(1989) - discrete-time modeling and control of robotic manipulators

7/30/2019 (1989) - Discrete-Time Modeling and Control of Robotic Manipulators

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Journal o f Intelligent and Robotic Syst em s 2 : 4 1 1 - 4 2 3 , 1 9 8 9 . 4119 1989 Kluwer Academic Publishers. Printed in the Netherlands.

Discrete-Tim e Modeling and C ontrol of RoboticManipulators

S . N I C O S I A , P . T O M E I

Dipartimento di lngegneria E lettronica, Seconda Universit~ di Ro ma , Tor Vergata, Via Raimondo,

00173 Roma, I ta ly

a n d

A . T O R N A M B I ~

Fondazione Ugo Bordoni, Via Baldassarre Castiglione 59, 00142 Ro ma , I taly

(Rece ived: 15 February 1989; revised: 22 June 1989)

Abstract. A ge ne ra l a pp roa c h i s p re se n t e d t o de r i ve d i s c re t e - t i me mode l s o f robo t i c ma n i pu l a t o r s . S uc h

mode l s a re ob t a i ne d by a pp l y i ng nume r i c a l d i s c re t i z a t i on t e c hn i que s d i r e c t l y t o t he p rob l e m o f t he

m i n i m i z a t i o n o f t h e L a g r a n g e a c t i o n f u n c t i o n a l . A l t h o u g h t h e se m o d e l s a r e i n i m p l i ci t f o r m , t h e y o w n a

dyn a m i c s t ruc t u re t ha t a l lows us t o de s i gn d i s c re t e - ti me fe edba c k l i ne a r i z i ng c on t ro l la ws. T he p ro pose d

mode l s a nd c on t ro l a l go r i t hms a re va l i da t e d by s i mu l a t i on wi t h r e fe re nc e t o a t h re e l i nk robo t .

Key words. R ob o t s , d i s c re t e -t i me mode l i ng , d i s c re te - t i me c on t ro l , de c oup l i ng c on t ro l .

1 . In t ro duct io n

The dy na m ic s o f r obo t i c ma n ipu la to r s ha s r e c eive d c ons ide r a b le a t t e n t ion in t he

robo t ic s l i t e ra ture . V ar ious ap proa ches , such as Lag rangian [1] , r ecurs ive Lagran gian

[2 ], a nd r e c u rs ive N e w ton - Eu le r [3] ha ve be e n p r o pos e d f o r t he f o r mu la t ion o f the

m a n i p u l a t o r d y n a m i c s . R o b o t m a n i p u l a t o r s h a v e c o m p l i c a t e d b e h a v i o r i n c l u d i n g

in te rac t ions a m on g m ul t ip le jo in ts , n onl inea r e f fec ts such as Cor io l i s and cent r i fuga l

forces , and v a ry ing ine r t ia depen ding on the a rm conf igura t ion . A s h igher pe r form ancein te r ms o f spe e d a nd a c c u r a c y is pu r sue d , t he se c om pl i c a t e d dyna mic s b e c om e m or e

p r omine n t . Mode l - ba se d a lgo r i t hms c a n be u se d to ob ta in a c c u r a t e c on t r o l sy s t e ms

whic h t a ke in to a c c oun t mode l c omple x i t i e s [ 4 ] .

M o s t o f th e c o n t r o l l a w s, b a s e d o n a c o n t i n u o u s - ti m e m o d e l o f t h e m a n i p u l a to r ,

a r e u sua l ly im p le me n te d by me a ns o f m ic r oc omp u te r s . S o , in p ra c t ic e , t he c on t inuou s

c on t r o l l a ws a re d i sc r e ti z ed to a l l ow thei r imp le me n ta t ion o n m ic r oc omp u te r s . O n the

othe r hand , s imula t ions and rea l te s t s have shown tha t the sampl ing t ime i s c r i t ica l

t o t he f e e dba c k pe r f o r m a nc e o f r obo t s [ 5 ]. A poss ib l e w a y to ove r c om e th i s d i ff ic u lty

could be f i r s t to de r ive a d isc re te - t ime mode l and then to synthes ize a su i tab led isc re te - time c ont ro l law. T his w ay i s sure ly mo re e f fec tive in the case of l inear t ime

invar ian t sys tems [6] .

