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ReviewDynamic analysis of exible manipulators, a literature review

Santosha Kumar Dwivedy a , Peter Eberhard b, *a Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781 039, Indiab Institute of Engineering and Computational Mechanics, University of Stuttgart, 70569 Stuttgart, Germany

Received 9 September 2005; received in revised form 14 January 2006; accepted 26 January 2006Available online 20 March 2006

Abstract

In this paper a survey of the literature related to dynamic analyses of exible robotic manipulators has been carried out.Both link and joint exibility are considered in this work and an effort has been made to critically examine the methodsused in these analyses, their advantages and shortcomings and possible extension of these methods to be applied to a gen-eral class of problems. Papers are classied according to modeling, control and experimental studies. In case of modelingthey are subdivided according to the method of analysis and number of links involved in the analysis. An effort has beenmade to include the works of a huge variety of researchers working in this eld and a total of 433 papers created in theyears 19742005 have been reviewed in this work.

2006 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7502. Modeling of flexible manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513. Single-link manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

3.1. Assumed mode method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.2. Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7533.3. Lumped parameter models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7533.4. Other studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754

4. Two-link manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7545. Multi-link manipulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7566. Flexible joint manipulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7577. Inverse dynamics and computational programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7598. Experimental investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7599. Control of flexible manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760

9.1. Single flexible link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7619.2. Two-link flexible manipulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

9.2.1. Multi-link flexible manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

0094-114X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2006.01.014

* Corresponding author.E-mail addresses: [email protected] (S.K. Dwivedy), [email protected] (P. Eberhard).

Mechanism and Machine Theory 41 (2006) 749777www.elsevier.com/locate/mechmt

MechanismandMachine Theory
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10. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763Acknowledgement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

1. Introduction

Robotic manipulators are widely used to help in dangerous, monotonous, and tedious jobs. Most of theexisting robotic manipulators are designed and build in a manner to maximize stiffness in an attempt to min-imize the vibration of the end-effector to achieve a good position accuracy. This high stiffness is achieved byusing heavy material and a bulky design. Hence, the existing heavy rigid manipulators are shown to be inef-cient in terms of power consumption or speed with respect to the operating payload. Also, the operation of high precision robots is severely limited by their dynamic deection, which persists for a period of time after amove is completed. The settling time required for this residual vibration delays subsequent operations, thusconicting with the demand of increased productivity. These conicting requirements between high speedand high accuracy have rendered the robotic assembly task a challenging research problem. Also, many indus-trial manipulators face the problem of arm vibrations during high speed motion.

In order to improve industrial productivity, it is required to reduce the weight of the arms and/or toincrease their speed of operation. For these purposes it is very desirable to build exible robotic manipulators.Compared to the conventional heavy and bulky robots, exible link manipulators have the potential advan-tage of lower cost, larger work volume, higher operational speed, greater payload-to-manipulator-weightratio, smaller actuators, lower energy consumption, better maneuverability, better transportability and saferoperation due to reduced inertia. But the greatest disadvantage of these manipulators is the vibration problemdue to low stiffness. For instance, it has been estimated that many cumulative hours would be spent in order todamp down the vibration within 1 in. in the remote manipulator system in a Space Station-assembly Shuttleight, Dubowsky [101]. Due to the importance and usefulness of these topics, researchers worldwide are now-adays engaged in the investigation of dynamics and control of exible manipulator. In the following, some

literature dealing with the application aspect of exible manipulators is cited.The study on the control of a exible arm manipulator started as a part of the space robots research, as aspace manipulator should be as light as possible in order to reduce its launching cost, Book [35,36]. Tzou [364],Uchiyama et al. [367], Alberts et al. [5], Krishnamurthy and Chao [192], Dubowsky [101], Cyril et al. [82],Mavrodis et al. [237], Caron [51], Kovecses et al. [186], Nagaraj [269], Gouliaev and Zarrazhina [135] and Miy-abe et al. [247] also studied exible manipulators used for space applications. Shi et al. [321] discussed somekey issues in the dynamic control of light weight robots for space and terrestrial applications.

Powerful and large robotic manipulators are needed in nuclear maintenance, e.g., to perform decontami-nation tasks. The nozzle dam positioning task for maintenance of a nuclear power plant steam generator isan example of a task that requires a strong manipulator with very ne absolute positioning accuracy [243].Jansen et al. [149] studied the long-reach-manipulator system for waste storage tank remediation. Kresset al. [188] studied modeling and control of waste tank cleanup manipulator.

Kumar et al. [196] studied exible manipulators used for micro-surgical operation and Riviere et al. [298]describes research in active instruments for enhanced accuracy in micro-surgery. Meggiolaro [244] analyzedthe patient positioning system used for cancer patient treatment at Massachusetts General Hospital, and Flanzstudied the same at the Northeast Proton Therapy Center [116]. Lin and Fu [214] and Chang and Fu [60] ana-lyzed the exible manipulator system for automated deburring operation. Munasinghe et al. [263] studied thehigh speed precise control of robot arms for trajectory generation. Yang and Sadler [406] used the nite ele-ment method (FEM) to study the dynamics of high speed machinery. Pfeiffer and Bremer [285] studied a sur-face polishing operation. Gruber and Schiehlen [138] and Schiehlen [313,314] studied the biped walking andwalking machines using the multibody dynamics approach.

Some other topics arising while using exible arms are collision control and contouring control which maybe used in grinding robots, painting or drawing robots and pattern recognition with soft touching and manysimilar applications, e.g., Fukuda and Kuribayashi [117] and Fukuda [118].

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As pointed out before, exible manipulators can nd many applications but since the main problem is tocontrol their vibrations, many researchers have tried to solve this problem by improving the dynamic modelsand incorporating different control strategies. In the following sections a review of the literature on modelingand control of the exible links and joints is carried out. Though one may nd even more references availablein this eld, here a total of 433 publications have been reviewed. It may be noted that a previous review in this

work was carried out by Gaultier [122] in 1989. Benosman and Vey [29] carried out a partial survey on thecontrol aspect of exible multi-link manipulators. This review will show the state-of-the-art in this researchto engineers, manufacturers, and scientists working on exible manipulators.

This review is divided into 10 sections. While Section 1 deals with a brief introduction of applications of exible manipulators, Section 2 describes the different modeling techniques for exible manipulators. Sections35 deal with the works on single, two and multi-link manipulators. In Section 6 the exible joints, in Section 7the inverse dynamics and some computational programs, in Section 8 the experimental investigation, and inSection 9 control of exible manipulators are described. Finally the work is summarized in the last section.

2. Modeling of exible manipulators

In this section different modeling techniques used in the analysis of exible manipulators are brieydescribed mentioning just a few references and detailed works on these techniques are cited in Sections 36.

There are two kinds of errors introduced if the exibility effect is not considered in the mathematical model.The rst kind of error is introduced in the torque requirement for the motors and the second kind results in thepositioning inaccuracy of the end-effector. The positioning of the end-effector for precision jobs should involvevery small amplitudes of vibration, ideally no vibration at all. Therefore, to achieve greater accuracy one hasto start with very accurate mathematical models for the system.

