Gimbal modeling

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Transcript of Gimbal modeling

Kinematic Conti01 Experiments with Tkussarm, a VariableGeomet ryTi:uss Manipulator
Regina S u n  K m Lee
A thesis submitted in coaformity with the requirements for the Degree of Doctor af Philo8ophy
University of Toronto Institute for Aerqace Studies
@Copyright by Regina SimKyimg Lee, 2000

uisitii and Acquisitions et 8' bgraphie Sewicm services bibliographiques 3
The author has grauted a non exclusive licence ailowing the National Library of Canada to reproduce, loan, distfl'bute or sel copies of this thesis in microform, paper or electronic fonnats.
The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts f3om it may be printed or odierwise reproduced without the author's pamission.
L'auteur a accord une licence non e x c l ~ v e permettant B la Bibliothque nationale du Canada de reproduire, prter, distribuer ou vendre des copies de cette thse sous la forme de rnicrofiche/film, de reproduction sur papier ou sur format lectronique.
L'auteur conserve la proprit du droit d'auteur qui protge cette thse. Ni la thse ni des extraits substantiels de celleci ne doivent tre imprims ou autrement reproduits sans son autorisation.

Abstract
'Lhwarm is a VariableGeometryaims (VGT) manipidator, developed and constriicted at the University of Toronto Institute for Aerpace StiHes. The objective of this dissertation is to examine this new technoIogy to determine its kinematics and conho1 characteristics. In order to examine the kinemat ics properties of Thia9ann, a kinematics model is developed. An algorithm to represent the macro8copic geometry of 'Ihisasrm as an equivaient discrete model (a serial open chah of simple 3DOF gimbal modules) is presented. Redundancy resolution based on the quivalent discrete mdel is developed, and,the kinematica and dineraitid kinematics d a l h m a m module ara de+ rived. Through experiments and simuiations, the control c h a r ~ i c s of 'Pnissarm including workspace, dexterity, speed, and tracking accuracy, are determiaed. Erthermore, a a m contd algorithm ie developed to improve the control pedormance of 'lhissarm. The experimental results suggest that Th pommsee many advantages over conventional manipulators although its acctuacy d m fiom its phpicai size and complexily. However, the accu racy may be improved with c i ~  1 o o p poeition conttoL

Acknowledgments
I wodd Iike to express my sincere gratitude to many people who made this work possible.
First of dl, 1 w d d like to t h d my supervisor, Prof. Peter C. Hughes, for his academic and technical advice throughout my graduate studies. 1 also would like to thank my doctoral cornmittee members, Prof. Gabriele MT. D'Eleuterio and Dr. Rank Naccarato, for th& technical advice in all aspects of this work. Much of the work on the kinematics developrnent in Chapters 3 and 4 is the r d t of conversations with Dr. Kourosh Zanganeh. He patiently taught me everything about hyperredundant dpuiators .
The collvematio~~ with my cdleagues in the Spsceaaft Dynamics and Robotice gmup have been invaluable to me. 1 am eepecially indebted to Thierry Cberpillod for his technical support on the design and constmction of 'km.' Much of the work on the design of Ihissarm was also providecl by many previous students, including Stephen Oikawa and Roger Hertz. A b , many th& to Andrew Allen for proofkacihg pette of this document.
1 a h would B e to acknowledge the Naturel Science and Engineering Research Councii (NSERC), ISIS (now CresTech), University of Toronto, and Dynacon Inc. for hancial support duting rny doctoral progm.

1 also wish to thank rny parents, SangWook and JungJa Lee, and my sister, Sophia, for theh love and support. Their constant encouragement and patience have been and will always be appreciated. Lsstly, 1 wodd like to take this opportunity to th& my hiuband, Thierry* Without his love and support and most of all, his faith in me, this work wodd not have been posgible.
July, 1999 Maple, Ontario

Contents
List of Figues
Cbapter 1 Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background 1
. . . . . . . . . . . . . 1.1.1 Hyper Redundant Manipulators 1 . . . . . . . . . 1.1.2 Variable Gbmetry 'Ihies Manipulatom 2
1.2 Thissarm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . 1.3 Motivation and Prnpose .. 4
. . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Literature Eteview 6 1.4.1 Kinematics Analysis of HyperRedundant VGT
Manipulatom . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 Design of and Ekperiments with VGT Manipulators . . 8
. . . . . . . . . . . . 1.4.3 Kinematia Analpis of 'Zhissarm 14 . . . . . . . . . . . . . . . . . . . 1.5 OverviewofthisDocument 15
Chapter 3 System D d p t i o n 17 2.1 Actuators and Senromechanianis . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . 2.1.1 lhsmnnActuator 17

2.1.2 sexvomechanism . . . . . . . . . . . . . . . . . 19 2.2 Siipporting Stnictiue . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 M o . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Passive Members . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 'Ihissarm Coordinate System . . . . . . . . . . . . . . 21
2.3 Tnissarm Control Environment (TCE) . . . . . . . . . . . . . 22 2.3.1 Tkajectory Planning . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . 2.3.2 HyperRedmdancy Redution 23 2.3.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.4 GALIL corxuna~ds . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . 2.3.5 Motion Execution 24
. . . . . . . . . . . . . . . . . . . . . 2.3.6 Cartesian Control 24
Chapter 3 Kinematics 25 . . . . . . . . . . 3.1 Kinematics of an Equivaient DDiactete Model 26
. . . . . . . . . . . . 3.1.1 3DOF Extensible Gimbai Module 26 . . . . . . . . . . 3.1.2 Chain of Gimbal Modules .. .. . . 28 . . . . . . . . . . . . . . . . 3.2 Kini?maticsofa'lkussarmModule 31
. . . . . . . . . . . . . . . . . . . . 3.2.1 InverseKinematics 31 . . . . . . . . . . . . . . . . . . . 3.2.2 Forward Kinematics 35
. . . . . . . . . . . . . . . . . 3.3 JointLimit Avoidance Problems 37
Chapter 4 Dinerential Kinematics 42 . . . . . . . . . . . . . . . . . . 4.1 Resolveci Motion Rate Control 43 . . . . . . . . . . . . . . . . . . 4.2 Jacobian Matrix and Hessian 44
. . . . . . . . . 4.3 Dinerential Kinematics of a Twmum Module 50 . . . . . . . . . . . . . . 4.3.1 Differential Inverse Kinematics 51

Chapter 5 Paformmce Measurements 58 5.1 Workspace Analysis . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . . 5.1.2 Workspace Measmement Methods 60
5.1.3 'hssarm Workspace Experiments . . . . . . . . . . . . 68 . . . . . . . . . . . . . . . . . 5.1.4 Controllable Workspace 71
. . . . . . . . . . . . . . . 5.1.5 Reach H i e r d y Algorithm 73 5.2 Dexterity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 75
. . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Deanition 75 . . . . . . . . . . . . . . . . . . 5.2.2 Dexterity Measmement 76
. . . . . . . . . . . . . . . . . . . . . . 5.3 RepeatabiityAnalysis 82 . . . . . . . . * . . . . . . . . . . . . . . 5.3.1 Definition .. 82
. . . . . . . . . . . 5.3.2 Repeatabii Measurement Method 82 . . . . . . . . . . . . . . . . . . 5.3.3 Ihisarm Repeatabiliw 84
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 SpeedAnalysis 85 . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Actuator S p d 85
. . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 TipSpeed 85 . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Accmwy Anal@ 88
. . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Definition 88 . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Poeition Control 89
. . . . . . . . . . . . . . 5.5.3 Kinematc ('Ii.acking) Control 90 . . . . . . . . . . . . . . . . 5.5.4 OnLine Cartesian Contd 96 . . . . . . . . . . . . . . . . 5.5.5 Velocity Tkacking Control 97
Chapter 6 Contcol System Design

6.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 SimulationTest . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Chapter 7 Conclusion 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 FinalRemarks 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Cont~bl~tiom 112
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 FlltureWork 113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 115
Appendix A An Example of a Contrained Optimization Probleml2l

List of Figures
A Discrete Morphology Hyperreduudant Manipuiat or . Calt ech Tkussarm . . . . . . . . . . . . . . . . * . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 3DOF VGT at NASA . . . . . . . . . . . . . . . . . . . 'Lhrssann Mark 1 at UTIAS
Tapered Twobay OctahedraIQCt&edral VGT at NASA LRC Serpentine 'Ihiee MdpuIator, NASA Kennedy Space Center .
. . . . . . . . . . . . . . . . . . . . . . . Plansr VGT, Caltech Tetrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Payload Inspection end Processing System . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .  M u e Actuator Geometry . . . . . . . . . . . . . . . . . . . . . . . . Hinge Assembly . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . Th Control System
. . . . . . . . . . . . . . . . 3DOF extensible @bal module . . . . . . . . . . . . . . . . . . Kinematia of a discrete mdei
Vectors in the kinematica of a Ihissarm module . . . . . . . .

