A nonlinear extension of model reference adaptive control (MRAC) technique is used to guide a double arm nonlinearizable robot manipulator with flexible links, driven by actuators collocated with joints subject to uncertain payload and inertia. The objective is to track a given simple linear and rigid but compatible dynamical model in real, possible stipulated time and within stipulated degree of accuracy of convergence while avoiding collision of the arms. The objective is attained by a specified signal adaptive feedback controller and by adaptive laws, both given in closed form. A case of 4 DOF manipulator illustrates the technique

Skowronski, J. M.

English

NASA Technical Reports Server

Algorithms for Adaptive Control of Two-Arm 
Flexible Manipulators Under Uncertainty 
rc) ("'" Lp705'i 
University of Queensland 0 9  
J.M. Skowronski 
St. Lucia, Australia 4067 
-- --- ~ 
i 
' c <A 1. Abstract f 5 . /  
-2 non 'near extension of model reference adaptive 
The objective is t o  
control (MRAC) techKque t/ t a  guide a double arm nonlinearirdble robot 
manipulator with flcxible links, driven by actuators collocated with 
joints subject t o  uncertain payload and inertia. 
track a given simple linear and rigid but compatible dynmical modcl 
in rcal, possibly stipulated time and within stipulated degree of 
accuracy of convergence while avoiding collision of the arms. 
objective is attained by a specified signal adaptive feedback controller 
and ty adaptive laws, bcth given in closed form. 
nunipirlator illustrates th-. technique. 
2. Introduction 
The 
A casc oi 4 W F  
Thc MILK technique becomes popular proposition for guidance of recenvrobot manipulators, with demand for 
precision pointing in difficult Conditions, under the action of full scafe dynamic forces, and subject t o  
uncertainty in paramcters. 
non1incari:able structures (geometric nonlinearity of elastic links, large angle articulation, nonlinear coupling 
of DOF's. nonignorable gyro and Coriolis forces. several equilibria). while c1assi;al MRAC is linear and 
applicable t o  rigid bodies only. 
links, see 121. 
require at least two arm systems. Thus the tracking has to,'bc a double MRAC (mutual reference adaptive control) 
which secures tracking thc same model by two arms while qvoiding mutual collision - cf. [SI, [ 4 ] .  If adaptive 
(self-organi-ing) control is intended. the tracking relates not t o  a given path but to  a given dynamic target- 
inodcl with prescribed target-parameters. We take the modcl simple thus rigid and linear, but locally compatible 
with the nonlinear arms regarding equilibria. 
characteristics m d  coupling, elastic links, drivcn by actuators collocated with joints, under uncertain inertia 
parameters and uncertain payload. 
adaptivc feedback controllcr and integrable adaptivc laws in thc s t a t e  space, while avoiding collision between 
arms of a11 the joints (and elastic nodcs) in Cartesian configdration space. 
3 .  Motion Equations 
Such manipulators, particularly these used on spacecraft are highly nonlinear and 
Thus the extension is needed,for handling nonlinearity.see [ l ] ,  and flexible 
On the other hand many robotic objcctivcs, &ain particularly these in difficult space conditions 
Each arm is represented as an open chain with n W F ,  nonlinear 
The tracking is done in real possibly stipulated time by a dcsigncd signal 
I.ayrange motion equations g i v e  the rigid dynamics of the arms in the general format 
AJ(qJ,sJ)qJ + rJ(qJ,sJ,,\j) + fiJ(qJ,AJ,sJ) = BJ(qJ,~j)uJ , j 1 , 2  , 
where q J ( t )  . .lq .- Rn , t : to = 0 , is the configuration vcctor of the joint variables q: ,...,qi of the 
j-th arm varying in thc known bounded work region ' of the configuration space Rn ; i ( t )  is the correspond- 
ing vcctor of joint velocities in the specified bounded subset 
u J ( t )  - U +. Rn 
are the vectors of adjustable system parametcrs in bounded bands of values A , and s ' ( t )  F S c Rk' is an 
uncertainty psr:imctcr within the hnown hand 5 . ?lorcovcr AJ(qJ,sJ) :ire the Lncrt ia  n x n  matrices obtained in 
the known way from the quadratic form of kinetic energy. The vectors I IJ  = (Ilj,  ....!I:)' represent pctential 
forces (gravity, spring) whilc I'J = (7;. ... ,i'J)T represent the internal nonpotential acting forces (Coriolis, 
gyro, ccntrifugal, damping structural or  viscozs, etc.) and BJ 
n m  matrix. The control vectors u J ( t )  arc selected for the objectives of tracking and avoidance by odaptive 
feedback control programs 
Aq x A *  x ,\ . 
