Effect of Friction on Legged Robots

8/11/2019 Effect of Friction on Legged Robots

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Undergraduate project thesis titled

Study of the effect of friction on feet forces and

joint torques of a legged robotSubmitted in partial fulfillment of the requirement

for the degree ofBachelor in Technology

inMechanical Engineering

ByAmit Datta (10/ME/32),

Prateek Dugar (10/ME/39)and

Ronit Nandi (10/ME/41)

Under the supervision ofDr. Nirmal Baran Hui

Associate Professor

Department of Mechanical EngineeringNIT Durgapur

Department of Mechanical EngineeringNational Institute of Technology Durgapur

West Bengal, India

May 26, 2014


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NATIONAL INSTITUTE OF TECHNOLOGY DURGAPUR(Deemed University)

CERTIFICATEThis is to certify that Amit Datta (10/ME/32), Prateek Dugar (10/ME/39) andRonit Nandi (10/ME/41) have carried out the research work presented in this projectreport entitled Study of the effect of friction on feet forces and joint torques of alegged robot for the award of Bachelor of Technology degree from National Instituteof Technology Durgapur in the academic session 2013-2014, under my supervision.The project embodies result of original work and studies carried out by studentsthemselves and the contents of project do not form the basis for the award of anyother degree to the candidate or anybody else.

Dr. Nirmal Baran HuiThesis Supervisor and Project Guide

(Associate Professor),

Department of Mechanical Engineering

NIT Durgapur


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ABSTRACT

In this study, we take a look at how the equation of motion, feet forces and jointtorques of a legged robot change with different surfaces. As different surfaces havedifferent coefficient of friction, the reaction forces and torque for each leg changesaccordingly. In future the results of this study can be used to compute the energyconsumption changes for a legged robot across different surfaces.

Authors:

Amit Datta (10/ME/32)Prateek Dugar (10/ME/39)Ronit Nandi (10/ME/41)

Thesis Supervisor:

Dr. Nirmal Baran HuiAssociate Professor,Department of Mechanical EngineeringNIT Durgapur


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ACKNOWLEDGEMENT

We would like to take this opportunity to thank our mentor and project supervisorDr. Nirmal Baran Hui for his support and patience, without which this project wouldnever have taken shape. We would like to thank the Head of the Department, Prof.Anup Kumar Saha for so graciously accepting this project thesis. We would like tothank all the professors of our department for instilling in us curiosity and the thirstto learn from diverse fields. We would like to acknowledge the contributions of our

fellow students, whose support has been irreplaceable in providing an environmentconducive to academics and healthy growth. Above all we would like to thank ourparents and families without whose uninterrupted nurturing neither we nor thisproject would ever come to be.


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CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Paper 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Paper 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Paper 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 Paper 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9 Paper 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Dynamics(also known as multibody system) . . . . . . . . . . . . . . 26

4. Formulation and Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1 Generalised kinematic equation . . . . . . . . . . . . . . . . . . . . . 304.2 Acceleration equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Torque equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Torque equation with friction . . . . . . . . . . . . . . . . . . . . . . 35

5. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


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LIST OF FIGURES

2.1 Top view of the six-legged robot walking in a circular path . . . . . . 102.2 Six-legged walking robot model . . . . . . . . . . . . . . . . . . . . . 15

3.1 Denavit-Hartenberg frame assignment for the PUMA robot . . . . . . 233.2 A robotic leg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Degrees of freedom in a robotic arm . . . . . . . . . . . . . . . . . . . 27

4.1 A 3 degree of freedom manipulator . . . . . . . . . . . . . . . . . . . 31

5.1 Variation of torque with friction . . . . . . . . . . . . . . . . . . . . . 36


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1. INTRODUCTION

Research on legged robots has attracted many attentions during the last century.The main purpose of the development of legged robots lie in the advantages of leggedrobots that they are able to go to the place that inaccessible for wheeled or trackedvehicles.

Legged robots are more suitable for complex environments than wheeled or

trucked ones. Legged robots include single-legged hoppers, two-legged humanoidrobots, quadrupeds, hexapods and others with more legs. Hexapod, especially hy-brid hexapod robots can easily accomplish static stable walking and therefore attractthe attention of the research community. Kinematics and dynamics analyses andplanning of the gait are key elements to achieve legged walking. These topics havebeen widely investigated.

A mobile multi-legged robot is a parallel-serial hybrid system. Kinematics anddynamics of multi-legged robots can be complicated and have generated a lot ofdiscussions. Many studies as examined the kinematics of a walking robot body as

a parallel mechanism. However, the kinematics and dynamics of walking machinesare much different from that of parallel mechanisms: a walking machine is morecomplicated due to its many degrees of freedom; the topology of a walking machineis more complicated because of mechanism changes while robot legs change onlybetween transferring phases and supporting phases. In the body of a walking robotwas dealt as another link of each leg, but the coupling effects among supportinglegs, body and swing legs were not completely taken into account. It is necessary toconsider the entire kinematics of a robot for a certain locomotion type.

This report presents an analysis on how legged robots behave on different terrainshaving different coefficients of friction. Mathematical formulation has been doneand the results have been collected from multiple computer programs. Differentparameters such as angular velocity, torque and joint forces have been computedover a range of value of coefficient of friction. Respective graphs have been plottedas required.


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2. LITERATURE REVIEW

This section covers the different published papers that were studied, in order tounderstand robot kinematics, dynamics, gaiting and related energy consumptions.

2.1 Paper 1

Shibendu Shekhar Roy, Dilip Kumar Pratihar, Effects of turning gait parameters

on energy consumption and stability of a six-legged walking robot, Robotics andAutonomous Systems, Volume 60, Issue 1, January 2012, pp. 72-82.

Problem Studied

Energy consumption, which plays major role in locomotion of a robot, is muchmore for a legged robot than a wheeled robot due to driving and control units,other than the trunk and the payload. But for terrain with irregularities a leggedrobot is always preferred to a wheeled robot. The given journal paper focusses onthe analysis of the dynamics and energy efficiency for generating turning motionin a six legged robot. Turning motion or gaits are more complex as compared touni-directional gaits.

Methodology Applied

In this study, an attempt is made to minimize energy consumption of a six-legged robot during its turning on flat terrain. A power consumption model isderived for statically stable wave gaits by minimizing dissipating power for optimalfoot force distribution and minimizing total energy expenditure for optimal selectionof gait parameters, namely angular velocity, angular stroke and duty factor. Twoapproaches, namely Approach 1 (that is, minimization of norm of feet forces) and

Approach 2 (that is, minimization of norm of joint torques) are developed and theirperformances are tested through computer simulations for generating the turninggait of a six-legged robot with four different duty factors. The effects of angularstroke and duty factors on the minimum value of NESM over a locomotion cycle arealso analyzed.