5.2 'Ihisearm Workspace by J3xtmsion Method. zzplane . . . . . 62 . . . . . . . . . . . . . . 5.3 Workspace Boimdary Approximation 65
. . . . . . . . . . . . . . . . . . . . . . 5.4 workspace of 1 Modiile 66 . . . . . . . . . . . . . . . . . . . . . 5.5 Workspace of 2 Mod~des 66 . . . . . . . . . . . . . . . . . . . . . 5.6 workspaceof 3 Modules 66 . . . . . . . . . . . . . . . . . . . . . 5.7 Woxkspace o f 4 Modules 66
. . . . . . 5.8 Tkussarm Workspace by Sweeping Method, szplane 67 5.9 Thissarm Workspace by S w i n g Method. xzplane . . . . . . 67
. . . . . . . . . . . . . . . . . . . . . . . 5.10 Home Cod&uration 68 . . . . . . . . . . . . . . . . . . . . . . 5.11 JShape Coafiguration 68
. . . . . . . . . . . . . . . . . . . . 5.12 Contracted Configuration 69 . . . . . . . . . . . . . . . . . . . . . 5.13 Extended Configuration 69
. . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Pitive x Reach 69
. . . . . . . . . . . . . . . . . . . . . . . . . 5.15 NegativexReach 69 . . . . . . . . . . . . . . 5.16 Wore of PUMA761 mruiipulator 72
. . . . . . . . . . 5.17 Ihisearm Controllable Workspace, xzplane 73
. . . . . . . . . . 5.18 'Ihissarm Controllable Workspace. zyplane 74 . . . 5.19 Controllable Wor with Reach Hierarchy Algorithm 75
. . . . . . . . . . 5.20 Dexterity Measma dong a c w e (xzplane) 78 . . . . . . . . 5.21 Dexterity Measures dong a straight line (2axb) 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22Paetiuel 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23 Ptiue2 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.24Ptu.re3 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25Poetiire4 80 . . . . . . . . . . . 5.26 DexterityMeamesatPOehne1,2, 3and4 81
. . . . . . . . . . . . . . . . . 5.27SetupforRepeatabilityAnalysis 83 . . . . . . . . . . . . . . . . . . . 5.28 RepestabityTestResultl 84

529 RepeatabityTestResult2 . . . . . . . . . . . . . . . . . . . &q 530 Repeatability Test Result 3 . . . . . . . . . . . . . . . . . . . 84 5.31 Vibration During Acceleration and Deceleration . . . . . . . . 86 5.32 The 12ich PickaadPlace Maneuver. . . . . . . . . . . . . . 87 5.33 Square Tkajectory in Amiracy Analysis. . . . . . . . . . . . . 90 5.34 'Pradcing Accuracy A d p i s = Square Qajectory, n=2, xzplane 91 5.35 Configuration of Ihissarm: Square najeetory, n=2, xzplane . 91 5.36 Configuration d 'lhissarm: Square Trajectory, 724, xyplane 92 5.37 Tkacking Aawscy Analysk Square Tkajectory, 71==4, xgplane 93 5.38 'Ikacking Accuracy Analysis: Square 'hajectory, r4, xzplane 93 5.39 T1:acking Aa?uracy Analysis: Ciicular Tirajectory, n=4, xzplane 94 5.40 Velocity Tkacking Control Simulation Results, nt = 20 . . . . . 98 5.41 Velocity Tracking Contml Simulation Results, n( = 1000 . . . 99 5.42 Second nom of the enor vs. number of trajectory points . . . 99 5.43 Velocity Thckhg ControI, nt = 50, xzplane . . . . . . . . . . l2
Augmenteci Kinematic Control Algorith . . . . . . . . . . . 105 &or obtained h m the Simulation of ClosedLoop Control, Square Tkajecbry, n=2, xzplane . . . . . . . . . . . . . . . . 106 Simulation Wts h m C1osedbop Contr01, Square Tkajec tory, n=2, xzplane . . . . . . . . . . . . . . . . . . . . . . . . 107 Simul~tion R d t s from Closedhop Contr01, Circula 'Ikajec tory, n=2, xzplane . . . . . . . . . . . . . . . . . . . . . . . . 107 'Ikacking Error vs. Number of lteration amund Fine Poeitioning Control . . . . . . . . . . . . . . . . . . . . . ,. . . . . . . . 10s Results kom cimeclhop Contml, Squm 'hajectoryt n=2, xz pIane . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . 109

Chapter 1
Introduction
1.1 Background
1.1.1 Hyper Redundant Manipulators
Hyperteciundant manipulators are those that have very large degr of kin* matic redundancy. These manipulators are often r d to perform tada that conventional robots are mcapable of accomplishing. Exemples Uiclude manip ulatom in a highly comtrained environment, snakelike maneu, and the grasping of ob jects. As Immega and Antonelli (1995) pointed out, "the pomi biities [of the applications of hyperredundant menipulators] are limited only by the imagination."
On the other haad, an dcient hyperredundant manipulation highly depends on the available methode for remlving the hypertedundant kinemat ics. Most known rnethods of redimdancy resolution Iead to serial computa tions, hence their computational complexity gr with the manipulatot's d e grees of fieedom. Due to the computational difficulties asaociated with hyper redundancy, hyperredundsnt mmipuiators have been cooeidered inapplicable

Figure 1.1: A Diacrete Morphology Hyperredu11ciant ManipuIator, Caltech
to industrial tasks until very recently. Hyperredundant manipulafors are generally grouped into t h e c a b
gories accordhg to their morphologies: continuous morphologr, ckrete mor phology, and Variabl&eometry'Lhiss (VGT). A tentacle is an example of a continuou8 morphology hyperdudant manipulator. A diactete morphol ogy hyperredundant d p u l a t o r is a long chain of serid h h g e s r d t i n g in a manipuletor with a very large number of degrees of f idom. A spatial hmerredundant Mpula to t of diecrete morphology (Figure 1.1) is under d e velopment at the California Mitute of Technology. 'Ihiss8rm (Figure 1.2) is an example of a hm=redundant VariableGeometxyThrss manipuletor.
1.1.2 Variable Geometry Truss Manipulators
The origin of ViableGeometryTh= (VGT) structures &tes back to the w l y 8's when deployable and stackable etnictme8 were firat coneidered for space applications. The VGT concept was dbved kom the concept of 'adip tive', or 'smart' [Rabertshaw and Reinholtz, 19881 structures. Since then,

Figure 1.2: 'Ihieearm
numemus applications for VGT structures have been exBrnineci, inciuding the manipuiation of payloads.
VGT manipulators comist of a series of selfantained modules. Each module, in tum, is a parailel manipulator with thabldength members. The adwmtage8 of a VGT manipulator, compared to a anventional manipula tor of the same volume and mass, were realiaed early in the development: stiffness, modularity, flexibiity, and dexkrity. buse of these advantages, VGT rnanip~ilators have been considered for applications in space and under sea explorations and inspection. Examples of proposeci applications for VGT manipidators incliide waste storage tank remediation [Salerno and ReinhoItz, 19941, onsite equipment r e m 4 w i a m s and Mayhew, 1997a]. and payload inspection [\N.fiama and Mayhew, 1997bl.
More recentIy, VGT teEhaology has been ale0 amiderd for manufac

tiiring and automation applications. Due to the modiilarity in VGT maaip idators, they have adwtages for tasks that require changes in configuration, or tasks in a constrained environment.
The term '%mm' refm to a modidar VGT manipulator design developed at the University of Toronto Inetitute for Aerospace Studies (UTIAS). Labors tory prototypee have been b d t to support the study of structures, manipdator design, and the kinematics, dynamics, and control Iissociated with VGT me nipulators. The &t ?husarm, 'Ihissarm Mark 1, was built in the early 90's as a proof~fancept VGT mAnipulator. As sown in Figure 1.4, 'Ruassrm Mark 1 is a 3 degreeoffreeom, doubleoctahedral manipulator.
More rently, a lZDOF prototype n~mely 'Ihieserm Mark TI (which will be simply refmed to es b' hereiaefer), was buiit. 'Ihiesarm con sists of four identical 3 d e ~ f  ~ o m (3DOF) modules that are connected in series. Each module is a doublmtahedral patallei manipulator coneisting of two identicai octahedral modules ehaRng a cornmon trianguiar facet. More over, each doubleoctahedral paranel mruiipuiator hes tkee extensible links that can be adjusted in lengths continuously, mueing three priematic actuators.
1.3 Motivation and Purpose
The main objective of this disseitstion is to develop a khematics control scheme for a hyperredudsnt VariableGeometzy'Lhiss (VGT) madpuiator, such as 'hssann, and to investigate its control characteristifs through exper iments and simulations. The r d t s obtained kom this cbertation can be

iised to explore indiietnal applications. In order to develop an efficient control scheme for h m , the follow
ing research objectives have been p r o p d
a to efnciently model the kinematics of 'Ihuisarm
O to analyze the perSommce characteristics of 'Ihissarm
0 to design and implement control schemes for Thusam
The kinematia of 'Ihisearm hm been previously presented in (Sallmen, 1993). However, the formulation of the kinematics used here t h admtage of the stmcturai symmetry of 'Lhissam, thus providing a physically meaniaghil model with a vety simple mathematical presmtation. One of the contribu tions of this dimertation in the field of VGT kinematics is the development of a kinematics model at the velocity laval. The model pnsented hem is compu tstionally simple and easy to imp1ement.
Many hyperredundant VGT d p u l a t o t e remained latgdy at the ex perimental level, primarily due to the difEiculties in the implernentation of control schemea. As a r d t , many chamcteristics of VGT manipulators have been examineci only through simulations and experimental verifications have not been performed. Throughout thia dissertation, a number of expesiments designed to examine the control characteristics of 'Ihisearm are presented. The experimental r d t s are compared with simulation r d t s and suggestions for progressive improvements are made. The experiments presented here may be adopted for other VGT manipidators to exsrnine their appiicability to practical applications.
A control algorithm based on the hematics control experimentst is p r o p d to improve tradting accuracy The control design employa a

layer appraach b the kinematics controL The method is implemented and verified through a series of simulations and experhents.
1.4 Literature Review
1.4.1 Kinematics Analysis of HyperRedundant VGT Manipulators
A cornmon a p p d to the kinematics problem for a hyperredundant VGT manipiiiator is mn/igumtion wnbrd In this approach, the shape of the back bone curve  a curva that captuma the ~ o ~ c o p i c geometric features of the manipulafor  is amtrolled throughaut the motion in order to Bchieve a set of predefined objectives. These objectives can depend on the kinematics, dynamics, or other relevant featues of the eystem. Many papers have been written on codguration control.
$ h o , Reinholtz and Dhande (1990) used Eubic pammetric curvcs to capture the geometry of a long chain of VGT maiiipulators. They explaineci the adwmtages of the cubic c m m : "the pararnetric description allm for curve dopes to pass through the vertical, without numerical dScultiesn and aiso "aows for more flexibiity in controlhg the curve shape."
Nacauato and Hughes (1991) intrduced the concept of efmce shape curves, which is equident to the backbone curves. To model the kinematics of the reference shape c m , several spa curves were mggestecl inciuding Bezier cimes, Esplines, and Betasplines. Nscclvato (1994) i d Bezier akes to model the kinematics of 'Ihtsssrm.
The above h o examples are both extrinsic schernes to parameterize a backbone c m The backbone curve is modeled bg mathematic qace curves

mich as splines. An alternative is to adopt an intrimic scheme where a back bone cime is defined by its intrinsic representation in terms of a natimal p* rameter. Chirikjian and Bwdick (1990  1994) used this scheme to define the backbone curve by using four independent shape fimctions. Their method was refmed to as the continuum approach and has been applied in numerou8 applications of hyperredundant manipulators.
Zanganeh and Angeles (1995) argued that "the contin~nun approach requires some intuition in the choie of the modes and is not a straightforward task." An &rinsk scheme was p r e f d and it was suggested to use splines for parameterization of the backbone cuves. Regarding the reference shape curve method suggested by Naccmato and Hughes (1991), Zangmeh and Angeles suggeste that Bezier curvee and BapLine8 do not provide a direct means for incorporating the prescribed orientation ata of the ende$ect~r in the curve. Instead, they proped a parameterization of the backbone cum using cubic and quintic splines. One of the advantages of this method is that the user can directly incorporate the endeffector pose constrainta into parametric equations of the backbone c m ,
Zanganeh, Lee and Hughe8 (1997) d a de13 of 3DOF eztensible gimbal modules (ah referred to as "equivalent serial mechanismn) to reptesent a backbone c m . The method was d e r r d to as a discrete mode1 approach and applied to 'Lhunurrm. Wiams and Mayhew (1997a) applied a similar approach to another VGT manipulator and the method was referred to as the virtiial serial manipulator approach.
In peral, configuration control is used because of computation & ciency. The procedure is a b often weilsuited for paralle1 procasng. Haw ever, configuration contro1 does not permit opthnhation of otha &ormance criteria such as joint limit amidance and siquhity avoidance williams and

