structural Mechanics. 1993 ISBN 0-7923-2492-725. J. Herskovits (ed.): Advances in Structural Optimization. 1995 ISBN 0-7923-2510-926. M.A. Gonzalez-Palacios and J. Angeles: Cam Synthesis. 1993 ISBN 0-7923-2536-227. W.S. Hall: The Boundary Element Method. 1993 ISBN 0-7923-2580-X28. J. Angeles, G. Hommel and P. Kovacs (eds.): Computational Kinematics. 1993
ISBN 0-7923-2585-029. A. Curnier: Computational Methods in Solid Mechanics. 1994 ISBN 0-7923-2761-630. D.A. Hills and D. Nowell: Mechanics of Fretting Fatigue. 1994 ISBN 0-7923-2866-331. B. Tabarrok and F.P.J. Rimrott: Variational Methods and Complementary Formulations in
Dynamics. 1994 ISBN 0-7923-2923-632. E.H. Dowell (ed.), E.F. Crawley, H.C. Curtiss Jr., D.A. Peters, R. H. Scanlan and F. Sisto: A
Modern Course in Aeroelasticity. Third Revised and Enlarged Edition. 1995ISBN 0-7923-2788-8; Pb: 0-7923-2789-6
33. A. Preumont: Random Vibration and Spectral Analysis. 1994 ISBN 0-7923-3036-634. J.N. Reddy (ed.): Mechanics of Composite Materials. Selected works of Nicholas J. Pagano.
1994 ISBN 0-7923-3041-235. A.P.S. Selvadurai (ed.): Mechanics of Poroelastic Media. 1996 ISBN 0-7923-3329-236. Z. Mroz, D. Weichert, S. Dorosz (eds.): Inelastic Behaviour of Structures under Variable
Loads. 1995 ISBN 0-7923-3397-737. R. Pyrz (ed.): IUTAM Symposium on Microstructure-Property Interactions in Composite
Materials. Proceedings of the IUTAM Symposium held in Aalborg, Denmark. 1995ISBN 0-7923-3427-2
38. M.I. Friswell and J.E. Mottershead: Finite Element Model Updating in Structural Dynamics.1995 ISBN 0-7923-3431-0
39. D.F. Parker and A.H. England (eds.): IUTAM Symposium on Anisotropy, Inhomogeneity andNonlinearity in Solid Mechanics. Proceedings of the IUTAM Symposium held in Nottingham,U.K. 1995 ISBN 0-7923-3594-5
40. J.-P. Merlet and B. Ravani (eds.): Computational Kinematics ’95. 1995 ISBN 0-7923-3673-941. L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Mech-
anics and Inverse Problems. 1996 ISBN 0-7923-3849-942. J. Mencik: Mechanics of Components with Treated or Coated Surfaces. 1996
ISBN 0-7923-3700-X43. D. Bestle and W. Schiehlen (eds.): IUTAM Symposium on Optimization of Mechanical Systems.