Claim/Achievement

It is seen that Approach 2 is more efficient compared to Approach 1 for all dutyfactors. The variations of average power consumption and energy consumption perunit weight per unit traveled length with angular velocity and angular stroke are

studied. The most energy-efficient gait is obtained for the minimum value of duty


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2. Literature Review 9

Fig. 2.1: Top view of the six-legged robot walking in a circular path

factor and maximum value of angular velocity. For a particular value of duty factor,specific resistance is seen to decrease with the lower values of angular stroke, and

it is found to decrease further with the higher values of duty factor. In order tominimize total energy consumption, the velocity should be as high as possible for aparticular duty factor. As angular velocity increases, the maximum reachable dutyfactor is reduced due to the dynamic constraints of joint actuators.

Readers Observation

This paper presents an analysis on energy consumption of a six-legged robotduring its turning motion over a flat terrain. An energy consumption model isdeveloped for statically stable wave gaits in order to minimize dissipating energyfor optimal feet forces distributions. The effects of gait parameters, namely angularvelocity, angular stroke and duty factors are studied on energy consumption, asthe six-legged robot walks along a circular path of constant radius with wave gait.The variations of average power consumption and energy consumption per unitweight per unit traveled length with the angular velocity and angular stroke arecompared for the turning gaits of a robot with four different duty factors. Computersimulations show that wave gait with a low duty factor is more energy efficientcompared to that with a high duty factor at the highest possible angular velocity.


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2.2 Paper 2

Mustafa Suphi Erden, Kemal Leblebicioglu, Free gait generation with reinforce-ment learning for a six legged robot, Robotics and Autonomous Systems, Volume

56, Issue 3, 31 March 2008, pp. 199-212.

Problem Studied

For free gait generations, two models - the Reflex model and the Central PatternGenerator (CPG) model have already been developed previously. However, theCPG models are not as adaptive as the reflex models considering the changingenvironment and system conditions. The reflex model, on the other hand, canmanage such situations. It gets feedback from the legs and determines the commandsaccordingly. The problem with the reflex model, on the other hand, is the lack of adurable walking pattern, and the necessity of processing the feedback from all legsfor every step. The approach presented here is an attempt to conciliate CPG andreflex models using the ideas of central generation of free gaits and reinforcementlearning for state transitions, respectively. Free gait generation learning are done onthe discrete model of stepping.

Methodology Applied

In the given paper, for free state gait generation mainly two rules are used. Thefirst is known as the rule of neighborhood which has five theorems in it. It says thatif one of the legs does not find a proper position to place the tip on the front side

of its stroke, it places it somewhere in the middle or rear of the stroke. The gaitof the machine is modified immediately depending on the current stroke positionof the legs. In the free gait generation the authors make use of the stability rule,which is adopted as the rule of neighborhood. The second rule used is the freestate generation algorithm. Based on the theorems of the rule of neighborhood, thisalgorithm is developed. This algorithm guarantees that the rule of neighborhood issatisfied and avoids any -1 or -2 states. It takes the current state and the speed ofwalk in units per iteration as the input, and generates the next state as the output.

Claim/Achievement

In this paper the problems of free gait generation, continuous improvement ofgaits, and adaptation to the unexpected condition of a rear-leg deficiency are ad-dressed for a six-legged robot. While the free gait generation corresponds to thecentral part of gait generation, the reinforcement learning incorporated with thiscorresponds to the reactive part. The possible walking patterns are generated bythe free gait generation and among these the ones most successful for adaptation arememorized and utilized by the reinforcement learning. In this way the approachesof central pattern generation and reflex model are somewhat conciliated and theirefficiency are utilized in a single gait generation scheme.


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Readers Observation

In this paper the problem of free gait generation and adaptability with reinforce-ment learning are addressed for a six-legged robot. Using the developed free gait

generation algorithm the robot maintains to generate stable gaits according to thecommanded velocity. The reinforcement learning scheme incorporated into the freegait generation makes the robot choose more stable states and develop a continuouswalking pattern with a larger average stability margin. While walking in normalconditions with no external effects causing instability, the robot is guaranteed tohave stable walk, and the reinforcement learning only improves the stability. Theadaptability of the learning scheme is tested also for the abnormal case of deficiencyin one of the rear-legs. The robot gets a negative reinforcement when it falls, and apositive reinforcement when a stable transition is achieved.

2.3 Paper 3Xiaowen Yu, Chenglong Fu, Ken Chen, Modelling and control of a single legged

robot, Procedia Engineering, Volume 24, Issue 3, 2011, pp. 788-792.

Problem Studied

A large number of researchers have done many fabulous works on legged systems.Many other works have been done to improve the original legged system of Raibert.But despite merits of Raiberts three part control scheme, it still suffers from thevagueness of forward speed control part, in which he didnt provide the specification

in determination of several parameters. This paper solves this problem in a mathe-matic manner that is able to give a simple way to determine the parameters in thespeed control.

Methodology Applied

This paper deals with a planer single-legged robot that is capable of hopping.The model of the robot is constructed as an inverted pendulum with an additionalflywheel. The phases of each hopping cycle are anglicized and are treated withdifferent control scheme. By adopting the three part control scheme that proposedby Raibert, the speed control part by solving the equations of motion of the modelis improved and introduced before which provides a better way to specify controlparameters which are not addressed by Raibert. Simulation results of rate regulationshow that the control scheme is able to regulate the rate of single-legged Robot ata desired speed with high stability thus show the control is valid and feasible forrealizing rate regulation.

Claim/Achievement

This paper designs a simple single-legged robot that is capable of hopping. Themodel of single-legged robot is constructed. Compared to the simple linear speed

control that adopted by Raibert, three part control scheme to realize rate regulation


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in a mathematic manner is used after analyzing the characteristics of the hoppingcycle. The simulation results show that the control scheme is valid and feasible. Also,the control scheme is easy to achieve and of relatively high precision. However, itis worth noting that speed of the robot cannot be stabilized under a wide range of

time, which requires more accurate controller and stability analysis of the model infuture work.

Readers Observation

In this paper, the author has divide the control scheme into two parts: the stancephase and the flight phase. The control goal of the system is to maintain the single-legged robot hopping at a desired speed with stability. As the constraints on thesystem are different from stance and flight phases, these two phases are controlledseparately. The model has been simulated on a computer as an inverted pendulum

and results have been plotted as a relationship between xo and x (the displacementfunction with respect to time).

