Table Of Content]974008732
NATIO,_AL AERONAUTICS AND SPACE ADMINISTRATION
Technical Memorandum 33-669
Robot Arm Dynamics and Control
A.K.Bejczy
(_A_A-C_-I_oC3-5) ._"=_-'_ A_'_ _Y' _._.iCS ,,_L _,7, lh_45
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(_.'-_.,_" CALIFORNIA INIITITUTE OF TECHNOLOGY .
i_:'; PAIADENA, CALIFORNIA
-_(_'& FebruaryIS, 1974 .".
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1974008732-002
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PREFACE
The work described ia this report was performed by the
Guidance and Control Division of the Jet Propulsion Laboratory.
JPL Technical Memorandum 33-669 iii
1974008732-003
CONTENTS
I. INTRODUC rION ................................. 1
II. DYNAMICAL MODEL AND CONTROL SYSTEM DESIGN ....... 5
III. GENERAL MODEL FOR MANIPULATOR DYNAMICS ......... 9
' A The General Dynamic Algorithm 9
,: B. Dynamic Equations for the JPL RRP Manipulator
Expanded in General Terms ...................... 14
0)"
,_ IV. RESTRICT_',D DYNAMIC MODELS ..................... Z4
_ A Alternative Model Restrictions Z4
...., B. Applications of Dynamic Model Restrictions ......... . . Z5
V. RESTRICTED DYNAMIC MODEL FOR THE FIRST THREE
• LINK JOINT PAIRS Z6
;_ A Gravity Terms Z9
1. For joint #1 ............................ 29
Z. For joint #Z ............................ 30
: 3. For joint #3 ............................ 30
B. Acceleration-Related Dynamic Coefficients • . . . . . , .... 31 '
I. Diagonal Coefficients D I"I' DZZ' D33 ........... 31 _:"
Z. Off-Diagonal Coefficients DIZ, DI3, DZ3 ......... 33
VI. COMPLETE DYNAMIC COEFFICIENTS FOR ALL SIX _:
LINK-JOINT PAIRS . , . . ........... . ............... 34
A. Gravity Terms in Complete Form .... . ...... . ...... 35
I. For joint #I ................ .......... ,. 35
Z. For joint #_ ......... ,. . . ,. , ..... ,. , .... 36
3. For joint #3 . . . . . , . ...... . • . . . • • . . , .., .. 37
4, For joint #4 . . ...... , . ........ ,......... 37
5, For joint #5 . . . , ,, . . .... ..... . ,, . , . , .... 37
mm
6. For joint #6 • • ,, • • • ,,. • • .... ,. • • , . • , ,, • • 38
JPL, Technical Memorandum 33-669 YK/_JSi)_(} _._._/_/.M_/& _U_ __ v
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1974008732-004
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CONTENTS (contd) i'
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B° Acceleration-Related Uncoupled Terms in
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Complete Form ............................... 38 i,
1 For joint #I 39 "
{
Z For joint 02 41 i'
: 3. For joint #3 ............................. 43
4. For joint#4 ............................. 44
, 5. For joint05 ............................. 44
6. For joint06 ............................. 45
Remark .................................... 45
' Vll CONCLUSIONS 47
• • • • • • • • • • • • • • • • • • • • • • • • • • Q • • • • • • • • • •
A. Variations in Total Inertia at the Joints ............... 48
I '
1. At joint #1 .............................. 48 z
i ' Z• At joint #Z • • • • • • • • • • • • • • • • • • • • • • • • • Q • • • • 5Z
: ' • • • • • • • • • • • • • • • • • • • • • • • • • • t • • • •
i J 3 At joint #3 56
Y";1 4 At joint 04 57
Xl 5 At joint 05 61
6. At joint 06 .............................. 6Z
¢
B. Maximum Gra_ty Load Variations .................. 63
j -
I. At joint #1 . ............................. 66
Z At joint #2 66 !:
