A technique is presented for controlling multiple manipulators which are holding a single object and therefore form a closed kinematic chain. The object, which may or may not be in contact with a rigid environment, is assumed to be held rigidly by robot end-effectors. The derivation is based on setting up constraint equations which reduce the 6 x n degrees of freedom of a manipulators each having six joints. Additional constraint equations are considered when one or more of the degrees of freedom of the object is reduced due to external constraints. Utilizing the operational space dynamics equations, a decoupling controller is designed to control both the position and the interaction forces of the object with the environment. Finally, simulation results for the control of a pair of two-link manipulators are presented

Hayati, S. A.

English

NASA Technical Reports Server

Dynamics and Control of Coordinated Multiple Manipulators 
S.A. Hayati 
Jet Propulsion Laboratory 
California Institute of Technology 
._ I r--\ ' J d 
*,-" f 
-& c /  Pasadena, CA 91109 L- 
Isnucr 
3 trrA~lqunlor oomtrolllm# mmltiplr mrmlpmlrtorr 
vhloh rro hOldh# 0 rim810 ob,Got r d  thrroforr form I rlorrd k l ~ o m r t l r  rholm. 
IAr o b j r o t ,  v h l r h  may or moy mot be l a  oomtrot v l t h  0 r l i l d  rmvlrommrmt, 1 8  
rrrmrd t o  bo hold rl#ldly by 8 robot ond-oflootoro. Tho drrlvrtlom Ir boo04 om 
mrmlpmlotorr rroh hrrlm# a 1 1  j o l m t r .  Addltlomrl oomrtrrlmt rqmc t l o ~ r  rro 
oomrlkrod v k m  o w  or mor. or tho do#rrrr or lrrrdom or tho o b j r o t  10 rrdmord 
dmr t o  oxtorma1 romrtrrimtr. U t l l l r l m #  tho oprrotlomr1 rpror dymrmlor 
rqutlomr. r dooompllm# oomtrollrr l r  .'rrI#nod to oomtrol both tho pr l t lom omd 
tho latorootiom foroor o l  tho objoot with tho r m r i r o u o ~ t .  F I B O ~ ~ ~ , - ~ ~ ~ E E P S I  
rlmmlotlom rromltr lor tho ooatrol of a pair o l  t v w l h k  m r m l p 8 l r t o r r ~  
O O t C i m #  mp OOSOtrOlBt O q 8 O t l O O O  .hioh tOd.00 ChO 48s do#rror O l  ~ r O O d O m  01 m 
2J*  P-- 
1. IwRowcrIoI( 
Imtoreotiom , l o r a r r  betvooa tho robotr.  P I o o 1 l y .  r rimmlrtlom r t8dy lor  r p r l r  of tvo-lImk o o o p r r a t i r #  
m O . i p m l r t O Z S " i S  prO#r8trd t o  T 0 l i d . t .  t b  r U l y * l r .  
i, 
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Il#U. 2.1. 
Tho do~lvr t1080 of tho rqutlom of mot108 rrr oomrldorrbly rlmpllflod 
b t  80 br#la 81th tho w r l l  how8 rqutiom of motto8 for r r l 8 # l r  m 8 l t l - l l a k  rrm ts.101 
a) wo 8ro tho C l f t O l h 8  # t a t 0  
dy.OB100 r q u t l o a r  [5,101, rmd b) vr l u g  tho obJoot morr/lurtlr h t o  t h t  of t b  o i x t h  1iak of tho W D &  
N(p) 1 + ycn.i, + P(n) - t (1.1) 
whoro RD drrotrr tho Jolat r ta to  vrrlrblo, M(P)CRUm l r  tho l a o r t l r  mrtrlx w h h h  10 oymmrtrlo rad p o r l t l v o  
drflmito, Y(n,i)cEr. 1s tho oratrlf8grl rad Corlollr f o r o r / t o r q n o  vootor. g(s)cRa 10 tho gravity rootor, rad 
sCIa 10 tho Jolmt f o r o o / t o r q u  vrotor .  Ia ordrr t o  r l r p l l f ~  tho wordlmg o f  tho popr ,  80 ~ 1 1 1  oolmoo that a11 
tho j o i r t r  rro rrrol8to rad bomor m a r  tho t o r s  " t o r q d  whoa rofrrrlmg t o  g r m o r a l l o o d  J o l a t  f o ~ o o / t o r q m o  
vootoro. Tho rbovo oqut loa  rppl lor  omly t o  ldro l lxod f r l o t l o 8 l o o r  r l g l d  rrma. 