Different schemes for modeling of the manipulators are studied by a number of researchers as describedbelow. The mathematical models of the manipulators are generally derived from energy principles and fora simple rigid manipulator, the rigid arms store kinetic energy by virtue of their moving inertia and storepotential energy by virtue of their position in the gravitational eld, but the exible arms store potentialenergy by virtue of the deections of its links, joints, or drives. Joints have concentrated compliance which

may often be modeled as a pure spring storing only potential energy. Drive components such as shafts or beltsmay appear distributed but store little kinetic energy due to their low inertia, and a lumped parameter springmodel often succeeds well for them. Links are subjected to torsion, bending, and compression. Torsion of alink stores potential energy but little kinetic energy due to the low mass moment of inertia about the longitu-dinal axis of the beam and is thus well represented as a massless spring. Compression stores little potentialenergy due to high compressional stiffness and the dynamics along this axis is often well described by a rigidmass. Links subjected to bending store potential energy by virtue of their deection as well as kinetic energy byvirtue of their deection rates and a good model must include this distributed nature. To include bending onemay often use the EulerBernoulli equation which ignores shearing and rotary inertia effects. These two effectsmay be incorporated using a Timoshenko beam element which generally must be used if the beam is shortrelative to its diameter. But, since links may be considered as being rigid, Book [37], in most models of exiblemanipulators EulerBernoulli beams are used. The original dynamics of a exible link robot, being describedby partial differential equations and thus possessing an innite dimension, is not easily available to be useddirectly in both system analysis and control design. Most commonly the dynamic equations are truncatedto some nite dimensional models with either the assumed modes method (AMM) or the nite elementmethod (FEM).

The robotic systems with exible links are continuous dynamical systems characterized by an innite num-ber of degrees of freedom and are governed by nonlinear coupled, ordinary and partial differential equations.The exact solution of such systems is not feasible practically and the innite dimensional model imposes severeconstraints on the design of controllers as well. Hence, they are discretized using assumed modes, nite ele-ments or lumped parameter methods.

The assumed mode method and the nite element method use either the Lagrangian formulation or theNewtonEuler recursive formulation. In assumed mode model formulation, the link exibility is usually rep-resented by a truncated nite modal series in terms of spatial mode eigen functions and time-varying mode

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amplitudes. The main drawback of this method is the difficulty in nding modes for links with non-regularcross sections and multi-link manipulators [347]. Many authors used the nite element method where the elas-tic deformations are analyzed by assuming a known rigid body motion and later superposing the elastic defor-mation with the rigid body motion. In order to solve a large set of differential equations derived by the niteelement method, a lot of boundary conditions have to be considered, which are, in most situations, uncertain

for exible manipulators, Hastings and Book [139]. Using the assumed mode method to derive the equationsof motion of the exible manipulators, only the rst several modes are usually retained by truncation and thehigher modes are neglected. In the lumped parameter model, which is the simplest one for analysis purpose,the manipulator is modeled as spring and mass system, which does often not yield sufficiently accurate results.

Book initiated in 1974 research on the dynamic modeling and control of the exible link manipulators. Hemodeled an elastic chain with an arbitrary number of links and joints [35]. His model was limited by theassumption that the mass of the manipulator is negligible compared to the mass of the load. He used theassumed mode method in his analyses.

Classical analytical techniques can be employed to derive the dynamic equations of motion for exiblestructures. Due to the complexity of these equations, some kind of discretization technique is typically usedto construct a nite dimensional system of ordinary differential equations. In this context two families of nat-ural modes, i.e, the unconstrained and constrained modes of vibration, are considered [16,75]. The uncon-strained mode solution is dened as the natural motion obtained in the absence of all external inuences.In this case the structure as a whole is allowed to vibrate and the solution involves inertia properties of therigid and exible parts. In contrast, the constrained mode solution is dened as the natural motion obtainedin the absence of all external inuences and the rigid body is constrained to be xed or attached to an inertialreference frame [16]. Book [37] presents a tutorial on exible manipulators, where he examines the mathemat-ical representations commonly used in modeling exible links and joints and discussed the design consider-ation directly arising from the exible nature of the arms.

In the following Sections 36 the literature related to the different methods described above are groupeddepending on the number and type of links.

3. Single-link manipulators

In this section the literature related to the modeling of single exible link manipulators is discussed and thecontributions are grouped under the categories assumed mode method, nite element method, lumped para-meter models and other studies. In the assumed mode models also the works with distributed parametermodels are included.

3.1. Assumed mode method

In the assumed modes of model formulation, the link exibility is usually represented by a truncated nitemodal series, in terms of spatial mode eigen functions and time-varying mode amplitudes. Although thismethod has been widely used, there are several ways to choose link boundary conditions and mode eigenfunc-tions. Cannon and Schmitz [47], Sakawa et al. [306], Bayo [18], Chalhoub and Ulsoy [57,58], Hastings and Book[139,140], Wang and Wei [371], Tomei and Tornambe [353], Wang and Ravani [372,373], Sasiadek and Srini-vasan [311], Krishnamurthy [189], Low [223], Bellezza et al. [24], Chang and Gannon [59], Chen and Meng [66],Chapnik et al. [61], Chiang et al. [70], Feliu et al. [108], Tadikonda and Baruh [346], Wang and Vidyasagar[374,377,378], Zuo and Wang [433], Li and Sankar [209], Ravichandran et al. [293], Yoshikawa and Hosoda[413], Ankarali and Diken [6], Jnifene and Fahim [152], Karray et al. [163], Diken [97], Wedding and Eltimsahy[391], Zhang and Zhi [424], Karkoub and Tamma [162], Nagaraj et al. [269], Martins et al. [227,228] and Tsoet al. [360] studied single-link exible manipulators using Lagranges equation and the assumed mode method.

Rakhsha and Goldenberg [289] used a NewtonEuler formulation to model a single-link manipulator. Bar-bieri and Ozguner [16] used an extended Hamiltons principle to derive the equation of motion and studied theunconstrained and constrained mode of vibration. Singh [330] also used the same principle to derive the equa-tion of motion and further studied the nonlinear phenomena using a perturbation technique. Using the samemethod Krishnamurthy et al. [190] studied single-link robots fabricated from orthotropic composite materials.



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They have shown that the magnitude of the control spillover effects, an issue of great concern in designingcontrol systems for exible structures, is very small for the composite robotic arms.