2.4 Paper 4

Shibendu Shekhar Roy, Dilip Kumar Pratihar, Soft computing-based expert sys-tems to predict energy consumption and stability margin in turning gaits of sixlegged robots, Expert systems with Applications, Volume 39, Issue 5, April 2012,pp. 5460-5469.

Problem StudiedDifferent methods have been applied in the past to study the energy consumption

and stability margin in turning gaits of six legged robots. The problem of optimalturning gait generation of a six legged robot had been solved previously using acombined genetic algorithm and fuzzy logic approach. It was extended for optimalpath and gait generations of a hexapod walking robot, but it considered a simplifiedmodel of the robot. It did not consider a detailed kinematics and dynamic behaviorof the leg and trunk body, although this contribution to gait generations was signif-icant. In the present study, attempts have been made to develop expert systems forthe said purpose using the principle of soft computing.

Methodology Applied

The prime aim of this study is to develop an expert system for predicting spe-cific energy consumption and stability margin for turning motion of a six-leggedrobot. Four different soft computing based expert systems (that is, Approaches 1through 4) have been developed. In Approach 1 and 2, optimal values of premiseand consequent parameters of multiple ANFIS have been found using BP algorithmand GA, respectively. The GA has been used to determine optimal structure ofCANFIS and BPNN in Approach 3 and 4, respectively. A batch mode of traininghas been adopted. The performances of four approaches have been compared for 50


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test cases in terms of percent deviation in predictions of specific energy consumptionand NESM.

Claim/Achievement

After four different approaches have been used to develop soft-computing basedexpert system to study energy consumption and stability margin in turning gaits,the comparison indicates that Approach 2 has achieved a slightly better accuracyin predictions compared to other approaches. The better performance of Approach2 could be due to the use of two separate ANFIS structures for the two outputsand that of a GA in place of BP algorithm. The solutions of BP algorithm may getstuck at local minima, whereas the chance of GA-solutions for being trapped intothe local minima is less.

Readers ObservationTurning gaits are the most general and very important ones for omni-directional

walking of a six-legged robot. Soft computing-based expert systems have been de-veloped in the present work to predict specific energy consumption and stabilitymargin of turning gait of a six-legged robot. Besides back-propagation neural net-work, three approaches based on adaptive neuro-fuzzy inference system have beendeveloped and their performances are compared with each other. Genetic algorithm-tuned multiple adaptive neuro-fuzzy inference systems are found to perform betterthan other approaches. This could be due to a more exhaustive search conductedby the genetic algorithm in place of back propagation algorithm and the use of two

separate adaptive neuro-fuzzy inference systems for two different outputs.

2.5 Paper 5

Shibendu Shekhar Roy, Ajay Kumar Singh, Dilip Kumar Pratihar, Estimation ofoptimal feet forces and joint torques for on-line control of six-legged robot, Roboticsand Computer-Integrated Manufacturing, Volume 27, Issue 5, October 2011, pp.910-917.

Problem Studied

With todays technologies, the legged systems have the disadvantages of low pay-load to weight ratio, and poor energy efficiency. An autonomous walking robotcannot function satisfactorily with poor energy efficiency, due to the fact that it hasto carry all driving and control units in addition to pay-load and trunk body. Longduration missions are also subject to power supply constraints. The minimizationof energy consumption plays a key role in the design of an autonomous multi-leggedrobot. A reduction in energy consumption results in robots, which not only travelmore but also require the smaller actuators that typically yield a reduction in theirweight and cost.


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Methodology Applied

Two approaches, namely Approach 1 (that is, minimization of norm of feetforces) and Approach 2 (that is, minimization of norm of joint torques) have been

developed. On the other hand, both the approaches developed in the present studymay be suitable for on-line implementations. Moreover, Approach 2 has yieldeda less variation of joint torques of the six-legged robot in comparison with thatobtained by Approach 1. Computational time of both the approaches is found tobe low and thus, they are suitable for real-time implementations. In practice, asix-legged robot is to be controlled on-line by consuming minimum energy aftermaintaining its dynamic balance, particularly when it generates non-periodic gaitsduring locomotion. The authors will be working on these issues in future.

Fig. 2.2: Six-legged walking robot model

Claim Achievement

An attempt has been made in the present paper to obtain optimal distributionsof feet forces and values of joint torques of a six-legged robot on-line. The results ofthe two approaches are found to be in-line with those of some published papers. It isimportant to mention that Approach 2 is seen to be more energy efficient comparedto Approach 1. Approach 2 makes a good use of friction between the tips of the

supporting legs and terrain.

Readers Observation

The present study aims to estimate optimal feet forces and joint torques necessaryfor real-time control of a six-legged robot. Two approaches have been developed,such as minimization of norm of feet forces and minimization of norm of joint torquesusing the least squared method. Results of these two approaches have been comparedwith each other, and with those of available literature. As both of these approachesare found to be computationally fast, these are suitable for real-time control of thesix-legged robot.


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2.6 Paper 6

Shibendu Shekhar Roy, Dilip Kumar Pratihar, Dynamic modeling, stability andenergy consumption analysis of a realistic six-legged walking robot, Robotics and

Computer-Integrated Manufacturing, Volume 29, Issue 2, April 2013, pp. 400-416.

Problem Studied

A multi-legged robot possesses tremendous potential for maneuverability overrough terrain, particularly in comparison with conventional wheeled or tracked mo-bile robot. It introduces more flexibility and terrain adaptability at the cost of lowspeed and increased control complexity. This paper may be considered as an exten-sion of the previous paper (i.e. only one type of gait pattern, alternating tripod gaitwas considered to determine energy efficient feet force and torque distributions).Here a more detailed parametric and systematic study related to energy consump-tion and stability of a realistic six-legged robot has been conducted.

Methodology Applied

Two different approaches are developed to determine optimal feet forces. In thefirst approach, minimization of the norm of feet forces is carried out using a leastsquare method, whereas minimization of the norm of joint torques is performed inthe second approach. The second approach is found to be more energy efficientcompared to the first one. The maximum values of feet forces and joint torques areseen to decrease with the increase of duty factor. The effects of walking parameters,

namely velocity, stroke and duty factors have been studied on energy consumptionand stability of the robot.