S om e d i sc re t e m ode l s o f r ob o t s a r e a va i la b l e in t he li te r a tu r e . T om iz uka , Ho r ow i t z

a nd La nda u [7] d i s cr e ti z ed d i r e c t ly t he e qua t ions o f mo t ion ne g lec t ing non l ine a r



412 s. NICOSIA ET AL.

terms. Nicosia and Tomei [8] simply used a forward Euler discretization of the

complete motion equations and obtained encouraging results with a discrete adaptive

control law. Neuman and Tourassis [9] derived implicit discrete models from a

discrete formulation of Lagrange equations, imposing the conservation of energy over

each sampling period. The model proposed by Monaco and Normand-Cyrot [10] was

based on a Volterra series expansion of the solution of the differential equations.

In this paper, a general approach is presented to derive implicit discrete models,

starting directly from the Lagrange action functional. In fact, as is well known, the

motion equations o f mechanical systems can be obtained by minimizing the Lagrange

action functional [11]. This fact can be exploited to derive discrete models by applying

numerical discretization techniques to the functional minimization problem. The

so-derived discrete models are validated by simulation. They exhibit a satisfactory

dynamic behavior, especially with respect to models obtained via discretization of theLagrange differential equations. Moreover, such models own a structure that allows

us to design discrete-time feedback linearizing control laws. These algorithms are

obtained without making explicit the implicit discrete models. The remainder of the

paper is devoted to illustrating the proposed modeling and control approaches by

application to a three-link robot.

2 . B a c k g r o u n d t o R o b o t M o d e l i n g

We consider robotic manipulators with N degrees of freedom whose spatial position

is completely defined by the generalized coordinate vector q = (ql . . . . . qN) T, where

qi denotes the relative displacement between the adjacent links i and i - 1.

The motion equations may be derived by a direct application of the Lagrangian

method. Let L ( q , O ) be the Lagrangian function defined as

L ( q , q ) = T ( q , ?1) - U ( q ) , (1)

where T ( q , ( 7 ) is the kinetic energy and U ( q ) is the potential energy of the manipulator.

The kinetic energy can be expressed as

, i i( q , q) = 89 O = ~ b i j ( q ) q ~ q j , (2)i=1 j=l

where the inertia matrix B ( q ) is symmetric positive definite for any q.

The Lagrangian equations may be obtained by minimizing the action functional

[ I I ]

S = f i l / L ( q , O ) d t (3)

with respect to q ~ C2( [ t~ , t : ] ) , in which q( t~ ) and q(t:) represent, respectively, the

initial and final positions of the robot.

The solution of this minimization problem, that is the trajectory followed by the

robot, is derived by solving the Lagrange differential equations

d tSL tSL- 0. (4)

dt 8q 8q



D I S C R E T E - T IM E C O N T R O L O F R O B O T I C M A N I P U L A T O R S 413

I f ex te rna l genera l ized forces ui a re appl ied to the jo in ts , the L agran gian equ a t ion s can

be de r ive d in a s imi la r ma nne r by a s soc i a t ing a su i t a b l e po te n t i a l e ne r gy to e a c h f o r c e

u~ [11]. D en ot ed by u = (u l , 9 9 9 UN) r , t he se e qua t ions t a ke the f o r m

d d L 0 L - u . ( s )

dt ~r ~q

Put t in g Eq ua t io n (2) in to (5), we ge t

= u, (6)(q) i~ + a(q , q)

w h e r e

c32La(q , q) -

~q ~( t

~L- - 0 O q . ( 7 )

W h e n t h e r o b o t is c o n t r o ll e d b y m e a n s o f a c o m p u t e r , t h e in p u t u ( t ) is upda te d a t e a c h

sa mpl ing in s t a n t b y a d ig i t a l- t o - a na log c onve r t e r ( DA C ) a nd a ho ld ing de v ic e so t ha t

the inpu t i s p iecewise con s tan t

u ( t ) = u ( k T ) = u ( k ) , k T <~ t < ( k + 1) T, (8)

where T represents the sampl ing t ime .

The inp uts to the com pute r a re suppl ied by sample r s and ana log- to-d ig i ta l conver te r s

( ADC ) wh ic h a c t on the r obo t - me a su r e d va r i a b l e s .