Mayhew, 1997aj. More direct methods of solving the kinematics problem for hyperreduudant VGT manipulators include closeciform inverse bernatics analysis [Padmanabhan, et al., 1992) and ni~merical solutions to the Jacobian matrix [Cdeld, et al., 19961.
1.4.2 Design of and Experiments with VGT Manipula
tors
While meny papera have been written on the design and kinematical analysis of VGT manipul8tora, only a handhil of VGT manipulatom have been actudy constructeci and experimentdly andyzed. TO date, VGT manipulators that have been constntcted are:
a SDOF Doublmctahedral VGT at NASA Langley Research Chter (LRC) W i , 1994 '
h b a y , Octahedraloctahedral VGT at NASA LRC [Robertshaw and Reinholtz, 19881
%perd b b a y , Octahedraloctahedral VGT et NASA LRC, * [William and Mayhew, 1997al
zhissarm Mark 1 [Hughee, et al., 19911
a Serpentine Thss Manipulator (STM), a 16DOF tetrahedral VGT, at NASA Kennedy Space Center Piams and Mayhew, 1997bI
'Som figures shown in this chapter have been coilected lrom the Id wide web. Copy rights d tha figures rue reswad by tha authors of the web pages. 21n aiabomtion with Departmat of Ewgy, Battelle, Pacific NO&.. Laboratory,
and Oak Ridge National Laboratory.

Figure 1.3: 3DOF VGT at NASA Figure 124: 'Ihmm Mark 1 at UTIAS
0 3@DF planar VGT at California Inetitute of Technology [Chirilcjian, 1994, 19951
TETROBOT system Badin and Sandn, 1994,199?j The 3DOF VGT at NASA (Figure 1.3) and the Ihuissrm Mark 1 (Figure 1.4) are both SDOF VGT manipulators. In fact, they may be utilized as building blocks for a larger and psibly redmdant VGT mrrnipulator. The 3DOF NASA VGT bears a atrong resernblance to the TRissMn Mark 1. In hia study, William (1994) modeled a 3DOF VGT as an extensible gimbal, which later extended to the deveiopment of the V g i d Serial Manipulator Approach to the h w b y VGT bematics. Hm, the term 'bay' d e r e to a module. This is similar to the discrete d e l appoach to Ihisssrm kinematics tibcmd in

Figure 1.5: Tapered W b a y O c t a h e d r a l e a l VGT at NASA LRC
Figure 1.6: Serpentine Ihiss Mluiipulator, NASA Kennedy Space Center
10

Figure 1.7: P k VGT, Caltech Figure 1.8: Tetrobot

Chapter 3. The 3DOF NASA VGT, hbay VGT (figure not available) and the
tapered twebrry VGT (Figure 1.5) seem to belong to the same family of me nipulators dedopeci by LRC, which has Leen uivolved with numerom expar iments relateci to the analyeje of VGT mruiipulators.
Robertsbaw and Reinholtz's twobay VGT is one of the fin& VGT ma nipulators developed by LRC to dernonstrate deplqrment concepts. This spi+ tial VGT is a b b a y , statically detamiriate, Octabedraloctahedd structure with three actuators, providing 3 degrees of fieedom. In totai, the structure is nearly 4.3 meters in length. Robertshaw and Reinholtz (1988) demonstrateci the accwacy of this long manipulator by drawing on a surface with a pan at the end of the manipulator.
The t a p d hmbay VGT is a part of the Selective Equipment Removai System (SERS). SERS was designeci for removing radioactive equpment and nuciear materiaie h m wciear waste sites. One of the subsysteme of SERS is an &DOF depbyment mluiipulator, whieh is a tepered ~ b a y VGT mounted onto a ZDOF &ai joint. The dimensions of the f h t VGT module are iden tical to the 3DOF VGT in Figure 1.3. The seamd module is d e r in size.
A 30DOF planet VGT (Figure 1.7)was ddgned and conet~ucted at the California Institute of T&oIlogy (Caltech) to validate th& analytical nork in kinematics and dynamics of hyperredundant manipulators. Chirikjian and Burdick (1993) wrote that the VGT was chosen among several morphologies of hmredundant ma,aipuiators for "practical implementationn and the p b ner manip~dator instead of spatial manipulator Tor simpIicity's saken. This planar VGT mruiipdator m118iSt8 of 10 identiccri modules. Each d u l e is, in turn, a 3DOF planar pardel manipulatoc. The manipulator arn mntract to a minimum leiigth of 3.66 meters and extend to a mmchum 1 of 5.49

Figure 1.9: Payload Inspection and Processhg System
meters . The manpulator weighs lass than 55 kilograum. The S D O F planar VGT manipulafor hae the edvantage of king moulair and extmnely easy to assemble and disassemble; however, it share~ with Thrssarm an accurq prob lem of enddector positioning. Frthermore, &ce it is a plenat manipulator, it would have Iittle practicality for mart industrial applications.
'Serpentine' is simply another term ueed to describe a hmredundant rmnipulator. 'Snake,' 'tentade,' Wghly articulatecl' and 'highly redundant ' also denote the same f d y of manipulatom. The serpentine d p u l a t o r (Figure 1.6) described by Williams and Mayhew (1997a) is a 16DOF tetrahe &al VGT manipulator developed at the NASA Kennedy Space Center. This manipulator is a mbsystem of the Payload Inspection and Processing System (Figure 1.9).
The kinmatics modeiing and configuration contro1 algorithm, ref'ed to as the followtheleader algorithm, has been developed and applied to this manipulatot. Based on the eXpenen leamed h m this exercise? s e v d rec ommendations were made by Williams and Mayhew including reducing the actuator noise, and ensuring that the control system aiIowa all motors to reach

each commandeci set in the same time interval. The TETROBOT (Figure 1.8) developed at Remdaer is a modular
and hyperredundant VGT mechanism. The TETROBOT famy of robots use a joint mechaniwi called Concentric Multilink Spherical Joint, that sup ports modularity in not only the hardware configuration but also the software architecture. The system is reconfigtuable with different applications. Three examples given by Hamlin and Stindemn (1997) are the doiibleoctahedral and tetrahedral VGT manipulators and the sixlegged walker. The double octahedral TETROBOT has the same configuration as the ?hrsaarm Mark 1 and the 3DOF NASA VGT. The main diffaence is that 12 of the 21 links in the doub1wxtahedral TETROBOT are actuatom The d p u i a t o r is 1.5 me in length, and can m m verticaily alm08t 1 meter. The p e y l d is 80 kilogram and the peak end4ector speed is 6 m / s . The tetrahedral manipu lator TETROBOT is corn& d 4 modules with all membem actuated. The tip velocity is approxhately 8 cm/s and the theoretid payload is 40 kg.
1.4.3 Kinematics Analysis of Tkussarrn
Two approachts have been taken ta the bernatics problem of Thssarm: the linklength metho and the peta angle m a .
The petal angle method wm used by Hughes and Naacarato (1991), Sall men (1993), end Naccarato (1994). In all three cases, a dosedform solution was achieved for the i n . kinematics, but not for the fimuard kinematics.
Oikawa (1995) briefly corisidered the diflerentid imerae kinematics of a ?ihusarm module using the petal angle method. He demonstrated how compcated the solution of the Merentid kinematics problem is.
The linMength method wris exsmined by Hughep, Sincarsin and Carroll

(1991), and Naccarato (1994). Instead of the petal angles, the linklength method rises the coordinatee of the node points as variables.
The diff&rential kinematics based on the linklength method was a b stiidied by Naccarato (1994) and Oikawa (1995). The input to the differential inverse kinematics problem coneists of the coordinates of six moving nodes of B Thmm module, resdting in 18 independent coordinates. Accordingly, the size of the system Jacobian is 18 x 18. This requires over 60 times more computation than the 3 x 3 matrix in the petalangle method. Thiis, the petal angle method is computatiody more eflrcient than linklength methods.
1.5 Overview of this Document The system description of Thssarm is presented in Chapter 2. Components of 'Lhissarm hardware are divided acoading to th& functions into three groups: aewornechanism, supporthg stmctiire, and control eystem. The physical lay out of each component is desxibed uul their hctions SUII1II18fiZed. Several technologies related to 'ihiassrm geometry and kinematics are introduced. For instance, the definition of Ihissrrnn joint coordinates is state. Mhermore, the ciifference between joint coordinates and motor coordinates are given. The frame of rderence used in the following chapters is also definecl in this chapter in terms of the physical components of a h a r m module.
In Chapter 3, the kinematics model of ' Ihsarm is derived; a modifieci discrete model approach is used. A configuration optimization technique is applied to resolve the redtmdancy in kinematics. The forward and inverse kinematics of each module are solved in terms of joint variables (8, 4 and I ) related to the discrete modeL The joint limit avoidance problem is crleo briefly considemi.

Chapter 4 outlines the formulation of the differential kinernatics of Tnmsarm. A Jacobian matrUt and Hessian are derived using a discrete model. For n modules (a 3nDOF system) the system Jacobian matrix is 372 x 3n. EL ements of this rnatrix are expresseci in terms of (8, A 6). The Jacobian mat& of the lhmsarm module is &O solved.
In Chapter 5, engineering performance data are discussed. Experimmts designeci to measure the workspace, dexterity, repeatability, speed, and accu racy are describeci. Through the series of sirnulatian tests, these measu~ements are predicted, then the d t s are verifid with expesiments.
Chapter 6 summarizes a kinematic control systern design. The proposed control acheme is to apply coame control to brbg the endefktor cl to the target pOBture, then to apply fine position mntrol baseci on the Beneor idormation. Simuletion r d t s and experiments p r m that SUfficiently high accuracy in poeitioning can be achieved by using tbie propmed control methd
The conclusions are & a m and suggestions for future aueSarm research are made in Chapter 7.