Claim/Achievement

This paper deals with a detailed analysis on kinematics, dynamics, stability andenergy consumption of a realistic six-legged robot. The aim of this study is to extenda previous work of the author, in order to estimate optimal feet forces and jointtorques of the six-legged robot generating wave-gaits with four different duty factorsand deal with its stability issues. The variations of average power consumption andspecific energy consumption with the velocity and stroke are compared for four

different duty factors. Wave gait with a low duty factor is found to be more energyefficient compared to that with a high duty factor at the highest possible velocity.

Readers Observation

Besides kinematic analysis, two approaches, such as minimization of norm of feetforces and minimization of norm of joint torques have been developed to estimateoptimal distributions of feet forces during straight- forward motion of a six-leggedrobot with four different duty factors. The effects of walking parameters have beenstudied on energy consumption and stability margin of the said robot. It has been

observed that approach II provides more energy efficient distributions of feet forcescompared to approach I. Moreover, approach II has yielded less variation in joint


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torques of the six-legged robot in comparison with that obtained by approach I. Itis also important to notice that the joint torques of the supporting legs have turnedout to be much more compared to that of the swing legs.

2.7 Paper 7

S.V. Shah, S.K. Saha, J.K. Dutt, Modular framework for dynamic modelling andanalysis of legged robots, Mechanism and Machine Theory, Volume 49, March 2012,pp. 234-255.

Problem Studied

Fundamentally, legged robots are floating-base, variable-constraint, and tree-typesystems moving with high joint acceleration. Thus, dynamics plays a vital role in

their control. A way to define contact is to use a compliant model. In this method,ground reactions are computed from feet positions and velocities, where the groundis represented as a system that can exert vertical conservative and dissipative forces,and horizontal frictional forces. This is referred to as the configuration independentapproach. This method allows for a single set of dynamic equations of motion forthe legged robots irrespective of their phases, double or multi supports in stance.By varying the parameters of the ground model, a variety of walking and runningsurfaces can be simulated. The dynamic formulation proposed in this paper will beapplicable to any type of legged robots.

Methodology AppliedIn this paper, a legged robot is modeled as a floating-base tree-type system where

the foot-ground interactions are represented as external forces and moments. Dy-namic formulation thus obtained is independent of the configuration or state of thelegged robot. Framework for dynamic modeling is proposed with the concept ofkinematic modules, where each module is a set of serially connected links. A veloc-ity transformation based approach is used to obtain the minimal-order equations ofmotion, and module-level analytical expressions for the vectors and matrices appear-ing in them. Recursive algorithms for inverse and forward dynamics are proposedby using inter- and intra-modular recursions for the first time. Analyses of a planar

biped and spatial quadruped are presented using the proposed methodology.

Claim/Achievement

The constrained equations of motion were then obtained using the module-DeNOC matrices. Empowered with the proposed modular framework, the inverseand forward dynamics algorithms with inter- and intra- modular recursions havebeen developed for the simulation of controlled legged robots, which were illustratedwith a planar biped and a spatial quadruped. Inter-modular recursion has been con-ceived in this work for the first time. The algorithms in terms of the computationalcounts were found to be better than the algorithms available in literature, whereas


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in terms of CPU time they were found to be 16% to 30% more efficient than anO(n3) algorithm.

Readers Observation

This paper presents a general framework for the dynamic modeling and analyses ofthe legged robots using the concept of kinematic modules where each module is a setof serially connected links only. Legged robots having general module architecturehave been considered. Recursive kinematic relationships were obtained between anytwo adjoining modules and the concepts like module-twist propagation, module-level Decoupled Natural Orthogonal Complement (DeNOC) matrices, etc. wereintroduced. Such concepts helped to treat a system with large number of linksas a system with smaller number of modules. This generalizes the concept of theDeNOC-based formulation originally proposed for a serial-chain system consisting

of link to multi-modular tree-type systems, which is one of the basic contributionsof this work.

2.8 Paper 8

L.S. Martin-Filho, R. Prajoux, Locomotion control of a four-legged robot em-bedding real-time reasoning in the force distribution, Robotics and AutonomousSystems, Volume 32, Issue 4, 30 September 2000, pp. 219-235.

Problem Studied

This paper discusses the application of rule-based reasoning to manage in realtime the force distribution computation within a locomotion control of quadrupedrobots. The control uses inputoutput linearization in the attitude subsystem, andoptimal linear control in the overall locomotion system. The force distribution ap-proach provides more adaptability and flexibility to the locomotion control, becausethe system is capable of fast adaptation to a wide variety of situations. Rules defin-ing the knowledge about how to deal with walk events and feet forces calculation arepresented. The rule-based reasoning is made using the KHEOPS system (LAAS).

Methodology Applied

This work is done in the framework of an extension to legged robots of the generalapproach of operational and decisional autonomy that we are using at LAAS/CNRSfor wheeled mobile robots. The kind of robots that is considered is intended typicallyfor intervention tasks in unstructured or partly structured environments, indoors oroutdoors. The paper discusses issues around the problem of locomotion control andadaptive force distribution in legged robots. Although some of its results may applyto legged robots in general, and also to other kinds of closed kinematic chains withfriction contacts, the present work is done more specifically for quadruped robots.


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Claim/Achievement

In the domain of walking robots, and more specifically quadrupeds, the paperpresented a locomotion control and a force distribution approach, within an overall

architecture for a walk supervisor, based on classical control theory together withrule-based reasoning on events and situation. The methodology is well adapted toreal-time applications because it allows a direct calculation of the force distributionin most cases, and with very few iterations at worst. Besides, the rule-based rea-soning is carried out specifically by means of a real-time dedicated implementation.The FD calculations, in spite of a loss of optimality sometimes, provide solutionsthat are always sound from an engineering point of view.

Readers Observation

To summarize, the dynamic simulation has proven the soundness of the main

concepts of this work that the paper recalls here:

A fully compliant leg system via force control, while the robots body is con-trolled in position/speed.

The use of a simplified model in the control loops.

A flexible force distribution system, embedding multiple ways of solving thedistribution.

A rule-based reasoning system functioning in real time to supervise the opera-

tion of both the control loops and the force distribution, and designed to reactto the various walk events.

2.9 Paper 9

Yi Sun, Shugen Ma, Yang Yang, Huayan Pu, Towards stable and efficient leggedrace-walking of an ePaddle-based robot, Mechatronics, Volume 23, Issue 1, February2013, pp. 108-120.