q 
while using thc Rit:-Kantorovitch scries :xpansion 
3 
A* 
q 
of the space tangent t o  Rn ; 
are the control vectors in given compact set  of constraints IJ ; h J ( t )  E A c R L  t S 2n , 
is the actuator transmission (gear) nonsingular 
1 2  
uJ ( t )  = pJ(ol(t) ,q2(t) ,:'(t) ,i'(t) , 4  ( t )  , I  ( t ) )  on corresponding products of 
For convcnicncc the superscripts "j" will be dropped until they are needed t o  avoid ambiguity. 
CansiJcring the links c l a s t i c  he intrducc the Jrfonatiori coordinates f o r  the i-th link as shown in Fig. I ,  
https://ntrs.nasa.gov/search.jsp?R=19890017193 2020-03-20T02:06:39+00:00Z
m 
ri(yi , t)  - 1 r i (u i ) ry ( t )  - r i (y i ) r i ( t )  (2) 
v- 1 
and f o r  v (y t )  , w ( y . , t )  analogously, with t h e  exact so lu t ion  expected f o r  m * - . We take m l a r g e  i i' i l  I I 
1 
/ -length- 
Figure 1. Flexib le  l i nk  
enough so tha t  the  Kantorovitch l i nea r i za t ion  is physically j u s t i f i e d .  
stepwise subdividing the  l i nks  between gr id  a s  long a s  the d i t f e rence  of r e s u l t s  fo r  successive a's b e c a e s  smrll. 
ttaving spec i f ied  ( 2 )  we form the  vector o; t )  ? [o l ( t ) , . . . , on ( t ) )T  , where o i ( t )  4 ( r i ( t ) , v i ( t ) ,w i ( t ) IT  and 
following [j] wri te  the  hybrid system as 
The technical way about it i s  to 
where A (o,s) , 
Dc(q,4,0,n) , Pc(q,ri) and t h e  in te rna l  damplng D ( * l , q , r l , r ) )  a s  well as  t h e  hybrid res tor ing  coe f f i c i en t s  
P(q,n) 
tha shape functions,  see IS]. Let t ing  
ro(o,ti) , I1 [o ,s )  a re  the c l a s t i c  correspondents of A ,  r, I1 while Ac(q,Q) , 
Q. r i  
a r e  matrices coupling thc e l a s t i c  and j o i n t  coordinates. These matrices a r e  formed by in tegra ls  over 
A(q.n,s) - 
t o  be t h e  hybrid i n e r t i a  matrix which i s  nonsingular positiv,. dcfinite,we i n e r t i a l l y  decouple ( 5 ) :  
[q,6)'r + D(q,G,r i , ; i , \ ,sJ + P(q ,r t , , \ , s )  = B(q , { . s )u  ( 4 )  
A 
where D = A-'(Dc;l+I',D;+I* ) T  and P 
forces and the meaning of  th.? matrix 8 is obvious. 
T .: 
x ( t )  = ( x l ( t )  ,..., x , ( t ) )  
convenience (4 )  may be then wr i t ten  in the  genera! sta te  form 
A - l ( P c r : + l l . P n + i l  )T are successively vec tors  of  nonpotcntial and po ten t i a l  
rl ri 
The vectors q ,  (1, '1, r i  form the s t a t 5  vector 
[ q ( t ) , n ( t ) , G ( t ) , ; , ( t ) ) T  .\ 8 A A -  x .I- 2 A ': RN , N = Jn , for  each a m .  For 
q r i  q r i  
i = E ( X . U . i , S )  ( 5 )  
with 
contingent form: 
f I ( f l ,  ..., fN) of the  shape specified by ( 4 )  i n  an obvious way. Formally ( 5 )  may be written in t h e  
i * i f ( x , u , \ , s J  1 s 6 S '  i 5 ) '  
which for su i t ab le  f ( . ) ,  p ( - ) ,  , A [ * )  has so lu t ions  x ( t J  = k ( x o , t )  , t : 0 , absolutely continuous c u n e i  
through each xo = x(0) in A . We sha l l  consider the c l a s s  of such s o l u t i m s  K(xo) by exhausting a l l  values 
o f  s(t) in (5) a t  each t . 