Chapter 2
System Descript ion
In general, a mmipulator pcissesses three attribut: actuators, supporthg stmctures, and eome intelligence to control the actuators. Some appiicatione require workqace feedback eeneore to achieve the higher performance. In the m e n t codguration, Tkmam poeseinies only the joint level feedback. This chapter describes the t h attributes of Ttvsaarm
2.1 Actuators and Servomechanisms
2.1.1 Ihissarm Actuator
The Thmsarm actuators are the most basic substructures of the Thissarm con trol systern. They are independent linear actuators, each dnven by a DC motor and a gear reduction assembly (pulleys and a drive belt). The motion of each motor is monitored and controiled by an optical encoder and senromechaniism, respectiveiy. The date of each actuator, Le., the distance that the Iead screw

has traveled h m the zem position, is reiated to the encoder counts (M) as
where M, is the number of e n d e r counts per motor revolution, G is the gear ratio, and L the lead screw travel ratio. The factor 4 is included because the Tkussarrn enders operate in the qttadrature mode. In the nirrent configurac tion, the ratioe are 2,771,638, 1,422,878, 1,767,652, and 2,660,782 for module 1,2,3 and 4, mpectively. The specification of the actuatots b given in Table 2.1.
The trawmhion ratio, caldate in Table 5.1, ie utilized to transf' joint coordinates (actuator lengths) to motor amdinates (encoder cou&). For example, the mBlfimum extension of a 'Ihniearm actuetor is 0.0762 m e tars. For actuator 1, the maximum extension in encoder counts is 0.0762 x 2,771, 48 = 211,199 counts. F'or actuator 12, the same length equds 0.0762 x 1,351,734 = lO3,2 counts.
Table 2.1: Thmam actuator apecincations
v
I  Motors
Mluiiit8cturer
Tkansnrission Ratio Encoder Gear Ratio Lead Screw Factor Counts n a d

Table 2.2: 'Ihieserm sewornechanisai PID gains
The enornechankm of each actuator is controUed by GALa motion con trollers. Two DMC1080 controlks have been installeci on the wntrol oom puter to provide PID control. DMCl080's operate in the biie stmcture.
Eech DMC1080 is capable of providing control up b eight axes. For Ihiesarm, only eix axes are servoed by each DMC1080. With two DMCl080's, the conttol system provides ~ e c h ~ to all twelve axes.
The performance of the aervomechanism has been adjusteci using the software, GALIL Motion Control  SDK100, provided by GALE communic& tion Inc. The gains selected for Thssarm actuators are ahown in Table 2.2 and are btued on the step responae of each motor.

Figure 2.1: Ihissarm Module
2.2 Supporting Structure
2.2.1 Modules
Each 'Ihisami module is a doubleoctahecai pardel mAnipulator and pos sesses three tctuators of the bear electric leadmew type. The lead mew assembly acts as a bearing th& converts rotary motion to linear motion. Each actuator is fasteneci at one end directly to the hinge amembiy that connecta the actuator to the other end of the next actuator. As a result, tbree actuators form a triangular actuator plane (see Figure 2.2). When fUlly extendeci each actuator m m 46.4 cm h m the to ter of the binge mechanim. The length of a fully retracted actuator is 31.2 cm. Thus, Ihisrcarm actuators have a 149 % extension ratio.
2.2.2 Passive Members
Each corner of the actuator plane is comected to a pinjoind hinge, which in turn is connecteci to a 1ZMegree aliiminum piece. Ekom the 12Megree connecfor, two dtirninum passive manbers are connected 60 degrees apart.

Figure 2.2: Actuatot Geometry Figure 2.3: Hinge Assembly
See Figure 2.3. This aasembly provide8 the rotetional degrees of keedom needed at the juncture between the passive structure and the actuator plane. The aliimin~un passive members make up six l a t d triangles. In the m e n t configuration, each member is made of 0.95 cm aluminum tubing.
At the base of the lateral triangles, three pacsive members, made of 0.64 cm cidl rod, form a base or top trimgth plane. A top plane of one module and a base plane of the next module are connected with a passive comector to connect contiguous modules.
2.2.3 h s a r m Coordinate System
Consider the base plane of module 1, shm in F i e 2.1. The three cornara of the plane are denoted hi, L, b, in clockwiee direction. The geometric center of the base triangle, 00, is denoted the amter of the base Bame, FO. Moreaw, we define the xaxis of F0 as a unit vector through L, and the zaxis as the vector out of the base plane. The yack is d&ed as the vector orthonormai to the x and zaxea The actuator above bot is denoted actuator 1 and actitators 2 and 3 are numbered in clockwiae direction.
The geometnc centei: of the top plane, 01, is the cent of 4. The

.   
Figure 2.4: 'Itusearm Control System
position of Ol e x p d in Fo and the orientation of of Fi with respect to describe the codguiation of module 1. Fi eleo coincides with the baee kame for module 2. Therefore, for n modules, there are n + 1 kames in total.
In the actual manipulator, the base plane of module i + 1 and the top plane of module i are aeparated by a srnail ofkt in the passive connecter. To simplify the kinematb, the Fi are definecl in the midplane behmen t h e two physid planes.
2.3 Tkussarm Control Environment (TCE) The purpose of the control unit is to eervo dl the actuators simult8neo~~ly in such a way that the manipdator performs the desired task. One of the simplest tasks proped for h a r m is trajedory tracking, such as trachg a prescribed endeffector trajectory
The control architecture is s h m in F i e 2.4. The main unit for the Thssarm control system is the 'huniarm Control Environment (TCE). TCE was designeci in M i ~ o 8 o f t Visual C++, based on the DMC communication

sohare provided by GALE communication Inc. In addition to the basic huictionsliQ reqimed for simple joint contrd,
TCE is capable of nimber of fimctione.
2.3.1 Tkajectory Planning In order to demonstrate the kinematic controI, a simple path generatioa h c  tion has been added to TCE. A lin, circiilar, or rectangcdar path may be generated fmm the path generation menu. Base on geometric information, such as the circuiar path, the corresponding poeifion and orientation of the endeff'tor is computed in three dimensional space.
2.3.2 HyperRedundancy Rasolut ion
Given the Cattesian coorinate8 (pition and/= orientation) of the en6 effector, the configuration of each module can be mputed using an eqyivalent discrete mudel approach to hyperredundancy remlution.
2.3.3 Kinematics
TCE is capable of eolvhg both inverse and forward kinemafics of ' h s s a m modules. Once the configurations of d modules are obtained from the hyper redundamy resolution process, the information is then used to solve the inverse kinematics, wbich yielcb the desired mtuator lengths.
For a given end&ector path, two DMC filas are prepared by TCE. Each file contains a trajectory for six actuators (in mtor counts). The des id

actuator lengths, or joint coordinates, are converteci into motor counts. At the beginning of each DMC file, the servo apeed is adjusted.
2.3.5 Motion Execution
The TCE is designecl to communicate with 1 or 2 DMC 1080 boards. For motion control with 6 or fewer actuators, only one controk may be used. For motion control with 7 or more actuators, both boards need to be selected. For the kinematic control, two DMC files must be externdy downlosded to DMC1080 boards. Then, fkom the motion menu, the program on two boards are executed simuitaneously.
2.3.6 Cartesian Control
Using the terminal mode, the current Carmian position of the end4ectOt may be manitoreci and commandeci.

Chapter 3
Kinernat ics
The motion of a manipulator is often epecified by the mtion ad/or orien tation of i ts endeffector. If the manipulator ie redUILdllllt, soma functional specification is additionally regnired to resolve the rediindruicy. HHamn and Sanderson (1997) d&ed this problem as the kinematic control pmbIem. In their paper, the term "kinematic controln was used to describe Utlie stutic po sitioning of a monipdator, mociing slowly thmugh a set ofpdeflned trajectory points." In kinematic control of a hyperredundant manipulator, tthee tasks are i r r m 1 d redundancy remlution, manipulatot kinematias, and control. In this chapter, kt two topics in kinematic control are considered. The control methads are considered in Chapter 6.
A configuration optimization is an approach to the redundancy resolu tion problem. A proposed scheme for configuration optimization by discrete mode1 approach, first developed by Zanganeh, the author and Hughes (1997), is presented in Section 3.1. In Section 3.2, the forward end inverse kinematics of a IhL9881pl module are exBrnuleci. In Section 3.3, another issue in kinematic contrai, joint limit amidance, is considezed.

  
Figure 3.1: 3DOF extmsii1e gimbd module
3.1 Kinematics of an Equivaient Discrete Mode1
3.1.1 3DOF Extensible Gimbal Module A diecreta mode1 that crpprooMurtes the backbone curve in a p i e i n e a r W o n was inttodud by Zanganb, Lw and Hugha (1997). A d i s e mode1 of a byperredundant mAnipdator b obtaimd by a oonc8tmation of 3DOF extensible gimbd modules. Each module hs, in tum, three sp that p d d e ampliance Lot the three motion freedmie.
T h e g e o ~ d b ~ t i c a o f a ~ d u l e i e p ~ b y t h r e a independent vatiablcs, O,+, and 1. As shown in Figure 3.1 (i), a gimhl m d e wneisCB of a iuiiversa jad with tno rotational DOF and one prismatic joint. Taie mode1 is modifieri b taise advantage of the geometric qmmetry (about the actuator plane) of the module. Instead of one prismatic joint, two coupled priamatic joints are used (Figure 3.lCi)). The 1 4 kame is assigneci to the ith gimbal module such that its center is at the base of the nrSt prismatic joint and the 2 and y are paralle1 to the rotation axes of the universal joint. The orientation d the zth kame, Fi, with mpct to &i, is described

&O the position of the ongin of Fi, e x p r d ni 61,
where 1 ie the 3 x 3 identity matrix. The kiaematica of an extensible gimbal

3.1.2 Chain of Gimbal Modules
Let us now consider a chain of ngimbal modules, refmed to as the equivdent dLPmte modeL Expressxi in the base hune, FO, the position and orientation of the end &tor (or the tip of the manipulator) are
n
where t ami Qt represexst position and orientation of the end &ector with respect to Fn, and a h ,
r
where b = k+l = 0. Aleq E is a selection matrix and k is a unit vector,
The inverse kinemafics of the quivalent diacrete mode1 is much more complicated than the forward kinematics. This problem is oRen refmed to as the dundancy ~ o l ~ pmblem, since the number of unknown variables is much greater than the number of constrejnts. The redundancy rdution methode are often at the velocity level. They are baseci on the manipulator Jambian pseudc&verse, The main goal is to minimize a ct hction, Z(q), and to d v e for the joint variables, q.