Problem Studied

In this paper, we focus on the motion planning method of the race-walking gaitfor achieving a highly stable and efficient legged walking with an ePaddle-basedquadruped robot. In part 2, the eccentric paddle mechanism as well as its versatilegait modes is firstly introduced. In part 3 and 4, the planning method of the race-walking gait for a single ePaddle module and the method of tracking planar path foran ePaddle-based quadruped are respectively presented. Gait characteristics relatedto the stability and energetic efficiency of the race-walking gait, such as duty factor,static stability, and specific resistance, are discussed in part 5. Finally, the proposedidea to race-walking and walking planning methods are verified by simulations, andare discussed in part 6.


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Methodology Applied

This paper studies the motion planning method of this unique race-walking gaitfor an ePaddle-based quadruped robot. The standard gait sequence that consists of

four phases is firstly presented. The selection of wheel-center trajectory for achiev-ing the gait is then discussed based on kinematic models of the ePaddle module inthese phases. Two motion planning methods are presented for an ePaddle-basedquadruped robot to track planar path with the proposed race-walking gait. Stabil-ity and energetic performances of the proposed race-walking gait are discussed byevaluating duty factor of the ePaddle module, and by measuring stability marginand specific resistance of the robot. A set of simulations on tracking straight andcircular paths verifies the idea of the race-walking gait as well as its stability andefficiency.

Claim/AchievementIn this paper, a unique race-walking gait for achieving highly stable and highly

energetic efficient legged walking for an ePaddle-based robot has been presented.The standard gait sequence which consists of four phases, has been proposed. Mo-tion planning method based on wheel-center trajectory planning has been discussedfor both single ePaddle module and for an ePaddle-based quadruped robot. Twotracking strategies with and without rotation have been presented for tracking cir-cular paths. To prove the stability and energetic efficiency of the race-walking gait,measures of locomotion, such as duty factor of the ePaddle module, absolutely sta-bility margin, and specific resistance, have been applied to our robot. Simulations

on tracking planar straight-line and circular paths verified the idea of race-walkinggait as well as its performance.

Readers Observation

The work suggests that by applying appropriate landing and lifting trajectories,performance of race-walking gait can be further improved. On the other hand, theoptimal design of ePaddle mechanism will lead to a better locomotion performanceas well. We hope that a focused effort toward energy minimization will lead to fullyautonomous, highly efficient mobile robots in the near future.


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3. KINEMATICS AND DYNAMICS

Robot kinematics and dynamics refers to the study of the motion of element in akinematic chain with reference to different reference frames. It also deals with thedifferent forces and torques involved in producing such a motion.

3.1 Kinematics

Robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. Theemphasis on geometry means that the links of the robot are modeled as rigid bodiesand its joints are assumed to provide pure rotation or translation.

Robot kinematics studies the relationship between the dimensions and con-nectivity of kinematic chains and the position, velocity and acceleration of eachof the links in the robotic system, in order to plan and control movement and tocompute actuator forces and torques. The relationship between mass and inertiaproperties, motion, and the associated forces and torques is studied as part of robot

dynamics.

3.2 Kinematic equations

A fundamental tool in robot kinematics is the kinematics equations of thekinematic chains that form the robot. These non-linear equations are used to mapthe joint parameters to the configuration of the robot system. Kinematics equationsare also used in biomechanics of the skeleton and computer animation of articulatedcharacters.

Forward kinematics uses the kinematic equations of a robot to compute theposition of the end-effector from specified values for the joint parameters. Thereverse process that computes the joint parameters that achieve a specified positionof the end-effector is known as inverse kinematics. The dimensions of the robot andits kinematics equations define the volume of space reachable by the robot, knownas its workspace.

There are two broad classes of robots and associated kinematics equations se-rial manipulators and parallel manipulators. Other types of systems with specializedkinematics equations are air, land, and submersible mobile robots, hyper-redundant,

or snake, robots and humanoid robots.


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3. Kinematics and Dynamics 21

Forward kinematics

Forward kinematics specifies the joint parameters and computes the configu-ration of the chain. For serial manipulators this is achieved by direct substitution

of the joint parameters into the forward kinematics equations for the serial chain.For parallel manipulators substitution of the joint parameters into the kinematicsequations requires solution of the a set of polynomial constraints to determine theset of possible end-effector locations. In case of a Stewart platform there are 40configurations associated with a specific set of joint parameters.

Forward kinematics refers to the use of the kinematic equations of a robot tocompute the position of the end-effector from specified values for the joint parame-ters. The kinematics equations of the robot are used in robotics, computer games,and animation.

The kinematics equations for the series chain of a robot are obtained using arigid transformation [Z] to characterize the relative movement allowed at each jointand separate rigid transformation [X] to define the dimensions of each link. The re-sult is a sequence of rigid transformations alternating joint and link transformationsfrom the base of the chain to its end link, which is equated to the specified positionfor the end link,

[T] = [Z1][X1][Z2][X2]...[Xn1][Zn], (3.1)

where [T] is the transformation locating the end-link. These equations arecalled the kinematics equations of the serial chain.

In 1955, Jacques Denavit and Richard Hartenberg introduced a conventionfor the definition of the joint matrices [Z] and link matrices [X] to standardize thecoordinate frame for spatial linkages. This convention positions the joint frame sothat it consists of a screw displacement along the Z-axis.

[Zi] =T ransZi(di)RotZi(i), (3.2)

and it positions the link frame so it consists of a screw displacement along the X-axis,

[Xi] =T ransXi(ai,i+1)RotXi(i,i+1), (3.3)

Using this notation, each transformation-link goes along a serial chain robot, andcan be described by the coordinate transformation,i1Ti= [Zi][Xi] =T ransZi(di)RotZi(i)TransXi(ai,i+1)RotXi(i,i+1), (3.4)

where i, di, i,i+1, ai,i+1are also known as Denavit-Hartenberg parameters.

The kinematics equations of a serial chain ofn links, with joint parametersi are given by

[T] =0 Tn=ni=1

i1Ti(i), (3.5)

where i1

Ti(i) is the transformation matrix from the frame of link i to link i 1.In robotics, these are conventionally described by DenavitHartenberg parameters.


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The matrices associated with these operations are:

TransZi(di) =

1 0 0 00 1 0 0

0 0 1 di0 0 0 1

,RotZi(i) =

cos i sin i 0 0sin i cos i 0 0

0 0 1 di0 0 0 1

(3.6)

Similiarly,

TransXi(ai,i+1) =

1 0 0 ai,i+10 1 0 00 0 1 00 0 0 1

, (3.7)

RotXi(i,i+1) =

1 0 0 00 cos i,i+1 sin i,i+1 0

0 sin i,i+1 cos i,i+1 00 0 0 1

(3.8)

The use of the Denavit-Hartenberg convention yields the link transformationmatrix, [i1Ti] as

i1Ti=

cos i sin icos i,i+1 sin isin i,i+1 ai,i+1cos isin i cosicos i,i+1 cos isin i,i+1 ai,i+1sin i

0 sin i,i+1 cos i,i+1 di0 0 0 1

(3.9)

known as Denavit-Hartenberg matrix.

