• ..- . '> ,,i, ,v -, J Dynamic& of Flexible- Link . Xavier, Cyril - .... ,.- - 1 B. E. ( Bangalore University ) M. E,ng. ( McGill Umversity ) 1985 Departmel\t<of Mechanical McGill • .. / '. A thesis submitted to the Faculty of Graduate Studies and Research \ in partial fulfillment of the requirements-for the degree of . " f_'/ Doctor of Philosophy ., July 1988 @ Xavier Cyril 1..
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J Dynamic& of Flexible-Link . M~rlipqlators,
Xavier, Cyril - ....,.- -1 B. E. ( Bangalore University ) ~
M. E,ng. ( McGill Umversity ) 1985
Departmel\t
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my Parents, wife Patricia. -
'son Alla.n and unde Lawrence
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natur:al orthogonal complement tO'~btain the independ"nt d~namical equations. TheS!' equations are then splved f~T the acceler~~ions using the Cholesky decomposition. 'Thr
: \,:;, "-integration ~f the\ ~lerations is performed using Gear 's stiff method of ba.ckward
differentiation.
./ 1'0 ascertain the effects of Iink flexibili~y on the overall prformance of tiH'
manipulator, the dynamic simulation of the flex,ible-link manipulator is compared wit h
Jhe dynamic simulation of the sarnNmanipulator considering ail lihks to be rigid. Thf'
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inverse dynamics, to obtain the nominal joint torques or forces for the dynamic sim- # , ~ , ,'--'"
ulation of manipulators, and the dynarnk simulation of rigid-Iink manipulators, can
be accompIished using the hybrid approach; however, a second method of for.mulat~n, ., that ,of NE is presnted. The reason f~r doing this is that the N ~o'tmulation is corn\.
pU,tat10nally more efficient, for rigid-lin.k rnanipulators and also it provides a limited, yet./
-' ess~ntial verification, on~ the validity of the hybrid approach.
In the NE form~1'ation, the NE equations of each rigid link are written with
,_ ~espect to the new body-fixe'd coordinate frame" as opposed to the usual practice of
-writing the equations with respect to a coordinate frame attached either at the join~ or the 'mass centre of the body. In doing 50., the computational efficiency of the formulation
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is improved. The inverse dynamics of rigid manipulators is then solved as a series of
, forward and backward recursions along the prescribed joint trajectories. In the forward
... recursio
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dynamic simulation, rangirrg from a one':'link- to a six-link-manipulator, with ail their . . " l'
\links flexible, is presented. The results of\~imulation for ail these manipulators indicate
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that the effects of structural flexibiIity on ,the motion of the manipulaior is considerable
even at I~w speeds. The simulation reJ~ts also bring to light the affects of various joint torq~es op the elastic behaviout ~r. tre manipulator .. FinallY-4 it was found that
the sta.bility of the nunterical integration schemes used depend largely on t'4e type of
t input joint torques. t , .
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" J, L'auteur de ~ette thse prsente une mthode gnrale pour la formulation
, , :.des quations dynamiques d'un manipulateur robotique de type srie architecture ar-
bitraire comportant N articulations reliant des membres rigides ou flexibles. -Il propost' 0
deux formulations diffrentes des quations: la premire est une formulation hybridf' ,..
Newton-Euler (NE)/Euler-Lagrange (EL) pour, l'tude des manipulateurs membrd , .
flexibles, ia seconde, de type NE, est utilise pour modliser les manipulateurs mem-, ,
" , bres rigides. La prcmir
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ces quations par le complment orthogonal naturel, ce qui conduit aux quations dy-
namiques indpendantes. Ces quations sont res?lues grce la dcomposition de
ChoJesky qui permet de calculer les acclrations. Celles-ci sont alors intgres dans
le temps en utilisant la mthode d'intgration inverse de Gear.
Afin de dterminer les effets de la dformation des membres sur la perfor-
mance globale d'up manipulateur dOfm, l'tude compare les rsultats de la simulation . . dynamique ceux obtenus avec le mme manipulateur dont tous les membres sont con-
sidrs comme rigides. Le problme dynamiques inverse --c'est--dire, l'obtention des
forces ou couples moteurs- et la simulation dynamique des manipulateurs membres ~ \
rigides peuvent tre resolus en utilisant la formulation hybride des quations. Cep en-
qant, il a t fait appel une seconde IJl~thode, celle de NE qui, juge plus effic~e , . ....~
lorsque les rnembres sont considrs rigide,s, permet, mme de faon limite, de vrifier
la validit des rsultats obtenus avec l'approche hybride.
Dans la formulation de NE, les quations dynamiques de chaque membre
rigide ~ont poses dans u~ repre de coordonnes fix ce membre. Cette faon de
formuler les quations est plus efficace, don~rapide, que la mthode traditionnelle
consistant utiliser des repres fixs aux articulations ou au barycentre de chacun des
corps. La solution du problme dynamique inverse des manipulateurs rigides est alors
calcule par une srie de rcurrences directes et inverses le long de la trajectoire prescrite.
Dans les rcurrences directes, les vitesses et acc1rations soift'calcules la (orce ou le
couple des articulations tant, eux, calcule dans -les rcurrences inverses. Par ,ailleurs,
la simulation dynamique des manipulateurs rigides se base sur la mthode dcrite plus
haut pour les manipulateurs flexibles. L'intgration des acclrations est cependant
"-~ effectue l'aide de la mthode de Runge-Kutta (Sme et 6me ordres). l'
L'auteur inclus des exemples numriques tirs de publications afin de
dmontrer la validit des quations dynamiques et des algorithmes utiliss pour les vu
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rsoudre. Il prsente des exemples de simulation dyna.mique de divers manipulateurs,
allnt du cas le plus simple . un seul membre, au cas le plus complexe, six membres
flexibles. Les rsultats obtenus indiquent que l'effet de la flexibilit des membres sur,
le mouvement du manipul~teu~ est considrable, mme . vitesse rtduite. -Les ~sultat8
de la simulaon permettent galement de dceler influence des divers couples moteurs ,. ~ur le comportement lastique du manipulateur. Enfin, les rsultats dmontrent aussi
que la stabilit numrique des algorithmes. d'intgratj est intimef!1ent lie aux types
de couples moteurs.
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) Acknowl~dgement8 "
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" 1 would like to express my deepest gratitude to IVY research supervisors
Profs. J. Angeles and A. ~isra for their continued'support and guid'ance. Working in, /'
association with them was a pleasant experience.
1 sincerely thank Clmlnt Gosselin for translating the abstract into French.
Thanks are also due to ail my' col(eagues with f whom 1 had usefuI discussions
dCs the course of this research u'nde'rtak)ng, especially Ou Ma, IQna Shan and Meyer
Nahon. 1 would like to express rny heartfelt thlnks to ail my relatives and friends for
their encouragement and the good times, 1 had with them. This research was possible
under I:'CAR Grant No. AS2517 and NSERC Grant No. A4532 granted to Prof. J.
Angeles, NSERC Grant No. A0967 granted to Prof.. A. Misra and the Actions Struc-
turantes (FCAR) Grant awarded to McRCIM.
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To the best of the author's knowledge, the formulations and algorithms pre-
sented in this thesis are original and have not been presented elsewhere. The simulation
package -FLEXLINK- developed for the dynamic simulation oro rigid- and Hexible-link
manipulators was solely written by the author.
, The original contributions in this thesis can be categorized under two 'Bec-
tions, namely, mathematical modelling al'\d algorithms. First, the contributions towards
the mathematical modelling of robotic manipulators: The hybrid NE/Er; method is a
new approach for modelling the dynamics of elastic-multibodies, and the use of the
natural orthogonal complement to eliminate the constraint wrenches is yet another new
technique. Additionally, the introduction of the so-called neW coordinate frames for
describi~g the manipulator architecture aiffers from the currently existing approaches.
~ Next, the contributions in the area of algorithms to solve the inverse and
direct dynamics problems of robotic manipulators: The 'algorithms presentd here for
the inverse dynamics of rigid-Iink manipulators and the dynamic simulation of rigid-"
and Hexible-link manipulators are the most efficient to date. The solution technique
presented for the accelerations of both the rigid- and flexible-link manipulators is al80
an origiIJal contribution. Perhaps the greatest contribution would b~ the simulation
package -FLEXLINK-, which has been tested and results presented unlike previou8
studies.
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.. Contents
List of Figures . .................. , , ............................. '\ . , . . .. xv - .
List of Tables . ... , ............. , ............................... :. . . . .. xxiii
Nomenclature ........... , ............ , ..... ,.,........................ xxv
Chapter 1 Introduction ....................... ',' ....... :. .. . .. ... 1
1.1 Robo);ic Manipulators ..... , ................ , ........... ~ ... , ..... , 1 [
. , 1.2 Kinernatic~ of Robotic Manipulators . , .. , , ... , , , , , , ..... , ....... , .. , 4
1.3 Dynarnics of Robotic Manipulators .,., ....... : ..... , ....... .'.:..... 6
1.4 Objective and Motivation ................................ , .. , . . . . . 11 ,
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3.3 Dynamical Equations of Rigid-Link Ma.nipulators for
Simulation .. ' ..... ~ ............... , ... .- ............................. 42 3.4 Solving for the Joint Accelerations and Integration of the
Dynamical Equations .............. ' .... ~ . . . . . . . . . . . .. . . . . . . . . . . . .. 47 \/
3.5 Algor~thm and Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . .. 49
3.6 Example................................ . . . . . . . . . . . . . . . . . . . . . .. 52 --,
Chapter 4, Kinematics of Flexible-Link
Manipulators ...... " ................. '.' . . . . . . . . . . . . . .. 54
4.1 Introduction........ . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
4.2 Coordinates for the Rigid-Body Motion of Link i . . . . . . . . . . . . . .. . . . .. 56
4.2.1 Euler Parameters ............................ ~ . . . . . . .. . . . .. 57 'J
4.2.2 Linear Invariants .......................................... 59
4.3 Cobrdinates of a Flexible Link ........................... .... . . . .. '61
4.4 Kinematic Descrtption of a Point on the Flexible link .......... ,..... 64
4.5 Rotation Matrices ...................... ' .... -; . . . . . . . . . . . . . . . . . . .. 68
4.6 Recursive Relations for Velocity and Acceleration of the
Body-Fixed Coordinate Frame ..................................... 69
Chapter 5 Dynamical Equations of Flexible-Link 1
Manipulators .... '. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
5.1 Introduction ......... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
5.2 An Outli~e of the Formulation Methodology ............. : ...... ,... 13
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5.3 Dynamical Equations of a Flxible Link ....... : . . . . . . . . . . . . . . . . . . .. 76
5.4 Constraint Equations and Orthogonal Complement .. ~ . . . . . . . . . . . . 85
t 5.5 ~dependent Dynamical Equations of the System ... ' ....... '. . . . . . 88
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Chapter 6 ' Sim\llation and 'Computational Complexity '. of Flexible-Link Manipulators . '. ........... " . . . . . .. 89
6.1 Introduction............................... .................... 89 6.2 Solving for the Accelerations and Integration of the Dynamical .