F o r s imu la t ion a nd /o r c on t r o l pu r po se s , wh a t we ne e d is a mo de l o f the d i sc r et e -

t ime sys t e m c ons i st i ng o f the c a sc a de c onne c t ion o f DA C , ho lde r , r obo t , s a mple r , a nd

ADC. We re fe r to th is d isc re te - t ime sys tem as the d i s c r e t e - t i m e r o b o t .

Usua l ly , a pp r ox ima te mo de l s o f t he d i sc r e te - time r o bo t a r e de rive d by a pp ly ing

nume r i c a l d i s c r e t i z a t ion p r oc e dur e s t o ( 6 ) . The mos t popu la r a pp r oa c he s a r e t he

bac kw ard and forw ard Eule r a lgor i thm s [7 , 8 , 1 1]. Us in g these m ethod s , the fo l lowing

e qua t ions a r e ob ta ine d :

b a c k w a r d m o d e l

1)) ( q ( k + 1 ) - 2 q( k) + q ( k - 1 ))B ( q ( k

T z +(9 )

J o r w a r d m o d e l

( q ( k + 1 ) - 2 q(k ) + q ( k - 1).~

B ( q ( k+

1)) T 2 ) q '-

+ a ( q ( k + 1), q ( k + 1 ) - q ( k ) ) u(k + 1),

where the s impl i f ied nota t ion q ( k ) i s used in p lace of q ( k T ) ,

( l o )



414 s. NICOSIA ET AL.

3 . T h e P r o p o s ed D i s c re t e -T i m e M o d e l

A n a l te r n a ti v e a p p r o a c h t o d e r iv i n g a p p r o x im a t e m o d e l s o f t h e d i s c re t e- ti m e r o b o t

cons i s t s o f d i r ec t l y ob t a in ing approx imate so lu t i ons o f t he min imiza t i on o f t he

func t ional (3)9

Let the ini t ial s tate q ( t i ) , ; t (t ~ ) be fixed. Let the t ime interval [ t i , t ] be f ixed and

decom posed in M sub in te rva l s o f a con s t an t wid th T .

D e n o t i n g b y

~ k (k ) = q ( t i + k T ) (11)

and approx imat ing t he t ime-der iva t ive wi th t he bac kw ard d i f fe rence

if(k ) - ~k(k - 1)q ( t , + k T ) , (12)

T

t he i n teg ra l (3 ) can be app ro x im ated b y

S ~ S ( W ) = T ~ L ( ~ k ( k ) '~ k ( k ) - ~ b(k 1).), (13,

i n wh ich

~(o)V = @(1) .

~k(M)

Thus , t he p rob l em o f min imiz ing t he func t i ona l (3 ) i s r educed t o t he p rob l em o f

min imiz ing fun c t i on (13 ).

As i s we ll know n , t he so lu t ion o f such a p rob l em is ob t a ined by so lv ing t he

fo l lowing d i f ference equ at ions

d S ( ~ ) M d L ( ~ ( k ) ' ~ k ( k ) - ~ k ( k -) ) T

= T ~ dW = 0 (14)

dW k=ltha t a re equ iva l en t t o

aL (~b(1), ~b(1) - ~( 0)- ~

a~(o)d L ( ~ k ( k )' ~ ( k ) - ~ b ( k 1 ) )

= 0, (15)

+a~(k)

O L ( ~ k ( k + 1),

a L ( ~ b ( M ) ' ~ b ( M ) - ~ b ( M - T 1))

t~b (M)= O .

~k(k + l) - ~k(k)'~

J

a~(k)= O ,

(16)

(17)



D ISC R ETE-TIME C O N TR O L O F R O B O TIC MA N IPU LA TO R S 415

E q u a t i o n s ( 1 6 ) , t o g e t h e r w i t h t h e i n i t i a l c o n d i t i o n s 4 ( 0 ) a n d 0 ( 1 ) , c o n s t i t u t e t h e

d e s ir e d a p p r o x i m a t e m o d e l o f t h e d i s c re t e -t im e r o b o t .

I n t h e c a s e w h e r e e x t e r n a l f o r c e s a r e a p p l i e d t o t h e j o i n ts , ( 16 ) b e c o m e s

+ = - u ( k ) ,d O ( k ) d O ( k )

w h e r e u ( k ) = u ( ti + k T ) .