Ia tho f l ro t  part of tho dorirrtloa, 80 w i l l  ooarldor ra  objrot-flrrd o o o r d l u t r  frrmo rolrtiro t o  r h l o h  
tho dymrmloo r q m r t l o m o  w i l l  be writtom. Wo 8111 r l r o  oronme t h a t  tho objoot  Lor boo. portitlomod I n t o  a 
p o t t o .  Eroh of t h o r o  p r r t r  w i l l  tho. bo ooaoldorod r e  r part  of tho l a r t  link of o r o h  arm. Dlguto 1.2 
1118otrrtor om0 of thoro porta togothor  with CAI l a s t  I lak  of arm 1. . 
, c - p" 
Sfmoo tho rrmo oro oomridorrd to bo r l g i d l y  r t t r o h r d  t o  tAo lord, thoro 1s r oomrtrat trrarfor8otiom rOlOtiR8 
thr 1 oad E fromoi, p. It tho8 followr thlt  
whoto '1 i t h o  piondo-Lnortir m a t r i x  of t h o  l o o t  1l.k of tho  l t h  arm oxproorod l a  tho  E from.. L o t  ma mow 
r a r u r  thot  L b  Jel i r  tho proudo-Laortla matrix o i  tho Lth prrtltiom of tho lord. TAir mo 
194 a 
i 
a 
(1.4) 
I8 ordrr to alrpllfy tho wtotloa,  l o t  o. drilao 
NOT, r q u t i o 8  (1.12)  om8 bo writto8 I 8  tho lilmplrr form 
(2.11)  
(1 .13)  
(1 .14)  
(1 .1s)  
(2 .16)  
3. DXNMICS OF WLTIPU OooPgAnwa m o ~  IIAWIIPLA'IDIS r m  ~ ~ g p u t  amsn~m 
. I8  t h l r  r o o t 1 0 8  wo -111 dor lro  the eqartlomr of motlom whoa tho o b j r o t  t h a t  l r  bela#  Lo14 by 8 robot  
mamlpalatorr I0 18 oomtrot wlth r f r l o t i o n l e r r  r161d emrlrouemt. h t  C be r coordlnato frrmo OD tho objoot  t o  
roprero8t the 8r8Or~1laOd ooartrrlmt fororr. Thio o o o r d l u t r  frrmr, lm #oaerrl, w i l l  be dlr t lmot  from tho g- 
frame. Wltb r r l l g h t  modlftoatloa. equ t lom (1.10 01. bo trrmrformd BO that tho # r r . ? a l l r e d  forre r o o t o r  1 
s a 8  bo r ep ro  omtod 1. the  C framo. If ET 18 r Lomo#rnoomr traarformrtlom t h a t  ro l r t , r  the 2 frame t o  tbe I 
frame, Chon 6 Or8 bo obtrlmed t o  r r l a t r  tho ~ r u r r l l t o d  foror r o o t o r r  11 thoro two frrmor. Tho Jrroblra mrtrlr 
18 airom from 
(3.2) 
and hone. oqmtloa (2.16) i r  
fho statemeat of v i r t u a l  work 18 
196 
( 3 . 8 )  
(1.9) 
4. aMLIoL OF YllLTIPU Q)OPEMTINO XOBOTS 
In t h l r  rrot lon wo w i l l  lntroduae a ooatrol trohnlqne whloh 10 brrod om tho oprrotlomrl opaoo formnlatio8 
to  oont ro l  tho )oritLon of tho obfeot rad l n t r r ao t lon  fororr  of tho objoot  wlth tho oxtrraol omvlron~rat. 8. 