Equations ...... ' ................................ -................ , 89
6.3 Aigorithm and Computational Complexity ..... : . . . . . . . . . . . . . . . . . . .. 93
Chapter 7 Simulation Results and Discussion. . . . . . . . . . . . . . .. 99
7.1 Introduction ....................................... ~ . . . . . . . . . . .. 99'
7.2 ' Example 1 ~ One Link Flexible Manipulator . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Example 2 - ~"'Link Flexible'- M,:mipulator . _ . . . . . . . . . . . . . . . . . . .. 103 7.4 Example 3 - CANADARM in Planar Motion .......... ~.......... 164
7.5 Example 4 - CANADARM~jn 3D Motion. . . ............... ....... 105
7.6 Example 5 - Six Link Flexibl& Manipulator ........ _ ............. : . 107
Chapter 8 Conclusions......................................... 172
8.1 Conclusion ................................................ ,.... 172
8.2 Suggestions for Further. -Work -.................................... - 174
Re~enCts .................................................... -. . . . 176
Appendix A. Hartenberg-Denavit Parameters ......................... 184
Appendix B. Detailed Derivati
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Appendix D. Detailed Derivation of Equations (S.15), (5.18) and
(5.23a) .............. ', ..... , ......... , ....... , ~ ... , . . 193
D.1 Derivation of Equation (5.15) .......... ', ........... \ . . . . . . . .. ,193
/' D.2 Derivation of Equation (5.18) ........................... ' . . . .. 194
D.3 Derivation of EquatJon (5.23a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Appendix E. Proof to Equation (5.28) . . . . . . . . . . . . . . . .. . . . . . . . . . . .. ... 197. . -. .
Appendix F. FLEXLINK - Program Description .................... -; 100 . , F.1 . Program Outhne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
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'List of Figures
1.1 Robotic System ................................................. 2 \.2 Kinematic Structures: a) Closed-loop, b) Open-loop Chain
1.3
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2.2
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2.4 2.5
Configuration, and c) Open-Ioop Tree Co!figuration ............... :. .. 3
Dynamic Simulatio~ of Flexible-Link Ma~iPul.ators .................. 13
N -axis seriai link ma'nipulator. . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . 18
Link Parameters ..... " .......... , ........................... ,.. 19 ,
P 't' V t ' 21 OSI Ion ec ors ............. ~ .................................. .
Joint Actions of the Stanford Manipulator .. ~ . . . . . . . . . . . . . . . . . . . . . . . 33 Joint Trajectories for the Inverse Dynamics of Puma-600
Manipulator .......... " ......................................... , 34 2.6 Joint Actions of the Puma-600 Manipulator obtained from the
! Inverse Dynamic Simulation ........ , .. _ .. , ........... , .......... ~: .. 35 1
3,f1 Simulation Error for the Puma-600 Manipulator ... , .............. ,.. 53 \
4.1 Motion of a Flexible Link ..... , , .... , .... , . . . . . . . . . . . . . . . . . . . . . . .. 55
4.2 P~ition Vector of a Point on the Flexible-Link . . . . . . . . . . . . . . . . . . . . .. 66 ~.1 Comparison of Joint Angle and Joint Rate Responses of the
One-Link Manipulator for a 5 N-m Step Torque(- - - Rigid, -
Ffexible). ...................................................... 108 7.2 Degradation in Joint Angle Response of the One-Link Manipulator
due to Link Flexibility for the Step Torque. ........................ 109 7.3 Link Tip Deflection due to Flexibility and its Time Derivative, of
the One-Link Manipulator for the Step Torque .................... " 110 7.4 Comparison of Joint Angle and Joint Rate Responses of the
One-Link Manipulator for a t N-m Ramp Torque ( - - - Rigid, ~ ,
Flxible). ........ ".' ................................ , ............. . 111
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/' ,> 7.6 Nominal Joint Torque Required for the One-Link Manipulator to Il
Accomplish the Prescribed Trajectory of Eq.('7.1) ....... , ............ 1l:l
7.7 Dynamic Simulation Error of the One-Rigid-Link Manipulator for
,," the Prescribed Trajectory of Eq.(7.1). , ......... , .............. " " .... , 114 7.8 Comparison of Joint Angle and J~int Rate Responses of the
One-Link Manipulator for the Given Nominal Torque(- - - Rigid,
-, Flexible) .......... : .......................................... 115 r
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Link Tip Deflection due to Flexibility and its Time Derivative, of 7.9
the One-Link Manipulator for the Given Nominal Torque ............. 116 --
7.10 A Two-Link Manipulator Hanging Freely Under Gravit y, Shown
Here in its Initial Configuration of 91 ::;;: _90 and fJ2 ::;;: 5. "' 1. 117
7.11 Response of Link 1 of the Two-Flexible-Link Manipulator for the
Initial Condition of fh ::;;: _90 and (J2 ::;;: 5 with One Bending
Mode ......................................................... 118 . 7.12 Response of Link 2 of the Two-Flexible-Link Manipulator for the
Initial Condition of fJ1 ::;;: _90 'and (J2 = 5 with One Bending , -Mode. Il It l'' Il.'.' l'' ,. '" 119
7.13 Response of Link 1 of t,he Two-Flexible-Link Maniptilator for the
Initial Condition of fJ1 ::;;: _90 and (J2 = 5 wi~h Two Bending L
Modes . tJ. 1 , 120
7.14 Response of Link 2 of the Two-Flxible-Link Manipulator for the
Initial Condition of fJ I ::;;: _90 and fJ2 = SO with Two Bending ~ . Modes. ................................. , f .... ,. " ......... 121
7.15 Response of Link 1 of the Two-Flexible-Link Manipulator for the
Initial Condition of 81 ::;;: _90 and fJ2' = SO . with Two Bending
e . Modes,and a Payload of 1 kg. f , ............................... 122 - xvi
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. 7.16 Response of Link 2 of the Tw~Flexible-Link Manipulator for the >
Initial ~9ondition of 01 = _90 and 02 = 5 with Two Bending /
Modes and a Payload of 1 kg. ................................... 123
7.17 Schematic Diagram ~f the CANADARM. '.' ................... .. 1' 124 7.18 Nominal Joint Torques Required for the Three-Link Manipulator
to Accomplish the Prescribed Trajectory of Eq.(7.2). ............... 125
7.19 Comparison of Joint Angle Responses of the Three-Link
Manipulator for the Given Nominal Torques(- - - Rigid, -
Flexi ble) . . .................... '................................ 126
7.20 Comparison of Joint Rate Responses of the Three-Link l' \
Manipulator for the Given Nominal Torques(- - - Rigidj -
Flexible) ........................................... -....... , . . .. 127
7.21 Link Tip Defieetions due to Flexibility and its Time Derivatives,
.:::, of the Three-Link Manipulator for the Given Nominal
Torques. . ........... ; ............................ ! . . . . . . . . . . .. 1~ 7.22 Nominal Joint Torques Requi~d for the Three-Link Manipulator
ta Accomplish the Prescribed Trajectory of Eq.(7.2) with a
Payload of 100 kg. ............................................. 129
7.23 Comparison of Joint Angle Responses of the Three-Link
Manipulator with Payload for the Given Nominal Torques(- - -
Rigid, - Flexible) .......................... ,' . . . . . .. . . . . . . . . . . .. 130
7.24 Comparison of Joint Rate Responses of the Three-Link
Manipulator with Payload for the Given Nominal Torques(- - -
Rigid, - Flexible) ............................... ~ ............ " 131
7.25 Link Tip Defiections due to FlexibiIity and its Time Derivatives,
of the Three-Link Manipulator with Payload for the Given
Nominal Torques. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132
7.26 Nominal Joint Torques Required for the CANADARM to
Accomplish the Prescribed Trajectory of Eq.(7.3). . . . . .. . . . . . . ... .. .. 133
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7.27 Comparison of Joint Angle Responses of Links 1 and 2 of the
CANADARM for the Given Nominal Torques( - - - Rigid, -
Flexible). ..................................................... '134
7.27 Cont. Comparison of Joint Angle Responses of Links 3 and 4 of
the CANADARM for the Given Nominal Torques(- - -,
Rigid, - Flexible) .......................... -. . . . . . . . . . . . . . 135
7.27 Cont. Comparison of Joint Angle Responses of Links 5 and 6 of
the CANADARM for the Given Nominal Torques(- - -
Rigid, - Flexible). . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . 136
7.28 Comparison of Joint Rate Responses of Links 1 and 2 of the
CANADARM for the Given Nominal Torques(- - - Rigid, -
Flexible) ....................... -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
7.28 Cont. Comparison of Joint Rate Responses of Links 3 and 4 of
the CANADARM for the Given Nominal Torques(- - -
Rigid, - Flexible).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 '
7.28 Cont. Comparison of Joint Rate Responses of Links 5 and 6 of
the CAN ADARM for the Given Nominal Torques(- - -
Rigid, - Flexible).. . . . . . . . . .. . . . . .. . . ... .. . . . . . . . . . . . . . .. 139
7.29 Link Tip Defieetions due to Flexibility, of the CANADARM for
the Given Nominal Torques .............................. .'. . . . . .. 140
7.30 Time Derivatives of the Link Tip Defieetions due to Flexibility, of
the CANADARM for the Given Nominal Torques. ................. 141
7.31 Nominal Joint Torques Required for the CANADARM to
Accomplish the Prescribed Trajectory of Eq.(7.3) wit~ ,R Payload
of 500 kg .. f , , , , , 142
7.32 Comparison of Joint Angle Responses of Links 1 and 2 of the
CANADARM with Payload for the Given Nominal iTorques(- --
Rigid, - Flexible) ........................................... :.. 143