3. At joint f3 .... ••••. ............. ........ 68 i_',
4. At joint #4 .... .. • ...... ................. 69 __ •
[
5. At joint #5 ...... .. ........... . . ......... 70 _:_:_:/
6. Atjoint,6.......... ............ .... ....
C. Relative Importance of Inertial Torques/Forces Versus ::'_'_ii__'..:i_
Acceleration-Related Reaction Torques/Forces ......... . 71 "_=_/::',_,'
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vt _PL Technical Memortndum 33-669 _"
1974008732-005
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CONTENTS (contd) i;
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D. Simplification of Torque/Force Equations ............. 8g t
I. Inertial Terms ........................... 8Z !,
* Z. Gravity Terms ........................... 90 ,
., E. Relative importance of Gravity Terms Versus
_o- Inertial Terms .............................. 91
References .......................................... 101
_/• Appendix A: CFuonmcptiloenteally Set EoxfpliPcaitrtialForDmerivfaotrivethe MJPaLtricResRP Uji in •
Manipulator .............................. A- 1
Appendix B: Mass Center Vectors aald Pseudo Inertia Matrices
for the JPL RRP Manipulator ..................... B-1
Appendix C: Manipulator Dynamics with Load in the Hand .......... C-1
Appendix D: Simplification of the General Matrix Algorithm
for Manipulator Dynamics ...................... D-1
LIST OF TABLES
I. Variations in Total Inertias (Exact " _Lues) ................. 64
2. Maximum Gravity Load Variations ...................... 7Z
3. Simplified State Equations for Inertia and Gravity Loads
at the Six Joints .................................. 9Z
4. Parameters in the Simplified State Equations for Inertia
and Gravity Loads ................................. 93
LIST OF FIGURES :*;,
I. -Mani-pulator Servo Scheme. .... . . • . . . • . . • . , , • • • • ..... • -
2. Reference Frames for Link-Joint Pairs of Arm ............. 27 _::i_I
4. Relative Importance of 01/0"Z Coupling as Seen at Joint #I . . . . . . 76
3. Relative Maximum Variations in Total Link Inertias........... 65 __
i JPL Technical Memozandum 33-669 vii ;-
1974008732-006
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CONTENTS (c ontd) t_
5. Relative Importance of 0 /¥ Coupling as Seen at JointH1 • 77
1 3 .... i
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6. Relative Importance of 01/0Z Coupling as Seen at joint #2 ..... 79 I
7. Relative Importance of 01/i;3 Coupling as Seen at Joint 03 ..... 81 :
8. Relative Importance of Gravity Versus Inertia Torque at 1
• Joint #2 ....................................... 95
9, Relative Importance of Gravity Versus Inertia Force
' at Joint #3 97
; 10. Relative Importance of Gravity Versus Inertia Torque
at Joint #4 ..................................... 98
i 11, Relative Importance of Gravity Versus Inertia Torque
at Joint #5 ..................................... 99
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viii _PI, _echnicsl Memorandum 33-669 _,;
1974008732-007
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ABSTRACT
This report treats two central topics related to the dynamical aspects of
the control problem of the six degrees of freedom JPL Robot Research Project
{RRP) manipulator" (a) variations in total inertia and gravity loads at the joint
outputs, and (b) relative importance of gravity and acceleration-generated reac-
_ tiontorques or forces versus inertia torques or forces. The relation between
_ the dynamical state equations in explicit terms and servoing the manipulator is
briefly discussed in the framework of state ,ariable feedback control which also
forms the basis of adaptive manipulator control.
f
Exact state equations have been determined for total inertia and gravity
,. loads at the joint outputs as a function of joint variables, using the constant
i, ' tiinveertiallink ancdoordgienoamteetric framepsa.rametersThe raonfgethe ofindmivaixdiumalum linvkasriatidoenfsined inintotthael rineseprteica-
and gravity loads at the joint outputs has been calculated for both no load and
load in the hand. i
The main result of this report is the constructlon of a set of greatly sim-
plified state equations which describe total inertia and gravity load variations i
at the output of the six joints with an average error of less than 5%. The sim-
plified state equations also show that most of the time the gravity terms are
more important than the inertia terms in the torque or force equations for joint
numbers 2, 3, 4, and 5. Further, the acceleration-generated reaction torques
or forces, except from extreme arm motion patterns, are shown to have very
low quantitative significance as compared to the straight inertial torques or
forces in the dynamic equations restricted to simultaneous motions at the first
three joints.
The results are summarized in four tables and nine figures. The report
also contains all analytic tools and byproducts needed to arrive at the outlined
conclusions. An important analytical byproduct is the simplification of the
general matrix algorithm tor manipulator dynamics.
