The problem of Cartesian trajectory control of a closed-kinematic chain mechanism robot manipulator, recently built at CAIR to study the assembly of NASA hardware for the future Space Station, is considered. The study is performed by both computer simulation and experimentation for tracking of three different paths: a straight line, a sinusoid, and a circle. Linearization and pole placement methods are employed to design controller gains. Results show that the controllers are robust and there are good agreements between simulation and experimentation. The results also show excellent tracking quality and small overshoots

Nguyen, Charles C.

Pooran, Farhad J.

Premack, Timothy

English

NASA Technical Reports Server

ORIGINAL PAGE IS 
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TRAJECTORY CONTROL OF ROBOT WANIPUWTORS 
WITB CU)SBD-MWQUTIC CHAIN M8CHJUf1811 
Charlos C. Itguyen and ?arhad J. Pooran 
Contor f o r  Artificial Intolligonco and RobotSc0 (CAI11 
Tho Catholic Un%vorsity of A~orAca 
Washington D.C. 
and 
Timothy Premack 
NASA/Goddard Space ?light Center (GSFC) 
Greenbelt, Maryland 
IlbsRACT: nis paper considers tbe problea of Cartesian 
mtPt 
I D i P L 1 a C C Y I ~ t l y b u i l t a t C U R t o s ~ ~ ~ y a f l l R S A  
budware far the future space statim. lbe stu& is perf- & 
&th crqputer s iulat im and experimentation for tracking of 
Mm differmt paths: J straight line, a s i n m i d  rod a circle. 
tiasuLzatian mi pole plamrett wecbods up to &sign 
aatmller painr. Rerults rrhow tbat tbe antrallers u e  d u s t  
and there are good agree0ents between simulation and 
e*periplpntatiaa. Yby ahw d*w @en& t r a w  qudity ud 
sail ove&lLvts. 
trvktcxy c r n M  of a Japed-kimmtic d u i n  Pmanrs * 
1. lnlmwrm 
The amtrol poblem of lpbot mnipulatar an be mid t red  
to mnrist of tm ahrent ra\bprablents: trajectav plamiw and 
trajectory amtrol. Based m the coordinate system wed i n  
pluming and car tml l iq  the robot hand trajectory, control 
schemes can be carvenicntly divided into two classes: joint 
rpace and carteaiao space ontrol d a r e s .  In joint spce ccmtrol 
mtimds, tbe urn r tuat ing signal to the joint actuators is 
cqruted based cm the crra be- the desired joint pxition 
md tbe actual joint poaition of tbe maniplator hand. (h the 
other W, the cartesian spaceoriented mthcds me the erxuc 
between the desired and actual Cartesian position of the 
d p l a t o c  to -te tbe urn actuating signal to the joint 
actuators 131 while enjoying the simplicity in trajectory 
planning, the joint spaceoriented method auffers f r a  the 
d i f f icu l ty  in determining link locations of tbe robot hand 
during mtion, a task requiring to ensure obstacle avoidance 
a l q  the tmjecta-y 131. mtkucre, joint cmniinates are not 
ntitable as a wakimg omrdinate systsp bemuse the joint axes of 
Dst manipulators are mt athoocoal and they & not warate  
positicm fran orientatim [lo]. The Cartesian space-oriented 
rbose path plannisq requires intedve cmputaticas f a  
tmsfawitioar between cartesian and joint asxdinates i n  real 
tiae. has tbe advan- of assuring a certain degree of acaraq 
8law tbe desired path ami behw llcce adaptable to the wer. 
l bue  b s  been rnrerolrs interest i n  developing Cartesian 
rpoe-aieated amtrdl scfrepes. amsidering that paths are made 
up by straight line sepmts, aDanected by timoth tmsiticm, 
Paul (101 and Taylor [15] proposed schemes that control the 
d e r a t i m  at the transitim in ader to &eve the desired 
Cartesian path. Cartesian control of joint compliance 
mnipllatas was rtudied in 141. The r d  in [61 and [17l 
sixam that artesian d e r a t i o n  and farce can be amtrollel 
suuxssfully. Recently a closed-kinematic chain * robot 
wan built at NASA (Galdiud Space Flight Center) to study the 
mbotic d l y  of N&sA hardware fa the future apace statim. 