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the CANADARM with Payload fOf-the Given Nominal
Torques(- - - Rigid, - Flexible) .. :' .......... " . . . . . . . . . . . . .. 144 ;
7.32 Cont. Comparison of Joint Angle Responses of Links 5 and 6 of
the CANADARM with Payload for the Given Nominal ~
, Torques(- - - Rigid~ - Flexible) ......................... ,. 145
7.33 Comparison of Joint Rate -Respon~es,of Links 1 and 2 of the 1
CANfiDARM with Payloa? for the Given Nomimil Torques(-, - -1
R' id, -,- Flexible) ............................................ ,. 146
7.33 ont. Comparison of Joint Rate Resp.onses of Links 3 and 4 of
the A~ADARM with Payload for the Given Nominal ,
Torques(- - :' Rigid, - Flexible) ........... , . ... . ..... . . . . .. 147
7.33 Cont. Comparison of Joint Rate Responses of Links 5 and 6 of (
the CANADARM with"Payload for the Given Nominal ,
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Torques(- - - Rigid, - Flexible) ........... : ......... ,.,.: .... 148 ,
"7.34 Link Tip Deflections due to Flexibility, of the CANADARM with
Payload fo~ the Given Nominal Torques. .......................... 149
7.35 Time Derivatives of the Link Tip Deftections due to Flexibility, of
the CA:NADARM with Payload for the Given Nominal
,Torques. . .......................................... ~ . . . . . . . . .. 150
7.36 Comparison of Joint Angle Responses of Links 1 and 2 of the
CANADARM with Payload and .5% Structural Damping for the
Given Nominal Torques(- - - Rigid, - Flexible) .. , . . . . . . . . .. . . . . . .. 151
7.36 Cont. Comparison of Joint Angle R~sponses of Links 3 and 4 of
the CANADARM with Payload nd .5% Structural
Damping for the Given Nominal Torques(- - - Rigid, -
Flexible). III 152 (
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7.36 Cont. Comparison of Joint Angle Responses of Links 5 and 6 of
the CAN ADARM with Payload and .5% Structural
Damping for the ~iven Nominal Torques(- - - Rigjd. -
Flexible) ... ; ...... ' ...... : ............................... 153 "-7.37 Comparison of Joint Rate Responses of Links 1 and 2 of the "
CANADARM with Payload and .5% Structural Damping for the
Given Nominal Torques(- - - Rigid, - Flexible) .................... 1,. 154 7.37 Cont. Comparison of Joint Rate R~spons~ of Links 3 and 4 of
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the CANADARM with Paylod and .5% Structural
Damping for the Given---Nominal Torques(- - - Rigid, -
7.37 Cont. ~~:~!::;o~ ~; ~~;~; .~~~ R.;.~~r:.~; . ~f i;~~; .; ~d' ~ . ~f .. .' . .. 1.. \ the CANADARM with Payload and .5% Structural
t; Damping for the Gi~en Nominal Torques{- - - Rigid, -
Flexible) .... ~ . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. ,156 7:38 Link Tip Deftections due to Flexibility, ef tbe CANADARM with
Payload and .5% Structural Damping for the Given Nominal
Torques ........................ , .......... ,................... 157 7.39 Ti~e Derivatives of the Link Tip Defiections due to Flexibility" of
the CAN ADARM' with Payload and .5% Structural Damping for
the Given Nominal Torques .................... " .. , . . . . . . . . . . . . .. 158 '.
7.40 Nominal Joint Torques Required for the Si-x-Link Manipulator to
j'ittto~plish the Prescribed Trajectory of Eq.(7.3) .... , . . . . . . . . . . . .. 159 7.41( Comparison of Joint Angle Responses of Links 1 and 2 of the
Six-Link M~ipulator for the Given Nominal Torques(- -'- Rigid,
- Flexible).. . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160
7.41 Cont. Comparison of Joint Angle Responses of Links 3 and 4 of
the Six-L~nk Manipulator for the Given Nominal Torques(-
- - Rigid, - Flexible). ........... .. ....................... 161
-
7.41 Cont. Comparison of Joint Angle R~sponses of Links 5 and 6 of,
the Six-Link Manipulator for the Given Nominal Torques(~
- - -Rigid, - Flexible) ......... , ....................... J. 162
7.42 ~ompar'son of Joint Rate Responses of Links 1 and 2 of the
Six-Link Manipulator for the Given Nominal Torques(- - -/igid,
~- Flexible} ....................................... , . . . . .. . . . . .. 163
7.42 Cont. Qomparison of Joint Rate Responses of Links 3 and 4 of \
;.' the Six-Link Manipulator for the Given Nominal Torques(-
7.42 Cont.
- - Rigid, - Flexible). .. . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . .. 164
Comparison of Joint-Rate Responses of Links 5 and 6 of
the Six-Link Manipulator for the Given Nominal Torques(-
: -'-Rigid, - Flexible) ........................ : .- .... ' ..... " 165
7.43 ' Link Tip Deflections of Links 1 and 2 due to Flexibility, of the 1
1 Six-Link Manipulator for the, Given Nommai Torques_ ...... ' .... ~ . .. 166
7.43 Cont. Link 'Tip Deflections of Links 3 and 4 due to Flexibility, of
)he Six-Link Manipulalor for the Given Nominal Torques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
7.43 ~nt. Link Tip Deflections of Links 5 and 6 due to Flexibility, of )
the Six-Link Manipulator for the Given Nominal
, Torques. ................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 168
1 7.44 Time Derivatives of the Link Tip Deflections, of Links 1 and 2 '"
due to Flexibility, of the Six-Link Manipulator for the Given
Nominal Torques. ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169
7.44 Cont. Time Derivatives of the Link Tip Deflections of Links 3
and 4 due to Flexibility, of the Six-Link Manipulator for
the Given Nominal Torques .............. ,'. . . . . . . . . . . . . . . .. 170
7.44 Cont. Time Derivatives of the Link Tip Deflections of Links 5
and 6 due to Flexibility, of the Six .. Link Manipulator for
the Given Nominal Torques ............. ' .................. '. 171
-
e A.l Traditional H-D P~ameters 184 1
, ,
-
"
~ ... , r
"
}
.J ,.
List of Tables
2.1 Computations Required to Solve for the Joint Actions, f' .... . . . . . . . .. 28 .' "
2.2 Computation~ Required to Obtain the Angular Velocities, w ...... '. . .. 28 2.3 Computations Required to Obtain the Angular Accelerations,
2.4
2.5
w .............................. ' ................................ 2~
Computations Required to Obtain the Accelerations, Pt
com'ations Required to Obtain the ACC~leration~, ",
...
29
.............. ~ ~ \-~ '\ ,
2.6 COIIlP.u1Mions Requi~ed to Obtain the Moments, ~ t-l . . . . . . . . . . . . .. 29 ~. .;'
2.7 Comparison of Inverse Dynamics Complexity ,. 30 2.8 C~mprison of Inverse Dynamics Complexi,ty for N = 6 .. . . . . . . . . . . .. 30
2.9 Parameters of St.anford Manipulator ... ~ .......... J ........ : :. 31 2.10 Comparison af Customized Algorithms for the Stanford ' .
~lanipulator ........... .- .................... '. . . . . . . . . . . . . . . . . . .. 31
2.11 Parameters of Puma-600 Mnipuator .................. ',' . . .. . . . . . 32
:. , 3.1 Computations Required to Solve for the Joint Accelerations, (J 51 ..,.
/
3.2 Comparison of Dynamic Simulation Complxity for N = 6 ....... :-. . . . 52 "
6.1 Computations Required to Solve for the Accelerations f/J ...... '" 97 6.2 Comparison of Dynamic Simulation Complexity for N = 6, n == 2
and m = 1 ...................................................... 98
7.1 Parameters of One Link Flexible Manipulator . . . . . ... . . .. .. . . . . . . . . 101-
7.2 Parameters of Two Link Flexible Manipulator ...... '.' ... 4 104
7.3 Parameters of 3 d.o.f CANADARM ............ : ................ 4. 105
xxiii
-
1
r~
-.
tIIIi....~
-""
7.4 Parameters of 6 d.oJ CANADARM. .. .. ....... ...... . . .. . . . . .. ... 106
7.5 -Parameters of-Six LiJ?k Flexible Manipulator .. :. . . . . . . . . . . . . . . . . . . lOt
"
(
J l, 1 1
-
-- Nomenclature 1
In this thesis ail bold-face, lower-case, latin letters denote vectorsj all bold-
face, upper-case, latin and greek letters denote matrices. Lower-case greek letters de-
noting vectors are supplied with an arrow on top.
al the distance between axes Zt and Z1.+ l' It is positive.
At : area of cross section of li~ i. '"
hl : coordinates associated with bending in link i.
o B, : shape function associated with bending in link i .
... c, -= coordinates associated with tqrsion in link i.
~ the coordinate of the intersection or' axes X'-1 and ZI in the XI' YI' ZI coor-
dinate frame. It can be positive or negative. o
el unit vector parallel to the axis of rotation of_QI'
E, modulus of elasticity of link i.
f, : total force exerted on the centre of mass of link i.
-
r".~ 1 force exerted on link i by link i - 1. - ,
g : gravitational acceleration.
Gi sbou modulus of Iink i. ('
xxv
-
~ shape function associated with torsion in link i.
11 : moment of inertia of the cross-sectional area of link 1 about its neutral axis
JI : polar moment of inertia of.the cross-sectional area of link i.
JI inertia tensor of link i about its centre of ma.ss.
'-t length of link i.
L\ : matrix transforming the time dedvative of ftexible screw ql to twist VI' - - - .
ml mass of Iink i.
Mt : concentrated mass attached to the distal end of lirik i.
~ : total moment exerted on link i.
~,I-l moment exerted o~"link 1 by Iink i - 1.
PI ; position vector of the origin of XI' YII Z" defined for rigid manipulators.
PI : position vector of the origin of X" YII Z" defined for flexible manipu)ators.
CL : flexible screw of Hnk i.
Q, : tensor .defining the orientation of X Y" Zi .coordinte frame with resp~t to
Xo,Yo,Zo
Jl,. -: tensor defining the orientation of Xi! Y" Z, coordinate frame with respect to
X'-l'Y'-l,Z'-l' defined for liigid manipula~ors.-: , -
xxvi
-
/ ,
;
/ ;
/ ;
/
c
Rs : tensor defining the orientation of X" YI' Z, coordinate frame with respect to
X,-l' Yi- b Z,_ b defined for flexible .manipulators.
v, : twist of link i.