4. The Reference Model 
We l e t  the given Cartesian "world" coordinates representation of the rcference sodel i n  general t e rms  
'rn = F [ArnJ :  , b)  
3.31 with 2n WF, [ ( t )  c R , and F ( I m )  s u i t a b l e  matrix, be o f f - l i n e  recalculated to  the  jo in t  coord ina te  
format of the r ig id  l i nea r  system 
;im + Dm(im)yn + P m ( f r n N m  = '1 1-1 
with the  Zn-vectors qm, Grn of joirit coordinates and ve loc i t l c s ,  b ta te  'cm(t)  = (q , ( t ) ,Gn( t ) )T  . '\ - RN , and 
246 
Om, P, s u i t a b l e  matrices, while Am (Aml ,..., Ami) const c A c Re , e = n . Moreover 
Pm(Am)(qe#ne) = 0 ( 8 )  
E.(%.\) = iiT< + P,,,(Am) da (9) 
with (qa,rle) denoting t h  equ i l ib r i a  of (3) on t h e  surface i = 0 fi 0 . The t o t a l  energy of t h e  model 
w i l l  be denoted by Em(t&,fm] i n  t h e  world coordinates and E m ( \ , i )  i n  t h e  j o i n t  coordinates,  obviously q u a l  
t o  one another. Then 
% 
< 
and subs t i t u t ing  (7) ,  
(10) 
i,~%.i,,~ = -vp,,,i(i,,,) 2 .
tim($,4J 5 0 P v% rl 0 
The model is se lec ted  such as t o  allow achieving o f  a s t ipu la t ed  t a rge t  behavior i n  t h e  s t a t e  space. 
a t t en t ion  on something spec i f i c  and ye t  general enough, l e t  it be s t a b i l i t y  o f  the  o r ig in ,  guaranteed by the 
nonaccunnnulation of t he  t o t a l  energy i .e.  non-negative damping 
To focus 
(11) 
while 
VE,,,($,$J > 0 (12) 
in-the-large i .e.  on same CAL = A - AL where AL is the set i n  RN enclosing a l l  t he  equ i l ib r i a .  
5. Object ives 
Fig. 2 .  
N o w  we consider both arms J = 1 . 2  and the  model together. The block scheme of t he  system i s  shown in 
Figure 2 .  Block scheme of  the systcm 
Define two product 2N-vectors XJ( t )  = ( ~ ~ ( t ) , x , ( t ) ) ~  
which vary i n  A*K', generating the  product t r a j ec to r i e s  
?I b 4 A' and two . .  2-vectors ( x J ( t )  = AJ(t)  - i m  , 
(XJ(XJO, t ) ,  , x J ( , J O , t ) ) ,  t 3 0 ,  X J O  = XJ(0) , 
(aJo = ~ ~ ( 0 )  . Then we define thc  "diagonal" s e t s  
MJ = ( (XJ ,xJ)  I S'X! 1 xJ  = X m ,  = 0) 
and given s t ipu la t ed  uJ  > 0 , t h e l r  nelghbourhoods 
1 1  M; = t ( ~ * , i - j  , A:* .  , 1 x I - x  + J ,  , , J i  4' 
Moreover we l e t  1" be a dezired subset of S where we want t h e  tracking 
rn' 
24 7 
, ) = 2  . 
t o  occur, and l e t  t C  be the 
s t i p u l a t e d  time a f t e r  which the  tracking is a t ta ined  with accuracy pJ . 
P i n t  Objoctivw The manipulator ams ( l j  a r e  n r t r u i l y  p-tracking the t a rge t  (7) on A. 
such tha t  f o r  each solution 
i f  there  Is a pair o f  
c o n t r o l l e r s  p 1 ( e )  , J - 1.2 
A 2 x A is pos i t i ve ly  invar ian t :  (XJO,aJO) e A: x A -(X'(t),ca'(t)) c do 2 x A and given tC, f o r  each 
k 3 (.) c K(xj0) t he  product t r a j e c t o r i e s  s a t i s f y  
k'(xJo,t) , t 2 0 of (4) i n  K(xJo) , the set 
The convergence is i l l u s t r a t e d  i n  Fig. 3. 