W pper pwents part of the cmgoh robotic research at CNli 
fa the antrol ad the NASA mbot. In puticulu, w aDasider the 
cartesian trajectay antrol of a Wl rule ,  mnipllator with 2 
&grim of fmdm, rcsablinu part of the m e t e  )RsA robot. 
Qntl.Qllem M deaigned tiuougb the lineariza tica of the robot 
dynmics and pale alvpapnt mth0Q. l h e  tracldng perf- of 
tbe cartroller will tba be rtudied by both simulation urd 
experhntatioa for three different trajectories: a rtraight 
line, a s inmid  rrd a circle. This paper is structured as 
fallars. First *c present the bardwans of the mnipla ta  uder 
s M y  topether with thc dctrelqment of ita kinematic md w c  
d imwed  in aajmction with the amtroller desim thrarElh the 
pale platxmmt atbod. Finally no rcport the results of cequter 
rinulation and experimental study and Bake comparative 
dultiaa. 
eqUatiCXB. lhen tbe lh&Zat ion  .bout m a9errti4 @It iS 
2. mIsmmLmaILDB, 
! 
The d n  parts of the rdwt manipulator under study are 
pnsented ip w. 1. It L a clami-kirmatic chain medmim 
manipulator w i t h  two wees of frdcm. lin~. of t k  
dpu la toc  are of tm ball- -tors, driven by 
tuodcmtasardbubelar af i roedplatfambymmof pin 
jaints. Two llneat voltage diffaeotial trarrrfarPrs (Lvar) are 
mmted rlm the links tom!asure tbeir leJqbts. The other ads 
of the links u e  j o h d  tcvether by a pin joint on which a 
griwt?r is uamted. gripper mtim is eqmsbed with KeSpEt 
to a refereoa x-q OartcJian Eoadinate syntea attached to the 
fixed platform as seen. Tbe Cartesian variables x and y are 
related to the joint variables 1 p . I ~ .  the m le!mJhts by 
[91: 
(1)  x=(l12-12 2 2  +d ) / ( a d )  
(NASA-TM-89305) TEAJECTORY CONTROL UE' h O B U T  N88-13605 
M A N X  PULATORS W ITH CLOS ED-K IN EM ATXC CII A X N  
6ECHANISM (NASA) 5 p A v a i l :  NTIS Ht'  
A02/MF A01 CSCL 131 Unclas  
G3/37 0 114226  
https://ntrs.nasa.gov/search.jsp?R=19880004223 2020-03-20T09:22:36+00:00Z
t 
-Dl is t h e m  d t b e m p a f t o f  the l h k ,  D h tbe 
total .ass of the link, md 1,repsentlt tbe fimd lcnptb of tbe 
actuatas. Both actuatm are assuaed to be identical. In tbe 
derivaticm of the equations of atim the gripper is not 
isclded. 
The Jaoobian relaw joint variable velocities ilad i to 
cartesian velocity variables i and i t given by 
J= [11d2+lll$-li) 1 (qd) (12d2+121$-1$) / (qd) (1 1 
Ihe trajectory oartral sdrer  employed i n  t h i s  paper is 
presented in rig. 2 rrbere cartesim positim fsedbact is abtainsd 
through the forward -tic transform [MS. (1) and (211. 
Actual velocity is abt.ined by differentiating tbe actual 
positim, which is Wemmted by the data aquisiticm software. 
Ih emxs of cartesian positim ard velocity are anverted into 
corres- joint variable errors by means of the inverse 
Jaoobian matrix. Ih joint variable e~mt8 w i l l  then serve a8 
ipplts to tbe H) antmllers that i n  turn p d u c e  required forces 
for the actuators to track the rabot gripper a l a  a desired 
trajectay with a desked velocity profile. 
