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6.1991-2789

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ADAPTIVE ITERATIVE LEARNING CONTROL OB ROBOTIC MANIPULATORS IN TASK SPACE
B Porter and S S
ent of Aeronautical and
University of Salford
Salford M5 4WT
England
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2789
Abstract
two-link robotic manipulator operating in a
gravity-free environment.
It is shown that digital iterative learning
controllers can be readily rendered adaptive by
identifying in real time the step-response
matrices of plants.
In particular, by
identifying the step-response matrices of
robotic manipulators along their trajectories in
task space, it is shown that such manipulators
are amenable to adaptive iterative learning
control. These general results are illustrated
by the presentation of numerical results for the
adaptive iterative learning control of a
two-link robotic manipulator operating in a
gravity-free environment.
1.
Analysis
2.
The dynamics of n-link non-redundant robotic
manipulators are governed on the continuous-time
set by non-linear differential equations of the
form
M(6)8 + N(6,h)
B
=
and
n
Introduction
rr(6)
=
where 8 a Rn is the vector of joint
M(6)
s the
6 Rnxn is the inertia matrix, N(6,e
vector of Coriolis and gravitational torques,
a Rn is the vector of joint torques, n ( 6 ) a Rn
is
the
vector
of
forward
kinematic
relationships, and n e Rn is the vector of
It is well known
end-effector co-ordinates.
that
the
linearised
dynamics
of
such
manipulators in the neighbourhood of an
operating point P = ( F ) , S , n ) are governed on the
cor*tinuous-time set by linear differential
equations of the form
Robotic systems, in space and other domains
of application, are often required repeatedly to
track pre-specified trajectories rapidly and
accurately in the manner of industrial robots.
I n recent years much attention has therefore
to the design of iterative
been devoted(1)(i)(3)
learning controllers that progressively improve
their performance in tracking such repeated
trajectories. Such controllers embody iterative
learning algorithms in which the (k+l)th input
to the plant consists of the kth input together
with an increment formed from the difference
between the pre-specified desired trajectory and
the kth output from the plant.
In this way,
"better
iterative
learning
controllers
themselvesttC2) in ways "which make mechanical
~
L , e + L i e + L,e
=
E
(3)
and
w =
manipulators
learn
without
help
of
human
operators a prescribed form of motion through
or
automat i ca 1 1y
repeated
self - training
opera ti o n s 1 1 ( 4 )
Je
?
(4)
t w, .
ia Rnxn
where 6 = 6 + 8 , B = 2 + E , n =
is the non-singular Jacobian matrix evaluated at.
the operating point, and the constant matrices
L , a R1lxn, L , a Rnxn, and L , a Rnxn depend upon
the operating point.
However, the existing theory of iterative
requires that the plants
learning control(')
under
control
have
known
time-invariant
state-space models. Furthermore, this existing
theory yields analogue controllers which are
then digitalised for purposes of implementation.
In order to remove these limitations, it was
shown by Porter and MoI~amed(~) that digital
iterative learning controllers for possibly
irregular plants can be directly designed using
only input/output models of such plants in the
Therefore,
form of step-response matrices.
since the step-response matrices of plants can
be readily identified in real time, it is shown
in this paper that such digital iterative
learning controllers can be readily rendered
adaptive.
In particular, by identifying the
step-response matrices of robotic manipulators
along their trajectories in task space, it is
shown that such manipulators are amenable to
adaptive iterative learning control.
These
general results are illustrated in this paper by
the presentation of numerical results for the
adaptive iterative
learning control of a
Copyright 0 1991 American Institute of Aeronautics and
Astronautics, Inc. All rights reserved.
a1 Engineering
Now the end--effector force F E Rn is related
to the joint torque E: a Rn by the equation
B
=
JT(6)F
(5)
so that
where F = $ + f and i is the end-effector force
at the operating point.