F r o m ( 1 8 ) , b y u s i n g ( 1 ) a n d ( 2 ) , w e o b t a i n

l

r - 1) , ~k (k)) - -~5 ( B ( O ( k

- B ( O ( k ) ) (O ( k ) - O ( k - 1)))

i n w h i c h

1 ~ ~ d b o ( O ( k ) )

r - 1) , ~k(k) ) = 2 T 2 i=1 j=l d 0 ( k )

+ 1 ) ) ( 0 ( k + 1 ) - 0 ( k ) ) -

= - u ( k ) ,

du(0(k))

08)

(19 )

f r o m w h i c h

r O ( k + 1)) - T1---5 B ( O ( k ) ) ( O ( k

- B ( O ( k - 1 ) ) ( 0 ( k ) - O ( k - 1 ) ) )

w h e r e

~oe (O (k), O (k + 1 )) = 2 T 2 i = , j = , dO (k)

x ( 0 , ( k + 1 ) - - O , ( k ) ) ( @ ( k + 1 ) - - 0 j ( k ) )

a0(k)= - u ( k ) ,

(22)

+ 1 ) - - 0 ( k ) ) -

= - u ( k ) ,

(23 )

au(o(k))

d0(k)

( 2 4 )

d i f f e r e n c e

O ( k + l ) - 0 ( k )i t( t ~ + k T ) .

T

F o l l o w i n g t h e s a m e p r o c e d u r e , w e o b t a i n

d L ( O ( k ) ' O ( k + I ) - 0 ( k ! ) T

+dO(k)

d L ( O ( k - 1), 0 ( k ) - - O ( k -T 1).)

(21)

• (O , (k ) - O i ( k - l ) ) (O j (k ) - O j ( k - I ) ) d O ( k ) ( 2 0 )

a n d Oi ( k ) s t a n d s f o r t h e it h c o m p o n e n t o f v e c t o r 0 ( k ) .

A n a l te r n a ti v e d i s c re t e -t im e m o d e l c a n b e d e r i v e d a p p r o x i m a t i n g q b y t h e f o r w a r d



41 6 S . NICOSIA ET AL.

4 . T h e D e e o u p l i n g C o n t r o l L a w

T h e i m p l i c i t d i s c r e t e - t i m e m o d e l s d e r i v e d i n t h e p r e v i o u s s e c t i o n a r e d e s c r i b e d b y

e q u a t i o n s o f th e f o l lo w i n g f o r m

q(~ , (k - 1 ) , ~b (k ) , ~b (k + 1 ) ) = u ( k ) . ( 2 5 )

D e n o t i n g b y ~ ___ R N x R N x R N t h e s e t o f th e p o i n t s ( ~ b( k - 1 ), ~ ( k ) , ~ ( k + 1 ) )

s u c h t h a t

d e t I t ~ ( k O q + 1 ) ] :/: 0 , ( 2 6 )

s y s t e m ( 2 5 ) c a n b e l o c a ll y m a d e e x p li c it , f o r a n y p o i n t o f ~ , a s

~ ( k + 1 ) = g ( d / ( k - 1) , ~b(k) , u ( k ) ) . ( 2 7 )

L e t u s d e f i n e t h e s t a t e v a r i a b l e s

X l ( k ) = ~ b ( k - 1 ), x 2 ( k ) = ~ O (k ) ( 2 8 )

a n d t h e o u t p u t v a r i a b l e s

y ( k ) = ~ b ( k - 1) . (29)

E q u a t i o n s ( 2 7) c a n b e r e w r i tt e n a s

x ( k + 1) = f ( x ( k ) , u ( k )) , y ( k ) = h ( x ( k ) ) , ( 3 0 )

w h e r e

x = [ x r , x ~ ] r , f = [ x ~ , g r ] r , h = x l .

V a r i o u s w o r k s h a v e b e e n c a r ri e d o u t o n t h e p r o b l e m o f l in e a ri zi ng , b y s t a te f e e d b a c k ,

d i s c r e t e - t i m e n o n l i n e a r s y s t e m s s u c h a s ( 3 0 ) [ 1 3 - 1 6 ] .