Though many researchers studied manipulators with revolute joints, only few works are reported on pris-matic joints. Tabarrok et al. [345] developed the nonlinear equation of motion of an axially moving beam andthen linearized these equations to obtain one closed-form similarity solution and a semi-analytic solution for

specic axial velocities. Baruh and Tadikonda [17] established the dynamic model of exible manipulatorswith the elongation deformation and gave the relationship between the transverse and elongation deforma-tion. Tadikonda and Baruh [346] developed the dynamic model of a translating exible beam with a prismatic joint and studied different issues related to control in this case. Chalhoub and Ulsoy [58], Chang and Gannon[59], Krishnamurthy et al. [191], Wang and Vidyasagar [378,379], Buffinton and Kane [45], Buffinton [46], The-odore and Ghosal [348], Kar and Dwivedy [160], and Dwivedy and Kar [102,103] also dealt with single-linkexible manipulators with prismatic joints. Kress et al. [188] and Love et al. [220,221] carried out modeling of a hydraulically actuated single-link exible manipulator with prismatic joint. Coleman and McSwency [79]studied a Cartesian manipulator with roller support at one end.

3.2. Finite element method

Nagarajan and Turcic [270] derived elemental and system equations for systems with both elastic and rigidlinks. Bricout et al. [42] used the FEM to study elastic manipulators. Moulin and Bayo [260,261] also usednite element discretization to discuss the end-point trajectory tracking for exible arms and showed that anon-causal solution for the actuating torque enables tracking of an arbitrary tip displacement with any desiredaccuracy. Chang and Gannon [59] developed an enhanced equivalent rigid link system (ERLS) model usingnatural-mode shape functions for exible manipulators and an experimental validation of the model was per-formed for a single-link manipulator. The Lagrangian dynamics and the nite element methods are used toderive the equation of motion. Tokhi et al. [350352,227,228] developed dynamic models for a single-link ex-ible manipulator using the nite element approach and compared the modal frequencies found experimentallyto validate the FE modeling in some cases. They used bangbang type of torque to study the dynamicresponse. Also they have applied a command shaping technique to control the vibration of single-link manip-

ulator. Ge [127] derived a nonlinear dynamic model using the Lagrange approach. Theodore and Ghosal [347]give a very good comparison between the assumed mode method and the nite element method used for ex-ible manipulators. Alberts et al. [5] used FEM analysis to study the effectiveness of viscoelastic passive damp-ing augmentation to active control of a large exible space manipulator. They have shown very low frequencymodes due to joint exibility and high frequency modes due to bending in the booms which results in signif-icant end-point motion. Mohamed and Tokhi [254] derived the dynamic model of a single-link exible manip-ulator using FEM and then studied the feed-forward control strategies for controlling the vibration usingcommand shaping techniques based on input shaping, low-pass and band-stop ltering. Chung and Yoo[78] carried out the dynamic analysis using FEM.

Lee and Wang [202] studied a single-link exible manipulator in a 3D work space using FEM. Liu [218]used a geometrically nonlinear nite element dynamic model to study the large deection of a three-dimen-sional, three-link robot manipulator with a exible prismatic link fabricated from composite material operat-ing at a high speed. He observed unstable behavior when the axial sinusoidal motion frequency is close totwice of any transverse oscillation frequency of the exible link.

3.3. Lumped parameter models

Zhu et al. [431] considered a lumped model to simulate the tip position tracking of a single-link exiblemanipulator. Khalil and Gautier [166] used a lumped elasticity model for exible mechanical systems. Mega-hed and Hamza [239] used a variation of the nite segment multibody dynamics approach to model and sim-ulate planar exible link manipulator with rigid tip connections to revolute joints. The formulation employs aconsistent mass matrix in order to provide better approximation than the traditional lumped masses oftenencountered in the nite segment approach. Simo and Vu-Quoc [325,326] used a oating or shadow frameand inertia-frame methods.



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Raboud et al. [288] studied the stability of very exible cantilever beams to show the existence of multipleequilibrium solutions under a given load condition. Tzou et al. [365] used multi-layered piezoelectric polymersfor the control of exible manipulators and also studied the distributed modal identication and vibrationcontrol [366]. Nissing [271] studied the use of a spring and damper attachment to damp out vibration in a ex-ible single link manipulator. Saravanos and Lamancusa [307] considered the optimal structural design of

robotic manipulators with ber reinforced composite materials.3.4. Other studies

Under the assumption of the slender beam, Bayo [22,23], Soon and Jaw [335] and Morris and Madani [257]considered the inuence of the shear deformation to nd the inverse dynamics of the exible manipulator.Wang et al. [385] studied the effect of shear deformation and rotary inertia and Wang and Guan [381] dis-cussed the effect of shear deformation, rotary inertia and payload in frequency domain. Bayo [18,22], Spectorand Flashner [336] examined the application of Euler and Timoshenko beam theories to exible link modelingfor controller design. Golnaraghi et al. [134] and Tuer et al. [361] used the concept of reducing vibration in theexible structure by internal resonances due to modal interactions. Kar and Dwivedy [160] considered modalinteraction between rst and second modes and Dwivedy and Kar [102,103] consider modal interaction amongthe rst three modes. They have shown many nonlinear effects including chaotic behavior in a base excitedcantilever beam with an attached mass which can be considered as a manipulator with prismatic joint.

Some investigators, e.g., Chalhoub and Ulsoy [58], Wang and Vidysagar [375,376,379], Choi and Krishna-murthy [74], Zuo and Wang [433], Rossi et al. [302], Choura and Yigit [80] considered only one link of themulti-link manipulator to be exible.

Though most of the works presented for the analysis and control of exible manipulators deal with a xed-size/shape manipulators, some researchers have tried to nd the optimum shape of the manipulator by max-imizing the fundamental frequencies of the manipulators. These work include those of Asada et al. [11], Wang[380], Wang and Russel [382,383], Kumar et al. [197,198]. Cui and Xiao [83] obtained the optimized shape of single-link exible manipulators with weight constraints using genetic algorithms.

Choi et al. [73] addressed the dynamic modeling and control of a single-link exible manipulator fabricated

from composite laminates and compared the results with that of aluminum. They have shown that the manip-ulator fabricated from composite laminates has superior performance characteristics such as faster settlingtime, smaller input torque and smaller overshoot relative to the manipulator fabricated from aluminum. Theyused both Hamiltons principle and FEM to develop the dynamic model.

4. Two-link manipulators

Manipulators with some exible links are attractive as they avoid the severe control problems associatedwith the large inertia forces generated when the large-mass, rigid links in the conventional robots move at highspeed. In fact, often only two of the links of a typical six degree of freedom industrial robots cause signicantinertia forces and hence these two should be exible, Morris and Madani [259]. In the following paragraphsmodeling aspects of two-link exible manipulators are reviewed in the sequence of application of assumedmode-method, nite element method and lumped parameter methods similar to that carried out for single-linkmanipulator.