oppromoh prooomtod Lor. Is booed om d i o t r l b u t l n #  tho  ootratlor foro.. omoms tho  ormo suah t h o t  ooah or-  
oont r lb8 t ra  t o  tho motlon of tho obJrot ond to it. 0-0 i n o r t l s l  fororr I 8  r prodotOrDl8od m r W r .  kt nr b o l l s  
by l n t r o d n o l n ~  a d o o o n p l i n ~  ~ o n r r a l l r r d  foror r8oh thot 
~ h o r r  Icd is tho d o r l r r d  i n t r t r o t l o n  foror rootor brtwroo th t  o b j r c t  rad tho r i t r r a r l  envlroomat. 
W r  mnrt uow show t h a t  t h o  .bore oh0100 for e, dorr r e r m l t  l m  t h r  m u l t l p l r  r o b o t l o  r y r t o r  fo l lowin6 t h o  
r p r o l f l o d  t r r J e o t o r y  r o d  rpplyla8 t h r  d r r l r r d  fororr on t h o  o o n r t t r l n t  rnrfror .  I m  o r d r r  t o  S h P l l f r  t h o  
d r r i v a t i o n  and with0o.t 1080 of ~ e m o r r l l t y ,  lot uo s o o u o  that  tho pOrltlom rad f o r o r  rnbrpacrr arm rrorbrod 1. 
tho o q u r t l o n r  rnoh t h a t  tho f i r s t  6 - ~  e l r ~ r n t r  of t h e  p o r i t i o m  r e c t o r  o o r r r r p o u d  t o  tho p O 8 l t l O O  c o a t r o l l e d  
rabrpaor and thr l a r t  r l rmra t r  bolona to  the force oomtrollod rubopacr. Thlr morn# that. Cor riamplr.  
197 
(4.4) 
( 4 . 0  
whoro Nl lcE6-m 6-.. 
3.11) oan bo writtom a0 
With tho tboT0 dooompooltlo8 of tho foroo-goritlom onbopaorr, tho U 8 t t r i x  (roo r q u t i o .  
Mtor a m  mtnlpnl.riur ;h r q u t l o n r ,  080 obtolnr 
198 
/ 
(4.12) 
(4.13) 
L 
n l o  moono thot In tho p o r l t i o n  rnboporo. tho m u l t i p l o  mooipulotor  ayatom 000 bo c o o t r o l l o d  by tho proper  choico 
of tbo a d  Lpp matricoo. 
To rumorfar, a # o u r o l l x ~ d  foro. roctor oo #lvom by oqrutloo (4.1) woo found oooh t h o t  both tho p o o i t i o o  
of tho  o b j o o t  ood i t o  l o t o r a c t i o n  foroor T i t h  oo o n r l r o ~ o o t  coo bo coa t ro l lod .  Not. t h a t  t h i o  rqoot loo  door 
not opoclfy br tho #ooorol laod foro. r o c t o r  w i l l  bo rool irod by 0 ayrtom of r rdoodaot  a c t o o t o r r  of m u l t l p l o  
c o o p o r o t l n #  m o n i p ~ l o t o r r .  000 ohoico  I o  t o  l o t  ooch  orm r o p p l y  oooo#h t o r q ~ o r  o t  l t o  j o i n t r  t o  p r o v i d e  t h o  
n o o o o a r y  o o t u t l o o  fo r  I t a  010 l i n k s  ploa  0 portloa of tho objoct ' r .  Simllorly,  ono coo dotrrmioo a p r l o r l  how 
m8.L o a o h  arm m u r t  c o n t r t b a t o  t o  t h o  r o o l l s a t l o 8  of t h o  I n t o r o c t i o n  forcoo botwooo t h o  o b j o c t  and t h o  
o s ~ l r o u o o t  (for mor0 doto i lo ,  p looro  ooa h f .  I). n o  i n t o r e a l  forcor o t  tho orl#lo of tho E-from. eon 0100 bo 
e~mtrollod in tho pooltion robrpaco b odd181 foro. voctora  to El (are o q l u t i o o  2.17) ooch t h ~ t  000 hor 
or 
nh moons tho t  tho t o t o 1  o f f o c i  of addin8 tho Pi door oot  a f f o c t  tho motioo of tho  m u l t i p l o  mra ipolo tor  ryr tom 
~ L l l o  rool lxtn#  tho doairrd l o t o r o o l  forcor. 