Xz : unit vector parallel to axis Xz.
Xt : axis perpendicular to Z, and Z~+ l' and directed from the formr to the latter.
Yz : unit vector parallel to axis Y,.
- -Yt : axis perpendicular to X t and Zt'
Zl : unit vector parallel to axi~ Zz.
ZI : axis oriented along the ith pair.
~ : angle between axes Zz and Z'+l measur.ed along the positive direction of axis
X t
~ : anlgular rotation of link i due to torsion, measured at a distance x, from the
origin of X" Y" Z,.
~ : angular "small" rotation 'vector of the tip of link i with respect to ?Ci, Y" Z" due to structural deformation of link .
'i : angle ,between axes XI and X.- 1 measured along the positive direction- of axis. ~ .
Pi : mISS per unit length o~ link i
~f : kinematic constraint wrench assodated wit~~~ink i. xxv
-
~! : external wrench associated with link i.
i>! : system wrench associated with link i.
CPt : angle 'of rotation of Qt measured p~e}~ the direction of vctor el'
angular velocity of Xtl Ytl Zi with respect to Xo, Yo, Zo, defined for rigid
manipulators.
angular velocity of X" Yt , Zi with respect. to X01 Yo, ZOI 'defiried for flexible
manipulators.
Onm : denotes the n x m zero matrix.
't'
On : denotes the n-dimensional zero vector. 1
Inn: denotes the n x n identity tensor.
o : denotes the tensor product .
( ) : relative time derivative (in a rotating framl!).
) () : absolute time derivative (in an inertift) frame).
)
~ \ ' . ..
-~---
i, ,
..
-
c
j
, ,
Chapter 1 Introduction
1.1 Robotic Maniplllators
The word robot first entered the English vocabulary when Karel apek
(1923) published his play RUR (Rossum's Universal Robots), whi~h in his native Czech
meant 'serf' or 'worker'. Since then, novelists, piaywrights, and scientific investigators
alike have used this word in their works but, to one's amazement, from these different
areu, a unified definition of the word robot does not emerge. Furthermore, contrary to
the portrayal of robots in movies and in science-fiction literature, -as being human-Iike
or 'android', present-day actual robots are mostly mechanical systems under computer
control. One of the popular definitions of a robot, which is als~ widely accepted in the - , -scientific community (Heer 1985), is:
"A programmable multifunction de vice designed to move objects through Variable prt;>grammed motions for the performance of a variety of tasks. "
Robotic systems consist of the following basic physical components: the me-
chanical system, the sensors and, the controller ~ _ ! simple robotic manipulator of the
seri al type is illuatrated in Fig. 1. 1. Its mechanical system consists of the open kinematic ) ,
-
\' .,
1. Introduction
chain with links interconnected at the joints and the gripper (end-effector). The ma--- , 1
nipulator provides the moti9n required for the task at ha~.d and the gripper provides
the means to grasp and hold the manipulated object, during the computer-prog~ .. mmed
motion to accomplish that task. The manipulator itself tan be of open- or c1osed-loop
kinematic structure, or a combinat ion of both (Fig.1.2). The actuating mechanism
\. consists of the power so~rce(s), the acJuators (e1ectric, h)ld~aulic" pneumatic) and the d , .
drive mechanisms (belts, chains, ge~s). While the power suppl y is the motive P9wer
to t~e robotic system, it is the actuators that convert them to useful work via the
drive mechanisms. The sensors (force, tactile, visual, acousti) meuure and determine
the state (positio"n, orientation, velocity and angular velocity) of the manipulator links, , the gripper and the workpiece. They also easure and determine the .forces and mo-
. ments exerted by the workpiece,on the manipul or. The controlJer is the device which
supervises and regulates the programmed motion.
1
. EndEffector .
" , .
~~ ,
. ( . ... Actuators, -..
" Manlpulator ~
.
Task ~ - Controller Senlorl -.. -,
~ .. .
Flpre 1.1 Robotic S)'lkm
2
-
c
1., introductIOn
(a) (b)
(c)
'laure 1.2 Kinematic Str.ucture.: a) Cloeed-loop, b) Open-Ioop Chain ConfifU-~ion, and c) Op.n-Ioop 'IrH Configuration --
Robots are being used in many areu such as manufacturing, spa.ce, under-
water, nuclear and mining indu8tr~ are gaing to have many more applications in
the yeUl ahead. Behind the ~uccessful utilization of today's-robots, and th06e which
are gaing to he used in the future, lies the backbone atudy of kinematics, dynamics and
3
... \
-
...,./ l introduction
conJ.rol of robotic manipulators. It should be pointed out that the ahove rnentiont>d
areas of study are becoming more crucial today than it was a decade ago, and that is
because of two reasons, namely, i) the growing need to improve the performance of th('
robots and li) the increasingly powerful capability of the computer hardware required
to control that performance.
\ The kinematics of robotic rnan~ulators deals with the geometry of the ma-,.,
nipulator motion without regard to forces or moments that cause the motion .. ln othN
words, it de aIs with the spatial cQpfiguration of the manipulator as functions of time;
in particular, kinematics deals with the relations betweerl the joint vS:riables and the
position and orientation of the end-effector.
~amics of robotic manipulators deals with the relations between 8.C-
t!lator tor,ques or forces and the link motions, upon consideration of the inertia of thE'
links. Equations of motion are useful for the determination of joint forces or torques
required to prod\!ce a specified end-effector motion, the computer simulation of ma-
nipulator motion, the de,sign of suitable control algorithms for the manipuJator and .-.,
the Evaluation of the kinematic design and structure of the manipulator. The control
of robotic manipulators deals with the study of how to maintain the prescribed ma-
tion of the end-effector in the presence of uncertainties in the model parameters a!ld
unpredictable disturbances.
1.2 Kinematics of Robotic Manipulators
../ .:;/
From the mechanics point of view, manipulators are composed of coupled
The coupli~gs provide the relative motion between adjacent links. The ID08t
c mmon couplings found in manipulators are: a) the simple hinge or tuming joint,
7
-
,.
....
1 introductIOn
usually -called the revolute joint; b) the sliding or rectiIinear joint, usually called the
prismatk joint. While a revolute joint provides rotation al motion between adjacent ~
links, the prismatic joint provides translational motion. Discussion in this thesis will be
restricted to tHe said two types of joints. \
The ~rst step in kinematics, given the geometric structure of the manipulator,
a180 referr~d to as its architecture, is to define a set of coordinate frames, which will then
be the basis for the mechanical description of the manipulator ... The next logical step in
kinematics is to decide upon a set of kinematic parameters that will fully and uniquely
describe the configuration of the manipulator. Following this, the most essential step . ~ in kinematics is to set up the correspondence amongst coordinate frames and, as a
re8ult, the correspondence amongst vector and tensor omponents in different coordinate
frames, through transformation matrices. It might be worth mentioning at this point
the work of Hartenberg and Denavit (1955 and 19(4) towards the ahove-mentioned
objectives, which has been faithfully followed by many researchers in their analyses.
Although this work has set a trend, as will he pointed out in a Jater chapter in this
thesis, a modified Hartenberg-Denavit notation will prove heneficial.
Once the kinematic description of the manipulator has been dealt with, the
next issue is the kinematic analysis. Kinematic analysis is not the subject of this thesis.
However, a brief discussion on this issue is in store, and this is in order to ensure the continuity of this chapter and the greater appreciation of the chapters ahead. Kinematic
a.nalysis can be subdivided into three sections, namely, trajectory planning, inverse kine-
maties and direct kinematics. Trajectory planning is the determination of the trajectory
ln the 6-dimensional configuration space, along which the end-effector has to move in
order to execute the desired task. The trajectory, which is represented in terms of
the time histories of the screw (position and orientation), twist (velocity and angular
, -
-
/
1 L Introduction vel city) and the twist's time derivative (acceleration and angular acceleration) of thE'
. en -~ffector, is to he determined taking into consideration the constraints on the maxi-
allowable velocity and acceleration at each joint and the constraints arising from
ma! ipulator geometry and workplace. Inverse kinematicst-is the matheniatical mapping
of ~he end-~ffector trajectory to the joint trajectories. ln simple words, inverse kinE'-
maFics provides ,~he meanS to determine the joint t~~ectories that would produce the
deJired etid-effec or trajectory. Direct kinematics, in turn, is the mapping of the joint.
trajec~ories to th end-effector trajectory. Direct kinematics is needed in the iterative
5 lut ion of the' verse kinematic problem, ~ weIl as in simulation (Fig.1.3).
\ .3 Dynamics of Robotic Manipulators \
The study of the dynamics of rohotic manipulators can be broadly, cJaasifled
into three categories, namely, the formulation of the dynamical equations of motion,
inverse dynamics and direct dynamics. The formulation of the dynB:f!1ical equatio-ns
of motion is the process of deriving the nonlinear differential. equations of motion of ")
the robotie manipulator, utilizing either the equilihrium between inertia and external
wrenches (forces and moments) or a variational principle. Inverse dynamics consists of
determining the joint actions (driving torques or forces), required to produce the desired
joint trajectories, which would in turn produce the desired end-effector trajectory. On
the other hand, direct dynamics, also referred to as dynamic simulation, is the proces!,
of determining the joint trajectories given the time histories of the joint actions, as weil
as the initial 'state of the mapipulator. \
Generally, in the formulation of the dynamica) equations oLmotion of robotic
manipulators, the joints and links are considered rigid, which &ives rise to rigid-link ma-
nipulators. A rigid joint is one which produce8 rotational or translationaJ diaplacement
6
-
c
1. Introduction
... ~ithout any slippage and/or restraint. Slippage occurs in the presence of joint friction,
and restraint occurs when the motor shaIt that drives the joints exhibits deformation. It ,---(
is obvious from the above discussion and from our knowledge of the real world structures,
that these consideratiens are approximations of the actual joints and links. Flexibil-
ity in joints and .links is taken into account when their effects become significant in
the successful operation of the robotic manipulator. The foregoing statement by itself
justifies and warrants a thorough study and understanding of the subject of joint and
link fte~jbility. In applications where robotic manipulators are required to have long
lin~s and as ao result Iink Aexibility is of major concern, the effects of joint flexibility are
relatively smal!. Furthermore, modelling joint ftexibihty is much easier than mo8elling
link flexibility. Hence, in this thesis attention will be focussed on link flexibility alone.