I n o u r c a s e , i t i s i m m e d i a t e l y r e c o g n i z e d t h a t b y i n t r o d u c i n g t h e s t a te f e e d b a c k

u ( k ) = r / ( x t ( k ) , x 2 ( k ) , v ( k ) ) , ( 3 1 )

w h e r e v is t h e n e w i n p u t v e c t o r , s y s t e m ( 3 0 ) i s t r a n s f o r m e d i n t o N d e c o u p l e d l i n e a r

s y s t e m s

x l ( k + 1) =

t h a t c o n s i s t o f t w o

N o t e t h a t , s i n c e

t r a j e c t o r y c o u l d b e

v ( k ) = q , ( k

x 2 ( k ) , x 2 ( k + 1) = v ( k ) , y ( k ) = x , ( k ) ( 3 2 )

c o n s e c u t i v e d e l a y e l e m e n t s .

y ( k ) = v ( k - 2 ) , t h e t r a c k i n g p r o b l e m f o r a g i v e n r e f e r e n c e

s o l v e d b y a d o p t i n g

+ 1) . (33 )

I n o r d e r t o p r e v e n t t h e a c t i o n o f d i s t u r b a n c e s a n d p a r a m e t r i c v a r i a ti o n s , i t m a y b e

m o r e c o n v e n i e n t t o s h i ft th e p o l e s f r o m t h e o r ig i n b y u s i n g a s th e c o n t r o l l a w

v ( k ) = A o x l ( k ) + A t x z ( k ) + % ( k ) , ( 3 4 )

w h e r e A 0 a n d A 1 a r e c o n s t a n t d i a g o n a l m a t r i c e s a n d v r ( k ) s u i t a b l y d e p e n d s u p o n t h e

r e f e r e n c e v a r i a b l e s .



D ISC R ETE-TIME C O N T R O L O F R O B O TIC MA N IPU LA TO R S 417

' x 3 I - 0 . 4 : 0 . 6 ; 0 . 4 51 .

)

x2

(~ [0.9:0:01, ~ x I

Fig. 1.

u ( t) u ( k ) u ( k T )

- - ~ S a m p l e r I J Z e r o ~ J] h o l d e r 1 7

Robo t and t rajectory considered n the case study.

q ( t )

R o b o t H S,

+

)1 M o d e l ( i 0 )

~I M o d e l ( 1 9 )

I

S a m p l e r

q ( k )

1Fig. 2. Blo ckdiagram o f the simulated system.

5. C ase Study

In order to validate the discrete-time models and the control algorithms derived in the

previous sections, some simulations were carried out for the robot schematically

represented in Figure 1. It corresponds to a real three-degrees-of-freedom manipulator

having links 0.5 m long.

Models (10) and (19) have been taken into consideration in the first set of simulation

runs. The inertia matrix B and the vector a that completely characterize the continuous-

time model, are reported in the Appendix.

The aim of the simulation was to compare the dynamic behavior of models (10) and

(19) with the behavior of the discrete-time robot, when they are excited by the sameinputs (see Figure 2). The tests were performed with reference to the straight line

trajectory illustrated in Figure 1, assuming a trapezoidal velocity law with a maximum

velocity of 0.75 m/s and a maximum acceleration of 0.75 m/s2; the sampling time was

10ms.



418 S . N I C O S I A E T A L .

C a r t e s i a n

c o o r d i n a t e s

[irrjI

J o i n t

c o o r d i n a t e sO p e n l o o p

c o n t r o l i n p u t

[ q lr

R o b o t i n v e r s e I ~ J R o b o t i n v e r s e I1 3

Fig . 3. Scheme o f the open loop con t ro l com pu ta t ion .

r a d ~ / ~ ~ . 5 q2qZ

i i I

1 2 3 s--.5

"~ ~ \ \ \ \ . . . . . . . . . .......................q 3

Fig. 4 . Outp uts of the d iscrete- t ime robot .

The de s i r e d t ra j e c to r y , e xp re s sed in C a r t e s ia n c oo r d ina t e s , w a s u se d to p r od uc e the

o p e n - l o o p c o n t r o l i n p u t u ( t ) , v ia t he r obo t i nve r se k ine ma t i c s a nd dyna mic s ( s e e

Figure 3) .