Environment 
F i e .  S . 1  Schomotic Drrwi.8 of a Poir of n o - L i n k  C o o p o r a t i o ~  Yaaipola tora  
199 
5. WlKS 
Ia o r d e r  to  v i l l d i t e  t h e  propoird theory, a i l m a l i t l o n  i t u d y  w e 1  n n d r r t a k e n .  The m o d e l  c o n i l r t e d  01 
ilmplr planer robotic s y s t e m  composed of i pelr  of two-1l.L revolute manlpulitore. n e  object w i i  i s s u e d  to 
i p o l a t  ~ a r i  attichid t o  tho  apper llmk of reoh of the manlpulator i .  ace each m i a l p n l a t o r  h i s  only t 
d e g r e e i  of freedom. the ob jec t  w e i  r i r u e d  to  be In c o a t i c t  8 1 t h  the 8pp.r l l n k i  only  through lnteractloa foro 
richer Cham throagh forcei  r a d  ' o r q u r .  
h o  c a i e i  were c o a i l d e r r d .  Ia the f l r i t  ceee; I t  w e i  e i i u e d  that t h e r e  wece no eavlroamentil coiitrilnl 
T h l i  c i i e  r e p r e i e a t r  e parr  trraiport problem r h o r e  two  m o a l p a l i t o r r  c o o p e r a t e  In  movlng i n  o b j e c t .  11 t 
iecoad c e ~ e ,  i a  e a r l r o u e a t  we8 e i i u e d  whlch r e i t r l a t e d  t h e  motlon of the object  la the x d i r e c t l o r  Ia bo 
c i i e i .  the d e i l r e d  motlon l a  the 7 d l r e c t l o m  w e i  o b t r l a e d  from e c o o i t i a t  rcce lorat lon trijoctory. The drilr  
motloa 10 t h e  x d l r e c t l o a  w i i  1.1 oqaal  to  the l n l t l a l  x v i l n r  (Le., x - 0)  In the f l r i t  c i i e .  and the d e i l r  
l n t e r e c t l o n  force between the  object rad the ravlroamrmt w e i  1.1 equal t o  xero la  the second coco. 
The ilmulitlon of c l o s e d  klnrmitlc cha la i  c i a  elther be performed by compntlng t h e  l n t e r e c t l o o  forcei b i i  
on tho dynimlc i  eqoatloui  ( inch a i  those developed la t h l r  p i p e r )  or by riiumlng the exlitince of s t i f f  i p r l n  
(or rpr lng-dashpot)  i t  the contact point s .  Slncr la a c t u a l  rxperlmenti, one n i e i  forcr/torqne i e n i o r r  t o  obtr 
t h e r e  l n t e r i c t l o n  f o r c e r  ( t o r q u e r ) ,  t h e  i e c o n d  a p p r o a c h  I i  n i e d  l a  t h i s  s t u d y .  F l g u r e  S.2 l l l u i t r i t r r  t 
m o d e l l a g  o f  t h e  c o n t a c t  mechanl rm b e t w e e n  t h o  npper l i n k s  a n d  the  o b j e c t  e n d  between  t h e  o b j e c t  r a d  t 
envlronment. Plgure  S.3 shows the  overal l  i lmnl i t lon  block dlagrem. 