Proceedings of the IUTAM Symposium held in Stuttgart, Germany. 1996ISBN 0-7923-3830-8
44. D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky: Solution of Crack Problems. TheDistributed Dislocation Technique. 1996 ISBN 0-7923-3848-0
45. V.A. Squire, R.J. Hosking, A.D. Kerr and P.J. Langhorne: Moving Loads on Ice Plates. 1996ISBN 0-7923-3953-3
46. A. Pineau and A. Zaoui (eds.): IUTAM Symposium on Micromechanics of Plasticity andDamage of Multiphase Materials. Proceedings of the IUTAM Symposium held in Sevres,Paris, France. 1996 ISBN 0-7923-4188-0
47. A. Naess and S. Krenk (eds.): IUTAM Symposium on Advances in Nonlinear StochasticMechanics. Proceedings of the IUTAM Symposium held in Trondheim, Norway. 1996
ISBN 0-7923-4193-748. D. Iesan and A. Scalia: Thermoelastic Deformations. 1996 ISBN 0-7923-4230-5
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49. J.R. Willis (ed.): IUTAM Symposium on Nonlinear Analysis of Fracture. Proceedings of theIUTAM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4378-6
50. A. Preumont: Vibration Control of Active Structures. An Introduction. 1997ISBN 0-7923-4392-1
51. G.P. Cherepanov: Methods of Fracture Mechanics: Solid Matter Physics. 1997ISBN 0-7923-4408-1
52. D.H. van Campen (ed.): IUTAM Symposium on Interaction between Dynamics and Control inAdvanced Mechanical Systems. Proceedings of the IUTAM Symposium held in Eindhoven,The Netherlands. 1997 ISBN 0-7923-4429-4
53. N.A. Fleck and A.C.F. Cocks (eds.): IUTAM Symposium on Mechanics of Granular and PorousMaterials. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997
ISBN 0-7923-4553-354. J. Roorda and N.K. Srivastava (eds.): Trends in Structural Mechanics. Theory, Practice, Edu-
cation. 1997 ISBN 0-7923-4603-355. Yu.A. Mitropolskii and N. Van Dao: Applied Asymptotic Methods in Nonlinear Oscillations.
1997 ISBN 0-7923-4605-X56. C. Guedes Soares (ed.): Probabilistic Methods for Structural Design. 1997
ISBN 0-7923-4670-X57. D. Francois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume I: Elasticity
and Plasticity. 1998 ISBN 0-7923-4894-X58. D. Francois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume II: Visco-
plasticity, Damage, Fracture and Contact Mechanics. 1998 ISBN 0-7923-4895-859. L.T. Tenek and J. Argyris: Finite Element Analysis for Composite Structures. 1998
ISBN 0-7923-4899-060. Y.A. Bahei-El-Din and G.J. Dvorak (eds.): IUTAM Symposium on Transformation Problems
in Composite and Active Materials. Proceedings of the IUTAM Symposium held in Cairo,Egypt. 1998 ISBN 0-7923-5122-3
61. I.G. Goryacheva: Contact Mechanics in Tribology. 1998 ISBN 0-7923-5257-262. O.T. Bruhns and E. Stein (eds.): IUTAM Symposium on Micro- and Macrostructural Aspects
of Thermoplasticity. Proceedings of the IUTAM Symposium held in Bochum, Germany. 1999ISBN 0-7923-5265-3
63. F.C. Moon: IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics inMechanics. Proceedings of the IUTAM Symposium held in Ithaca, NY, USA. 1998
ISBN 0-7923-5276-964. R. Wang: IUTAM Symposium on Rheology of Bodies with Defects. Proceedings of the IUTAM
Symposium held in Beijing, China. 1999 ISBN 0-7923-5297-165. Yu.I. Dimitrienko: Thermomechanics of Composites under High Temperatures. 1999
ISBN 0-7923-4899-066. P. Argoul, M. Fremond and Q.S. Nguyen (eds.): IUTAM Symposium on Variations of Domains