Fig. 3.1: Denavit-Hartenberg frame assignment for the PUMA robot

Inverse kinematics

Inverse kinematics specifies the end-effector location and computes the asso-ciated joint angles. For serial manipulators this requires solution of a set of polyno-

mials obtained from the kinematics equations and yields multiple configurations for


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the chain. The case of a general 6R serial manipulator (a serial chain with six revo-lute joints) yields sixteen different inverse kinematics solutions, which are solutionsof a sixteenth degree polynomial. For parallel manipulators, the specification of theend-effector location simplifies the kinematics equations, which yields formulas for

the joint parameters.

Inverse kinematics refers to the use of the kinematics equations of a robot todetermine the joint parameters that provide a desired position of the end-effector.Specification of the movement of a robot so that its end-effector achieves a desiredtask is known as motion planning. Inverse kinematics transforms the motion planinto joint actuator trajectories for the robot.

The movement of a kinematic chain whether it is a robot or an animatedcharacter is modeled by the kinematics equations of the chain. These equations

define the configuration of the chain in terms of its joint parameters. Forwardkinematics uses the joint parameters to compute the configuration of the chain, andinverse kinematics reverses this calculation to determine the joint parameters thatachieves a desired configuration.

For example, inverse kinematics formulas allow calculation of the joint pa-rameters that position a robot arm to pick up a part. Similar formulas determinethe positions of the skeleton of an animated character that is to move in a particularway.

Kinematic analysis is one of the first steps in the design of most industrialrobots. Kinematic analysis allows the designer to obtain information on the positionof each component within the mechanical system. This information is necessary forsubsequent dynamic analysis along with control paths.

Inverse kinematics is an example of the kinematic analysis of a constrainedsystem of rigid bodies, or kinematic chain. The kinematic equations of a robot canbe used to define the loop equations of a complex articulated system. These loopequations are non-linear constraints on the configuration parameters of the system.The independent parameters in these equations are known as the degrees of freedom

of the system.

While analytical solutions to the inverse kinematics problem exist for a widerange of kinematic chains, computer modeling and animation tools often use New-tons method to solve the non-linear kinematics equations.

Other applications of inverse kinematic algorithms include interactive manip-ulation, animation control and collision avoidance.


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Approximating solutions to IK systems

There are many methods of modelling and solving inverse kinematics prob-lems. The most flexible of these methods typically rely on iterative optimization

to seek out an approximate solution, due to the difficulty of inverting the forwardkinematics equation and the possibility of an empty solution space. The core ideabehind several of these methods is to model the forward kinematics equation usinga Taylor series expansion, which can be simpler to invert and solve than the originalsystem.

The Jacobian inverse technique is a simple yet effective way of implementinginverse kinematics. Let there be m variables that govern the forward-kinematicsequation, i.e the position function. These variables may be joint angles, lengths,or other arbitrary real values. If the IK system lives in a 3-dimensional space, the

position function can be viewed as a mapping p(x) :R

m

R

3

. Let p(x0

) give theinitial position of the system, and p(x0 + ) be the goal of the system. The Jacobianinverse technique simply attempts to generate iteratively improved estimates ofto minimize the error given by ||p( x0+ ) p( x0)||. Each one of the intermediateestimates can be added to x0and evaluated by the position function to animate thesystem.

For small -vectors, the series expansion of the position function gives:

p( x0+) p( x0) +Jp( x0) (3.10)

Where Jp( x0) is the (3 x m) Jacobian matrix of the position function at x0.

Note that the (i, k)-th entry of the Jacobian matrix can be determined nu-merically:

pixk

pi(x0,k+h) pi( x0)

h (3.11)

Where pi(x) gives the i-th component of the position function, x0,k+ h is simply x0with a small delta added to its k-th component, and h is a reasonably small positivevalue.

Taking the Moore-Penrose pseudoinverse of the Jacobian and re-arrangingterms results in:

J+p ( x0)p (3.12)

Where p = p( x0+) p( x0). It is possible to use a singular value decompositionto obtain the pseudo-inverse of the Jacobian.

Applying the inverse Jacobian method once will result in a very rough estimateof the desired-vector. A line search should be used to scale this to an acceptablevalue. The estimate for can be improved via the following algorithm (known asthe Newton-Raphson method):

k+1=J+

p ( xk)pk (3.13)


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Once some-vector has caused the error to drop close to zero, the algorithm shouldterminate. Existing methods based on the Hessian matrix of the system have beenreported to converge to desired values using fewer iterations, though, in somecases more computational resources.

Fig. 3.2: A robotic leg

3.3 Dynamics(also known as multibody system)A multibody system is used to model the dynamic behavior of interconnected

rigid or flexible bodies, each of which may undergo large translational and rotationaldisplacements.

Introduction

The systematic treatment of the dynamic behavior of interconnected bodieshas led to a large number of important multibody formalisms in the field of me-chanics. The simplest bodies or elements of a multibody system were treated by

Newton (free particle) and Euler (rigid body). Euler introduced reaction forces be-tween bodies. Later, a series of formalisms were derived, only to mention Lagrangesformalisms based on minimal coordinates and a second formulation that introducesconstraints.

Basically, the motion of bodies is described by their kinematic behavior. Thedynamic behavior results from the equilibrium of applied forces and the rate ofchange of momentum. Nowadays, the term multibody system is related to a largenumber of engineering fields of research, especially in robotics and vehicle dynamics.As an important feature, multibody system formalisms usually offer an algorithmic,

computer-aided way to model, analyze, simulate and optimize the arbitrary motionof possibly thousands of interconnected bodies.


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Concept

A body is usually considered to be a rigid or flexible part of a mechanicalsystem (not to be confused with the human body). An example of a body is the arm

of a robot, a wheel or axle in a car or the human forearm. A link is the connectionof two or more bodies, or a body with the ground. The link is defined by certain(kinematical) constraints that restrict the relative motion of the bodies. Typicalconstraints are:

cardan joint or Universal Joint; 4 kinematical constraints

prismatic joint; relative displacement along one axis is allowed, constrainsrelative rotation; implies 5 kinematical constraints

revolute joint; only one relative rotation is allowed; implies 5 kinematical con-straints; see the example above

spherical joint; constrains relative displacements in one point, relative rotationis allowed; implies 3 kinematical constraints

Fig. 3.3: Degrees of freedom in a robotic arm

There are two important terms in multibody systems: degree of freedom and con-straint condition.