If the manip~iiator possesses a large number of degrees of redundancy? Le., if the manipiilator is a hyperredundant manipulator, its kinematics can not be e d y formdated iising an inverse Jacobian since it is ciifficuit to buiid a mathematical model. An alternative technique to resolve hyperredundaucy is configuration control.
The method proposeci by Zangaaeh, Lee and Hughes (1997) is to mini mize the cost fimction, 2, of the sy~tem, aven as
while satg the kinernatic constraints:
where x, ami n, are the desireci position and orientation of the endeffector. The Lagrangian approech is adopted to h d the solution set, q where
Ifwe define w(q) and u as
then the Lagrangian is

and thus, the normality conditions are
where X is the vector of Lagranpian multipliem. For n gimbal modules @ = 3n DOF system) with m kinematics constraints (m = dim(g) + dim(h)), eqs. (3.14) and (3.15) lead to p + m nonhem qations in te of q and A.
The w term in q(3.14) hm alreaciy been derived in eqs. (3.3) and (3.4). Now let us consider the tem in eq. (3.15). Rom eq.(3.7), we have
Next, consider term. Rom eqs. (3.3) and (3.4), the foliowing equations may be derived:
where
*
for i = 1 ,... ,n with P,,, = ki + t and =Q&. By d e t i o n , $ is

for i = 1, . . . , n. For each i, the fimctions can be differentiated with respect
Equ&ions (3.14) and (3.15) may be dved using any numerical method for noniinear systems of equatiom. Examples include the muitidimensional secant methods euch as Newton's method, quasiNewton method, and the Broyden's method. in the contder design, Broyden's method was emplayed Chapter 9 of Pr, et al. (1992) may be referred to fol more detae.
3.2 Kinematics of a Tkussarm Module
3.2.1 Inverse Kinematics
In this section, the inverse kinematics probiem for a ' I h saxm module is con sidered. The objective is to solve for the actilator lengths of the jth module, kit when the corresponding gimbd module's joint able qj aie specined. Since ody one ' I h a m ~ ~ module is considemi at a t b , the subscript j will be omitted in this and the folowing section.

Consider the geometry of a modide shown in Figure 3.3. It can be observed that the vector d is the mm of 4 and 9. Le.,
d = I ~ + f 2 = l ( ~ + ~ ) = d i i l (3.29) where nI is the unit normal of the actuator plane. The unit direction vector of the top triangle, fi2, may be cornputeci using 0 and 4:
where
By definition, coL(0, O, 1) sine dl the vectom me defineci in 3ji, wha~e zaxis is pazailei to &. Hence,

Let us defme this scalar as a. Then, iil is
sin t#
a
Since d is perpendicular to the actuator plane, dis perpendic~dar to any vector lying on this plane. Using this relation, we an calculate the petal angles, 71, n, and 73. Rom Figure 3.3, it is shown t h t 62 (the o h t of the actuator plane fkom the plane formed by petal nodes, ri, r~ and Q) lies in the direction of iil. Aence, for i = 1,2,3, we have
where is the ith column of the matrix e that tepresent the the base plane,
lm is the side hgth of an ieometric petai triangle and hl is the side length of

the equiiateral base or top triangles. For Ihissarm II:
hl = 29.14 cm (3.44) lm = 27.33 (345)
til = 0.95 (3.46) 62 = 4.83 (3.47)
Refer to Oikawa (1995) and Lee (1995) for more details of h a r m geometric constants.
Substituting q(3.40) into eq.(3.39), we obtain the following equation for the petal mgles:
[(CI + Q C Q B ~ ~ ) ~ + (61 + ~ ~ i n r i ) i i o  (1%  b&)lT d = O (348) Eq.(3.48) may be simplified:
where
Thus the petal angles are
Once the petal angles are cdcttlated, nociai coordinates are found h m eq.(3.40). LBPtly, the actuator leogths are computed from the nodal coordinates:

The remaining nodes on the base triangle and the top triangles are determineci by simple trigonometrie relations. The base nodes are
and, for i = L,2,3,
(3.58)
The top nodes for i = 1,2,3, are cornputeci using symmetric propertias, i.e.,
b, = Rolbo,+d (3.59) & = to,6ih (3.60)
and
3.2.2 Forward Kinemat ics
The fornard kinematics problem deals with solving for enhffector pmition when the actuator lengths are given. ki this section the solution for q = (O, 0,l) are considered when & = (kl, ly, 4) is piven.

Upon substitiiting eq(3.40) into eqs. (3.54), (3.55), and (3.56), we get
These equations can be simplifieci to
These nonlinear equations are mlve for 71, .%, and numericaliy. Then,
Moreover, using eq.(3.38), we can write
or, a = 2ni,. Also, fiom eq.(3.38), we can derive the foIlowing relations for fi2:

for O 2 8, $ 2 $. Given the workspace of 'Ihusarm module, this restriction is remonable.
Equation (3.48) can be written as follows:
Thus, we can solve the above equation for 1, Le.,
The inverse kinematia solution preaented here is b a d on the petal angle method. By introducing a new set of input variables, d , q5 and 2, the requirement to solve the forward kinematiica was r e m d . The algorithm is now in c l d form, and also codd be computed in pardel fashion. The p r o p d forward kinematia could also be solved by pardel computation. Udortunatdy, the expiicit c l d  f o m solution for the forward kinmtics problem h a not yet been obtained The solution for the differentiai forward kinernatics is presented for the 6rst tirne in Chapter 4.
3.3 JointLimit Avoidance Problems The redundancy resolution scheme presented in Section 3.1 does not guaran tee that the resulting optimal confisuration remaim within the actuator lim its. Hence, the next Logical step is to develop a hredundancy resolution scherne that indudes the actuator limits in the configuration control.

The existing Zhisssrm actuators are capable of hear motion between f0.0762 m fiom their [email protected] This, for i = 1,. . . ,3n,
Let iis also define k,, a t h ens si on al vector whose components are three actuator lengtbs of the jth bay. Hence, k j is a function of Bi, #j and lj. h o , we define a new vector k as col{kt, kz,  , ka%).
The objective of the prob1em ia now to rninimize the cost function, 2, while the folowing kinematic constraints and the following inequality con straints are met:
w  v = O (377) ( q )  5 O i  1 ,  ,3n (3.78)
A nonlinear optirnization pmblem with both equality and ineqdty constraints is &en mlved numerically. h shown in an example in Appendix A, it is very difficidt to fhd a computationally &cient algorithm for the problem. For more details on the optiniization techniques, the readers can d e r to Strang (1996) or Vaderplats (1984).
Vanderplaats (1984) exBrnineci a general approach to the constrained optimization problem called aeqwntid uneonstrcrined minimizution techniques (SUMT) . The classical approach to iising SUMT is to create a pseudoobjective function of the form
where Z(q) is the original objective function, P(q) is an i m p d penalty func tion, and the scalar r, is a multiplier which determines the magnitude of the

penalty. The siibecript , is the unconstrained minimization number. For more details, see Vmderplaats (1984) and Fiacco and McCormick (1968). Three traditional approaches to SUMT are "the exterior penalty fimction method," "the interior penalty hinction method," and "the extendeci interior penalty hinction method." The nret method is to penaze the objective fimction only when constraints are violateci. It is the &est method to incorporate into the design algorith. The second method penaiizes the objective function as the design oppmache~ a constraint, and the last method is the combination of the fArst two*
In Tbsarm's kinematics analysis, the exterior penalty hinction method waa applied. The peeudwbjective fiuiction for the probkm can be defineci as
1  &. The problem is solved in three stepe:
1. Minimize @(q) with initial guess for q, a mail value of r, and a constant r ( W ~ Y 3)
2. If the solution convesges  that is, if the 801ution q satisfies the kine matic conetraint, w(q) = O  end the procedure.
3. If wt , the penalty parameter r, ie in to "~r, and the procedure is repeated.
In order to rninimlze O(q), a set of 3n nonlinear equatiom are obtained and solved iising Broyden's method:

T [m=[O.h]  m=[O,Q] ] (3.82)
The solution for $ was presented in Section 3.1. To evaluate g, let us define p, a~ the joint variables of the jth gimbai module, Bi, 4j and S. Then,
is the i n . e Jacobian of the jth 'IhLssarm module and its solution wiii be discussed in Section 4.3.
The algorithm is relatively easy to program and a h reamnably efficient.