Fukuda [118], Fukuda and Arakawa [119] studied the modeling and dynamic characteristics of two-linkexible robotic arms and controlled the vibration by taking into account the gravity, payload, and the coupledvibration between the rst and second arm. Only bending vibration is considered in the links. They derived thegoverning equations by using a homogeneous transformation matrix, Euler beam equations with properboundary conditions and modal analysis methods. Fukuda et al. [120] considered the modeling and controlmethod of bending-torsion coupled vibration in the system. Ower and Vegte [277] used a Lagrangianapproach to model the planar motion of a manipulator consisting of two exible links and two rotary joints.Buffinton and Kane [45] and Buffinton [46] developed equations of motion for exible robots containing trans-lational motion of elastic members. The specic system investigated is a two-degree of freedom manipulator



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whose conguration is similar to the well known Stanford arm and whose translational member is regarded asan elastic beam. The assumed mode method and an alternative form of Kanes method are used in the formu-lation of equation of motion. The assumed mode expansion method is also used by Sasiadek [310] and Greenand Sasiadek [136,137] for two-link manipulators. Tomei and Tornambe [353] also used a Lagrangianapproach and expanded the generalized coordinates describing the exact shape of the beams constituting

the robot in a limited number of terms. Morris and Madani [256,258,259] developed the equation of motionfor a large single-link manipulator including shear deformation, and extended the same to a two-link manip-ulator using the LagrangeEuler formulation and assumed mode method. Choi and Krishnamurthy [75] stud-ied unconstrained and constrained motion of a planar two-link manipulator. Li and Wang [211] used theiranalysis to simulate a planar elbow arm.

Lee [201] showed that the conventional Lagrangian modeling of exible link robots does not fully incorpo-rate the bending mechanism of exible link as it allows free link elongation in addition to link deection. Thiselongation causes modeling inaccuracy for links with rotation. To correct this he proposed a new dynamicmodel.

Baruh and Tadikonda [17] describe various issues in the dynamics and control of exible robot manipula-tors. An approach similar to substructure synthesis is used to model the system where each link is rst modeledindependently of the others. The joint displacements are then used as constraints to synthesize the equation of motion. De Luca and Siciliano [90], Sunada and Dubowsky [343,344], Dado and Soni [85,86] developeddynamic models for planar two-link manipulators using a Lagrangian based nite dimension model assumedmode method. Kalra and Sharan [156,157,319] extended the work of Sunada and Dubowsky [343] and Naga-nathan and Soni [265] to model the exible manipulator using the nite element method. In contrast to thework in the last two references where C 0 continuity in axial direction and C 1 continuity in the transverse direc-tion is taken, in the work of Kalra and Sharan [157] C 1 continuity is taken both in axial as well as in transversedirection of the manipulator. The nite element derivation is carried out for a multi-link manipulator and thenumerical example is carried out for a two-link planar manipulator with revolute joints. The results obtainedfor motor torque responses and the end-effector positioning accuracy clearly showed that the orientation effectintroduces signicant error in the result. Unlike Sunada and Dubowsky [344] where a lumped parameter FEMmodel was developed, Kovecses [185] developed a distributed parameter model for the dynamics of exible

link robots. Lee and Wang [203] also derived the equation of motion of a two-link manipulator.Rosado and Yuhara [300] and Rosado [301] developed dynamic modeling of planar exible robotic manip-ulator with two exible links and two revolute joints using a NewtonEuler formulation and the nite elementmethod. Milford and Ashokanathan [242] derived the exact partial differential equations governing the systemmodes of general two-link exible manipulators by matching the boundary conditions at the elbow and haveshown that eigenfrequencies may vary up to 30% as the manipulator sweeps across its range of motion. Theyalso carried out experiments to validate their modeling.

Meghdari and Fahimi [245] used Kanes method of multibody systems to decouple the dynamic equation of motion of the two-link exible manipulator. Cheong et al. [68] modeled the two-link manipulator as springmass systems to develop the controller.

Everett et al. [104] showed that it is possible to design a two-link exible manipulator that has a nearly posi-tion invariant rst natural frequency with wide separation between the rst two natural frequencies to have itsbehavior similar to that of rigid manipulators to avoid vibration. Singh [330] used Hamiltons principles andassumed-mode method to develop equation of motion for two-link manipulator. Zhang et al. [427] derived apartial differential equation model for a exible two-link manipulator using Hamiltons principle and thentransform this to a form suitable for the development of stable controllers.

Low and Vidyasagar [222] considered a two-link manipulator with only the last link as a exible memberand used the Lagrangian method to study its dynamics. Dogan and Iftar [98] carried out the modeling andcontrol of a two-link robot manipulator, whose rst link is rigid and the second link is exible. They appliedthe extended Hamiltons principle to obtain the equation of motion and the corresponding boundary condi-tions. A composite controller based on the singular perturbation method is designed in this work.

One may nd general design rules for building rigid-robotic manipulators in the work of Yang and Tzeng[399], Asada [9], Toumi and Asada [357], and Park and Cho [279].



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5. Multi-link manipulators

In this section, the literature review is carried out for manipulators with more than two-links. Similar to theprevious two sections, here also, the references are grouped according to assumed-mode method, FEM andlumped parameter models.

Cannon and Schmitz [47] recognized that unlike for single-link manipulators, multi-link manipulators can-not be considered by a linearized model. The difficulties involved in dealing with the nonlinearities and the factthat the mode shapes of a linearized model vary with the conguration of the nonlinear system, are the mainreasons which make this analysis a complicated one. Book [35,36], Cetinkunt and Ittoop [53], Siciliano andBook [323], Cetinkunt and Book [54,55], Cetinkunt and Yu [56], Chedmail et al. [62], Book and Kwon[38], Tosunoglu et al. [356], Arteaga [8], Caron et al. [51] and Yang et al. [407] used a Lagrangian basedapproach to model the exible robot arms. Khorrami [167], and De Luca and Siciliano [9092] analyzedthe multi-link manipulators using asymptotic expansion methods. Asada et al. [10] proposed a techniquebased on an assumed mode model for a general n-link case that solves the problem by using special movingcoordinate systems, so called virtual rigid link coordinates. Chen [67] developed a linearized dynamic modelfor multi-link planar exible manipulators which can include an arbitrary number of exible links. Flexiblelinks are treated as EulerBernoulli beams. Simulations are mostly carried out for a two-link manipulator.Boyer and Coiffet [40,41] developed dynamic equations of motion for multi-link manipulator using theNewtonEuler method.

Ahmad [3], Krishnamurthy and Yang [193], Matsuno and Hatayama [236], Yamano et al. [396398], Sun[341], and Zhang [426] carried out dynamic modeling and simulation of two cooperating structurally exiblerobotic manipulators. Sunada and Dubowsky [343,344], Dado and Soni [85], Gaultier and Cleghorn [123], andSharan and Kalara [156,157,319] developed nite element based dynamic equations of motion for multi-linkrobotic manipulators. Benati and Morro [25] developed a Lagrangian approach for the dynamics of a chainwith exible links. Each link is modeled as a system with a nite number of degrees of freedom, one of themdescribing the rotation, the other ones the exibility. The dynamic equations are written explicitly for a chainwith two links and a payload. Bayo [22] used FEM to deal with multi-link exible manipulators consideringTimoshenko beam theory and including nonlinear Coriolis and centrifugal effects for the elastic behavior. An

iterative solution scheme is proposed for nding the desired joint torques where the solution of each lineari-zation is carried out in frequency domain. Jonker [153,154] used a nonlinear nite element formulation to ana-lyze a exible three-degree-of freedom manipulator. Dubowsky et al. [100] studied the dynamics of exibilityin mobile robotic manipulators based on a FEM method.