An extensive hterature is available Qn the subject of rigid-link manipulator
dynamics, and in general, on rigid-body dynamics. It is not necessary, for the purpose
of this thesis, to give a comprehensive survey of the literature on this subject, but
important contributions in this area wilJ be mentioned.
Multibody dynamics in a modern context was first considered by Hooker and
MarguIies (1965). ln their paper, the authors studied a system of interconnected rigid
bodies in a tree configuration (Fig. 1.2) , and used the Newton-Euler formulation to d-
rive their dynamical equations of motion. ln their method of analysis, aiso referred to as
augmented-body method, each body was~ considered separately- as an augmented body
with masses of ail other bodies as being concentrated masses at appropriate joints_ The
rotational and translational equations were then written about the mass centre of the , /
body, which resuited in the explicit appearance of the constraint torques acting at the '-,
joints. Roberson and Wittenburg (1967) also used the augrnented-body method and em-
ployed Lagrange multipliers to obtain the constraint torques, while Hooker (1970) used a ~
7
-
1. Illtroduct 1011
:projection method to eliminate the constraint torques. Wittenburg (1974) later included
the relative translation between bodies in his formulation. A second method of &nalysis
that emerged during this period was the direct-path method, in which the rotational
equations of motion are written about the body hinges or joints. This method elimi-
nates the need for considering constraint torques at the joints. Huston and Passerello
(1976 aJ).d 1980) and Huston-et al.(1978) studied human-body dynamics and multibody
dynamics in generaI, using the direct-path method and Lagrange's form of d'Alembert's
principie. Williams and Seireg (1979) presented a computer-oriented procedure for mod-.
ellins open- and dosed-Ioop systems. Rane et al.(1983) and Kane and Levinson (1985)
studied space~raft dyn\Cs using Kane's dynamicai equations, where!U' Hughes (1986)
used Newton-Euler eqUa)iOnS in bis investigation of spacecraft attitude dynamics.
While moat of the above-mentioned works were on multibody dynamics di-
rected primarily towards spacecraft applications, parallel work was being carried out in
mechanism dynamics and, in particular, in manip.ulator dynamics. Uicker (1965) used
homogeneous coordinates and the Euler-Lagrange formulation to derive the equations
of motion of spatial cl08ed-Ioop linkages. This Was tater modified by Kahn (1009) to
include open-loop mechanisms. In the early seventies resea!chers were exploring the
possibility of using dynami~s in the control of robotic manipulators, which were then,
and still are, kinematically controlled. But the complex manipulator dynamita posed
a computation al bottleneck in the dynamic control of rnanipulators. This led sorne Te-
se archers to comider simplifications in the dynamical equations, namely, ignoring the
Coriolis and centrifugai forces (Paul 1972, Bejczy 1974). However, this simplification 1
is justifiable only for slow movements of the manipulator. Yet another approach to ,
ease the computational burden was co~idered by Atbus (1075a &. b), ~ibert (1977),
and Raibert and Hom (1978). They prop08ed a table-look-up method, whereby the
8
-
c
1. Introduction
confi'guration-dependent terms in the dynamica) equations for discrete points on the'
trajectory were computed in advance and tabulated. This method had serious limita-
tions, ftamely, large computer memory requirements, poor accuracy of the trajectory, the
most important limitation of all being that the trajectory had to be known in advance,
thereby preventing simulation applications.
At about the same time it was realized that the open-loop manipulators with
simple kine~c-chain structure, could be~nalysed using recursive methods. These re-
cursive methods especially prove useful in improving the" algorithmic computational
"-efficlency. Stepanenko and Vukobratovic (1976), moti~ated bJ: human limb dynamics,
deveJoped a recursive Nwton-Euler method of analysis. J Further 'vor~c with modifi-~
cati~ns to the above method was Jater reported by Vukobratovic et a~ (1977) and
Vukobratovic (1978). Armstrong (1979) proposed a recursive Newton-Euler method for .
the dynatnic siml,!lation, while Orin et al. (1979) proposed a recursive Newton-Euler for-
mulati'on for inverse dynamics. Waters (1979) on the other hand, introduced a reqursive ~ ~
Euler-Lagrange formulation, and further improvements on this method were reported
by Hollerbach (1980), although the simplifications in the complexity of his scheme were
still about three times as large as those reported by Luh et al. ~980). Motivated
by biomechanics as weil ~ robotics, Benat et al. (1982) proposed yet another form
of recursive Newton-Euler formuJation. Walker and Orin (1982) solved the dynamic
simulation problem for rohotic mecharlisms. Kane and Levinson (1983) demonstrated 1 ~
the use of Kane's equations for muipulator dynamics. In their paper, Angeles and Lee
(1988) demonstrated the use of the natural orthogonal complement in conjunction with
the Newton-Euler formulation for the dynamic modelling of rigid bodies, whereas Ma
and Angeles (1988) did the same with Kane's equations for serial-Unk manipulatQrs.
~here,s numerous investigations haie focused on multiple-rigi?-body dy-1
,1 9
-
1. bauoduction "
namics, the literature on multiple-elastic-body dynamics, especially on elastic- or flexible-
link manipulators, is rather searee. Ho and Gluck (1971), and Ness and Farrenkopf
(1971) developed an inductive method of analysis to obtai~ equations of motion of a
system of rigid bodies with flexible terminal bodies. Roberson (1972) extend(;'d his
earlier work on multiple-rigid-body dynamics to include flexible bodies. Howt'vt'r. hr
. did not present the equations of motion for th{' elastic coordinates. ThE' first p('rsoll
to consider flexibility in manipulators was Mirro (1972). His work involved the mod-
elling and control of a single-link device. Dynamic analysis of elastic meehanisms W8.'i
considered by Winfrey (1972). Likins (1972 and 1973) presented the dynamical analy-" t~
sis of appendages using finite elements and the Hooker-Margulies formulation (Hooklr
and Margulies 1965), respectively. Book (1974) and Maizza-Neto (1974) in'their Ph.D,
theses eonsidered the effects of linearly distributed flexibility on control-system perror-l'
mance of planar rms,. and planar nonlinear models with modal control of flexible arms\
respectively. Ho (1974 and 1977) used the direct-path method to describe the dynamics
of a system of rigid bodies with flexible terminal bodies. Further work on manipulator
fiexibility and control was published by Whitney et al. (1974), Book et al. (1915) and
Book (1976 and 1979) .. Hooker (1975), convinced by the advantages of the direct-path
method, reformulated his earlier wQrk on rnulti-rigid-body dynamics to include flrxiblr
terminal bodies. Hughes (1979) carned out a formulation for a chain of flexible bod-
ies )Jsi!lg the vector-dyadic approach ar,d applie it to thE' CAl"ADARM. Book (19RZ
and 1984) used recursive Lagrangian dynamics to model flexible-link manipulator armb.
However, the use of homogeneous transformations in his m~thod introduces computa-
tional inefficiencies,!and the symbolic derivation of the equations of motion becomes
cumbersome. Sunada and Dubowsky (1983), on the other hand, used finite-element
methods to model robotie manipulators with elastic members. In spite of the generality l'
q
of their approach, their model is basicaUy an approximation to the linearized dynamical
10
-
c
c
--- ._-_ .. _~--_._--_._----------------------------, -
1. Introduction
equations of the manipulator. Three-dimensional modelling of n-link-ilexible-link ma-,
nipulators and the simulation or a two-link manipulator in planar motion was carried
out by Cyril (1984), and Misra and Cyril (1985). Judd and Falkenburg (1985) formulated
the equations of motion for a two-link planar manipulator using the (4x4) homogeneous
transformations. Csoro et al.1\-~986) used a finite-element/Lagrange approach to model ,
a two-Iink planar tlexible-link manipulator and presented the simulatiOn results for its
free -vibration. Cyril et al. (1987) used the orthogonal complement in conjunction with
, the hybrid NE/EL formulation to model the dynamics of flexible-Iink manipulators.
Low (1987) presented the analytical derivation of the dynamical equations of robotie
manlpulators with elastic links using Hamiltons principle.
1.4 Objective and Motivation
The primary objective of this thesis is to devise a methodology for the formu-
lation of the dynamieal equat~ns of general N-axis, seriaI-type ~oboti~ manip'ulators, of
otherwise arbitr~ry architectuJe, with rigid and flexible links. The methodology has 'to
br conceptually simple, easy 1.0 implement, aH-d eomputationally. effici~nt. To this end,
thE' work is divided into the following tasks, in the given order: a) to introduce a set ~
of computatioaally efficient coordinate frames and kinematic parameters, such that a
rigid or flexible link cah be described with ease; b) to devise a computationally efficient \
method to solve the inverse dynamics problem for rigid manipulatorsj c) to devise a '"
computationally efficient method for the dynamic simulation of rigid manipulatorsj d)
to formulate an algorithm for the dynamic simulation of flexible-link manipulators to
ascertain the effects of flexibility on the ove rail performancej and finally, e) to develop
a general-purpose software package for the dynamic s~ulation of serial-link robotic
manipulators of arbitrIJY archittbre with rigid and flexible links.
11
-
",. '.'
1. Introduction
There is a growing number of applications of robotc manipulaton in space
(e.g., the CANADARM and the proposed servicing robots for the space station) and
a demanding need to improve the performanc of pt:'esent-day industrial robots with .
respect to speed and accuracy. Whereas the former application requres the use of
flexible-link robots. in the latter application, ftexibility appears as an undesirable effect
in the form of vibrations in the links. The motivation behind this work was the la
-
Direct Kinematies of Rigid-Manipulators
Desired End-Effector Trajectory
Inverse Kinematies of 1 Rigid-Manipulators
1. Int.toductlon
10.-. __ -- Joint Trajectories
Dynamic Simulation of
Rlgld .. Manipulators
~ Inverse ~!,amics of. Rigid-Manipulators
Nominal Joint ~Actions
Effect of FlexiblUty
Dynamic Simulation of
r_lexible-Manipulators
rlpre 1.1 Dynamic Sn\lla'~D of Flexib1e-Link Manipu~ton
13
-
o r
1.
simulation and then solved using a computat ionally efficient appro3.ch in 'Chapter 3. A
discussion on the coordinate alternatives for the representation of the ori~ntation of a
rigid body is carried out in Chapt,er 4, along with the introdu('tion of the gt'neraliz
-
Chapter 2
2.1 Introduction
~
Inverse Dynamics of
'Rigid-Link Manipulators
An essential tool in the analysis of any system s...the mathematical descrip-
tion of that system, also referred to as the mathematical model. For dynamical systems,
the mathematical models are referred to as dynamical equations or simp]y equations
of motion. A robotic manipulator is one \ such dynamical system and in this thesis the
formulation of its dynamical equations and their computer simulation will be discussed. Intuitivf'ly, one would expect that, for a given system, aU the different approaches of
formulation should result in the same or equivalent set of equations, as long as the
generalized coordinates are the same. This in its entirety is true, but there lies a great
difference between choosing one formulation from the other. The two major differences
amongst the various formulat1ons are their conceptual and their computational com-
plexities. Conceptual complexity refers to the difficulty of understanding and deriving
t~e model. On the other hand, computation al complexity is measured in terms of the ~ .