(ii) Vacm on the base p h e *Y
(iii) Vactors on the b a t o r p h e Figure 3.3: Vectors in the kinematics of a Thsami module

Chapter 4
Differential Kinematics
In the previous chepter, it was shcnun how mcult it is to h d an explicit cldform solution to the kinematim problem of hm. In genesal, e lutions to the kinematic problem require numerical approech. An alternative approach to the kinematic control problem is to examine the differential relk tionship between the enddfctor coatiguration, u (z, and fi), and the joint variablea (or design variables), q.
Previously, the bction w(q) was defineci as
If the above function is differentiatd with respect to time, we get
If the Jacmbian matrix, $, ie invertible, Le., if the manipulator is not redun dant and has no kinematic singtilarities, the joint velocity vector 4 can be easily obtained by

However, if the manipulator ia redundant, eome redundancy tesolution methoci needs to be employed. The objective of this cihapter is to find a solution to q, when O is specified.
The control technique describeci in q(4 .3) is commody rderred to as resolved motion rate control (RMRC). Schilling (1990) defined RMRC as a ''probim to expmss the actuator velocities as finctons of the veloctOCtties of the task coordinates."
In Section 4.1, the RMRC of ?hissarm is discussed. The Jacobian matrix and Hessian are derivecl in Section 4.2. In Section 4.3, the diffixential kinematics of a Tnissarm module is presented. The actuator vdocity vector, k , is relete to the equivaient dirwete model's joint velocity vector, q, and its inverse hction is presented.
4.1 Resolved Motion Rate Control
8 w T where W = (=) and r = (#)T. If we partition W and x thus,
where, dim(Wl)=m x m, dim(Wz)=@  n) x rn, dim(zi)==m x 1, and dim(a)=@  m) x 1, then we can write

Let iis defme the lahandside of the above equation as N. The problem in hand c m be written as follows:
Compareci to eqs. (3.14) and (3.15), the total number of equations hm been d e c r d &an m + p to p. Upon differentiating eq.(4.8), with respect to tirne, we obtain
where quivalent mtrix and vector We and v. are defineci as:
Thus, provideci We is not singuiar, we have
4.2 Jacobian Matrix and Hessian In this section, we concentrate on the solution for W. in eq.(4.11). Note that the andytical solution for $ is given in Section 3.1.2. The solution for is derived here. By dennition,

Fkom eq.(4.7), we can derive the relation bdow
where and are the fmt rn rows of $ ($f)T and & (c)=, respectively, while and are the remaining ( p  m) m. Rom eqs. (3.16), (3.17) and (3.18), the fohwing derivatives may be obtained:
for a, j E (1,  , n, i # j), where q, is one of the joint variables associateci with the j'th module, (O,, #j , 1 ,) . Moremer,
In rnatrix form,
O, fkom eq.(3.22), we can write in the fonn
Equations 3.23 thmugh (3.28) cm be used to calculate $ (e) and & (h) . Since Qi is a hrnction of Bi, q5i and 4 oniy, = 0. Fhthemore, we
define

Then, for j < i , Merentiate eq43.23) with respect to Bj to obtein
The
(3.26), eq(3.27) and eq(3.28) we get
saxne d e appiies to the derivateg with rsspect to #j and lj.

Next consider the second derivative of W, &, where i = j. First differentiiate eq.(323) with respect to Bj to get
Simibrily, the deri~8tive~ with respect to Bi are,
N e , c0f18jda the derivativee nth respect to &j

Findy, the derivatives with respect to t j are
So far, the elements in have been expressed in terme of the first and second derivatives of the rotation matrix, Qj, 2 and a. We now derive the expression for these derivatives. First consider 2.
The second derivatives of the rotation matrice8 Qj may be derived from the

Thus far we have derived the solutions for aml in eqe(4.14), aar which is part of RMRC equation, (4.12). We are still misahg the solution
8 ~  ' for w;' and in order to complete eq(4.14). Since the dimensions of Wl depends on the number of hematic constraints, rn, we can not find the general solution for the problern. However, an example case with m = 3 is
For rn = 3, Wl may be written as
The adjoint matrix of Wr is denotecl by w1, where
And, the determinant of Wl is obtained as
The corresponding dezivatives of Wl with respect to qj take on the forms

4.3.1 Differential Inverse Kinematics
The objective of this section is to derive the dgebraic solution to the iwme Jacobian matrix, G:
where, k = (ki, k2, ka), and q = (O,#, t ) . Hence,
Ekptiom (3.54), (3.55) and (3.56) may be written as followa:
Differentiating the above equations with respect to qi, for i = 1,2, and 3, yidds:
In order to consida derivatives with respect to al1 three independent variables, O , $ and 1, let us define an operator, $, as foilows:

Then, for for j = 1,2,3, we have
upon solvhg eqe. (4.67), (4.68) and (4.69) for 2, we obtain
A b , eq(3.40) can ba cMerentiated to fhd an expression for 2, namdy,
Hem, the only unknown variab are ge h m eq.(3.49), we have
Rearrange this equation to solve for $.
80 8 In order to solve this equation, we now need the s01utioas for f , &, and K . First, let us conaider $. Rom q(3.50): hi
If we dehe new vectom 4 = coi{q, 0 2 , a3) and f = col{ f I, f2, f3), their derivatives may be expresseci in the matrix form as

and similm for g. Equation (4.76) then can be written as
Recall that e ie defined in eq. (3.41). N d , for $ digerentiate eq(3.52).
where
an, 8% 81  = (Ro + h)T()l + g(p + () [iiT(ng + w] 6q a!? Bq Bq
Third, dilferentiate eq. (3.51) to hci $.
Up to this point, derivativee have been expressed in ternia of derivatives of unit normals, RI and t2. h m (3.32),

Fkom eq. (3.38), we have
where
4.3.2 DBerential Forward Kinematics
The forward Jambian, 3, is developed in this section. At the outset, we note that
Differentiating eqs. (3.66), (3.6?), and (3.68) with respect to k, yielde:

where and for i = 1,2,3 are known constants. For example,
where O is the 3 x 3 zero matrix, PI = diag(a1, al, al), Pl = diagm, Pl, A), and ao on, and k = col{kl, 0,0,0,4,0,0, O, ks). The above equation ie aolved for 2 for i and j = 1,2,3. Moreover, by dinerantiathg eq.(3.40) with respect to k, we can &O find 2 as
Rom the definition of the unit normal of actuator plane, fii, the following relation is derived.
where

Note that
aArl Brl Brz =
Bk a akd (4.99) eq. (4.97) may be expandeci using the properties of the cross prcxiuct ,' Le.,
Rearrange the above equation for %:
But, a = 2bi,. Thdore,

Similady, n2 is a function of 0 a d 4. Therefore, fiom eq.(371), we have
where
The Ja~obian is, in fact, the collection of eqs. (4.105), (4.104) and (4.106),

Chapter 5
Performance Measurement s
The robotic applications have been proposeci in numerous areas where the use of humane iii imprELCticai and undesirabie. The following meas are examples of the application of industriai dpulators:
O material trda taslee, such acl molding and stampng
O assemb1y tasks such aa weiding and grinding
taaks in hazardow m.nments such as undema and planetary expl* ration, and satellite r e t r i d and repair.
The perfotmance of an indiutrial manipdator is often determined by several criteria inciuding:
a repeatab'ity
O speed, and

Depending on the application of the manipulator, the signifimce of each prop erty may vary. For exsmple, for a positionhg application such as a pickand place maneuver, speed and accuracy are the m a t important characteristics while dexterity is not as signifcant.
In order to examine 'Lhtssarm's practicality as an industrial manipula tor, these properties must be determineci. This chapter is about such measure mente. Throughout this chapter, the control experiments designeci to measiue aome of the pmperties are outlined and performance characteristics are dia cusse* The measurement methode introduced hem may be ex~~flded to other VGT manipulators.
Sections 5.1 through 5.4 focus on the generd duracteristics of aUeearm. A more indepth discussion is pmvided for accuracy of the kinematic contd in Section 5.5. The kinemtia modal, developed in the previous chapters, are applied to implement Mnematic contrd algorithm In Sections 5.5.2 and 5.5.3, the f0d and i n . kinematics algorithm preeented in Chapter 3 are a p plied for position contrd and tracking control, respectively. In Section 5.5.5, resolvedmotion rate control developed in Chapter 4 ia applied to examine the veiocity contml.
5.1 Workspace Analysis
5.1.1 Definit ion
The wotkspace of a mmipulator is the total volume swept out by the end effectoc as the mruiipulator executes dl possible motions. Two types ofworkspace are o h considd. a reachab1e workspace and a dexteroua workepece. Spong

and Vidyasager (1989) de the reachable workspace ae "the entire set of points in space reachable by the manipidator," whereas the dexteroiis wrkspace is the set of pointe that "the manipidator can reach with an arbitrary orien tation of the end &ectorn In this section, only the reachable workspace of Thssarm is considered. The dexterous workspace of Ihiasarm wiU be consid ered in the following section.
5.1.2 Workspace Measurement Met hods
The methods to meernue mkspacea have been examinai by several authors including Korein (1985) and Kwon et. ui. (1994). One oommonly used method is the 'extension method.' A set of points in the workepace ie generated by stepping egch actuator through its range. For Ihissann with n modulee (that is, 3n ~ctuatom), comb'u18tions of joint coordinates have to be considered, where r is the number of dleaete joint settings. For exemple, when three set (maximum, center and minimum actuator lengths) are used for 1 2 DOF 'Russarm, 312=531,441 points have to* be considemi. It is extremely difncult to ewmine that many points to derive a mathematical description of the workspace.
The extension metho was first applied to 'Ihiaearm by SaIlmen (1993) for one module. S h e n d d b e d the wot.lrspace of a 'Ihisaann module as an umbrda with three corners. The workspace of 12DOF h a r m wes studied by Hughes et al. (1999). In both studiea, 'Ihissarm workspace was computed by solving fumard kinematics at each actuator lengtbs at discrete actuator displacements. The result found in Hughes et al. (1999) is shown in Figure 5.1. Similar simulation was performed with a set of BCtuator displacements that yield endeffector position^ on the xzpIane. Such joint states can be

Figure 5.1: Ihisearm Workspace by Extension Method

Figure 5.2: Thrss8fm Workspa by Extension Methd, xzplane

uniqiiely d&ed k3i+l = k3i+2 for i = O, 1,. . . ,n  1. To keep the num ber of discrete configurations amall, three joint states are chosem km=, 0, and &. The tesiilting joint sets are combination of the following settiags for IYUM~: (hnmc, h a x , &), (hmw k m , 0)7 ()max, kmax, krnac), (Q,O,kmax), @,O, O), (0, O, kw), (hl b9 krnax), (&, &, 0). and (kmin, k, kmin). For 4 modnlee, the total nimbe of diecrete configuration is g4 = 6,561. Result of the simulation test is shown in Figure 5.2.
Another method is the 'sweeping method.' This method was proposed for an openlink kinematic chah by Korein (1985). First, the k t joint in the Chain is ewept through its r w to generate Wi. Then, the second joint is m p t to generate Wg and eo on until the last joint is swept.
Since Thsam is not an openIink seriai dpula tor , the gweeping method can not be directly apped. The method p r o p d here is a c o m b b tion of the extension method and the sweeping method. First, the extension method is applied to the nth module to determiae the workspeca of the last module. Then, joints in the module n 1 are VIUjed to detamine the workspace of chsui of module n and n  1, and so forth. As in the pcevious section, the workspace on xzplane is considered h m . The boutldary of the workspace ie appraximated as foilm: (tefer to Figure 5.3.)
a Choose a point in space, 6.
a Cornpute the distance, ri and angle, Bi from e. to anend&tor pasition rithin the workspace7 pi:
c.,  Pi, 8. = tan'( ) C O $  Pi&

a Determine points pj where its radius is the mBximum and minimum arnong dl pi's auch that 8ji < Bi 5 ( Bj = W), j = 1,.* ,No N be arbitrarily chosen for the 8ccuracy of the woaicspace representation.
a The workspace b o d a r y can be approximated as the curve on which pj lie.
This method of approximation can be easily extendeci to the three dimensionai representation of the worirspace.
Figure 5.4 shows the workspace of one module generated by the sweep ing method. The nine actuator displacements that were used in the extaion math& are also marked on the aame gmph to permit cornparison between the two methode. Figure 5.5, 5.6 and 5.7 show the rorkepece of 2, 3 and 4 modula respectiveiy. The boundary of workspace is marked with a dotted iine. The ehaded region an each plot ie the workspce of chah 1, 2 and 3, respectively. Figure 5.8 is the coilection of the results lrom Figure 5.5, 5.5,5.6 and 5.7. It can be e d y aeen how quickly the workspace of Tivssazm grows with the number of modules.
Lady, Figure 5.8 is the combination of results lrom the extension method and the sweeping method. Rom this coniparison, it is noted that the extension and 8weepi.q methods provide very emiiler redts. However, because sweeping is only the firstorder approximation, it does not yield all possible points. Instead, only a good approximation can be obtained. On the other hand, this method is more computatiomlly efficient and less tirne consiuning*