Yang and Sadler [405] and Usoro [369] developed nite element models to describe the deection of aplanar multi-link model. The effect of Rayleigh damping was incorporated in Yang and Sadler [405], whereasthe model in Usoro [369] was derived as a complete nonlinear system in the generalized inertia matrix.

The nite element models developed by Geradin et al. [132], Geradin and Cardona [133], and Naganathanand Soni [265267] incorporate a fully three-dimensional element to simulate manipulator motions with effectsof gravity and strain energies from torsional, axial and lateral deformation. Beres and Sasiadek [31], and Bereset al. [32], used an Lagrangian nite element approach to formulate the dynamic model of a exible manip-ulator system in three-dimensional space. Beam type nite elements with third order interpolating polynomialsand six generalized coordinates per nite element nodal point were used for the description of the link smalldisplacement eld. The DenavitHartenberg matrix method was used to describe the exible manipulatorkinematics. Sarkar et al. [309] used a numerical method developed by them [308] to minimize tracking errorof multi-link elastic robots where the dynamic equation of motion were developed using a Lagrangian basednite element discretization technique. Farid and Lukaiewicz [105] also used Lagrangian based FE modelingfor spatial manipulators with exible links and joints. Unbehauen and Gnedin [368] dealt with the problem of stabilizing vibrations in a system of interconnected multi-link exible beams by applying point-actuatorslocated at the joints. They derived the equation of motion using Hamiltons principle. Benati and Morro[26] provide a systematic thorough procedure for the derivation of dynamical equations of a chain of exiblelinks using Hamiltons principle.

Jonker and Aarts [155] developed a method in which the vibration motion of the manipulator is modeled asa rst order perturbation of the nominal rigid link motion. The method is based on a full nonlinear nite



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element formulation which treats the general case of coupled large displacements motion and small elasticmotion. A planar one-link and spatial two-link manipulator cases were studied to show that this method iscomputationally more efficient. Aarts and Jonker [1] also studied a planar multi-link manipulator using amodal integration technique.

Naganathan and Soni [264] developed a nite element based nonlinear model for a revolute joint spatial

manipulator using Timoshenko beam theory. Somilinos et al. [334] describe the design, dynamic modelingand experimental validation of a three-degree-of-freedom exible arm. Here the design and control is basedon the assumption that the mass of the arm is concentrated at the tip. Pedersen and Pedersen [282] derivedthe equation of motion for general 3-D exible mechanical systems using the multibody approach. Xuet al. [393] presented a FEM based distributed parameter model method for the exibility and kineto-elasto-dynamic analysis of robotic manipulator where only the generalized equation of motion was derived. No sim-ulation work was carried out to validate the model. Hermle and Eberhard [141] presented a hierarchicalcontrol concept for exible robot manipulators and derived the equation of motion using the multibody sys-tem method incorporating exible links equipped with surface bonded actuating and sensing devices. The con-trol strategy was veried using a SCARA robot with exible links. Khadem and Pirmohammadi [164]analytically derived the dynamic equations of motion for three-dimensional exible manipulators having bothprismatic and revolute joints using a perturbation method. Gouliaev and Zarrazhina [135] studied the problemof dynamic and kinematic control of the spatial movements of a exible multi-link manipulator, mounted on aplatform freely moving in the cosmic space.

Hussain et al. [144] considered the effect of shear force in addition to bending while modeling multi-linkexible manipulator. Gaultier and Cleghorn [124] formulated a spatially rotating and translating beam niteelement which incorporates the effects of longitudinal loads on lateral vibration, gravitational body force,actuator joint mass and end-effector payload, and Rayleigh damping into a tool for modeling the vibrationof exible manipulators of arbitrary geometry and motion prole. Theodore and Ghosal [347] made compar-isons between the assumed modes and nite element models for exible multi-link manipulators.

Yoshikawa et al. [411,412] introduced the concept of a macromicro manipulator system with a rigidmicro-manipulator mounted in the end-effector region of a large exible link manipulator. The fast and highaccuracy motion of this micro-manipulator is applied to compensate for the tip error of the macro manipu-

lator. This active vibration control has the potential drawbacks that additional actuators and their accessoriesare required and the weight of the robot system is increased. Lew and Trudnowsky [208], Book and Loper [39],Moallem and Patel [251], and Krauss and Book [187] also used similar micro-manipulator concept to dampout the vibration of the manipulator. George and Book [131] discussed some of the issues related to this typeof damping.

Saravanos and Lamancusa [307] used FEM to study a SCARA-class manipulator and showed that for alament wound structure using a high strength carbon ber in a thermoplastic matrix, the resultant optimumply congurations have increased the specic stiffness and specic load capacity by factors 1.5 and 16, respec-tively, in comparison to identically sized aluminum links.

Yoshikawa et al. [414] proposed a state estimation method and a parameter identication method of thedynamic model of exible manipulators based on the virtual passive joint model which they veried usingexperiments. Gasparetto [121] developed a dynamic model for exible link planar mechanisms using an equiv-alent rigid link model and experimentally validated the same for a ve-bar mechanism. Yue [418,419] studiedthe dynamics of exible robots with kinematic redundancy.

Pfeiffer and Gebler [283], Pfeiffer and Bremer [284,285], Wasfy [389], Pedersen and Pedersen [282], Hermleand Eberhard [141], and Dignath et al. [94,95] used multibody dynamics method to study the dynamics of ex-ible manipulators. A general survey of the multibody dynamics approach is well documented by Huston [143],Schiehlen [312,313], Shabana [316], and Wasfy and Noor [390].

6. Flexible joint manipulators

In modeling exible robots the accuracy of the dynamic model obtained from the analytical formulation ishighly dependent on the adopted mode shapes of the link deection. The mode shapes for exible link, ex-ible-joint robots are certainly different from the exible link, rigid joint robots, since the joint exibility may