~umbet of floating-point operations requird for the computer implementation of 'the . model.
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L! Q
/
. 2. Illvert
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2. Inver!e Dynamic8 of Rlgld-Link Manipulators
number of computations. Customized algorithms, and algorithms which ignore Coriolis
and centrifugaI forces in the dynamics formulation, have also been presented in order
to reduce the number of computations (Horn and Raibert, 1978, Hollerbac;h~and Sahar,
1983: Kaneand Levinson, 1983, and Horak, 1984) An extensive list of reference~ on the
computational aspects of robot dynamics can be found in Neurr1an and Murray ~198S).
In this chapter the dynamical equations wiIl be derived using the NE for-
mulation. Usually, the NE equations are written with reference to a coordinate frame
fixed either at the mass centre of the body or at the joint connecting the bodies. Here,
however, the NE equation!'> will be written with reference to an alternate body-fixed
coordinate frame (to be explained shortly), which exploits the architecture of the ma-
nfpulator.
The method presented here is a systematic and computationally efficient \
approach. The strength of the approach lies in the introduction of .a new set of body-
fixed coordinate frames and a modified set of Hartenberg-Denavit parameters to describe
the architecture of the manipulator. Computations are carried out with respect to the
body-fixed cooTpinate fTdrr)('~, thus avoidmg the need to transform vectors into the
base or inertial framC' Anoth
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\
2. lnvvae Dynamiu of Rigid-Link M.uipula"or~
- - -' \ -1 Link .\'
Z;".~I \,~k N - 1
~_ Joint N - 1
" ~X2 ,~r ,
Link 1 r
Link 0 Joint 2
i 1""" J' l '" omt l'(I
Figure 2.1 N -axu sen al Imk manipulaI or (.'
,~t
, .
Considered here is an N-axis seriaI Iink manipulator (Fig.2.1) of arbitrar)' , ,
architecture and com'posed of (N + 1) links which are cou pIed by revolute or prismatic
pairs. The links are numbered 8uccessively from 0 to N, letting 0 be the fixed link. Thf'
joints are numbered from 1 to N, such that the ith joint or pair couples link (1 - 1)
and link i. Next, a set of coordinate axes, X" ~,Z" is atlached to Iink i in 8uch a way
that axis Zt is oriented along the axis of the ith pair. Axis X, is defined as ~~e common
perpendicular to axes ZI and Zt+lt directed from -the form~r to the latter. The relative
position and orientation of two adjacent links are completely described by the set of
18
1
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c
2. Iuverse Dynarmcs of Rigid-Link ManipulatoTs
parameters {al' d,., ~, (Jt }f, al' d" ~ and 0t being defined as the disfanfe between
axes Z, and ZHl! the Z, coordinate of the intersection of axes X t - 1 and Z" the angle
between axes Zt and ZH 1 rneasured around the positive direction of axis Xt , and the
angle between axes Xl and Xt - 1 rneasured around the positive direction of axis Zt'
respectively (Fig.2.2). The variable ~sociated with link t is (Jt, if the lth pair is revolute , \'--f
and dt, if the tth pair is prismatic. These parameters are the 'modified Hartenberg-. -
Oenavit (HO) pararneters, a detailed description of the traditional HO parameters being
included in Appendix A.
~ __ .... ~XI-J "---'
o
l
1
Figure 2.2 Link Parameters
1.
l
1 ! 1
1
The matrix defining the orientation of axes Xi, ~,Z, with respect to axes ) -
19
-
[
1 0 ~ = 0 cos lt,:-l
_. _, 0 sin a.-l
2. lnvene Dynunl($ of Ririd-Link Manipulat.on
[
COS 'l si~',
- sin (JI cos o:._} cos (Ji sin 0:,-1 cos BI
- sin', cos,,
o .
(2.1 )
where ~ s represented in Xl'}~' ZI coordinate frame. Henceforth, note that v~tors
and tensors with subscript lare represented in Xl' Y., Z, coordinatE' frame,' unless statf'd "
otnerwise.
Let p; denote the position of the origin of joint (1 -t 1) with respect \'0 tlw origin of joint t, referred in Xp Yt Zl coordintes and given by (Fig.2.3):
(2.2)
2.3 Newton-Eulr Dynamical Equations
. r,~ Let ml denote the mass of link 2, 8: denote the position vector~ the centrt
of mass / of link t wfth respect to the origin of XI' } ~ , Z1 , and J. den ote the inertia tcnsor .,{
of link i about its centre of mass. ~otc that s; and JI have a constant r,presentatlOn
in t'h~tth cQwdin1 frameIXp~,ZI'
.~~:~ole. Ihe angular velocity of X" y,. Z, with respecl to Xo. Yo Zo. then the angular velocity of the Xl' Y" ZI frame can be compu~ed 'as follows:
- _ {W'-l + btzl , if joint i is rotational W
1 ~ w,-l' if joint i is translational (2.3)
where Si is the unit vector parallel to axis Z, and oriented along the ith pair. Differ-
entiating eq.(2.3) with respect to time gives the following expression for the angular .. , 20
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2. Inverse Dynamick of Rigid-Link Manipulators
al .. X , p.
1
1
1. p;'
~.
YO
---/
~ . .. -XO "" .... C '"
Zo ~-;,~~
t Figuri 2.3 position Vecto[$
acceleration: -'-
if joint i is rotational
if joint i is translational . ( (2.4)
The,position oVthe origin of X ~,Z, with respect to Xo, Yo, Zo is given by
(Fia2.3):
(2.5a)
where
(2.56)
21
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,
\
2. lnvene Dynamics of Ririd.Link Manipula~or.
in w,hich ~ is the unit vect.or parallel to a.xis X,. Difr~~nti&ting both sides of eq.(2.5a)
with respect to time, the foJJowing expression for the velocity of the origin of X" Y" Z,
with respect to Xo, Yo, Zo is obtained:
if joint 1 is rotational if joint 1 is translational
(2.6a)
where nt is the cross-product tensor defined as :
n =- a (!J, x k) t - k (2.6b)
for arbitrary k and for t = l, .... N.
Differentiatmg both Sicles of eq.(2.6a), thE' expression for the acceleration of
thE' origin of X" YI' Z, is obtained a." follows:
if joint t is rot.ational
(2.7a)
jf joint 1 is translational
Rewritting eq.(2.7a) gives
if joint i ie rotational (2.7b)
~
where i" the angular-acceleration tensor, is defined as:
(2.7l'}
H 8~ denotes the position vector of ~ntre of mM~ of link i w\~th resp~ct
to Xo, Yo, Zo then it can be written as: . u
(2.8a)
22
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c
c.
! . 2. In ..... Dynamic. 01 R;,pd-LUol MUlipula .... , "The recursive relations of its velcicity and acceleration are:
S :;t: (2.8b) . , {i'-1 - 0,-1S:_1 + O,-l~-1 + O,s:, if joint i is rotational l 'il-1 - 01-1S:_1 + Ot-l~-l + OIS: + cL.z" if joint t is translational
_ 1 8,-1 + "1-1 (at -l - 8:_ 1) + .,s; , if joint i is rotational s, = 81 -1 + .1-1 (at -] - 8:_1) + .,s;
+2d1 0t-1 It + dt 1
7, >, if joint i is translational
If"
(2.8c)
However, the acceleration of the centre of mass of Iink i can be efficiently
evaluated as:
Next the NE equations for hnk (c'an be written as :
\
(2.8d)
(2.9)
(2.10)
where f, is the total force exerted on the centre of mass of Iink i and ~ is the total
moment exerted on Iink z. The force and moment balance on link z result in the following
expressions :
(2) 1) r
(2.12)
where fi,'-l and Dj,i-l are the force and moment exerted on Iink i by link i -l, respec-\>..
tively and f.,I+l and ~,.+1 ~re the force and moment exerted on link i by link i + 1,
respectively. The torque or force exerted by the actuator at joint i is then given by :
1 {
1'[ ~ ,-1' 'if joint i is rotational Ti = l'[f'.~-l' if joint i is translational . (2.13)
23
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J,
2.4
/
.. 2. Inverse Dynanucs of Rigld.Lilllt Manipula~on
t-
Aigorithm and Computational Complexity
ln this section, the inverse dynamics algorithm to compute the actuator ,
torques or forces, referred to in general, as joint actions, given the time histories of the
joint positions, velocities and acelerations, is presented. Aiso included in this section is
the computational complexity measured in terms of the number of multiplications and q ~/
~..;
additions required to compute the joint actions.
lt may be noted here that the computations are carried out in Iink coor-
dinates, namely, in X\, Y\, Zt coordinates, for i = l, ... , N. In short, the algorithm ,
consists of computing the total force and moment exerted on ea.ch link, starting from
the base and proceeding towards the end effector. Next, the force and moment exerted
-on each link by the previous link aJ'fd the external torque applied by ea.ch a.ctuator are computed starting from the end effector and proceeding towards the base. The details
of the algorithm for revolute pairs are presented below; if the ith pair is prismatic, then
the algorithm and its computation al complexity become mueh simpler.
for i=1. ... N do
Step 1) Compute ~ using eq.(2.1).
RI takes on a simple form sinee ao :::: 0, thus saving 4 multiplications.
Step 2) Compute wt using eq.(2.3).
Since wo = 0, computing Wl requres no operations, thus savin, 8 multipli
cations and 6 additions. Additionally, sinee Wl ~as only one non-zero term,
in computing W2 there is a saving of 5 multiplications and 5 additiona. l "
) ... KT-2.1 wi-l +- 1: w,;-l' \
, '
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... 2. Invene Dynamics of Rigid-Lmk Manipulators
c Step 3) Compute ~t using eq.(2.4).