and Free-Boundary Problems in Solid Mechanics. Proceedings of the IUTAM Symposiumheld in Paris, France. 1999 ISBN 0-7923-5450-8
67. F.J. Fahy and W.G. Price (eds.): IUTAM Symposium on Statistical Energy Analysis. Proceedingsof the IUTAM Symposium held in Southampton, U.K. 1999 ISBN 0-7923-5457-5
68. H.A. Mang and F.G. Rammerstorfer (eds.): IUTAM Symposium on Discretization Methods inStructural Mechanics. Proceedings of the IUTAM Symposium held in Vienna, Austria. 1999
ISBN 0-7923-5591-1
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
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69. P. Pedersen and M.P. Bendsøe (eds.): IUTAM Symposium on Synthesis in Bio Solid Mechanics.Proceedings of the IUTAM Symposium held in Copenhagen, Denmark. 1999
ISBN 0-7923-5615-270. S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. 1999
ISBN 0-7923-5681-071. A. Carpinteri: Nonlinear Crack Models for Nonmetallic Materials. 1999
ISBN 0-7923-5750-772. F. Pfeifer (ed.): IUTAM Symposium on Unilateral Multibody Contacts. Proceedings of the
IUTAM Symposium held in Munich, Germany. 1999 ISBN 0-7923-6030-373. E. Lavendelis and M. Zakrzhevsky (eds.): IUTAM/IFToMM Symposium on Synthesis of Non-
linear Dynamical Systems. Proceedings of the IUTAM/IFToMM Symposium held in Riga,Latvia. 2000 ISBN 0-7923-6106-7
74. J.-P. Merlet: Parallel Robots. 2000 ISBN 0-7923-6308-675. J.T. Pindera: Techniques of Tomographic Isodyne Stress Analysis. 2000 ISBN 0-7923-6388-476. G.A. Maugin, R. Drouot and F. Sidoroff (eds.): Continuum Thermomechanics. The Art and
Science of Modelling Material Behaviour. 2000 ISBN 0-7923-6407-477. N. Van Dao and E.J. Kreuzer (eds.): IUTAM Symposium on Recent Developments in Non-linear
Oscillations of Mechanical Systems. 2000 ISBN 0-7923-6470-878. S.D. Akbarov and A.N. Guz: Mechanics of Curved Composites. 2000 ISBN 0-7923-6477-579. M.B. Rubin: Cosserat Theories: Shells, Rods and Points. 2000 ISBN 0-7923-6489-980. S. Pellegrino and S.D. Guest (eds.): IUTAM-IASS Symposium on Deployable Structures: Theory
and Applications. Proceedings of the IUTAM-IASS Symposium held in Cambridge, U.K., 6–9September 1998. 2000 ISBN 0-7923-6516-X
81. A.D. Rosato and D.L. Blackmore (eds.): IUTAM Symposium on Segregation in GranularFlows. Proceedings of the IUTAM Symposium held in Cape May, NJ, U.S.A., June 5–10,1999. 2000 ISBN 0-7923-6547-X
82. A. Lagarde (ed.): IUTAM Symposium on Advanced Optical Methods and Applications in SolidMechanics. Proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France,August 31–September 4, 1998. 2000 ISBN 0-7923-6604-2
83. D. Weichert and G. Maier (eds.): Inelastic Analysis of Structures under Variable Loads. Theoryand Engineering Applications. 2000 ISBN 0-7923-6645-X
84. T.-J. Chuang and J.W. Rudnicki (eds.): Multiscale Deformation and Fracture in Materials andStructures. The James R. Rice 60th Anniversary Volume. 2001 ISBN 0-7923-6718-9
85. S. Narayanan and R.N. Iyengar (eds.): IUTAM Symposium on Nonlinearity and StochasticStructural Dynamics. Proceedings of the IUTAM Symposium held in Madras, Chennai, India,4–8 January 1999 ISBN 0-7923-6733-2
86. S. Murakami and N. Ohno (eds.): IUTAM Symposium on Creep in Structures. Proceedings ofthe IUTAM Symposium held in Nagoya, Japan, 3-7 April 2000. 2001 ISBN 0-7923-6737-5