I 
Figure 3. Convergence of product t r a j ec to r i e s  
Suppose the  transformation f r a a  jo in t  t o  world coordinates (forward kinematics) is given  by 
C i  = ::(qJ,nJ) , c = 1,...,3*2n 
and denote Z ( t )  4 (x1 ( t ) ,x2 ( t ) )  . Then we l e t  the set 
A ( 2  . . A a  j l ~ ~ - C ~ l  L d. vo ,v  = I , .  
be the  c o l l i s i o n  set between arms t o  be avoided. We define C A ?  A d  - A 
AA 4 [ Z  <. Liz 1 d I{!-<:: L 
be t h e  "slow down" safe ty  zone. with i > 0 su i t ab le  constant.  
Second O b j e c t i v e :  
any p a i r  I t J ( * )  e K J ( x J O )  the corresponding product t r a j ec to ry  
The tracking arms (1) avoid co l l i s ion  i f f  there is  A,, such tha t  for  m y  :" ' CA , and 
z(zO,L) C A  , v t -  0 .  ( I S )  
with t h e  pos i t i ve  constants 
The f i r s t  r e l a t ion  obviously r 
6.  Suf f i c i en t  Conditions 
We return now t o  the f i r s t  ob jec t ive  and specify by N[ J ( J A x A ) ]  a neighborhood of t h e  boundary ;(:aAu,l) 
fi: of t h e  region "XA . Then l e t  N L  2 [ ) ( A i x , l )  11 I such t h a t  DJ n M J  = 6 . Further we consider four C -functions V i ( - ) :  5 * R , V J (  1 :  DJ R , j = 1 . 2  
, CMG 2 (;iA;,\) - MJ and introdi-:e open DJ 
qulres forming V i ( . )  from s u i t a b l e  , ( A  x A )  taken . isits l c v c l ,  or m v e r s e l y ,  
In the ldtter Ca>e a" :ho, .I zna l lc r  t h Jn  their des i r ed  wi l l  forming 3A , 3.4 from leve ls  of su i t ab le  V,(*) . 
be the  secuye choice. 
THEOREH i: ObjCCtiVe I i s  a t t a ined  i f ,  given 1, u t h e r e a r e  prog-;ins pJ - and fiinct ions 
V i ( - )  , V G [ . )  such tha t  for  a11 ( X J , t J )  I A Z  x :X , 
9 
248 
J J J  J J  vS(x ,a 1 6 v! . V(X .a I c wC , J 1.2 
for each ut 6 pJ(X7,X2) ; 
v i ( x J ( t ) , a J ( t ) )  < o , v ~ J  c s (17) 
a long  t h e  product t r a j e c t o r i e s  (XJ(X1O,t)aj(atO,t)) , t L 0 , j 1.2 ; 
o < v$J,aJ) 6 vJ* P , v(xj ,aJ )  6 G?, , j = I , Z  ; 
vJ(xJ,aJ)  5 v;-, v(x',aJ) c ~j n MG , j = 1.2 ; 
f o r  each u j  - p (X',X*) t h e r e  is a cons tan t  c 0 such t h a t  P j 
j 
(18) 
J l  1 1 O,,(X (t1.a ( t ) )  5 -cJ , vs 6 s 
along t h e  product t r a j e c t o r i e s  (XJ(XJO, t ) ,aJ (aJO, t ) )  , t 2 0 , J 1,2 . 
Rcaark 1: n i e  Objec t ive  1 holds  a f t e r  a s t i p u l a t e d  tC < i f  Theorem I is s a t i s f i e d  with c rralected by J 
v j *  
c l  G-LL , j = I,Z . 
I C  
THEOREM I :  
such t h a t  for t h e  t racking  p a i r  
Objcc t ivc  2 is a t t a i n e d  i f  Thoorem I holds and given d t h e r e  is a Cl-funct ion  V A ( * ) :  1 5 ~  + R 
p J [ * )  , for  a l l  2 ' C A  , 
( v i )  V A [ Z )  VA(r) , V z  c JA ; 
[ v i i )  f o r  each U J  8 pJ[Z) , 
i A ( Z ( t o . t ) )  : 0 , Z o c  AA , V s J 6  S ( 2 0 )  
along product t r a j e c t o r i e s  t(to,t) , t : o . 
PROOF. Suppose some Z(:",r.) , t 2 0 , Z 0 c  AA crosses  3A a t  t l  0 . Then by ( v i )  , V,(t(tl)) < VA(Zo) 
which c o n t r a d i c t s  ( v i i ! .  