Figure 5.3: Workspace Boiindary Approximation

L a 
u
u
Figure 5.4: Workepace d 1 Module Figure 5.5: Workspace of 2 Modults
sa
) ....
u+
u
U
La
t
Figure 5.6: Worirspace of 3 Modiiles Figure 5.7: Workspace of 4 Modules

Figure 5.8: Ihissarm Workspw by Sweeping Method, xzplane
Figure 5.9: 'ihiaearm Workspace by S w i n g Method, xzplane

5.1.3 Dussarrn Workspace Experiments
Next, the focus is placed on the experiments, designeci to demonstrate the c* pabiity to reach the predicted workspace limits. F M , a computer simulation with animation was devdoped to simulate the motion during the worirspace ex periments. Figures 5.10 through 5.15 show some exampIes h m the computer simulation. Table 3.1 Iists the achiator lengths in cm useci in the simulation tests and also the resulting endefllitor coordinates. Even though egch of the
Figure 5.10: Home Configuratim Figure 5.11: JShape Co&gwation
twelve actuators can extend aad antract up to & 7.62 cm, a f 4.0 to I 6.0 mi extension wse chosen for thie atpeRment, to allm for a safety mwgin.
Contracted 6.0 6.0 Extended 6.0 6.0
Positive X Reach 4.0 4.0 Negative X Reach 4.0 4.0
JS hape 4.0/4.0 4.0/4.0
Table 5.1: Tkuss81111 Actuator Lengths in Wor Demonstration
68

Figure 5.12: Contracted Configuration Figure 5.13: Esttendeci Contiguration
Figure 5.14: Positive z Reach Figure 5.18 Negative x R d

n e series of motions illustrateci in Figures 5.10 through 5.15 were per formed with ?hiasarm at a rdatively low speed. The rnotor coordinates were computed from the joint coordinates in Table 5.1. A set of DMC programs were used to perform the experiments. Using the high1evel control software (refmed to as the 'hssarm Control Environment in Chapter 2) as an interfk tool, the motor commmcls were exgnited for all twelve achiators simiiltan+ ously.
The following is an excerpt of the DMC programs for h a r m workspace tests:
Note that throughout the program, the DMC commans DT and CD are re peatedly d. These commands ere used for contour mode motions. The contour mode results in smoother motion of the rxtanipuiator. The contour is characterized by the pmition increments (CD) of ach actuator and the time increments (M) between each point on the contour.
During the experiment it was obeerved th& as Thissarm reached the edge of the workspace, the motion of the d p u l a t o r was sigdicantly ns strained due to the structural limitation. It seem that the 1 4

on some mernbers c81lSed this problem. For example, in the Pmitive 2Reach configuration, actuator 3 was unda a very large lord. It was noted that at thie point the ectiietor was, in fact, moving s l o w and in a jerky m m e r . When the commandai joint length exceeded 6 cm, the motion of the actuator came nearly to a stop. Since the gravitational &ect is one of the most aignificant factors on the stnictural limitation, the size of the fimctional workspace highly depends on the design and &e of the actuatom
W e the nonlinear bacteristics or backlash of the individuai ac tors m o t be modeleci quntitatidy at this tirne, it was clear that this un certainty had a significant && on the performance of 'Lhtssarm. This result auggests that 'Ihuiserm, or auy other 'Ihrsssrmlike VGT manipulatore, may have 'functiod' workspace within its theoreticai workspace. For 'Ihuisarm, this functiod workspace exteuds appraximataly 1.0 meters in radius and 0.5 meters in height. Recall tbat in home configuration, 'hssarm is 1.9 meters in Length.
The workspace of ' I hmm is comparable to that of conventiod in duetrial manipulators. For a~csmple, PUMA761 is an example of a typical industrial mruiipulator. As shuwn in Figure 5.16, PUMA761 is apprachatdy 1.5 meters in length and its workepace extends 1.2 meters in radius.
5.1.4 Controllable Workepace
So far, the workspace has been considered as a set of ali possible points in space that may be reached by solving the forwatd kurematics, given the joint coordinaixs. For a cornplex msnipulator such as lkussam, it is not guasanteed that every point in such eprce may be actudy reached when the invercur hc+ matics ie eolved. In fact, ody a suluet of points in the reacbable worirspace

.3 rn d u s
modd appl10BCh9) derai in Chapter 3. This m b e t of ~~tkspace bseed on the eolution of the in kineaintia pb is & r d to as the mtmUab1e worbpa' herq abce the po8tim amtroi of the enddecfor is pdble oniy within this
TheBl;edtbalemvarbnrIthmeriyfactasmchse redundancy ramiution ulpritam, joint limft and dqpkity avoichm md oda, and wmacy in the kincrnetia model. The ratio be&ween controilable workspa a d ieschable wotirspece may be ussd as a meseure of control al gonthm.
Figure 5.17 &ors the cut 88ction of Thmaun's conhoIlable aorkspa on the xtplane (y = O). Note thst maqy pointa P the ICBBChBble workspace are in contro11A)ile worbpsce. Haraves, there are pointa mmr the ege of the

Figure 5.17: 'Lhissarm Cooitrollable Workspace, xzplane
reachable workspace that cannot be reached by the discrete mode1 approllch. In Figure 5.17,5.18, and 5.19, 'os wee marked when the inseme kinematics was succmstully s01d with joint coordinates nithin the joint limits, and k' was marked d e n it failed. Apprcnchately 69 petcent of the r d b 1 e workspace on the xzplane is in the controlable workqacs.
The controllable workspace is mder when the cut section on qplme (r = 1.8 m) is considerd as shown in Figure 5.18. When z = 1.8 m, the workspace qans approWmately 1.6 meters in the x and ydirectiom.
5.1.5 Reach Hierarchy Algorithm
In order to utilize the entire workspace, a control algorithm that makes use of the workspace information needs to be implemented. One approach is the regch hierarchy algorithm, adopted fiom porein, 19851. An example of a modified reached hierarchy *rithm ie now given for n modules The concept of the reach hier8fdly algorithm is,
1. F m , define the workqa of chain i as Wi, where chim i consist~ of

Figure 5.18: Zhieerum Controllable Worlrspace, zyplane
modules i through n.
2. If the deswd end4ectob position is not in Wi, then halt immediately.
3. Otherwise, adjust module 1 enough to bring the g d point within W2.
4. Repeat the last step for each remeiaig chain, adjusting module i to bring the goal point within Wi+l for = 2,3,. .., n.
Figure 5.19 shows the amfrollable workspace siice at y = O, when the reach hieraxchy algorithm is applied. Appmximately 92 percent of the r d  able workspace can be reached wing this algorithm. Compared to 69 percent in the discrete model approach, 23 percent increase ie obtained. The 8 percent of the reachable workspace that can not be reached with reach hierarchy dg* rithm may be reached if an exm% model of workepace is adable. As shown in Figure 5.9, the woricsprrce d e l is only an approximation, thus dowing 8

Figure 5.19: Controilabie Worhpa with Reach Hierarchy Algorithm
5.2 Dexterity Analysis
5.2.1 Definition
One of the advantages of aredudant rnanipdator is its dexterity. The concept of dexterty c m be interprated in several ways. hr example, a commonly accepted physicai concept is one's ability to use one's hands, one's body or even one's mind. In robotiee, dexterity has been interpreted as a spdcation of the kinematic or dynamic response of a manipulator p'wa, 19851. A similm concept has ben referred to as 'manipulabity.' In thie dissertation, we adopt the dennition of daxterity by Park and BlOckett (1994). Dexterity is Uthe ability to m m a d apply forces in arbitrary directions as easily as possible.'' As impiied in tbe definition, dexterity can be considered in both its kinematic aspecta and in its dynamic aspects. The focus here rernahs on the kinemtic aspects of dexterity.