87. W. Ehlers (ed.): IUTAM Symposium on Theoretical and Numerical Methods in ContinuumMechanics of Porous Materials. Proceedings of the IUTAM Symposium held at the Universityof Stuttgart, Germany, September 5-10, 1999. 2001 ISBN 0-7923-6766-9
88. D. Durban, D. Givoli and J.G. Simmonds (eds.): Advances in the Mechanis of Plates and ShellsThe Avinoam Libai Anniversary Volume. 2001 ISBN 0-7923-6785-5
89. U. Gabbert and H.-S. Tzou (eds.): IUTAM Symposium on Smart Structures and Structonic Sys-tems. Proceedings of the IUTAM Symposium held in Magdeburg, Germany, 26–29 September2000. 2001 ISBN 0-7923-6968-8
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
90. Y. Ivanov, V. Cheshkov and M. Natova: Polymer Composite Materials – Interface Phenomena& Processes. 2001 ISBN 0-7923-7008-2
91. R.C. McPhedran, L.C. Botten and N.A. Nicorovici (eds.): IUTAM Symposium on Mechanicaland Electromagnetic Waves in Structured Media. Proceedings of the IUTAM Symposium heldin Sydney, NSW, Australia, 18-22 Januari 1999. 2001 ISBN 0-7923-7038-4
92. D.A. Sotiropoulos (ed.): IUTAM Symposium on Mechanical Waves for Composite StructuresCharacterization. Proceedings of the IUTAM Symposium held in Chania, Crete, Greece, June14-17, 2000. 2001 ISBN 0-7923-7164-X
93. V.M. Alexandrov and D.A. Pozharskii: Three-Dimensional Contact Problems. 2001ISBN 0-7923-7165-8
94. J.P. Dempsey and H.H. Shen (eds.): IUTAM Symposium on Scaling Laws in Ice Mechanicsand Ice Dynamics. Proceedings of the IUTAM Symposium held in Fairbanks, Alaska, U.S.A.,13-16 June 2000. 2001 ISBN 1-4020-0171-1
95. U. Kirsch: Design-Oriented Analysis of Structures. A Unified Approach. 2002ISBN 1-4020-0443-5
96. A. Preumont: Vibration Control of Active Structures. An Introduction (2nd Edition). 2002ISBN 1-4020-0496-6
97. B.L. Karihaloo (ed.): IUTAM Symposium on Analytical and Computational Fracture Mechan-ics of Non-Homogeneous Materials. Proceedings of the IUTAM Symposium held in Cardiff,U.K., 18-22 June 2001. 2002 ISBN 1-4020-0510-5
98. S.M. Han and H. Benaroya: Nonlinear and Stochastic Dynamics of Compliant Offshore Struc-tures. 2002 ISBN 1-4020-0573-3
99. A.M. Linkov: Boundary Integral Equations in Elasticity Theory. 2002ISBN 1-4020-0574-1
100. L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Me-chanics and Inverse Problems (2nd Edition). 2002
ISBN 1-4020-0667-5; Pb: 1-4020-0756-6101. Q.P. Sun (ed.): IUTAM Symposium on Mechanics of Martensitic Phase Transformation in
Solids. Proceedings of the IUTAM Symposium held in Hong Kong, China, 11-15 June 2001.2002 ISBN 1-4020-0741-8
102. M.L. Munjal (ed.): IUTAM Symposium on Designing for Quietness. Proceedings of the IUTAMSymposium held in Bangkok, India, 12-14 December 2000. 2002 ISBN 1-4020-0765-5
103. J.A.C. Martins and M.D.P. Monteiro Marques (eds.): Contact Mechanics. Proceedings of the3rd Contact Mechanics International Symposium, Praia da Consolacao, Peniche, Portugal,17-21 June 2001. 2002 ISBN 1-4020-0811-2
104. H.R. Drew and S. Pellegrino (eds.): New Approaches to Structural Mechanics, Shells andBiological Structures. 2002 ISBN 1-4020-0862-7
105. J.R. Vinson and R.L. Sierakowski: The Behavior of Structures Composed of Composite Ma-terials. Second Edition. 2002 ISBN 1-4020-0904-6
106. Not yet published.107. J.R. Barber: Elasticity. Second Edition. 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6108. C. Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large