[ JPl. Technical Memorandum 33-669 ix
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1974008732-008
JPL Robot Research Project Manipulator _'
z JPL Technical lvlernorandurn 33-669
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1974008732-009
I. INTRODUCTION
b
The purpose ofcontrolistokeep fixedor alterthe dynamical behavior of
a physical system in accordance with man Is wishes formulated in terms of per-
formance requirements and goals. The nature of the control problem com-
prises two distinct parts: (a) quantitative description of the dynamical
: behavior of the physical system {in our case, the manipulator) _o be controlled
and (b) specification of a "scheme" or contro] law for car,-ying out the desired
_) controlled behavior (in our case, to accomp!ieh a vaiiety of manipulative tasks
_f with specified performancel. This report is mainly about the former part of the
_ manipulator control problem:
Modeling and evaluating the dynamical properties and
behavior of the JPL Robot Research Project (RRP)
, manipulator.
The fundamental idea of control is that the inputs should be computed ,
from _he state. Of course, this idea is known as feedback. Thus, the natural
J
1 framework for formulating and solving control problems is the state description
of the physical system. The state incorporates all information necessary to
'I'_":i determine the control action to be taken since, by definition of a dynamical
I
%: system, the future evolution of the system is completely determined by its
present state and the future inputs. The relation between explicit state equa-
7_" tions for manipulator dynamics and servoing the manipulator is briefly treated
_ in Section If.
The actual dynamical model for the six degrees of freedom $PL RRP
manipulator can be obtained from known physical laws (from the laws of the
Newtonian mechanics) and from physical measurements. This task amounts
to the de ,elopment of the equations of motion for the six manipulator joints
in terms of specified (measured) geometric and inertial parameters of the
links. :onventional procedures could then be applied to develop the actual
motion equations. Instead of using conventional procedures. *.he equations of
motion in this report are developed through the application _,f a general
algorithmic description of manipulator dynamics. The algorithm is based
on a specific representation of link coordinate frames in jointed mechanisms
Jl_l, Technical Memorandum 33-669 1
1974008732010
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and the formalism of the Lagrangian _echanics. The features of the general
algorithm together with the definitions of the involved functional symbols and _,
mathematical operations are described in Section III. Section III also provides
a general specification of the six equations of motion for the JPL RRP manipu-
lator as well as a condensed physical explanation of the different terms t
appearing in the equations. Section III concludes with a compact vector/matrix
description of the six motion equations.
The complete dynamical model of the JPL RRP manipulator is described
f
' by a set of six coupled nonlinear differential equations. Each equation contains
a large number of torque or force terms classified into four groups: (a) inertial
torque or force, (b) reaction torques or forces generated by acceleration at
other joints, (c) velocity-generated (centripetal and Coriolis) reaction torques
or forces, and (d) gravity torque or force. With few exceptions, each torque
i or force term depends on the instantaneous configuration (position) of several
links. To gainanalyticinsightintothe dynamical behavior ofthe manipulator
|.
• interms of explicitstateequations while keeping the analysis manageable,
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i well-defined and useful dynamical model restrictions are identified in Sec-
i .... tion IV. It is emphasized, however, that the model restriction,_ are introduced
only for analytic purposes.
InSectionV explicitstateequations are presented forinert_.al,gravity,
"'/"""i and acceleration-generated reactiontorque/force terms for manipulator
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motions rest-ictedtothe firstthree joints. The lastthree (wrist)jointsare
thoughttobe temporally atrestin a known configuration, While inSection V1
complete {unrestricted)explicitstateequations are presented for inertialand
gravitytorques or forces actingatallsixjointaxes, The exact stateequations
developed in SectionsV and VI form one part ofthe important resultsofthis
report.
Partial derivatives of the different link coordinate transformation _
matrices as well as the pseudo inertia matrices (together with numerical {:,_
values of inertial components) utilized in the development of the explicit state :!i-:
equations are compiled in Appendices A and B. Modifications of the explicit • _,=
and exact state equations for inertial and gravity terms when a load is __<_i
emplaced in the hand are treated in Appendix C.
2 JPL, Tochnical Memorandum 33-669
Description:SPACE. ADMINISTRATION. Technical Memorandum 33-669. Robot Arm Dynamics and Control. A. K.Bejczy. (_A_A-C_-I_oC3-5) ._"=_-'_. A_'_ _Y' _.