F i g u r e  J . 2  Deta i led  modeling of t h e  connect lonr  betweon ( a )  t h e  arms 
and the  o b j e c t .  and ( b )  the object  and t h e  e x t e t n i l  c o n s t r a i n t  
L I  
Figure  5 . 3  S i m p l i f i e d  block d i i g r r m  of t h e  r i m n l i t i o n  s tudy 
Although Chi8 p iper  doer not a t tempt  t o  a d d r e i s  the d i g i t a l  c o n t r o l  a s p e c t s  of the  problem, the r i m o l i t i o n  i t a  
w a r  made more r e i l i r t l c  by r e p i r a t i n g  the cont lnnonr  i n d  d i r c r e t c  p a r t s  1 s  8hOWn in Figure 5.3. 
200 
Plgnror 5 . 4  throagh 5 . 6  r h w  tho  rerponrer of tho ryrtam f o r  the f o l l w l n g  a r t  of p r r r m o t e r r  rad  c o n t r o l  g a l i r .  
l l n k  longthr  
l i n k  mrrror 
o b f a c t  maar 
Imltlrl l o l o t  a n g l e r :  
Porltloa loop  grlnr  
Pore. loop grins  
Spr ing  c o n s t a n t s  
Damping eonrtantr 
Sampl l n g  por lod  
- .04 [ m l  , o p u l  f o r  a l l  lrngthr 
- 4.0 [Kgl , f o r  lwrr Ilnkr - 2.0 [Kgl , f o r  wppor l l n k r  
- 2.0 [Kgl 
e l l  - 30' , e12 - 120° 
e21 - iso0 , 022 - -12uO 
kpp 4900.0 sd 98.0 
k i p  1.0 k f d  0 
500.000 [ N / m l  f o r  a11 of tha s p r i n g s  
140.00 [N/(m/roc)l 
1 =sac 
. 
P l g a r c  5.4 rhowr the t racking  c a p a b i l i t y  of  thr  objec t  moved by t ho  arms. Since tho  d l f f r r r n c e  b e t w e e n  tho 
p o r l t i o n  of  thr o b f e c t  and t h e  t l p  o f  o a c h  n p p e r  l l n k  1 s  I n r l g n i f l c a n t ,  t h o  p l o t s  rho+ t h e  d r s i r e d  VI. r c t n r l  
p o r l t i o n  of the npper l inks .  Fignre 5.5 shows the t r i ck ing  of tho o b f e c t  i n  the second care (motion c o n r t r a l n r d  
l a  t h e  x d l r e c t l o a ) .  P i g n r a  5.6 shows tho interaction f o r c e r  b e t w e e n  t h e  o b j o c t  a n d  the r n r i r o n m e n t .  The 
s i m n l r t l o a  r tndy i n d l c r t e a  t h a t  t h e  c o n t r o l  r l g o r i t h m  drvolopod i n  t h l r  paper  y i e l d s  o x c e l l e n t  r o r n l t r .  I t  mnrt 
bo anders tood  t h a t  s e v e r a l  Important  p r a c t l c r l  problems such a r  f r i c t i o n ,  f l e x i b l l l t y  of  the l i n k s  and  jo in t i .  
l i n k  p a r a m e t e r  e r r o r s .  e t c . .  were  not l n c l n d o d  I n  t h i s  r l m a l r t l o n  r t n d y .  Forther  r o r e r r c h  r a d  r t n d y  1r 
nocerrrry to  lnc lnde  snch e f f e c t s .  
PLOT of Load Position vs. TIME 
0.005 
0.000 
.0.005 
0.700 
0.550 
0.400 
ii 
. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 
TIME(SEC.) 
Plgnre 5 . 4  Pnrr Poritioa Coatrol  w l t h  Cooporat lag A r m  
20 1 
E z 
X 
a w 
I- 
W 
I 
t 
z 
W z 
X 
U 
0.005 
O.Oo0 
-0.005 
0.700 
0.550 
0.400 
PLOT of Load Position M. TIME 
L I 
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 
TIME(SEC.) 
Figure S.S Poai t ton  Tracki-; i n  Por i t ion /Porce  Control with Cooporatinn A n a  
LOAD-ENVIRONMENT INTERACTION FORSES vS. TIME 
0.000 
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 
TIME(SEC.) 