Since ~o = 0, computing .JI r~ires no operations, th us saving 10 multipli-.. cations and 8 additlOns.~ Additionally, since ~1 has only one non-zero term,
in computing .J2 there is a saving of 5 multiplications and 5 additions.
/
. 3.3) (3,1).+ (3.2) + OiZ;, . ,
, Step 4) Cmpute Pt using eq.(2.7b).
c Since. Pl ~ -Ri g, where g is the gravitational acceleration vector referred -
in Xo, l'e, Zo coordinates, there is a saving of 15 multiplications and 12 ad---- \
ditions. .t~ 1 used here is compul'd 'ror the previous lin~ in Step 5. Since , .
..... i 1 hac;; a simple form, 50 does q, l, thereby savmg 5 multiplications -and 5 .
\ additions in the computation of P2'
"
4.1) ompute at-l'
" 4.2} Compute .,-1~-1' --_/'0..
4.3) Pi-~ + (4.2).
Step 5) Compute ~ using eq.(2.8d). , l ' .. 25 . ,
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l .'
2. Inverse Dynanncs of RigidLink M~lIipulatol'$
Since .1 takes on simple form, there is a saving of 10 multiplications and 13 additions in the computation of SI'
5.1) Compute .,.
5.3) Pt + (5.2).
Step 6) Compute ft using eq.(2.9).
Step 7) Compute ~ using eq.(2.1O).
enddo
When the right-hand .side of eq.(2.10).i~ expanded, it will De noticed t.hat
mosf of the terms in: it have already been evaluated in il and hence, there
will be sorne saving in the computation ,of lit. Once again, because of the . .
simple structure of Wl and W1, there is a saving of,12 multiplications and 15
additions in the computation of Dl'
for i=N ..... l do
Step 8) Compute ~ t-1 using eq.(2.12). ,
'. N~te that the representation ;r p; in Xl; Y l , Zt has rendered it in a, simple .. , form, a.s_show~ in eq.(2.2), as opposed to the usual form (Appendix A). This . ~as introduced additional savings in-this algorithm. DN,N+l and fN,Nt! are
the external moment and force exerted on the end effecto! by the environment.
and RN +1 = 1, where 1 is the 3 x 3 identity matrix. Therefore, in computing
DN N-b there is a saving of 16 multiplications and 10 additions. Since the , , . . constraint moments and constraint fores-1it1Jlg at joint 1 .r~ irrelevant for'
'\ - ~
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c
~' ).
2. Invera~ Dynamicll of Rigid-Li Manipulatofs
the problem at hand, 'they need not be computed. Thus, in computing Dl,O
there 8 a saving of 17 multiplications and 15 additions.
8.1) ~,Hl +- ~+l~,t+l'
8.2) ft,HI +- Rl+l ft,t+l'
...j
8.3) p; x ft,t+l' "
8.4) dtz1 + s;.
8.5) (8.4)xfz
8.6) ~'- (8.1) - (8:3):+ (8.5) ..
Step 9) Compute f11 - 1 using eq.(2.U). ,
,
Once again, from the same argument as, In Step 8, the first and second
components of fl,O ned--not be comput~d, since they' re irrelevant for the
problem at hand.
Step 10) Compute Tl using e.q.(2.l3).
Since ZI' is parall~1 to both the {th pair andv
the Zt axis, the torque 'or force - ... 9
acting at th~ t'th pair is simply the zz-component of ~,t-l or f't l _ l ' respec-
tively. Thus, no operations are required for this step .
. enddo
The computationaJ complexity of the inverse dynamics problem cissociated
with an N-axis manipulatol, for ,N ~ 2, with the number of computations at each step
of the algorithm, is given in Table' 2.1. In Tables 2.2-2.6, the computational complexity
27
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6 '
1
2, _ Illvl!rse Dyllanllc~ of RlgidLillk Mal1lpulaLOr~
St'p No. Item Mult -Add
1 ~ 4N - 4 0 2 Wt SN -13 6N -11
3 wt ION - 15 SN - 13 --4 Pl 19N - 20 14N - 17
\ 5 St 15N - 10 IBN - 13
6 ft 3N 0
7 ~ 15N:- 12 15N - 15
8 n1.\-1 26N - 33 24N - 25 .' 9 - ft t-} o ' 3N - 2 .. 10 Tt 0 0
Total lOON -'107 8SN - 90 '\;
Table 2.1 Computtions Required t.o Solvl! for the Joint Actions, f
involved at each step of the algorithm are given in a greater detail. Th~ algorithm , 1
, just presented takes into account the arbitrariness of the manipulator's architecture, its '
physica1 parameters and its orientation with respect to the gravitational field.
- Item Mult Add 2.1 SN -13 SN -10
2.2 0 N - 1 -
Subtotal SN -13 6N -11
, Table 2.2 Computations Required \0 Ohtain the Angular Velocitiel, w
--Item Mult
~ Add
3,1 8N -13 SN - 10 . 3.2 2N -2 0
3.3 0 3N -'3
Subtotal ION -15 SN -13
Table 2.3 Computations Required to Obtain \ the Angular Acceleratio8l f iJ
A comparison of th inverse dynamics complexity reporte~ by various authofS
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1
2. Inverse Dynamics of Rlgld-Lmk Ma~lpu)ators
Item Mult Add
4.1 .' 2N -2 0
4.2 9N -14 6N -10
'4.3 0 3N -4
4.4 SN -4 SN -3
Subtotal .l9N - 20 14N -17
~ Table '2.4 Computatfonfl Required to Obtain t~e Accelerations, Pi
Item Mult Add
5.1 6N - 5 9N -9
5.2 9N -5 6N -4
5.3 0 3N'
Subtotal 15N -10 I8N"- 13
Tab)~:U Computatio:S ReqUlred to Obtam the AcceleratIOns, il
Item
8.1
8.2
8.3 .
8.4
8.5
8.6
Subtotal
Mult
SN -13
SN -13
4N - 3
0
6N - 4 '''; ~\
26N -f 33 '\ 1
Add
5N-S'
5N-8
N-l
N
3N -2
9N-6
24N - 25
Table 2.6 ComputatIOns Required to Obtam the Moments, ~,i-l
is gi~en in Table 2.7, and the same for a 6-axis manipulator is given in Table 2.8. As it
can been seen from the table the present algorithm (Cyril et al. 1988) is more efficient /
than others. /'
2.5 Examples
To demonstrate the validity of the above computational ~he1' the example ,
29
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2. Inverse Oynamiea of Rigid-Linlt rUlipul~tor' .
Method Multiplications Additiorts
Uicker fKahn (~)N4 + (!W)N3 + (~)N2 '25N 4 T (~)N3 + (~)N2 ~ +(~)N - 128 + (ljI )l'j; =~~- .~ .. --
Waters (~)N2+ (~)N - 512 82N2 + 514.\' aK4 .l- "" _
Hollerbach(4 x4) 830N - 592 675N - 464 --- -
HoUerbach(3 x3) 412N - 277 . 320N - 201
Walker and Orin 1 137 N - 22 10IN--l1 ,
-- --McInnis and Liu 134N - 47 113N - 38
Kazerounian and Gupta 136N 118!\' .. . Khalil et al. 105N - 92 94N - 86 - - ~-
Ma l09N - 109 95l\' tOR l00N - 10;~-'-1 -Present Algorithm 1 R~N - 90
Table 2.'1 Comparisoll of Inverse DYllalllir~ Complt'xit.y
Method MultiplicatJions Additions'
Uicker/Kahn 67,143 51,548
Waters 7,045 5,652 .-Hollerbach( 4 x 4) 4,388 3,586
Hollerbfch( 3 x 3) 2,195 1,719
Walker and Orin 800 585
McInnis and Liu 754 640
Kazerounian and Gupta. 816 "L08 --- 1 '1'_ - -- --~ ~ ... -Khalil et al. 53~ 1 478 ,
1 ~ ----1
- - -~-~ Ma 54;) 462
1 -- ~- -4 -Present Aigorithrn 493 1 43~
Table 2.8 Compan~oll of InverM! DYllallllcs COlllplexlty for JI,' ~J
dt:ed by Kane and Levinson (1983) was chosen. ln this example, the inverse dynamicB
was solved (or the Stan(ord manipulator for the following joint trajectories:
= u 0 + t - - sm -8 il () 8i (T) - 8t (0) ( T. 211't) 't T 211' T ' (2.14)
(or i::: 1,_ .. ,6 and with T::: 10 s, 81(Q) = 0,6,2(0) = 00, d3(O) :: ... = 86(0) :: , >
0, 6,(T) = 600 (i = 1,2,4,5,6) and d3(T) :::: O.lm. The architecture and inertial paral'l1-~ .
30
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c
(
...
c
2. Inverse Dynarnics of Rigld-Link MaDlpulators
Link
(m) 1 a 0.0 1 -90 ()I 9.6 0.0' 0.0 l -.1 1 0.0] 001 0.02 2 0 0.1 1 90 ()2 6.0 0.0 0.0 0.0 0.05 0.01 0.06
~--~-_.-4----~---4~~---+---+----T----r----.~-
3 0 d3 0 a 4.0 0.0 0.0 ! 0.0 1 DA 0.4 i 0.01 4 O! 0.7, -90 ji()4 1 1.0 00 00
1-.1 .001 .0005 1 .001
~---r---- ~--r---~-----+----4
5 0 1 0.0' 90 : 85 ! 0.6 1 0.0 - 06 ! 0.0 1 .0005 .0002 .0005 ~--4---~ ,
6 0 0.0 1 0 86 0.5 0.0 0.0 f .02 .003 .001 .oq2
Table 2.9 Parameters of Stanford Mampulator '-
eters of the Stanford manipulator are given in Table 2.9.
The joint actions obtained are shown ln Flg.2,4 and these are in perfect
agreement with the results obtained by.Kane and Levins~n (1983). A customized version
of the a)gorithm presented in this chapter resulted in a considerable saving in the nurnber
of computations. This is compared with other such custornized algorithrns in Table 2.10.
Method MultiplicatIOns Additions
Vukobratovit and Kircanski 384 152
Kane and Levinson 646 394
Horak 361 256 -----.-
Present Algorithm , ]60 J6R
Tabl(> 2.10 l'olt1panson of ('U;:toIll17td All!nJltlllll' for the Stanford Manlpulator
As a second example, the Puma-600 manipulator was chosen, whose atchiter-
ture ahd inertial parameters are given in Table 2.11. The joint trajectories are (Fig.2.5):
(2.15)
for i = 1, ... ,6 and with"'T = 10s.~ joint actions'pbtained are shown in Fig.2.6 and the correctness of this is tested in the ~xt chapter, where the joint actions obtained in
~
this chapter become the input and the joint trajectories are the output.