Strains. Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August2001. 2003 ISBN 1-4020-1170-9
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
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109. P. Stahle and K.G. Sundin (eds.): IUTAM Symposium on Field Analyses for Determinationof Material Parameters – Experimental and Numerical Aspects. Proceedings of the IUTAMSymposium held in Abisko National Park, Kiruna, Sweden, July 31 – August 4, 2000. 2003
ISBN 1-4020-1283-7110. N. Sri Namachchivaya and Y.K. Lin (eds.): IUTAM Symposium on Nonlnear Stochastic Dynam-
ics. Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 – 30 August,2000. 2003 ISBN 1-4020-1471-6
111. H. Sobieckzky (ed.): IUTAM Symposium Transsonicum IV. Proceedings of the IUTAM Sym-posium held in Gottingen, Germany, 2–6 September 2002, 2003 ISBN 1-4020-1608-5
112. J.-C. Samin and P. Fisette: Symbolic Modeling of Multibody Systems. 2003ISBN 1-4020-1629-8
113. A.B. Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisationin Problems of Mechanics. Proceedings of the IUTAM Symposium held in Liverpool, UnitedKingdom, 8-11 July 2002. 2003 ISBN 1-4020-1780-4
114. S. Ahzi, M. Cherkaoui, M.A. Khaleel, H.M. Zbib, M.A. Zikry and B. LaMatina (eds.): IUTAMSymposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior ofEngineering Materials. Proceedings of the IUTAM Symposium held in Marrakech, Morocco,20-25 October 2002. 2004 ISBN 1-4020-1861-4
115. H. Kitagawa and Y. Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of FractureProcess and Materials Strength. Proceedings of the IUTAM Symposium held in Osaka, Japan,6-11 July 2003. Volume in celebration of Professor Kitagawa’s retirement. 2004
ISBN 1-4020-2037-6116. E.H. Dowell, R.L. Clark, D. Cox, H.C. Curtiss, Jr., K.C. Hall, D.A. Peters, R.H. Scanlan, E.
Simiu, F. Sisto and D. Tang: A Modern Course in Aeroelasticity. 4th Edition, 2004ISBN 1-4020-2039-2
117. T. Burczynski and A. Osyczka (eds.): IUTAM Symposium on Evolutionary Methods in Mech-anics. Proceedings of the IUTAM Symposium held in Cracow, Poland, 24-27 September 2002.2004 ISBN 1-4020-2266-2
118. D. Iesan: Thermoelastic Models of Continua. 2004 ISBN 1-4020-2309-X119. G.M.L. Gladwell: Inverse Problems in Vibration. Second Edition 2004 ISBN 1-4020-2670-6120. J.R. Vinson: Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials,
Including Sandwich Construction. 2005 ISBN 1-4020-3110-6121. Forthcoming.122. G. Rega and F. Vestroni (eds.): IUTAM Symposium on Chaotic Dynamics and Control of
Systems and Processes in Mechanics. Proceedings of the IUTAM Symposium held in Rome,Italy, 8–13 June 2003. 2005 ISBN 1-4020-3267-6
123. E.E. Gdoutos: Fracture Mechanics. An Introduction. 2nd edition. 2005 ISBN 1-4020-2863-2124. M.D. Gilchrist (ed.): IUTAM Symposium on Impact Biomechanics from Fundamental Insights
to Applications. 2005 ISBN 1-4020-3795-3125. J.M. Huyghe, P.A.C. Raats and S.C. Cowin (eds.): IUTAM Symposium on Physicochemical
and Electromechanical Interactions in Porous Media. 2005 ISBN 1-4020-3864-X126. H. Ding, W. Chen and L. Zhang: Elasticity of Transversely Isotropic Materials. 2005
ISBN 1-4020-4033-4127. W. Yang (ed.): IUTAM Symposium on Mechanics and Reliability of Actuating Materials.