In bernatic control, dexterity measument may be used to determine optimal 'working position' for the manipulator, or to find the 'optimal posture' for a given endeffector position and/or orientation. In this section, examples for both applications are discusged.
5.2.2 Dext erity Measurement
A survey of methods to quantitatively represent local dexteriw was conducteci by Klein and Blaho (1987). F i , Ymhikawa (1a) measmeci local dexterity by the detaminrrnt of JJT:
1i = Jdet(JJT) (5.3) When q apprdes zero, the configuration of the manipulator is cansidered to be close to the singulatity.
Second, the condition number of the transpose of the Jecobian, c ( P ) was used as a measure of workspace qwdity [Saiisbury and Craig, 19821, [Gos selin, 19901:
w h e . o,, and a ~ n ste the largest and s d m t singular d u e s of the system Jacobian matrix. Thie is a memue of the eccuracy with which forces can be exerted. [Ybshikawa, 1985a] The optimal value for this memurement is 1.
Third, the minimum slgular value, u d was also wed as the measme of nearness to the ainpuienty [Klein and Blaho, lm. The singularity is marked by the niinim~un singular value approaching zero.
Finally, joint range availabity was used by Liegeois (1977):

where q: is the center of the range of each actuator. This method provides the 'naturalness' or 'evenness' of the manipdator'e posture [Klein and Blaho, 19871.
In order to compare these me88uTeB of local dexterity, two simple joint trajectorks were chosen for 'lhuisarm, The residting endeffector trajectories are a cuve on the XIpiane and a straight iine almg the zmis. A simulation test was petfiormeci dong these trajectories and fow d u e s (q, c, o and E ) were dculate dong the trajectory.
As shown in Figuree 5.20 and 5.21, the variation in these memures is si@c811t. Klein and Blaho conducted similar experiments and concluded that "it is not pomible to condude that one is superior to the others since the measuns have slightly dineient purp088?
In general, the Lesults suggest that the dexterous workspace of 'Ihiesarm at the center of its worhpace. Hem, 'IhiaeMn ie in the straight poehire with very 4 changea in actuator lengths. Therefore, Ihissarm is capable of perfonning more dexhous tasks in thk area.
In addition to the determination of the dextrous areas in the workspace, dexterity may be also u d 8s a ddgn criterion or to select the optimal (or the mt natual) partuFa for a given endleffector position and/or orientation. Figure 5.22 through 5.25 show four distinct pamirer of 'Ikuesarm, h m which the endefll'tor can be located at (0,0,1.85) m. Postures 1 and 2 are with constent positive and negative curvature. Posture 3 is when the changes in actuator Iengths are equai throughout d four modules. Poeture 4 is when the iength change of actiiators in module 1 are maximized and the otha modides remain unmodifieci. Jhter i ty Me88u~ements at four postures are shown in Figure 5.26. These measurements suggest that the second poshire is the mo& nafural posture with hi dexteriw

Danlllh=glrn+ i i i i i ! : i l ~ ~ i i ! l f i ! i t i ! i ! f i ! t f i 2 i ! ! i ! i i ! i i i : L I : .. :::;:::;....;:;::::::;..::::::::;;.:;::::::.:::::::::
..........................................................................................
................................ * ..........Li......... I ..................... a i r  . . S . . . . , . . . . . . . . . .
. ...\. : . . . . . . . . . . . . . . . . : . . . . . . . . i . . . . . . . . . . . . . . . . . . :. ........ :. . . . . . . : .......... : I : : : . i l . : . . : : : : . : : ; . . . . . . . . ...,, : : . : . i : ...................... . . . . . . . . . . . .  . . . .
......... L... .........
. . . . . . . . . . . . . . .
.......... S . . . . . . . . , ......... . . . . . .  . . ................... : \. . . . . . . . . . . . . . . .
......................
........ , . . . . . . . . . . .
Figue 5.20: Dexterity Memures along a curve (xzplane)

Figure 5.21: Dexterity Memues dong a straight line (2axi9)

Figure 5.22: Posture 1 Figure 5.23: Posture 2
Figure 5.24: Posture 3 Figure 5.25: Posture 4

Figure 5 . a Dexterity Measures et Pture 1,2,3 and 4
It should be noted that the measurements were made with the @v aient k e t e mode1 approach, d i s c d in Chapter 3 and 4. Thus the re sult refiects the micmecopic geometry of the gimbal modules, rather than the 'Ihisecum moddeg. In order to accurately meiiaure the dexterity of 'hmwm modules, exact kinematics shodd be applied.
5.3 Repeatability Analysis
5.3.1 Definition
Next, the repeatability of 'Ihisrrarm is examineci. The repeatability of a manip dator is a measure ofhow preasely the endeffector can be located, repeatedly.

5.3.2 Repeatability Measurement Met hod
In order to measure the repeatabiiiw, a very fine pen was meci as a tool. A white board was set up such that pen tip wodd touch the surface only when it cornes to a stop aRer the motion. Unf~rtmately~ due to the stnicturd vibration, short trams of the tool path were marked on some of the teet results. While cornputhg repeatability, these t r a m cf the endeffector were ignored. See Figure 5.27.
T h s a m was moved 6rom ib home configuration to a test pdtion and back to the home configuration. The motion was repeated 10 timea for three test points. The result wee scannai and the radius R of the cide aurrounding the scattered points memureci. The test points used are shonn in Table 5.2.
Table 5.2: Te& Points Use in Repeatability Analysis
w 2 ' io,i5,0 0.11 cm'
In order to measure the repeatability of the manipulator more accu rately, a better method to locate the end4ector tip position is desireci. For hitute referen, threedimewioonal position emor or a vision system are rec ornmended*
5.3.3 Trussarm Repeatability
0.10 cm Test 3
Based on the method used, it may be concludeci that Thissarmts repeatability is 1.4 mm. In other words, 'IhiseMn is capable of moving its end4ector to a Qven location with 1.4 mm preaeion. F i 5.3,5.29 and 5.30 ahow the
10.10.10

Figure 5.27 Satup for Repeatability A d @

Figure 5.28: Repeatability Teet Result 1
Figure 5.29: Repeatab'ility Test Remit 2
simple &ta b m th tests. On wmpMson, the repestabity of the PUMA 1W seria manipub
tom L M.2 mm. The repeatability of A645 manipulator (deeigned by CRS
Figure 5.30: Repeatab'ity Te& M t 3

5.4 Speed Analysis
5.4.1 Actuator Speed
As explaineci in Chapter 2, lkussann's actuaton, are independent lin (traas lational) actuators. Each is driven by a DC motor and a gear reduction as sembly. The motors are rated at 6,500 RPM (machuni speed with no load). However, amplifiers that supply the power to the motors Lmit the actuator speed to 2,925 RPM. Since quadrature encoders output 4 counts per motor rotation, 2,925 PRM conesponds to 4 x 2,925 = 37,440 counts/sec. This speed has been used as a guidaline for the rmuchum ectuator speed. 37,400 counts/sec comesponds to apprahatdy 1.4, 2.6, 2.1 and 1.4 cm/= for ac tuators in Module8 1,2, 3 and 4, respectively.
5.4.2 Tip Speed
Once the m8Icimum actuator speeds are determineci, the maximum tip speed can be either caidated using the forward kinematics mode1 or measured ex perimentally. End4ectOr (a pen at k(4.15, 0.28,O)m) paition is (0.15, 0.28, 1.675) and (0.15, 0.28, 2.109) m when all actu8tors are extendeci and retracted to 80 petcent of their mBMmum/ minimum extension (6 cm and 6 cm, respectively), This vertical distance of 43.4 cm nie used as a test path. At the d o m joint speed of 0.5 cm/s, a 43.4 cm path can be traced at the rate of 43.4 cm /(12 cm / 0.5 cm/s) = 1.8 cm/s. Or, at 1.4 cm/s joint speed, 5.0 cm/$ tip speed may be achieved.
For a horizontal path, a straight lre fiom (0,0,0) to (20,0,0) cm waa med, as shown in Figure 5.31. With a tool vector, t = (15,28,0) cm, an end efkctor poeition of (20, O, O) cm can be achieve with the motor coordinates

t L.J p j 2.1 cm 2.0 cm , Figure 5.3 1: Vibration Durhg Acceleration and Deceleration
shown in Table 5.3. The maximum joint speed is iimited by actuator 3 to
Table 5.3: Teet Points Used in Sped Analysis
36,432137,440 r~ 1 sec. Thue, the maximum tip speed for horizontal motian is 20 cm pet 1 sec = 20 cm/s.
It should be noted that at this speed, st~cfural vibration becornes evident during the d e r a t i o n and deceleration phases. This can be eeen in Figure 5.31 which is a scanneci image of the two end points of the horizontal teet trajectory. It may be o b e m d that the encEeffector m m almat f 1.0 cm when acce1erating and dderating.
A more standard teet in robotics for speed evaiuation is the '12inch pickandplace' maneuver. 'Ihuisarm is capable of this pickandplace maneu ver in many directions. For test purposes, the t h easiest paths were chosen: a 12inch path dong the x, y and zaxes. The maneuver is as follows: starth kom the home poeition @,O),
1. Move to the leR by 6 inchm, (6,O)

Figure 5.32: The 12inch PickandPlace Maneuver
2. Move d o m by 1 inch, (6, 1)
3. Move up by 1 inch, (6,0)
4. Move to the right by 12 inch, (6,O)
5. M m d m hy 1 inch, (6,1)
6. Mave up by 1 inch, @,O)
7. Mme back to the home poeition, (0,O)
In total, the endetlbctor path is 28 inches (n.1 cm), as ehown in Figure 5.32. Throughout the seria of pickandp1a maneuvers, the maneuver times
were determined. The sample r d t s are s h m in Table 5.4. The vertical and
x8Xi8
Table 5.4: 12inch Pi&AndPlace Test Mts
b
yaxis 16.9 cm/s 3.2 4 6.8 seconds
horizontai speeds are speeds of the endeffector dong paths 1, 4, 7 and.2, 3,
l h i z ~ ~ d speed, fi 16.9 cm/s
zaxk
5, 6. The total tirne, ttd in Table 5.4 corresponds to:
Vertid Speed, V. 3.2 an/s
3.2 cm/s
Total T i e , & , 6.8 seconds
13.5 cm/s 20.0 seconds

in conclusion, the maximum tip speed of'Ihissarm is 20 cm/s in horizo11 ta1 motion, and 5 cm/s in vertical motion. 'Ihuniarm is capable of perfog a standard 12inrb pi&andplace maneuver in less than 7 seconds horizontdy, or 20 seconds vertidy.
It appeare that h a r m is at dissdvantage in speed when compaied to conventional manipulators. The maximum hem tip speeds of both the PUMA 700 series robots and A465 are 1.0 m/s.
5.5 Accuracy Analysis
5.5.1 Definit ion
The aawacy of a manipulator is a me of how accurately the manipulator can be directed to a given Caztmian point or trajectory within its workspace. F M , the eccuracy of ?hissMn under kinematic control derived in Chaptg 3 is considered. The poeition accuracy and trsdring accuracy are examineci separately. Next, the mlvedmotionrate control, d e r i d in Chapter 4, is applied and the d t i n g accutacy is d d 8 t e d .
It is noted that the accuracy of a manipulator highly depend on the configuration of the mruiipulatot, and thdore, the aample resdts presented here only provides a foumlation for hirther reeeerch.
5.5.2 Position Control
A set of desired joint trajectories was generated and applied to Thssarm, and the paeition of the endefFector was recorded, using a pen as a tool. The setup is shown in Figure 5.27. The forwad kinematics was solved to compute the Cartesian amdinates of the endeffector that correspondeci to the desired

joint coordinates. Then, the results from forward kinematics and the a c t d location of the pen were cornparrd to compute the error in poeitioning.
Samples of the experimental data in [cm] are shown in Table 5.5. where
Table 5.5: Sample Experimental Data