7. C o n t r o l l e r s  and Adaptive Laws 
I.ct us s e t  up 
vA = l i t m ~ x ' J - E m ~ x ~ )  i ( 1 5 )  
whcrc aJ I [ s ign  IJ, ..., s ign  IJ) , j = I , ?  , and E,(xJ) 
N,  i n  CAI. , t h e  c h a r a c t e r  of E (.) s p e c i f i e d  a d d i t i o n a l l y  by ( 1 2 1 ,  s a t i s f i e s  t i ) ,  (iii) and ( i v ) .  
is E m [ * )  with x exchanged for  xJ . Choosing I 
To  .ice t h a t  ( V I )  holds ,  observe t h a t  E m ( x J )  = Em[SJ , *  :?J ) of ( 6 )  and t h a t  increas ing  t h e  d i s t a n c e  
A '  
F. - " '  0 for  a t  l c a s t  one J from i ts  ;A value i n c r e a s e s  t h e  value of V 
1 v 'J1 
where 
24 9 
To Lheck upon condi t ions  ( 1 1 ) .  ( v ) ,  ( v i i J  we d i f f e r e n t i a t e  (21)  - (23) with respect  t o  time 
t m ( x J )  - k,i.xm) aJIJ  , ( X J . i J )  i C*M: , 
) - b,(xJ) + a l l J  , [ X J , l J )  C-M; , 
a J i J  , ( x J , i J )  , M; ; 
i,C t )  = [ ER(x4)  -Em[x2) I * [ E,[X' ) - b R ( x 2 )  I , 
. .  
k , ( x J )  = ; E m [ ~ J ) * f J ~ - J  , u J ,  t J )  = (6u-o-P+pmqJ (q .  ) . 
I 
me b r a c k e t r  of  t h e  functions 8, 0 ,  P dropped f o r  c l a r i t y .  kreovcr C'n' are subse ts  of d defined by U P 
C*$: E,(&') 2 Cmix> 
C-$: Em(xl )  Em(x.) . 
With a s u i t a b l e  c h o i c e  o f  i n i t i a l  s t a t e s  t h e  fol lowing set of condi t ions  imples (i i) ,  (v) and ( v i i ) :  
(c 1 
Observe t h a t  f o r  
f o r  b 0 , 6 # 0 , j - 1.2 . In t h e  above C'h a r e  subse ts  of CA def ined  by: 
C*A: Em(xo) L Em(x') , 
C-A: Em(x4)  Em(x') . 
pJ;J I I x  ) - C aJ # 0 , J I 1 . 2  
m m J '  
1' I 0 t h e r e  is no need f o r  adapta t ion  and t h a t  the system ( 4 )  c r o s s e s  t h e  w r f a c e  4 0 .  
:I = 0 time i n s t a n t e n w s l y  ( v e r t i c a l l y )  so t h e r e  I S  no need f o r  controf i n  v i e r  of  t h e  smoothness of  t r a j e c t o r i e s .  
Condi t ions [ a ) ,  (b) are called cont ro l  condi t ions  he lp ing  t o  des ign  p ( a )  , condi t ion  ( c )  is c a l l c d  a d a p t i v e .  
helping t o  design adapt ive  laws. Let 11s check t h a t  ( a ) ,  (b) .  ( c )  indeed imply ( i i ) ,  (v ) .  ( v i i ) .  Consider f i r s t  
t h e  c a s e  Em(xJ)  # Em(xm) . S u b s t i t u t i n g  (c) i n t o  ( 2 5 )  i n  v i e r  of  ( i i )  we o b t a i n  
Boundcdness of t h e  work space necessitates 
( 1 1 )  i n t o  [ 2 4 )  with( l@.we s a t i s f y  ( v )  i n  s t i p u l a t e d  time tr . Note t h a t  t h i s  holds f o r  any i n i t i a l  s t a t e s .  