The degrees of freedom denote the number of independent kinematical pos-sibilities to move. In other words, degrees of freedom are the minimum number ofparameters required to completely define the position of an entity in space.

A rigid body has six degrees of freedom in the case of general spatial motion,three of them translational degrees of freedom and three rotational degrees of free-

dom. In the case of planar motion, a body has only three degrees of freedom withonly one rotational and two translational degrees of freedom.


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The degrees of freedom in planar motion can be easily demonstrated usinge.g. a computer mouse. The degrees of freedom are: left-right, up-down and therotation about the vertical axis.

A constraint condition implies a restriction in the kinematical degrees offreedom of one or more bodies. The classical constraint is usually an algebraicequation that defines the relative translation or rotation between two bodies. Thereare furthermore possibilities to constrain the relative velocity between two bodies ora body and the ground. This is for example the case of a rolling disc, where the pointof the disc that contacts the ground has always zero relative velocity with respect tothe ground. In the case that the velocity constraint condition cannot be integratedin time in order to form a position constraint, it is called non-holonomic. This isthe case for the general rolling constraint. In addition to that there are non-classicalconstraints that might even introduce a new unknown coordinate, such as a sliding

joint, where a point of a body is allowed to move along the surface of another body.In the case of contact, the constraint condition is based on inequalities and thereforesuch a constraint does not permanently restrict the degrees of freedom of bodies.

Equations of Motion

The equations of motion are used to describe the dynamic behavior of a multibodysystem. Each multibody system formulation may lead to a different mathematicalappearance of the equations of motion while the physics behind is the same. Themotion of the constrained bodies is described by means of equations that resultbasically from Newtons second law. The equations are written for general motion ofthe single bodies with the addition of constraint conditions. Usually the equationsof motions are derived from the Newton-Euler equations or Lagranges equations.

The motion of rigid bodies is described by means of

M(q)q Qv+ CqT= F, (3.14)

C(q,q) = 0 (3.15)

These types of equations of motion are based on so-called redundant coordinates,because the equations use more coordinates than degrees of freedom of the under-lying system. The generalized coordinates are denoted by q, the mass matrix isrepresented by M(q) which may depend on the generalized coordinates. C repre-sents the constraint conditions and the matrix Cq (sometimes termed the Jacobian)is the derivation of the constraint conditions with respect to the coordinates. Thismatrix is used to apply constraint forces to the according equations of the bodies.The components of the vector are also denoted as Lagrange multipliers. In a rigidbody, possible coordinates could be split into two parts,

q = [u ]T (3.16)

where urepresents translations and describes the rotations.


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In the case of rigid bodies, the so-called quadratic velocity vector Qv is usedto describe Coriolis and centrifugal terms in the equations of motion. The nameis because Qv includes quadratic terms of velocities and it results due to partialderivatives of the kinetic energy of the body.

The Lagrange multiplier i is related to a constraint condition Ci = 0 andusually represents a force or a moment, which acts in direction of the constraintdegree of freedom. The Lagrange multipliers do no work as compared to externalforces that change the potential energy of a body.

The equations of motion (1,2) are represented by means of redundant coordi-nates, meaning that the coordinates are not independent. This can be exemplifiedby the slider-crank mechanism shown above, where each body has six degrees offreedom while most of the coordinates are dependent on the motion of the other

bodies. For example, 18 coordinates and 17 constraints could be used to describethe motion of the slider-crank with rigid bodies. However, as there is only one de-gree of freedom, the equation of motion could be also represented by means of oneequation and one degree of freedom, using e.g. the angle of the driving link as degreeof freedom. The latter formulation has then the minimum number of coordinatesin order to describe the motion of the system and can be thus called a minimalcoordinates formulation. The transformation of redundant coordinates to minimalcoordinates is sometimes cumbersome and only possible in the case of holonomicconstraints and without kinematical loops. Several algorithms have been developedfor the derivation of minimal coordinate equations of motion, to mention only the so-

called recursive formulation. The resulting equations are easier to be solved becausein the absence of constraint conditions, standard time integration methods can beused to integrate the equations of motion in time. While the reduced system mightbe solved more efficiently, the transformation of the coordinates might be compu-tationally expensive. In very general multibody system formulations and softwaresystems, redundant coordinates are used in order to make the systems user-friendlyand flexible.


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4. FORMULATION AND CALCULATION

4.1 Generalised kinematic equation

Generalised kinematic equation of a 3-link manipulator is given below:

T1= 0A1

1A22A3 (4.1)

Where,

0A1=

cos 1 cos 1sin 1 sin 1sin 1 a1cos 1sin 1 cos 1cos1 sin 1cos 1 a1sin 1

0 sin 1 cos 1 d10 0 0 1

(4.2)

1A2=


0 sin 2 cos 2 d20 0 0 1

(4.3)

2A3=


0 sin 3 cos 3 d30 0 0 1

(4.4)

Where,i1Ai is the transformation matrix of the adjacent co-ordinate frames;i is the joint angle;i is the offset angle of the arm;di is the distance of origin of (i-1)th frame to (i)th frame;ai is the offset distance of each axis interaction;i1Tirepresents the arm matrix which specifies the endpoint of the manipulator withrespect to the reference co-ordinate frame of the base.


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4. Formulation and Calculation 30

Fig. 4.1: A 3 degree of freedom manipulator

4.2 Acceleration equation

t0= 0 (4.5)

t1= 0.5 (4.6)

D= (t t0)/(t1 t0) (4.7)

h1= /3 (4.8)

h0= 5/12 (4.9)

a1=h1 h0 (4.10)

= h0 a1 D2 (3 2D) (4.11)

2= /4 (4.12)

T2=

cos 2 sin 2 0 10 sin 2sin 2 cos 2 0 10cos 2

0 0 1 00 0 0 1

(4.13)

3= /6 (4.14)

T3=

cos 3 sin 3 0 5 sin 3sin 3 cos 3 0 5cos 3

0 0 1 00 0 0 1

(4.15)

b=

cos sin sin cos

(4.16)

A=

0 1 0 01 0 0 00 0 0 0

0 0 0 0

(4.17)


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=d

dt (4.18)

=d

dt (4.19)

T1=

cos sin 0 10 sin sin cos 0 10cos

0 0 1 00 0 0 1

(4.20)