I ll/d -12/d 
3. lnKmmmnm-IEsIQl 
Since tbe dyndcs of the manipulator is highly d i n e a r ,  
lirrearizatim .bout a selected opera* point should be q l q d  
in arder to &sign effective PD ontrollers. Acoording to the 
w a k  in (121. the lharizaticm muists of expndhg &s. (4) 
and (5) .bout a selected operatbg point using Taylor series 
expansicm, neplecting higher & terms and transfoming tbe 
resulting -iOa into a linear state equatico given by: 
2 = A z  + BF (12) 
w = cz (13) 
where z=(ll l2 i, i2iT 
and W=(ll 1,) T 
( 1 4 )  
(15 )  
%e devdammt d A, 8, and C can be fd in 1121. R a  
rig. 2 we cm write 
8 , I F-P ( V I - W )  +D (v2-W) ( 1 6 )  
(19) 
F r a  (18) it  M obvious tht the matriccs .nd D .bould be 
selectmi r u ~ h  that tbe eigenvalues ot (1-J 6 -IK) are stable, 
which can be reallzed by several available wthods of pole 
plremaat md softuarea. h r t i d a r l y  tbe poodums ppgosed in 
[Ulis-. & n ? w e ~ t h a t t b e r n t r i x ~ I + E l D C ) i s  -. 
Inardertoexamtheperfomanceofthecartaian antrol 
scbfme illustrated in fig. 2, we crnducted botb simlatian and 
-tal 8tUQ fa trackkg of three differeat trajectories: 
a straight line, a dwJaid a d  a circle. In the d a t i c m  
rttdy, tbe entire antrol  scheme including the manipulator 
Qaanics as sirulatad QL an US~JYJ the Systea skilaticm 
(sysu. Firat m antrolla gaim *en selected u9h 
tbe procedures discussed i n  Sect. 2 to  ensure the system 
stability. mar tk aMLined gaim yefc &ijurte!d fa eadl of the 
.booe three tracking CUKS Upti1 8 satisfactory tracking w a s  
achieved. According to  the riwlation results, the gain 
djustmmtsrerewithb 1Bof t h e a - i g h l  gaim dhined fm 
the linearizatioa and pole placement methods. I n  the 
-tal study, the robot nnipi lata  w interfaced with 
tb II# -ter t k q h  a data .crluLiticn eystm umsistb of 
mIBnboard ,andapterux lar fhnrcca l l e !d~Pl .  PD 
antmllem, hem JIoobian ud thc faward -tic transfm 
lm-8 irplaented using Labtech. ?be gain¶ &ta i l d  thruugh the 
riulatico stuiy - agplied to the real-th antrol of the 
mbot mniplata.  I n  the fo- rre will pesent tbe above 
stuQ &ta ud sake ccoparatlve evaluatiaa: 
.66 17.97 0 
3 
DR~WVAL PAGE LS 
.m W R  QUALITY 
lhckina a strridht be: 
In this sMy tbt mbot is to track a desired straight UIlC 
described by the eqtntica y(t) = *(t) - 42 cm with a &shed 
velocity of 1.7als.  Figure 3 presents tbe sbulatioo urd . 
eqxrimmtal results fu this case. ?be sindated mtiro follow 
the target pnth d d y  with a maximm &viatica of &. 
desired path with an excellent tracking quality n t h  some 
disturbamxs at thc starting p i t h .  After rea- the steaQ 
state, the robot t r r t r  the desired path very closely with a 
maxima deviaticm of h. Ibe simlatian and actual ve ld t ics  in 
x and y directions u e  presented i n  Figures 4 d 5,  
nspectively. As seen fm these figures, both sirulaticn md 
rtual velccities 1.0 behiad the desired cne. ?be siulat im cmd 
rtual veldt ies  8- mrstantly at rbart 76% ud 31% of the 
desired ale, respectirely. 