It therefore follows
from equations ( 3 ) , ( 4 ) , and ( 6 ) that the
linearised dynamics of robotic manipulators in
task space are governed on the continuous-time
set by linear differential equations of the form
This equation can obviously be expressed in the
standard state-space form
1614
X =
0
=
Ax + Bf
cx
I
under the action of the conkrol law
with
X =
E
fk(j)
=
2
- ;(l+a)Drk(j)
rk(j+l)
=
-ark(j) + sk(j)
sk+l(j)
=
sk(j)
+
2
+ (In + ;D)sk(j)
A{ek(j+l)-ek(j)}
R2n
where a E (-l,+l], D e Rnxn, and ek(j)
wk(j). Assume that
(i)
A =
=
v(j) -
2
(H(-r)(In + ;D)}-’
(ii) Ok+l(0)
=
wk(0)
= V(0)
(k=0,1,2, . . . )
.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2789
Then, when j e [0 , N],
as k
+
m.
and
In such iterative learning controllers, the
ultimate objective is to find an input f(j) that
produces a plant output w(j) which coincides
with the desired plant output v(j) over a fixed
time interval [ 0 , TI where T = Nr.
This
objective is achieved by producing a sequence of
outputs
(O,(j),Ol(j),Oz(j),
...,0k(j),. . .
on
[ O , N] corresponding to a sequence of inputs
(f, ( j1 f ( j1 f, ( j) , . . . fk(j 1 . . . I on [ o , N l .
But the dynamics of linear plants governed on
the
continuous-time
set
by
differential
equations of the form (8) are governed on the
discrete-time set by difference equations of the
form
I
I
x(j+l)
=
*x(j) + ‘uf(j)
1
I
I
However, in order to design iterative
learning controllers for large motions of
robotic manipulators it is necessary to render
such controllers adaptive by identifying the
,step-response
matrices of such manipulators
along the appropriate trajectories emanating
from an operating point.
Such adaptive
iterative learning controllers accordingly
embody standard parameter-estimation algorithms
(such as RLS o r E L S ) for ARMA models of the form
J
where
J
and T is the sampling period. The step-response
matrices of such plants have the form
H(T)
=
J:CeAtBdt
(12)
and characterise the responses of initially
quiescent plants after one sampling period.
Such step-response matrices can evidently be
measured directly from input/output data.
Then, since
H(r)
8,
e
RnX”
It is clear from equations (9b), (Sc), and
(9d) that robotic manipulators give rise in the
neighbourhood of
an
operating point
to
linearised plants with null first Markov
parameters
but
full-rank
second
Markov
parameters.
Such
plants
are
therefore
completely irregular and it is accordingly
immediately possible to use the following
theorem to design iterative learning controllers
for & motions of robotic manipulators in the
neighbourhood of an operating point:
the control law described in the theorem of
Porter and M~hamed(~> can be readily rendered
adaptive by using in the iterative learning
the
controller the
current estimate of
step-response matrix, H(7), from the parameterestimation algorithm. It is finally important
to note that it follows from equation ( 5 ) that
the actual vector of joint t,orques applied to
the robotic manipulator in the kth iteration is
___
Theorem
where j
sk(j)
(Porter and M~hamed)(~) In the case of
the completely irregular plant with discretetime governing equations
=
E
i
+ JT(e)fk(j)
[0 , N].
3.
Illustrative Example
These general results can be conveniently
illustrated by
considering
the
adaptive
iterative learning control of a two-link robotic
manipulator operating in
a
gravity-free
J
1615
and cx = 1 . 0 i s i n d i c a t e d i n F i g u r e s 3-8.
These
f i g u r e s show t h e a c t u a l t r a j e c t o r i e s of t h e end
adapt.ivt:
iterative
learning
effector
under
c o n t r o l , together with t h e i d e n t i f i e d elements
of t h e s t e p - r e s p o n s e m a t r i x o f t h e m a n i p u l a t o r ,
a f t e r 1 , 5 , and 9 i t e r a t i o n s .