T =A A T1 T2

0001

(4.21)

hp = /6 (4.22)

h2= /4 (4.23)

a2=h3 h2 (4.24)

2=h2 a2 D2 (3 2D) (4.25)

b2=

cos 2 sin 2sin 2 cos 2

(4.26)

=d

dt (4.27)

2=

d2dt (4.28)

=d

dt (4.29)

2

=d2

dt (4.30)

T1=

cos sin 0 10 sin sin cos 0 10cos

0 0 1 00 0 0 1

(4.31)

T2=

cos 2 sin 2 0 0.1 sin(+2)sin 2 cos 2 0 10cos 2

0 0 1 00 0 0 1

(4.32)

V =A A T1 T2 T3

0001

+T1 A A T2 T3

0001

2 (4.33)


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4.3 Torque equation

t0= 0 (4.34)

t1= 0.5 (4.35)

D= (t t0)/(t1 t0) (4.36)

h1= /3 (4.37)

h2= 5/12 (4.38)

a1=h1 h0 (4.39)

h3= /6 (4.40)

h2= /4 (4.41)

a2=h3 h2 (4.42)

= h0 a1 D2 (3 2D) (4.43)

2=h2 a2 D2 (3 2D) (4.44)

b=

cos(3/2 +) sin (3/2 +)sin (3/2 +) cos (3/2 +)

(4.45)

b2=

cos(+2) sin (+2)sin (+2) cos (+2)

(4.46)

m= 1 (4.47)

l= 0.1 (4.48)

A=

0 1 0 01 0 0 00 0 0 00 0 0 0

(4.49)

g=

0 9.8 0 0

(4.50)

=d(3/2 +)

dt (4.51)

2=

d(+2)

dt (4.52)

=d

dt (4.53)

2 =d2

dt (4.54)

J1=

ml2/3 0 0 0.5m0 0 0 00 0 0 0

0.5ml 0 0 m

(4.55)


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J2=

ml2/3 0 0 0.5m0 0 0 00 0 0 0

0.5ml 0 0 m

(4.56)

T1=

cos(3/2 +) sin(3/2 +) 0 0.1 sin(3/2 +)sin(3/2 +) cos(3/2 +) 0 0.1 sin(3/2 +)

0 0 1 00 0 0 1

(4.57)

T2=

cos(+2) sin(+2) 0 0.1sin(+2)sin(+2) cos(+2) 0 0.1sin(+2)

0 0 1 00 0 0 1

(4.58)

U11=A T1 (4.59)

U21= A T1 T2 (4.60)

U22= T1 A T2 (4.61)

D11= trace(U11 J1 U

11) +trace(U21 J2 U

21) (4.62)

D12=trace(U22 J2 U

22) (4.63)

D12=D22 (4.64)

D22=trace(U22 J2 U

22) (4.65)

U111=A A T1 (4.66)

U211=A A T1 T2 (4.67)U221=U212 (4.68)

U222=T1 A A T2 (4.69)

H1= trace(U111J1U

11)(j)(j)+trace(U211J2U

21)(j)(j)+trace(U221

J2U

21)(j)(j) + trace(U212J2U

21)(j)(j) + trace(U222J2U

21)(j)(j)

H2= trace(U211J2U

21)(j)(j)+trace(U221J2U

21)(j)(j)+trace(U212

J2 U

21)(j)(j)

c1=mg U11

0.5

001

mg U21

0.5

001

(4.70)

c2= mg U22

0.5

001

(4.71)

T rq= D11 D12D21 D22

11(j)22(j)

+ H1H2 + c1

c2 (4.72)


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4.4 Torque equation with friction

Ff=u m g(1, 2) sin(x) (4.73)

d1=l cos (4.74)

d2=l sin (4.75)

R1=Ff (d1+d2) (4.76)

R2=Ff (d2) (4.77)

T rq=

D11 D12D21 D22

11(j)22(j)

+

H1H2

+

c1c2

+

R1R2

(4.78)


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5. RESULTS

Fig. 5.1: Variation of torque with friction

The above graph is a plot of all the data derived from a computer simulation ofthe torque equation across various coefficients of friction. The top part representsthe variation of torque in the first degree link and the bottom part represents thevariation of torque in the second degree link.


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Bibliography

[1] Shibendu Shekhar Roy, Dilip Kumar Pratihar, Effects of turning gait parameterson energy consumption and stability of a six-legged walking robot, Robotics andAutonomous Systems, Volume 60, Issue 1, January 2012, pp. 72-82.

[2] Mustafa Suphi Erden, Kemal Leblebicioglu, Free gait generation with reinforce-ment learning for a six legged robot, Robotics and Autonomous Systems, Volume56, Issue 3, 31 March 2008, pp. 199-212.

[3] Xiaowen Yu, Chenglong Fu, Ken Chen, Modelling and control of a single leggedrobot, Procedia Engineering, Volume 24, Issue 3, 2011, pp. 788-792.

[4] Shibendu Shekhar Roy, Dilip Kumar Pratihar, Soft computing-based expert sys-tems to predict energy consumption and stability margin in turning gaits of sixlegged robots, Expert systems with Applications, Volume 39, Issue 5, April 2012,pp. 5460-5469.

[5] Shibendu Shekhar Roy, Ajay Kumar Singh, Dilip Kumar Pratihar, Estimationof optimal feet forces and joint torques for on-line control of six-legged robot,

Robotics and Computer-Integrated Manufacturing, Volume 27, Issue 5, October2011, pp. 910-917.

[6] Shibendu Shekhar Roy, Dilip Kumar Pratihar, Dynamic modeling, stability andenergy consumption analysis of a realistic six-legged walking robot, Roboticsand Computer-Integrated Manufacturing, Volume 29, Issue 2, April 2013, pp.400-416.

[7] S.V. Shah, S.K. Saha, J.K. Dutt, Modular framework for dynamic modelling andanalysis of legged robots, Mechanism and Machine Theory, Volume 49, March2012, pp. 234-255.

[8] L.S. Martin-Filho, R. Prajoux, Locomotion control of a four-legged robot em-bedding real-time reasoning in the force distribution, Robotics and AutonomousSystems, Volume 32, Issue 4, 30 September 2000, pp. 219-235.

[9] Yi Sun, Shugen Ma, Yang Yang, Huayan Pu, Towards stable and efficient leggedrace-walking of an ePaddle-based robot, Mechatronics, Volume 23, Issue 1,February 2013, pp. 108-120.

[10] Saeed B. Niku, Introduction to Robotics, Analysis, Control, Applications, Sec-ond edition.


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Effect of Friction on Legged Robots

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