thprimmtal results rb3w that the robot actually follow the 
Ttackim a sinusoid: 
In tbi. papcl w! h rmriQrsd the lrcblcm of urtcrirn 
trcijictay antral of a ched-kinmmtic chain .ectwie rc&t 
rmipllrtcc with two d fmukm. Liwu-izaticn md pole 
p l m t  mtbodr were e l o y c d  to design the PD controller 
wains. mc petiaraaoe of the oootml .dreoe Ira¶ QBDiDed hboth 
riulatioo and aperinentatiar. Study results showed good 
m t s  hem a i m l a t h  ad aprirentatica. 'hey dm 
rhared that rppropriate djwbaenta of a m t d l e r  gains desipred 
fm a linevired m&l of a robot nnipdator &les QT to 
design effective antrollers for this wnipulator. Since the 
LiDearitedpdcl is a l l y  valid in the D e i g ! & a k d  of a mekta.l 
werat- point, the tracking an be 4ruved i f  there exists I 
aiE3m to *to the antrollu g 8 . h  by rpQting the 1- 
Ddel as tbe manipilata mves a l ~  tbe path. current utive 
r aeu fha t  CAJR L f d  m the analpis md hlmtatb of 
m daptive a m t m l  lQep f a  the rbope rcbt d p r l a t c c .  
6. 
?be c w a t i t m  md experimental results for tracking a 'Ibe ruthrs of this paper risb to their thanb to 
(Goddatd Spsce Wt center) fcr * the research of .iaaoidal trajecttxy desQibed by the eqwtica y = 2 . 5 s i n ( ~ )  
-82.5 cm with a Qsired custant velocity of 1.7 4 s  are &cued 
in Figure 6. ?be skil.tioa a t i m  follow the desired trajectory 
very d d y  with a axstant ckiatica of h. Achral mticm 
d r n  gad tnckirg qualitywith llcoe dirt- durjlw the 
transient state. 
this papr rrder the reseaKh grant # NU3 5-780. 
7. - 
Trrkirp a cirdar  mth: 
In this case we r t d y  the performance of the robot in 
trv 8 T a r  trajectory described by the equ8tion (x- 
U.5) +(y+33) 14 with a dashed coacrtant relocity Ot 1.7ds.  
sirrlrticn mtim qreu  with the desired trajeaarU very cla#ly 
w i t h  insignifimt shifting. Dcpenmm . ta l resul tsalbl tbr t  
the rtualmticp of tbe mbDt resembles a cirnrlar path that is 
lhifted to tbe left -tally1 of the desksd trajeaav rrd 
is aapressed vertial ly.  %e baizcntal shifting md vertical 
aqressim are &out h md ha, respfftively. 
hraluatias and -icas: 
Ihe existence d the tracking errors in  the above study 
cases can be clarifiaiby tbe follaring -: 
(a) IbeeplayedmftwareauithemPpJterarewt capable 
of perforning fast  m t a t i m s  an3 high sawling rates. Because 
the data acquisiticm system has been using a sampling rate of 
saples/sec, the fee&ack infomaticn pwirkd by tbe position 
sensors (LVDT) cannot be updated quickly enough for the 
antroller to act aa the actuators. lhus the actual psitian of 
the! robot CIIYDt be pdated t M y ,  Cawirq mtim OVerdmta .  
7b? t r r c u  quality can be improved tremendously i f  a data 
,l. Bra&, Nichael, 'hajectory Planning", in Robot Hotion: 
Pluming and Control, Brady, N. et a1, &is, WIT Press, 
Cdxidgensswckaetts, 1983. 
2. Qrio, J., Intraductica to khtia, AMLar m y ,  
1986. 
3. m, K. S. et rl ,  Robotics: QnM, sensing, Visim, md 
IntelUgme, XcGraw-Wl, 1987. 
4. Bill, S. P., a d  Vaccaro, R. J., * Cartesian Cmtrol of 
Rabotic I(lmiprlatas with - a n t  CQpLigYY," Roc. rn 
Int. coafereoce m Rrrbatics cmd Autamticn, 1985, po.ll4-UD. 
5. Xhatib, 0. "A bifid Apprta& f a  lbtim and mrce Catrol 
of Robot nanipllataa: %e O p e r a t i d  Space Fomlatim," 
&mud of Robotics and Autmtim, Vol. RA-3, February 
1987, po. 43-53. 