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inuence the mode shapes. In the unavailability of the exact mode shape functions, researchers used the pin-ned-pinned, e.g., Asada et al. [10], or clamped modes, e.g., Book [36], Wang and Vidyasagar [374] or simplepolynomials, e.g., Tomei and Tornambe [353], to approximate the shape function. Few researchers used free free modes, e.g., Baruh and Tadikonda [17]. Hasting and Book [139] and Bellezza et al. [24] showed that if thebeam to hub inertia ratio is very small (order of 0.1 or less) the clamped end condition yields better result than

the pinned end conditions. Some researchers used experimental and nite element methods to determine themode shapes for exible link, exible joint robots, Karkkainen [161].Dado and Soni [86] carried out the analysis of a planar RR robot with rigid links and exible joints by

using the servo stiffness and damping along with the stiffness and material damping of the drive system. Itis shown that the servo damping plays an important role in the dynamic behavior of the system. Queirozet al. [287] also considered two-link rigid link and exible joints in their study. Spong [337,338] modeledthe joint exibility as a linear spring and proposed a globally feedback linearizable rigid link exible-joint(RLFJ) robot model that reduces to the standard rigid link robot model as the joint stiffness tends to innity.This RLFJ model has been widely used by the robotic control research community for the design of robotcontrollers, Queiroz et al. [287]. While Bridges and Dawson [44] studied RLFJ type robots with harmonicdrives, Bridges et al. [43] presented a survey of back stepping approaches in these types of robots. Yangand Donath [400] considered a single exible link with a exible joint and modeled the joint by linear spring.Chen and Fu [63] considered ( n + 1) rigid links connected by exible joints and actuated by DC motors. The joints are modeled by torsional springs and adaptive control scheme is used to control the vibration. Tzou[364] also considered elastic joints in the analysis of space manipulator. Yuan and Lin [415] considered n ex-ible links and joints and modeled the joints by torsional springs. Readman and Belanger [294] studied the sta-bilization of the rst mode of exible joint robots. Ahmad [2,3] addressed the hybrid position and forcecontrol of exible joint cooperative manipulators. One may nd a detailed survey on the work of exible jointrobots till 1993 in the book by Readman [295]. Tomei [355] considered manipulator similar to Chen and Fu[63] but considered friction in the elastic joints. Diken [96,97] consider the precise trajectory control of exible joint manipulator. Yoshikawa and Hosoda [413] and Yoshikawa et al. [414] studied exible manipulators bymodeling them as virtual rigid links and passive joints. The parameters of the model are identied by measureddata of the real arm. Kostine et al. [183] modeled the dynamics of an industrial robot KUKA IR 761 consid-

ering rigid links and exible electric drives. Yue et al. [417] proposed a nite element model for the link andtorsional spring model for the joint to analyze a planar 3 R manipulator. Ailon [4] considered the exibility of the electric drive for the analysis of exible manipulator.

Lozano and Brogliato [224], Colbaugh and Glass [81] and Tsaprounis and Aspragathos [359] carried outadaptive control of robots with exible joints. Massould and Elmarghy [229] and Massoud et al. [230] carriedout a hybrid control analysis of exible joint robot. Ankarali and Diken [6] analyzed exible manipulatorswithout control and discussed conditions to eliminate vibration. Feliu et al. [106,109,110] analyzed manipu-lators considering friction in the joints. Yue [418] studied kinematically redundant manipulators with both joint and link exibility for minimum deformation of the end-effector. Farid and Lukaiewicz [105] consideredthe torsional deformation of the exible joints in the dynamic modeling of spatial manipulator. Ider and Ozgo-ren [145] also studied spatial 3 R manipulators where the exible joints are modeled by torsional springs anddampers. It is shown that, in a exible joint robot, the acceleration level inverse dynamic equations are sin-gular as the control torques do not have an instantaneous effect on the end-effector accelerations due tothe elastic media. Wang et al. [384] also considered friction in the joint and studied the limit cycles and chaoticmotions of a single-link robot manipulator moving at slow speed. Smaili [333] analyzed a 2R planar manip-ulator with rigid and compliant joints using a three-node isoparametric nite beam element. In this formula-tion the joint compliances, the shear deformation and rotary inertia and the coupling effects of nonlinear grossmotion of the manipulator links with their distributed exibility and mass properties are included. Instanta-neous steady-state static response, modal analysis and transient response are obtained. Tomei [354] studiedrobots with elastic joints using a simple PD controller. Li et al. [210] present a systematic approach to dynamicmodeling and mode analysis of a single-link exible robot, which has a exible joint and a hub at the base endand a payload at the free end. They concluded that (i) even a small joint exibility can signicantly affect thesystem frequencies, (ii) the fundamental frequency is insensitive to the hub inertia or payload inertia, and (iii)for a given exible system, the fundamental frequency is mainly affected by the payload mass, while the second



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frequency is mainly affected by the payload inertia. Reshmin [296] used a semi-analytical approach based onthe method of averaging which allows one to separate the dynamics of the robot as a whole from its elasticvibration while dealing with multi-link manipulators connected by exible joints. Bahrami and Rahi [14]developed the dynamic equation of motion of an n-link articulated elastic joint manipulator subjected to sto-chastic base excitation using EulerLagrange equations. Subudhi and Morris [339] used a singular perturba-

tion approach for trajectory tracking of exible robot with joint elasticity. Korayem and Basu [181] studiedthe load carrying capacity of mobile base exible joint manipulators.

7. Inverse dynamics and computational programs

Bayo [19] introduced a method for the inverse dynamic analysis of a single-link exible robot to nd the tor-que to move the end-effector in a given trajectory in Cartesian space. Bayo et al. [20] made similar analysisincluding the nonlinear Coriolis and centrifugal effects. The experimental validation of the technique usingfeed-forward control yield good result. Deluca et al. [89], Asada et al. [10], Moulin and Bayo [260,261], De Lucaand Siciliano [92], Korayem et al. [180], Kwon and Book [200], Carrera and Serna [52], Damaren [88], Moallemet al. [248,249], Moulin and Bayo [262], Zou et al. [432], and Trautt and Bayo [358] also used an inverse dynam-ics method in the modeling and control of exible manipulators. Kanaoka and Yoshikawa [158] studied theinverse dynamics of planar mechanisms taking into account their singular conguration. Green and Sasiadek[136,137] studied the inverse dynamics model of exible two-link manipulators by coupling the nonlinear rigidlink dynamics with dominant assumed modes for cantilever and pinnedpinned beams.

The effect of the tip load on the dynamics of the exible manipulators were studied by Wang and Ravani[372,373], Parks and Pak [280], Wang and Russel [387], Korayem and Basu [179], Ryu et al. [305], and Yueet al. [420].

Shaheed and Tokhi [317] investigated parametric and non-parametric approaches for dynamic modeling of exible manipulator systems. The least mean squares, recursive least squares and genetic algorithms are usedto model linear parametric models and non-parametric models are developed using a nonlinear autoregressiveprocess with exogenous input structure with multi-layered perceptron and a radial basis function neural net-work. Bayo [22,23], Korayem et al. [180], and Dai et al. [87] studied the inverse kinematics of exible robot

manipulators.Only very few literature related to the programming aspects of the exible manipulators should be citedhere which may be used by the researchers to further develop the models. Most references given in the othersections are obviously also programmed for their numerical verication but this overview should not concen-trate on these aspects of exible robots and is therefore only very brief. One may nd algorithms for studyingexible manipulators in the work of Cetinkunt and Ittop [53], Tzes et al. [362,363], Skowronski [332], Ceti-nkunt and Book [54], Serna and Bayo [315], Kalra and Sharan [156], Jain and Rodriguez [147], Singh[330], Korayem et al. [179], Boyer and Coiffet [41], Tokhi et al. [350], Shyu and Gill [324], Zhang et al.[423], Kwok and Lee [199], Azad and Tokhi [13], and Dignath et al. [95].