The r e su lt s o f t he s imu la tion r uns a r e d r a wn in F igu r e s 4 - 6 . The o u tpu t s o f t he

disc re te - t ime robot , a long the cons ide red t r a jec tory , a re r epor ted in F igure 4 . F igures

5 a nd 6 show the d i f fe r enc e s be twe e n the ou tpu t s o f t he d i sc r e te - time r ob o t a nd the

ou tpu ts o f m ode ls (10) and (19), r e spec t ive ly .

The se c ond se t o f s imu la t ions wa s de vo te d to t e s t t he a c c u r a c y o f t he de c oup l ingcon t ro l laws de r ived in Sec t ion 3. In pa r t icu la r , w e re fe r red to the a lgor i thm ob ta ine d

by us ing the ba c kwa r d mode l ( 19 ) , na me ly

1u ( k ) = - ~ o . ( x , ( k ) , x ~ (k )) + - ~ x

x ( B ( v ( k ) ) ( v ( k ) - x 2 ( k ) ) + B ( x 2 ( k ) ) ( x 2 ( k ) - xt(k)) ) .

(35)

The c on t r o l v ( k ) was chosen as g iven in (34)

v ( k ) = X r z ( k + 1) + A o ( x , ( k ) - x r , ( k ) ) + A , ( x 2 ( k ) - x~2(k)), (36)

whe r e

A 0 = dia g [0.01], At = dia g [0.2],



D I S C R E T E - T I M E C O N T R O L O F R O B O T I C M A N I P U L A T O R S 419

r a d

.04

.02

-.02

-.04

e 3

F i g . 5 . E r r o r s w i t h t h e d i s c r e t e m o d e l (1 0 ).

r a d

.011 / e3

- o z / \ 3

82 81

F i g . 6 . E r r o r s w i t h t h e m o d e l ( 19 ) .

so tha t t he c lo se d - loop po le s a re e qua l t o 0 .1, f o r e a c h de c ou p le d se c on d- o r de r sy s t e m.

The va lue s o f x i (k ) a nd x~( k ) we r e c omp u te d , u s ing the s a m ple d s t a t e va r i a b le s o f the

r obo t a nd the s a mple d r e f e r e nc e t r a j e c to r y , a c c o r d ing to

x , ( k ) = x 2 ( k ) - T q ( k ) , x2(k) = q ( k ) ,

X r , ( k ) = x r 2 ( k ) - T q r ( k ) , x r 2 ( k ) = q ~ ( k ) .

H ow eve r , in prac t ica l cases , the necess i ty of the sam pled re fe rence ve loc i ty qr (k) can

be avoided . In fac t , a s exhib i ted by s im ula t ion te s ts , no s igni ficance va r ia t ions a r ise

i f x r j ( k ) is c o m p u t e d a s x r ,( k ) = x r 2 ( k - 1).

To r e p r odu c e the re a l ope r a t ing c ond i t i ons , a c on t inuo us m ode l o f t he r obo t ha s

be e n e mploye d . The a na log inp u t s o f t he r obo t s a r e c om pu te d b y u s ing the d is c r et e

con t ro l law (35), (36) and a ze ro-o rde r ho ld ing device . The b lock d iagram of the

s imu la t e d c on t r o l sy s t e m i s r e po r t e d in F igu r e 7 .Th e s imu la t ion resu l t s , r e fe r red to the t r a jec tory of F igure 1 and wi th a sampl ing

t ime of 10ms, a re drawn in F igures 8-10 . F igures 8 and 9 show, respec t ive ly , the

e r r o rs i n j o in t c oo r d ina t e s a n d the c o r r e spond ing to r que s a long the t r a j e c to ry . The

a bso lu t e e r r o r i n C a r t e s i a n c oo r d ina t e s i s r e po r t e d in F igu r e 10 .



4 2 0

J C o m p u t ."1 o f U ( k )

I

Tu ( k ) u ( t )

H z e r o - o r d e ~h o l de r I v l R o b o t

Fi g . 7 . Co n t ro l s y s t em s t ru c t u r e .

q ( t)q ( t ) J

H

S . N I C O S I A E T A L .

q ( k)q c k l ]S a m p l e r

m r a d e2 / e3

i ~ / j <r. " A f - ~

Fi g . 8 . E r ro r s in j o i n t co o rd i n a t e s .

N m ~ ~ 2 / u 3U 2

....