F i i u r r  5.6 Forcr Trackin6 i n  Poai t ion/Foree Control  w i t h  Cooperating Arms 
6. WNCLUSIONS 
This paper pr rsen ted  s theory f o r  the poaition and f o r c e  c o n t r o l  of mul t ip le  manipulators  holding an object  
w h i c h  i s  in c o n t a c t  w i t h  an r n v i r o n r e n t .  The derivation 11 f o r  II a a n i p n l a t o r a  e a c h  having a i x  d a g r r r r  o f  
f reedom. Tho c o n t r o l  i a  b a a r d  on t h e  C a r t e a i a n  foranlation of t h e  a rm dynsmicr  a n d  e x t e n d s  t h e  s ingle-arm 
b b t i d  p o r i t i o n / f o r c e  cont ro l  concept to  tho case  of a n l t i p l e  a r i a .  Simple but  r r a l i r t i c  aimnlst ion studies  
e m f i r m e d  t h a t  the developed cont ro l  concept r e s u l t s  in e x c e l l r n t  porition t rack ing  and force  control .  
20 2 
. .  
7. - 
Tho roooaroh doooribod 18 t h i r  doomreot wao porformod o t  tho J o t  ?ropmlaloo Laborotory, California 
Imotltmto of Toohsology. mdor oomtroot wlth tho ItatLou1 Aoroumtloo om4 S p o o  A l . l m l r t r a t l o r  
1. - 
(11 I o h l L .  T. mIoroo Coatrol l m  Coordlut lom of h o  4rm: h w o o d l o ~ o  of tho 9th Imcorucloul Cesforo~oo oo 
Artlflolal Imtolll~oooo. pp. 717-712, Am#mOt 1977. 
l l l  P m j l l ,  S. os4 Kuou, S.. "Coordlutod Compmtor Cemtrol of a h l r  of Yo01)mlatoro.~ k u t h  World Cos~rooo 
om tho n o o w  of k o h l u o  omd I.obamlro. O . l r o r a l t y  of Ik+ooatIo m p a  -0. k p t r k r  8-12, 1979. 
I31 Zhom8, T.P. mod L.L. J.T.1. gCoaotralud Rolotlmo Dotwoom h o  Coordlmotod Idmrtrlol Roboto," ? r w o o d l a ~ a  
of Ihehiao Iotolli~oaoo Coaforoaoo, Coobootor. Y l o ~ i ~ o o .  Apr11 13-24, 198s. 
ZL008. T.F. 0.1 Lmh. J.T.S.. "Jolmt Torqmor for Cootrol of Two Coordlmotod l o r l a 8  Cobota." ? r O O O O d 1 8 ( 0  of 
tho 1986 I~torutlomol Cooforomor om tobotloo 004 Amtowtlon, So* Prooolaoo. Collfor~lo, April 7-10. 1984. 
Khotlb. 0.. "Tho Oprrotlowal Sproo Pormalrtlom l m  tobot  Yoalpmlator Coatrol," 11th ISIR, Tokyo. Jopam, 
Soptombor 11-13. 1985. 
I41 
[SI 
[61 WcClomrooh, N.B.. " D y n a r l c o  of 0 Cloood Cholo Yaolpmlator." Proooodla8r of tha 1913  Amorloom Coatrol 
Coofrroooo, Sorttlo, Warhlo8too. 
I71 Eomaml, E. "Nodallag ood Cootrol of Coortrrlood D p a m l o r  Syotamo -111 AppllOotl08 to 8lp.d Looorot loa  l a  
tho Frootrl Plaoo," IEgg Traoorctlooo 00 Amtoutlo Cootrot. Vol. AC-24, No. 4, Am8sot in9. 
[ e l  Eayatl. 9.. "Eybrld Pooltioo/Porco Control of Yolti-Arm COOpOfatlD8 Coboto. Prooordla8a of tho 1CXX 
Cooforonoo 00 Pobotlca ood Aotoutloo, So0 Prooolooo, Calllorola, April 7-10, 1916. 
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20 3 


Dynamics and control of coordinated multiple manipulators

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