Proceedings of the IUTAM Symposium held in Beijing, China, 1–3 September 2004. 2005ISBN 1-4020-4131-6
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
128. J.-P. Merlet: Parallel Robots. 2006 ISBN 1-4020-4132-2129. G.E.A. Meier and K.R. Sreenivasan (eds.): IUTAM Symposium on One Hundred Years of
Boundary Layer Research. Proceedings of the IUTAM Symposium held at DLR-Gottingen,Germany, August 12–14, 2004. 2006 ISBN 1-4020-4149-7
130. H. Ulbrich and W. Gunthner (eds.): IUTAM Symposium on Vibration Control of NonlinearMechanisms and Structures. 2006 ISBN 1-4020-4160-8
131. L. Librescu and O. Song: Thin-Walled Composite Beams. Theory and Application. 2006ISBN 1-4020-3457-1
132. G. Ben-Dor, A. Dubinsky and T. Elperin: Applied High-Speed Plate Penetration Dynamics.2006 ISBN 1-4020-3452-0
133. X. Markenscoff and A. Gupta (eds.): Collected Works of J. D. Eshelby. Mechanics of Defectsand Inhomogeneities. 2006 ISBN 1-4020-4416-X
134. R.W. Snidle and H.P. Evans (eds.): IUTAM Symposium on Elastohydrodynamics and Microelast-ohydrodynamics. Proceedings of the IUTAM Symposium held in Cardiff, UK, 1–3 September,2004. 2006 ISBN 1-4020-4532-8
135. T. Sadowski (ed.): IUTAM Symposium on Multiscale Modelling of Damage and FractureProcesses in Composite Materials. Proceedings of the IUTAM Symposium held in KazimierzDolny, Poland, 23–27 May 2005. 2006 ISBN 1-4020-4565-4
136. A. Preumont: Mechatronics. Dynamics of Electromechanical and Piezoelectric Systems. 2006ISBN 1-4020-4695-2
137. M.P. Bendsøe, N. Olhoff and O. Sigmund (eds.): IUTAM Symposium on Topological DesignOptimization of Structures, Machines and Materials. Status and Perspectives. 2006
ISBN 1-4020-4729-0138. A. Klarbring: Models of Mechanics. 2006 ISBN 1-4020-4834-3139. H.D. Bui: Fracture Mechanics. Inverse Problems and Solutions. 2006 ISBN 1-4020-4836-X140. M. Pandey, W.-C. Xie and L. Xu (eds.): Advances in Engineering Structures, Mechanics
and Construction. Proceedings of an International Conference on Advances in EngineeringStructures, Mechanics & Construction, held in Waterloo, Ontario, Canada, May 14–17, 2006.2006 ISBN 1-4020-4890-4
141. G.Q. Zhang, W.D. van Driel and X. J. Fan: Mechanics of Microelectronics. 2006ISBN 1-4020-4934-X
142. Q.P. Sun and P. Tong (eds.): IUTAM Symposium on Size Effects on Material and StructuralBehavior at Micron- and Nano-Scales. Proceedings of the IUTAM Symposium held in HongKong, China, 31 May–4 June, 2004. 2006 ISBN 1-4020-4945-5
143. A.P. Mouritz and A.G. Gibson: Fire Properties of Polymer Composite Materials. 2006ISBN 1-4020-5355-X
144. Y.L. Bai, Q.S. Zheng and Y.G. Wei (eds.): IUTAM Symposium on Mechanical Behavior andMicro-Mechanics of Nanostructured Materials. Proceedings of the IUTAM Symposium heldin Beijing, China, 27–30 June 2005. 2007 ISBN 1-4020-5623-0
145. L.P. Pook: Metal Fatigue. What It Is, Why It Matters. 2007 ISBN 1-4020-5596-6146. H.I. Ling, L. Callisto, D. Leshchinsky and J. Koseki (eds.): Soil Stress-Strain Behavior:
Measurement, Modeling and Analysis. A Collection of Papers of the Geotechnical Symposiumin Rome, March 16–17, 2006. 2007 ISBN 978-1-4020-6145-5
147. A.S. Kravchuk and P.J. Neittaanmaki: Variational and Quasi-Variational Inequalities in Mech-anics. 2007 ISBN 978-1-4020-6376-3
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
148. S.K. Kanaun and V. Levin: Self-Consistent Methods for Composites. Vol. 1: Static Problems.2008 ISBN 978-1-4020-6663-4
149. G. Gogu: Structural Synthesis of Parallel Robots. Part 1: Methodology. 2008ISBN 978-1-4020-5102-9
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