I t i s s e e n , by
comparing F i g u r e s 1 , 3 , 5 , and 7 , t h a t -the
adaptive i t e r a t i v e learning c o n t r o l l e r rapidly
g e n e r a t e s t h e d e s i r e d t r a j e c t o r i e s of t h o end
e f f e c t o r i n t a s k s p a c e a n d , by comparing F i g u r e s
2 , 4 , 6 , and 8 , t h a t t h e r e c u r s i v e l e a s t s q u a r e s
i d e n t i f i e r obtains reasonably accurate estimates
of t h e s t e p - r e s p o n s e m a t r i x of t h e m a n i p u l a t o r .
environment.
The dynamics of t h i s m a t i i p u l a t o r
a r e g o v e r n e d on t h e c o n t i n u o u s - t i m e s e t by t h e
non-1 i n e a r d i f f e r e n t i a l e q u a t i o n s
J,+J,(1+3yz)+3J,ycose,
, J,(l+
23
ycose,)
J,
and
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4.
Conclusion
I t h a s been shown t h a t d i g i t a l i t e r a t i v e
l e a r n i n g c o n t r o l l e r s can be r e a d i l y rendered
a d a p t i v e by
identifying
i n r e a l time t h e
step- response
matrices
of
plants.
In
p a r t i c u l a r , by i d e r i t . i f y i n g t h e s t e p - r e s p o n s e
matrices of r o b o t . i c m a n i p u l a t o r s a l o n g t h e i r
t r a j e c t o r i e s i n t a s k s p a c e , i t h a s been shown
t h a t s u c h m a n i p u l a t o r s a r e amenable t o a d a p t i v e
iterative learning control.
These g e n e r a l
by
the
results
have
been
illustrated
presentation
of
numerical
results
for
the
adaptive
iterative
learning
control
of
a
two- link manipulator o p e r a t i n g i n a g r a v i t y - f r e e
environment.
are t h e l e n g t h s of t h e l i n k s ,
where 9 , and 9,
m, and m, are t h e masses of t h e l i n k s , J , =
( l / 3 ) m 1 9 : , J , = (1/3)m,t:,
Y = 9,/9,, and 0 , and
Ca,
are t h e C a r t e s i a n c o - o r d i n a t e s of t h e end
effector.
In case t h i s manipulator corresponds
t o t h e s e c o n d and t h i r d l i n k s o f t h e Unimation
PUMA 560 r o b o t , 9, = 9, = 0 . 4 3 2 m , m , = 1 5 . 9 1
k g , and m, = 1 1 . 3 6 kg.
I t is d e s i r e d t h a t t h e end e f f e c t o r p e r f o r m s
t h e following r e c t i l i n e a r motions i n t h e plane
of C a r t e s i a n c o - -o r d i n a t e s :
References
P(0.7375,-0.3055)m
and
moves
between
Q(0.7,-0.25)m
i n 1.5 s with equal periods
o f a c c e l e r a t i o n , c r u i s e , and d e c e l e r a t i o n
from r e s t t o 6 6 . 9 3 9 x
m/s and back t o
rest:
(1) M Uchiyama, " F o r m a t i o n of h i g h - s p e e d motion
p a t t e r n of a m e c h a n j c a l arm by t r i a l " , T r a n s
SICE, VOI 2 1 , pp 706- 732, 1978.
U
(2)
( i i ) t h e n moves i m m e d i a t e l y between
Q(0.7,
--0.25)m and P ( 0 . 7 3 7 5 , - 0 . 3 0 5 5 ) m i n 1 . 5 s
with equal periods of a c c e l e r a t i o n , cruise,
and d e c e l e r a t i o n from r e s t t o 6 6 . 9 3 9 x
10-3m/s and back t o r e s t .
S A r i m o t o , S Kawamura, and F M i y a z a k i ,
" B e t t e r i n g o p e r a t i o n of r o b o t s by l e a r n i n g " ,
.J R o b o t i c S y s t e m s , Vol 1 , pp 123--140, 1984.