\ 31
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: :
(~
2. Inverse Dynamics of Rigid.Link MAl\ipulator~
Link al ~ 0:, (Jl ml s x s y Sz J'X'Z jyy Ju (m) (m) (deg) (kg) (m) (kg m2)
1 0 0.0 -90 81 10.521 0.0 -.054 0.0 1.612 0.5091 1.612 2 .432 -.149 0 82 15.781 .14 0.0 0.0 .4898 8.0783 8.2672
3 .,0.02 0.0 90 83 8.767 0.0 -.191 0.0 3.3768 .3009 3.3768
4 1 0 -.432 -90 (}4 1.052 0.0 0.0 -.051 .181 .181 .1273 --5 0 0.0 '90 85 1.052 0.0 -.007 0.0 .0735 .0735 .1273 ----6 a -.056 0 86 .351 0.0 0.0 -.019 .0071 .0071 ~O141
Table 2.11 Parameters of Puma.6()() Manipulator
.. ..
r
',..
32
-
1 -e z "" ... ...
-Z ""
0.2
0.1
-0.1
-0.2 0 5 10
Time (sec) 0
-JO--------~------~--o 5 rime (sec)
10
1.4,...-----,.------. f
"'" 1.2 e z -1
0.8""------'-----....1 o 5
TIme (sec)
10
2. Inverse Dynarnlcs of Rigld-Lmk M,ampulators
18
-e z 16 -,.. ... 15
14 0 5 10
TIme (sec) 0
-e ! -0.5 ...
-e z -
-1~--------L-------~ o 5 TIme (sec)
10
.10-4 4~--------~--------
" -2~-----~--~--~ o 5 10 Time (sec)
Figure 2.4 Joint Action. of the Stanford Manipulator
33
-
2. Inverse Dynunic:. of Ri,id-Link MuplIlat.ou
l.5,..---,.---,.---.,.---.....--,.....-,.....-.,...--....... - ...... ......,
J
- 2.5 " " ....... "0 2 e - -" ....... "0 e -". -'0 e -ca ; .. -.. -. __ .. _-- ..... -,-' -..... -.. .. "
o
.... ;~ ~~
__ tt' ... 4............. ~, .... __ ..... , .... ;;-;;- .................... -.. -. ....... . ,- ....... ,...... - .... -... . ............ , .......... .. fi, _ ........................ J ........... :,., ............... .
. -0.5 '--_'---.-j~---''''''----.e ...... --'_--'_--'''_--''_--'_.....I o 2 345 6 8 9 10
Time (sec)
'Jeure 2.6 Joint Trajectorie. for the mverH Dynamic. 01 Pum~ Manipulator
J
, " '\
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c
-e z "" -...
-j' -~
-'E z -III ...
2. Inverse Dynamcs of Rigld-Link Malllpulat.ors
10~------~------~
-10 0 5 10
lime (sec) ..,.. 10
...
-5
-10 0 5 10
Time (sec) 0.2
0.1
-0.1------....-----o 5
Time (sec)
10
loo~------~------~
-e z -'" ...
-50
-100 0 .. 5 10
lime (sec)
/ 0.2
- 0.1 E z -.. ..
-0.1 0 5 10
Time (sec) 0.02
-E z -~ -0.02
-0.04------------.0 5
fune (sec)
10
flpre 2.8 Joint Action. of the Pum~ Manipulator obtained from the Inverse Oynamie SimulatioD ' ..
35
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;
1
Chapter 3 '
3.1 Introduction
Dynamic SimulatIon of
Rigid-Link Manipulators
Dynamic simulation is an inexpensive means of assessing the performance of
a given dynamic sy-stem, in this case the robotic manipulator. For robotie manipulators,
dynamic simulation amounts to obtaining the motion of the manipulator caused by the
given set of joint actions (torques or forces). In this chapter, the dyn~ic simulation of
rigid-link manipulators will be discussed.
Differentialequations of motion of multibody system~ are generally very com-
plex due to their nonlihearity and coupling, and hence their integration can only be done
numerically. Understandably, decades ago, when computers were not available, numer--
ical integration of the said equations were simply ignored. But, even today, with the
rincreasing use' of computera for dynamic simulation, there is a lack of fast algorithIll8
for the dynamic simulation of robotic manipulators. In recent years, researchers have
explored the possibility of improving the computational algorithms for robotic -UlAnipu-
lators, and during this process, they have discussed the merits and demerita of various
method~ of dynamic modelling and simulation (Hemami et al. 1975, Walker and Orin
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c
3. Dynamic Simulation of Rlgid-Link Manipulators
1982, Kane and Levinson 1983, Goldenberg and Kelly 1985, Wang and Ravani 1985,
Kazerounian and Gupta 1986).
3.2 General Formulation Metqodology of Dynamical
Equations
First, a few words on the terminology to be us~
-
1
3. Dynamic Simlliatton of RigidLink M&nipulator~
wh~re 1 represents the N n' x N n' generaJjzed exten,ded-mass matrix of the system. q
represents the N n'-dimensional generalized screw, while XE, XS , XC 'and i,K are the - Nn'-dimensional generalized externaJ wrench, generalizd system wrench, generalired
constraint wrench arising due to kinematic where M represents the N m' x N m' generalized extended-llJlUB matrix, ... repreeents , ..
the Nm'-dimensional generalized twist, while ~E, ~S and ~C are the Nm'-dimenaional-
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3. Dynamic Simulation of Rigid-Lmk Man!pulators
g~eraljzed externa/, generalized system and generaIjzed c:o~trajnt wrenches, respec-
t.iy~ly. Note that v and q do not necessarily have the same dimensions and hence, in
general, m' t- n'. The quantities in eq.(3.2a) are defined as:
'" M = diag(M}, ... ,MN) (3.2b)" :t
v= (3.2c)
whcrc Mt' v t' ;tE, ;j;S and ;j;C are the m' '" m' extended-mass matrix, the m' -dimension al t 1 .
scrw, external wrench, system wrench and constramt wrench, respectively, associated
with the lth bod).
Generalized speed and generaliz{) twist are one and the same for dynamic ~-
systems ln planar motion; in the case of 3D-motion they are related hy linear transfor-
mations. The advantage of choosing generalized twist, as opposed to generalize8 speed,
li('s in the eas(' of describing the dynamlcs of a system in termb of generalized twist. For
cxao:ple. in tht method of NE: .the equations arE' written in term~ of the generahzed j
twist clnd its time derivative. In the method of EL. Hl turn, the kinetic energy is simpler
to exprcs~ m terms of the generalizeft-wIst. and therefore the generalized mass matnx "'"
can he obtained by inspection.
1
The dynamical equations (3.2a) contain nonworking constraint wrenches due
to the physical coupling between adjacent bodies. These constraint wrenches i~oduce
addit.ional variables in the dynamical equati,ons and, as a result, the dimension of the /
system of equations for simulation, is increased. For the purpose at hand, the constra~t
. wrenches are not required and hence" will he :elimin.d from ~he dynamical equations
39
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2
3., Oynamic _ Simulation of -Rigid-Linl Manipulator.
hy following a-procedure descrihed next. First, the linear kinematic velocity cona~l'&ints
of the system ~d a natural orthogonal complement associated with the matrix of con-
straint coefficients will he derived. Following that, the natura} orthogonal comp)emt>nt f ~ .. 0 J
will be used tc?' eliminate the nonworking constraint wrenches to obtain 'the independE'nt
dynamical equations of"the system.
The kinematic velocity constraints of tht> system can be written as!
Av = Os (3.3) '.
where A = A(q,t) is an s y Nm' mat.rix.with.~ < Nm' and O~ i~/ the ,q.dimensional ZNO
veeotor. Let p be the tank 'of matdx A. Then, the degrees of f~eE'dom of the systt'llI,
henceforth denoted by r, is
r = Nm' - p (3.4)
'Hence, an r-dimensional vector denoted by ~ and containing indepcndent velocitie:- cali
be defined, which shaH also be referred to as the generalized veJocity.
The generalized twist v can hE' represE'ntE'd ac; thp following Iinear transfor-
mation of the r-dimensional vector ~'.
v = Nt/.! (3.5)
where N is a Nm' x r matrix. Upm substitution of v, as given by eq.(3.5), into flq.(3:3) '0 o. ,
the following is obtained:
AN;j; = 0" (3.6) ,
which holds for any vector ;fi. Since ;fi is r-dimensional and its components are linearly
independent, eq.(3.6} holds if, and only if,
AN = 0",. (3.7)
1\
.'
o
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C
c
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c
3. Oynamc Simulation of Rigid-Link Manipalatora
which shOW! that N is an N m' x r-dimensional orthogonal complement of A, where O"r
is the s x r zero matrix. Because of the fonn this orthogonal complement was defined,
it lS called here the natura! orthogonal complement 9f A.
By definition, the power n developed by ~c onto the generalized velocity .~ 'vanishes or, in other. words, the power developed' by ~(' onto the component of
the generalized twist v that is perpendieular to the velocity-constraint plane vanishes. " ..
Hence, one ean write the following:
< . (3.8)
Now, sinee ail. the eomponents of fi; are independent, eq.(3.8) implies that
(3.9)
which means that ~(' lies in the nullspace of NT. Howev~r, sincr N is an orthogonal
comple~ent of A, it follows that i>C lies in the range of AT, i.e., a.J8-~;nsional vector exists, say S, which is mapped by AT onto ;fF, namely,
(3.10)
Upon pre-multiplication of both sides of eq.(3.2a) by NT, the eonstraint
, wren"'h JC otihat equation is eliminated by virtue of q.(3.9), thus obtaining the foUowing equation:
(3.11)
From eq.(3.5), the time derivative of v can he readily obtained as:,
(3.12)
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\
3. Dyauoic Simula\ion of Rigid~Lnk Manipula\or,
When eq.(3.12) is substituted in eq.(3., Il. the following is obtained: \
(3.130)
A .". , '"
where M is the r x r generalized mass matrix of the system, which is symmetric and
positive definite, and given by:
(3.13b)
. Thus, equations (3.l3a) are0the independent -dynamita] equations of the m('-
chani
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