6. M, J. Y. S. et rl, "Resolved-Acceleration Cartrol of 
3 I k d M n i d  Ihnilallataa," rm: wars. whatic Qntml, crol.kc- 
15, no. 3, rp. 468474. 
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d ccllplter Interface of a nwdegmmf-fn!€!&m Ibbot", MSA 
~epat, &ant # m 5-780. a 1987. 
! 
8 .  Ilgrorcn, C. C., b a n ,  t. J. .nd Premack, t., "Modified 
Bybrid Coatrol of Robot thipulators  for Bigb Precision 
k&ly @eratias,' Xsm Pternat imrl  Conference Ccmputer 
Agplicaticm in Dcdga, Siul. t im and Analysis, Barolulu, 
RnrFi, Icb. l987. 
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Robot I(sniprlata Qgliance," in Recart Ttadr in Wia: 
Modeling, Ccmtrol m-3 Education, Chapter 2, edited by 1. 
JM, J. Y. s. ~ u b  ard n. shahinpoa , mtb milsnd, is. 
10. Paul, Rictrard, P. C., 'Ibnipllatur Cartesian Path Ccntml", 
I E  Trltuuactiam ao Systems, nan and Cybernetics, SK-9 
(19791, pp. 702-7ll. 
11. Satidis, G. N., Mvances in Autaraaticn ard Rabotiar, yol. I, 
JAI Press Inc., CamecLicut, 1985. 
12. scrrji, 8. , et rl, 'linear Ihltivariable Ccntrol of 
Lirlka kQtr," rlaupal of Robotic system, 3(4, 1986, pp. 
349-365. 
13. Sraji, 8. , P l m t  in Utivarieble mtcw Ikirr(l 
Ropottimal-Derivative Output Feedback," Int. J. Control, 
w. 31, m. 1,1980. W. 19s-m. 
14. ShFn, Uang G., "Robust Trajectory P l a n n b g  for Robotic 
Wanipulatora Under Payload Uncertainties," IEEE tran. 
AutavaticCcatrol, pol. AC-32, No. 12, 1987, pp. 1044-1054. 
15. Tayla, Russel 8.. "Planning and Executim of Straight  liac 
nanipulator Trajectories," i n  Robot Hotion, Planning and 
Ccntrol, Rady et rl,  &, KIT Ress, 1983. 
16. yhitney, D. E., "eolved Fbticm Rate Ccntrol of )hnipllatas 
and lhman Prastbeses," 
1%9, pp. 512-574. 
17. a, C. E., md hd, R. P. "Resolved Jbticm F a m  Ccmtrol d 
Rans. Automatic Control AC-14, 
Robot n?4rlirntaS," IEEE warn. systems, nan, cybern., vol.9c- 
12, w. 3, pp. 266-275. 
F i g .  1 :  The c l o s e d - k l n e m a t t c  c h a i n  
mechanlsm r o b o t  manlpulator 
ORJGINAL PAGE E3 
W R  QUALITY: 
I 1  
F i g .  2 :  C a r t e s l a n  t r a j e c t o r y  c o n t r o l  
o f  the robot  manipulator 
.... ') (i 
I 
h 
\ 
F i q .  3 :  Case s tudy o f  the s t r a i g h t  l i n e  
t r a j e c t o r y  
:l. . 5 
Time ( s e c . )  
F i g .  4 :  Horlzontal  v e l o c i t y  for the 
s t r a l g h t  l i n e  c a s e  
S 
h 
Y) 
\ 
E 
U 
Y 
Time ( s e c . )  
F i g .  5 :  V e r t i c a l  v e l o c l t y  f o r  t h e  
s t r a i g h t  l i n e  c a s e  
. X ( c m )  
r)RfGINAL PAGE I9 
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F i g .  7 :  Case s t u d y  o f  t h e  c i r c u l a r  
t r a j e c t o r y  
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F i g .  6 :  C a s e  S t u d y  o f  t h e  s i n u s o i d a l  
t r a j e c t o r y  


Trajectory control of robot manipulators with closed-kinematic chain mechanism

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