= negat ive  te ras  ( x J )  . 
t h e  p w e r :  i ( x ' )  S 0 t h u s  (11). S u b s t i t u t i n g  ( a ) ,  (E) ..and 
ll~e case t m ( x J )  = Em(xm) 
( v i i j .  Again f i r s t  l e t  Em(%') # L m ( x ' )  and observe t h a t  ( b )  s u b s t i t u t e d  t o  ( 2 h )  implies  ( V I I ) .  The c a s e  
iR[x*j i m [ x 2 )  is obviously t r i v i a l .  
is t r l v i a l  a s  then it 3Em(xm) . 0 , i t  * im(xm) - E - - E  . Final ly  we check 
J J  
Ohscrve t h a t ,  wi th  ( 1 1 ) ) .  I C )  is implied by t h e  ful lowing adapt ive  lads 
i: = - s i g n  ~ ~ ( 9 m i 4 ~ i  - 1).
for  L~ + 0 , I 7 I ,  ... .n . P h y s i c ~ l l y  thc  .elutions iJC iJo , t )  rcprcscn? t h e  d e l  energy flux which bccme 
poaitibe o r  ncgat ivc  Jcpending upon where i J o  is loca ted  Lbelow or abobe the  w r f a c c  I J  = 0 )  thus r e g u l a t i n g  
t h e  increment or 
8. Modular Oouble RP-Manipulator 
K O  zero iron Jnywlicre u u t % i J r  t h e  *urfdcr b J  A 0 . 
Our technlquc I Z  i l l u \ t r ~ t c r l  below on t h e  case sturly of t h c  fa tr  Ml' m.inipulJtor with two J ~ J  3houn in  
I1g. 1. 
thre As,  X4 a r c  d u p i n g  coe f f i c i en t s ,  A1,AA2 spring coe f f i c i en t s ,  g-gravity acce lera t ion ,  tlle rminl - . r  
of notations sham i n  Fig. 4. Ne takr the 
possible payload on the  gr ippers  as u n k n m  but within known bounds which makes 
The supe r sc r ip t s  "J", j 1.2 , a r e  ignored f o r  t h e  time being. 
m2 spec i f ied  b:r 
- . s m 2 5 i ,  
1 where E, pos i t i ve  constants.  Allowing s i n  q1 * q, - i;si , cos q1 = 1 - Iqf  , and subdividing tha  q u a t i d n s  
(29) by corresponding i n e r t i a  coe f f i c i en t s  we obtain: 
4, + ri + ni  iui , i - 1.2 
Choose ( X J O ,  lJ0) , C o d ,  , J a 1 . 2  .and Z0 . C*A . Then the cont ro l  conditions ( a ) ,  (b) hold  i f  successively 
and 
25 1 
Thus ne choose u; such that for 4; f 0 , 
and for such u; , re choose uf satisfying 
The procedure for 
symmetry of ams: 
i.J U J  , S! , i ,  
' ui and ui 
j = 1.2 , YE obtain the tracking controllers 
is identical utilizing the second inequalities of (M), (55). Assuming - m i  = nf = mI , m i  m i  = m2 C [!,a] , r4 = ra - r and substituting the exprcs3ions for 
! suitable const:int, '4: - (1 
( suit:iblc constant. Vq: = I) , 
which i m p l y  the control conditions ( a ) ,  (b) for our example. 'The adaptive laus 1 2 4 )  . ire 
1; -(sign , A J ) \  41 - :c  
;: = -[sign t J ) i  * z  I .  3 n3 nl J 
J d " m 2  - lc] 
Zlumerical '.J I , for J = 1 .2  . The first t w o  l : iws vanish identically, since by Jcsign = ' 1 a 1  ' 1 d 
simulation of OUT nodular case. uith the data 
t n 3  = 5, a4 = 2 , is 5hown in I i y .  5, .ind c o n f i n s  the convcrpcnce- voidance required. 
m I = ;Okg, 5 = j O h p ,  ii = JOhh. r = 0.hbm. i m l  = -0, lrnl  - 20. 
2 5 2  
. '  
9. References 
(1) J.M. Raronski, Control Dynamics of Robot Manipulators, Academic Pr-. C 
[Z] J .M.  Skowronski, Model Reference Adaptive Control Under Uncertainty of Nonlinear Flexible Manipulators, 
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Algorithms for adaptive control of two-arm flexible manipulators under uncertainty

The paper uses a nonlinear extension of model reference adaptive control (MRAC) technique to guide a double arm nonlinearizable robot manipulator with flexible links, driven by actuators collocated with joints subject to uncertain payload and inertia. The objective is to track a given simple nonlinear, rigid but compatible dynamical model in real, possibly stipulated time and within stipulated degree of accuracy of convergence, while avoiding collision of the arms. The objective is attained by a specified signal adaptive feedback controller and by adaptive laws. Both are given in closed form. A case of two degree of freedom (DOF) manipulator illustrates the technique

Algorithms for adaptive control of two-arm flexible manipulators under uncertainty

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