Tzes et al. [362] developed an algorithm for the generation of the kinematic and dynamic equations of multi-link rigid or exible manipulator. The kinematic equations are derived using homogeneous transforma-tion matrices and dynamic equations are obtained subsequently using EulerLagrange formulation. Kwokand Lee [199] and Azad and Tokhi [13] developed a Matlab based software package for the control of a sin-gle-link exible manipulator. Fisette and Samin [115] used ROBOTRAN for symbolic generation of multi-body system dynamic equations. Dignath et al. [95] presented a new software tool developed using theprinciple of multibody dynamics for rigid and exible mechatronics systems. One may nd more referencesrelated to software development using multibody dynamics method in Shabana [316], Betsch and Steinmann[33], Wasfy and Noor [390], and Schiehlen [313].

8. Experimental investigations

In this section studies on experimental investigations for exible manipulators are listed according to thenumber of exible links used in the experiments. Also the measurement techniques are mentioned in thesecases.



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Cannon and Scmitz [47] initiated experiments on single-link exible manipulator which was further carriedout by Rovner and Cannon [304] to control the vibration. Naganathan and Soni [266] also conducted exper-iments using a single link mounted on a stepper motor to validate to their FEM results. Bellezza et al. [24]conducted experiments to study pseudo-clamped and pseudo-pinned end-conditioned in slewing links. Yuhet al. [421] and Yuh and Young [422] carried out experiments on manipulators with prismatic joints. Luo

[226] used a strain based measurement technique to study the exible robot arm. Yang and Sadler [404]presented a modal data based and experiment-oriented method to predict the dynamic response of a robotmanipulator with elastic members. Case studies were conducted for robot trajectory planning of a pris-matic-and-revolute manipulator.

Meckl and Seering [238] demonstrated that force proles can be developed to control exible dynamic sys-tems with minimal vibration. By dening an appropriate cost function, a force prole can be derived that effi-ciently allocates kinetic energy so that excitation is minimized at the system resonances and maximum energyis used for system motion. Magee and Book [240] compared a modied command ltering technique that elim-inates the rst two modes of vibration in a large exible manipulator to track circular trajectory with that of pre-shaped command input. Kress et al. [188] and Love et al. [221] carried out experiments for a two-degreesof freedom single-link exible manipulator with hydraulic actuator. Tokhi et al. [351] perform experiments toverify their proposed FEM based analytical model for a single-link exible manipulator.

Using a micro-manipulator at the end of a large exible manipulator, Yoshikawa et al. [411,412], Lew andTrudnowsky [208], Book and Loper [39], Moallem and Patel [251], George and Book [131], and Krauss andBook [187] performed experiments to damp out the vibration.

Experiments for two-link manipulators are carried out by Khorrami and Jain [168], Khorrami et al. [169],Oakley and Cannon [272274], Yim et al. [409], Nagaraj et al. [268], Bai et al. [15], Milford [242], Moallemet al. [253]. Bai et al. [15] used some identication techniques to nd the generalized friction for an experimen-tal planar two-link exible manipulator. Experiments on exible joints were carried out by Dado and Soni[85], Queiroz et al. [287]. In the later case [287] experiments were carried out on two-link direct drive planarrobot manipulator.

Kumar et al. [196] studied the interaction between human and robot in micro-surgery experimentally. Tsoet al. [360] used an optical sensing system consisting of a laser diode, a position sensitive detector, for the real

time measurement of the dynamic deection. Utilizing a nonlinear, coupled and measurement based dynamicsystem model, they proposed a Lyapunov-type controller based on the deection feedback to damp out the tiposcillation of a single-link exible robot arm.

9. Control of exible manipulators

There are several control schemes such as modal reference adaptive control, self-tuning control, feed-for-ward control and regular PID control used to regulate the motion of the manipulators. In all these schemes anefficient and accurate mathematical model is necessary, Beres et al. [32]. In this section the literature on thecontrol aspects of the exible manipulator is reviewed only very briey since other state-of-the-art reviewsare available.

Cannon and Schmitz [47] initiated the experiment to control the end-effector of a exible manipulator bymeasuring the tip position and using that measurement as a basis for applying torque to the other end (joint)of the beam. However, they only considered a linearized model and also the arm can sweep only in the XYplane, so that it is not affected by the gravity. Since then many new control strategies are developed to controlthe exible link vibration. Recently, Benosman and Vey [29] presented a survey on the control of exiblemanipulators which mostly deals with the multi-link manipulators and mainly works between 1990 and2002 were cited. In this present work, many other publications are cited which ranges from 1974 to 2005and classications are mainly based on the number of exible links used in the study. For continuity purposedifferent methods used for the control of exible manipulator are briey described.

The control strategies for exible manipulator systems can be classied as feed-forward (open loop) or feed-back (closed loop) control schemes. Feed-forward techniques for vibration suppression involves developing thecontrol input through consideration of the physical and vibrational properties of the system, so that systemvibrations at response modes are reduced. This method does not require any additional sensors and actuators



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and does not account for changes in the system once the input is developed. On the other hand, feedback con-trol techniques use measurements and estimations of the system states to reduce vibration. For exible manip-ulators, Benosman and Vey [29] pointed out that the control objectives are mainly end-effector regulationproblems, end-effector to rest motion in a desired xed time, joint trajectory tracking and the end-effector trajectory tracking. The last one is the most difficult one due to the non-minimum phase nature of the system

dynamics. The control schemes applied to exible robots include proportional derivative control, computedtorque control, active damping control, adaptive control, neural network based control, lead-lag control, slid-ing mode control, stable inversion in the frequency domain, stable inversion in the time domain, algebraic con-trol, optimal and robust control, input shaping control and boundary control. Some works using these methodsare cited in the next subsections and are grouped with respect to the number of exible links in the manipulator.

9.1. Single exible link

Many investigators worked to control the position of the end-effector of the single-link manipulators. Herein this section, the methods used by different investigators are described very briey. While the computed tor-que control is used by Looke et al. [219], Kwon and Book [200], and Jnifene and Fahim [152], and Liu andYan [217] inversion based control schemes are used by Singh and Schy [328,329], Rattan et al. [290], Rattanand Feliu [292], Moulin and Bayo [261,262], and Deluca [93] for the end-point control of single-link exiblemanipulator. Sharf [320] used an active damping control scheme for a short-reach robot. Adaptive controlschemes are used by Menq and Chen [241], Sasiadek and Srinivasan [311], Feliu et al. [106,107,112], Chenand Meng [66], Feng and Palaniswami [114], Yang et al. [401], Yang et al. [402,403], and Cao and Xu [49]for tip position control of single-link manipulators. Korolov and Chen [182], Mishra [246], Chirinos et al.[72], Rattan and Feliu [291], Ravichandran et al. [293], Feliu et al. [111,113], Karray et al. [163], Ryu et al.[305], Karkoub and Tamma [162], and Caracciolo et al. [50] used robust control schemes for the single-linkmanipulator. The optimal control scheme is used in the work of Pal et al. [278], Lee [204], Onsay and Akay[275], and Xuemin et al. [395]. Trajectory tracking of the end-effector is studied by Bhat and Miu [34], Pondand Sharf [286], Sarkar et al. [308], Zhu and Ge [430], and Diken [97]. While self-tuning is used by Ji and Fam-iloni [150], the singular perturbation technique is used by Siciliano and Book [323]. Boundary control scheme

is studied by Morgul [255] and Lyapunov based control is used by Yong and Walcott [410], Shiffman [322],and Dadfarnia et al. [84]. Recently Shan [318] studied slewing control of single-link manipulators. Chenand Yeung [64] used sliding mode control to attenuate the vibration. The sensor based feedback controls werecarried out by Kotnik et al. [184], Geniele et al. [130], Xu et al. [392], Choi et al. [77], Li and Chen [212], Linand Lewis [215], Tso et al. [360], and Wang and Li [386] for single-link manipulators.