.... ~ ~ " ~ 3

1 1 1

Fi g . 9 . T o rq u es ap p l i ed t o t h e j o i n t s .

m l n

. 1

e t

i3

F i g . 10 . A b s o l u t e e r r o r in C a r t e s i a n c o o r d i n a te s .



DISCRETE-TIME CONTROL OF ROBOTIC MANIPULATORS 421

The e r rors a re o f the o rder o f 10 -4 rad in the jo in t co ord ina tes a nd of 10 2 mm in

the Car tes ian coord ina tes , showing tha t the d i scre te con t ro l l aw works qu i te wel l .

Indeed , the con t ro l a lgor i thm obta ined by d i scret iz ing the con t inuous- t ime decoupl ing

cont ro l law led to a ma xim um er ror o f 0 .8 mm along the same t ra jec tory [17].

6. Conclusions

Usual ly , d i scre te models fo r robo t ic manipu la to rs a re ob ta ined by the d i scre t i za t ion

of the d if feren t ia l equa t ions o f mot ion . In th is paper , a d i f feren t app roa ch has been

out l ined in which , s t a r t ing d i rec t ly f rom the Lag range funct ional , the d i screte models

are der ived by app ly ing nu mer ica l d i scre t i za t ion t echn iques to the funct ional min imiz-

at ion problem. Moreover, a discrete- t ime feedback l inearizing control law is presented

t h a t a l l ows u s an ap p ro x i m a t e d eco u p l i n g o f i n p u t -o u t p u t v a r iab l es o f th e ro b o t .Simula t ion t est s, p er fo rme d wi th reference to a th ree- l ink robot , have g iven sat is -

fac tory resu l t s fo r the d i scre te model ing and , even more , fo r the d i scre te con t ro l ,

especial ly i f co mp are d with those related to contro l laws obtained via the discret izat ion

of con t inuou s- t ime feedback a lgor i thms . Ou r fu tu re works wi ll regard the poss ib il i ty

of us ing mor e p rec i se models and the app l ica t ion to f lex ib le robots . Mo reov er , the

robus tness p roper t i es o f the p ropo sed app roa ch wi ll be inves t iga ted .

AppendixFo l l o wi n g t h e n o t a t i o n s o f (6 ) , t h e n o n ze ro e l em ent s o f m a t r i x B ( q ) an d v ec t o r a ( q , q),

refer red to the robot used in the s imula t ion t es t s , a re

bj l = h i + h2cos 2(q2) + h3cos2(q2 + q3) W h 4 c o s ( q 2 ) c o s ( q 2 + q3),

b22 = h5 + h4cos(q3),

b23 = h 6 + h7c os( q3 ),

b23 = b32 ,

b33 = h8,

al = (c102 + c2q3)ql ,

a2 = -~Clql + c3q3 q2 + ' ] - c4 ,

_ 1 .2 1 .2a 3 - - - - ~c 2q I - - ~c3 q 2 q - c5 ,

in which

c~ = - h 2 s i n ( 2 q 2 ) -- h 3 sin(2(q2 + q3)) - - ha sin(2q2 + q3),

C = - h 3sin(2(q2 + q3)) - h4co s(q2)sin(q 2 + q3),

C3 = -- h 4 sin(q3 ),



4 2 2

c4 = h 9 c o s ( q 2 ) + h l 0 C O S ( q 2 + q 3 ) ,

C5 = h l 0 c o s ( q 2 + q 3 ) .

T h e n u m e r i c a l v a l u e s o f t h e c o e f f i c i e n t s h i , i =

h i = 2 3 . 3 8 0 3 , h 2 = 1 0 . 4 5 6 3 ,

h 3 = 3 . 7 0 1 5 , h a = 7 . 9 ,

h 5 = 8 4 . 8 9 9 , h 6 = 3 . 8 7 7 4 ,

h 7 = 3 . 9 5 , h 8 = 2 7 . 0 2 7 8 ,

h 9 = 2 1 3 . 6 7 4 8 , h i0 = 7 7 . 4 3 2 6 .

1 , . . . , l O a r e

S . N I C O S I A E T A L .

A c k n o w l e d g e m e n t

T h i s w o r k w a s s u p p o r t e d b y M P I f u n d s .

R e f e r e n c e s

1. Paul, R.P., Robo t M an i pu l a t or s : M a t he mat i c s , Programmi ng and C on t ro l , MIT P res s , Cambr idge(1981).