( 3 ) T Mita and E K a t o ,
" I t e r a t i v e c o n t r o l and
i t s a p p l i c a t i o n t o motion c o n t r o l of r o b o t
arm - a d i r e c t a p p r o a c h t o s e r v o p r o b l e m s " ,
Proc
IEEE
Conference
on
Decision
and
C o n t r o l , pp 1393- 1398, F o r t L a u d e r d a l e , U S A ,
1985.
These d e s i r e d t r a j e c t o r i e s of t h e end e f f e c t o r
a r e shown i n F i g u r e 1 and t h e c o r r e s p o n d i n g
v a r i a t i o n s of t h e e l e m e n t s of t h e s t e p - r e s p o n s e
matrix of t h e manipulator corresponding t o a
s a m p l i n g p e r i o d of r = 0 . 0 1 s are shown i n
Figure 2.
The r a p i d l e a r n i n g of t h e a d a p t i v e
i t e r a t i v e c o n t r o l l e r with
( 4 ) S Arimoto, S Kawamura, and F Miyazalti, "Can
m e c h a n i c a l r o b o t s l e a r n by t h e m s e l v e s ? " ,
R o b o t i c s R e s e a r c h : The Second I n t e r n a t i o n a l
Symposium, pp 1 2 7 - 1 3 4 , 1985.
( 5 ) B P o r t e r and S S Mohamed, " D i g i t a l i t e r a t i v e
learning c o n t r o l of multivariable p l a n t s " ,
USAME/DC/101/90,
University
of
Report
S a l f o r d . 1990.
1616
z
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6.80:
t ( s > x10-1
End EFFector
X43
' i;. , , .,-,
, , ,
, I , ,
I , , , , , , , , ,, ,
,,, , , , I , , ,
~
-
~
.
(b)
,
-3.44
t( 5
)
I
x10-1
t(5)
x10-1
(d)
(C)
Fig.2:Actual Elements OF Step-Response Matrix OF Manipulator
ACong Desired Trajectory.
1617
-Y 32-
, , , , ,, , , , I , ,
I
~
-2
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-2
6.6
""'.
t( 5 )
0
xlO-1
t(5)
(a)
x10-1
(b)
Fig.3:Trajectories OF End EFF'ector Under Adaptive Iterative
Learning Control ( K=1 ) .
0
t(5)
0
x10-1
t ( 5 )
0
t( 5 )
-4.0
xlO-1
11.5+
xlO-'
t( 5 )
xlO-1
(d)
(C)
Fig.4:IdentiFied Elements OF Step-Response Matrix OF Manipulator
Under Adaptive Iterative Learning Control ( K = l > .
1618
-9.3im
I
, , ,, , , , , , , ,
I
~
,, , , ,, I
I
~
-2.45:
-2
A
E-2
-2
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6 .G0
.
8
0
~
~
i
I
t(
5)
- 1
"'"'?I
I % I I I I I IIIIIIIII
0
x10-1
t( s >
(a>
I , * I *
J0
x10-1
(b)
Fig.5:Trajectories OF End Effector Under Adaptive Iterative
Learning Control ( K = 5 )
.
0
t(s>
x10-1
t(s>
x10-1
(d>
(C)
Fig.6:IdentiFied Elements OF Step-Response Matrix OF Manipulator
Under Adaptive Iterative Learning Control ( K = 5 > .
1619
1
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I
Itl
tC5)
I I I I , I I I I I , I I I I I
13
x10-1
tC 5 > x10-1
Cb>
(a)
Fig.7:Trajectories OF End EFF'ector Under Adaptive Iterative
Learning Control (K=9).
- 1 .15
A
&-1 .72
c9
~ 2 . 2 9
-
,
r-T. 86
-3.43
0
t( 5 )
x10-1
(a>
0
t(
5)
x10-1
t(s>
x10-1
(d)
(C)
Fig.8:IdentiFied Elements OF Step-Response Matrix OF ManipulutotUnder Adaptive Iterative Learning Control (K=S).
1620
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