Singer and Seering [327] developed an impulse shaping method which transform each sample of desiredinput into a new set of impulses that do not excite the system resonances. The idea involves delaying a portionof the input by half the damped natural period of the system to cancel the vibration induced by the originalinput. Singhose et al. [331] used the same principle to damp out vibration using the vector diagram approach.Liu et al. [216] combined feedback linearization with input shaping technique to control the vibration of a sin-gle-link exible manipulator. The command shaping technique is used in the work of Rhim and Book [297]and Mohamed and Tokhi [254]. Jalili [148] proposed an innite dimensional distributed base controller forthe regulation of the angular displacement of a one-link exible robot arm while simultaneously stabilizingvibration transient in the arm.

Many researchers e.g., Patnaik et al. [281], Choi et al. [77], Ge et al. [128], Sun and Mills [340], Sun et al.[342] used smart material to control the vibration of exible manipulators. Patnaik et al. [281] studied the sta-bility of piezoelectric actuated Euler beam and Choi et al. [76] presented a robust control of a single-link ex-ible manipulator with two actuators: the motor mounted at the hub and a piezoceramic bonded to the surfaceof the exible link. The control torque of the motor activates desirable hub motion and the control voltage of the piezoceramic actively suppresses undesirable vibration of the exible link. Ge et al. [128] studied a modelfree controller design for exible smart material robot with one end of the link rigidly attached to the rotor of a motor and segmented piezoelectric materials covering both sides of the exible beam. They derived the con-troller from basic energy-work relationship. Sun and Mills [340] and Sun et al. [342] used PD controller andPZT actuators to control the vibration of the single-link exible manipulator.



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Chiou and Shahinpoor [71] simulated a simplied dynamic model for a one-link tip force controlled exiblemanipulator. They also carried out experiment to study the dynamic stability of this manipulator.

9.2. Two-link exible manipulators

Similar to single-link exible manipulators, in two-link manipulators also several control schemes are usedto attenuate the vibration of the manipulator. They are briey described in this section. Here, while computedtorque control is considered by Bayo [21], Khalil and Boyer [165], Cheong et al. [68], Morris and Madani[258], and Aoustin and Formalsky [7], adaptive control is studied by Skowronski [332], Lucibello and Bellezza[225], Chen and Etimsahy [65], Cao and Xu [48], and Lee et al. [206]. Optimal control has been used in thework of Lee and Wang [202] and Morris and Madani [259].

A PD controller is considered by Yigit [408] and Xu et al. [394] and sliding control is used in the work of Zhang et al. [425]. A self-tuning concept is used by Koivo and Lee [175]. Moallem et al. [249,252,253] carriedout an observer based inverse dynamics control strategy which results in small tip-position tracking errorswhile maintaining robust closed-loop performance for a class of multi-link structurally exible manipulators,they used an integral manifold approach in [250]. Konno et al. [176,177] found structural vibration uncontrol-lable congurations within a 2-link 3-joint type manipulators workspace and introduced the modal accessi-bility index. They also used a singular perturbation technique [178] to control the vibration. The trajectorytracking is studied in [90,7,27,276,137,30]. Hillsley and Yurkovich [142] designed a two stage control architec-ture to achieve accurate end-point position control for point to point movements. Khorami et al. [169] devel-oped rigid-body based controller with input preshaping for two-link exible manipulator. Kim et al. [173] usedPZT based smart material to control the vibration of two-link manipulators. Yuanchun et al. [416] developeda robust controller for two-link manipulators using neural network based quasi-static deection compensa-tion. Matsuno [233] and Matsuno et al. [231,232,235] studied the hybrid position and force control of two-dof exible manipulators. For measuring the vibration and contact forces, accelerometer and a force sensorare used. Output signals of the sensors are feedback to the driving motors for controlling the position of the end-effector and the contact force. Lee and Lee [205] used a hybrid control scheme for robust trackingof two-link exible manipulator.

9.2.1. Multi-link exible manipulatorsIn this section the control strategies used for the multi-link manipulators are briey addressed.Lin et al. [213] used nonlinear feedback, PID state feed-forward and Lyapunov based stabilization proce-

dure to control a six-degree-of-freedom exible industrial manipulator. Drapeau and Wang [99] considereda ve bar manipulator with one exible link and used a closed-loop shaped-input technique in conjunctionwith a rigid body LQR regulator to control the vibration. Khalil and Boyer [165] studied a two dof 2Rmanipulator and a three dof SCARA robot (PRR) using a generalized NewtonEuler formulation andcomputed torque control method. Rossi et al. [303] discussed the issues in the design of passive controller.Ge et al. [126] studied the asymptotically stable end-point regulation of a exible SCARA/Cartesian robotusing a PD controller. The trajectory tracking has been studied by Deluca et al. [89], Asada et al. [10], Zhaoand Chen [428], Yue et al. [420], and Sarkar et al. [309]. Yang et al. [407] considered the tip trajectory trackingcontrol of multi-link manipulators using an integrated optical laser sensor. Utkin [370] developed sliding modecontrol for a multi-link exible manipulator based on the pole assignment approach. Isogai et al. [146] usedneural network based controller for exible multi-link manipulator. Lee et al. [206] proposed an adaptiveenergy based robust control scheme for multi-link exible robots. But the simulations are carried out for atwo-link exible manipulator. Benosman et al. [28] describes different control schemes used for multi-linkmanipulator.

Some researchers, e.g., Zhu and Ge [429], Wang et al. [388], Ge et al. [125,129] and Lee et al. [206] have usedmodel free robust controller to avoid the problem of controller/observer spillover due to the truncated mod-eled based method obtained by using either assumed-mode method or FEM. Ge et al. [129] developed con-trollers for multi-link smart material robots. Khorrami et al. [170] and Kang and Mill [159] also usedpiezoelectric actuators to control the vibration in multi-link manipulators. Cheong et al. [69] applied singularperturbation technique to control the vibration of exible manipulators and the validity and effectiveness of



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is highly appreciated. Also the rst author wishes to thank IIT Guwahati for granting the necessary leave tocarry out the work at Institute B of Mechanics (since 2006: Institute of Engineering and ComputationalMechanics), University of Stuttgart, Germany.

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