2 . H o l le rbach , J .M. , A r ecu rs ive L agrang ian fo rm ula t ion o f ma n ipu la to r dynam ics and a com para t ive

s tudy o f dynamics fo rmula t ion complex i ty , IE EE T rans. Sy s t e ms , M an and C y berne t ic s S M C - 1 0 ,

730-736 (1980).

3 . Luh, J .Y.S ., W alker , M.W . and Paul , R.P ., On- l ine com putat iona l scheme for mec hanical man ipulators ,

A S M E J . D y n a m i c s S ys t e m s , M e a s ur e m e n t, a n d C o n tr o l 102, 69-76 (1980) .

4 . Freund , E. , Fas t non l inear control with arbi t rary pole-placement for indus tr ia l robots and m anipulators ,

In t . J . Robot ics Res . 1, 65-78 (1982).

5 . Chen , Y . , F requency r es pons e o f d i s c rete - time robo t s ys tems - L imi ta t ions o f PD con t ro l l e r s and

improvem en ts by l ag -lead compens a t ion , Proc . IE EE In t . C on f . Robo t i c s and Au t omat i on , Rale igh , pp .

464-472 (1987) .

6 . L andau , Y .D . , Ad ap t ive con t ro l t echn iques fo r robo t ic man ipu la to r s - The s t a tus o f the a r t , I F A C

Sy mp . Robo t C on t ro l , Barcelona (1985) .

7 . Tomizuka , M. , Horow i tz , R . and L andau , Y .D . , On the us e o f M R A C techn iques fo r mechan ica l

man ipu la to r s , 2nd l AS T E D In t . Sy mp . M ode l l ing , Iden t if i ca t ion , C on t ro l and Robo t i cs , Da vos ( I 982).

8 . N icos ia , S . and Tom ei , P . , A d i s c re te - time MR AS con t ro l fo r indust r ia l r obo t s , P r o c . 7 th I A S T E D I n t.

Sy mp . Robo t i c s and Au t om at i on , Lugano, pp . 83-89 (1985) .

9 . Neum an , C .P . and Touras si s , V .D . , D is c re te dynam ic robo t mode l s , IEEE T rans . Sy s t e ms , M an , and

Cyberne t ic s SMC-15, 193-204 (1985) .

10. Mo naco , S . and N orm and-C yro t , D . , D is c re te time mode l s fo r robo t a rm con t ro l , I F A C S y m p . R o b o t

Control, Barcelon a (1985).

1 I . Gold s tein , H . , Class ical Mechanics , Addison-Wesley (1950) .

12. Ko iv o , J .A . and G uo , T . , Adap t ive l inea r con t ro l l e r fo r robo t ic man ipu la to r s , 1EE E T rans. Au t om at i c

C on t ro l AC-28, 162-170 (1983).

13 . Monaco , S . and Normand-Cyro t , D . , Sur le comma nde n on in terac tive des sys tkmes non l ineaires en

temp s discret, Lectures Note s in Co ntrol an d I nform ation Sciences No . 63 , Spr inger-Ve r lag (1984) .

14. Grizzle, J .W., Fe e dbac k L i ne ar iza t ion o f Di sc re te T i me S y s t e ms , Lec tu res No tes in Con t ro l and

Information Sciences No. 83 , Spr inger-Ver lag (1986) .



D I S C R E T E -T I M E C O N T R O L O F R O B O T IC M A N I P U L A T O R S 4 2 3

15. Jaku bczy k, B. , Fee dbac k l inear izat ion of d iscrete t ime systems, Sy s t e ms and C on t ro l L e t t . 9, 4I 1-416

(1987).

16. L ee, H. and M arcus , S .I ., On in pu t-ou tpu t l inear izat ion of d iscrete- t ime non l inear systems, S y s t e m s

and C on t ro l L e t t . 8, 249-259 (1987).

17. Nicos ia , S . , Nic ol6 , F . an d Lent in i , D. , D yna mic al contro l of indus tr ia l rob ots w ith e las t ic and

diss ipat ive jo in ts , 8 t h 1FA C T r i enn i a l W o rm C ongres s, K y oto (1981) .

(1989) - discrete-time modeling and control of robotic manipulators

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