On robust decentralized control of a 5-axis robot

R~Jbotic s & ('t~mputer-lnte#rated Printed in Great Britain

•

Manulac turm~¢, Vol. 6, No. 4, pp. 309-316, 1989

0736-.5845.,89 $3.00 + 0.00 c. 1990 Pergamon Press pie

Paper O N R O B U S T D E C E N T R A L I Z E D C O N T R O L O F A 5-AXIS R O B O T M . J A M S H I D I * a n d C.-S. T S E N G CAD Laboratory for Systems/Robotics, Department of Electrical and Computer Engineering, The University of New Mexico, Albuquerque, NM 87131, U.S.A.

In this paper, a dynamic model of a robot manipulator is first derived based on the Euler-Lagrange equation and a state-space representation is derived for it. A number of non-adaptive robot control schemes are then reviewed. A robust decentralized control is then proposed for a 5-axis robot manipulator. Numerical simulation results are presented to verify the theory.

nature of manipulator dynamics and nonlinear dynamic terms such as Coriolis, centrifugal, and gravity torques. The design of an overall controller to accommodate interactions and nonlinearities in a multi-link manipulator is often very difficult and typically requires complete information about all the links in a multi-link manipulator. Most of the present day industrial robot control systems consist of a number of independent position servo loops with constant pre-defined gains to control each joint angle separately. ~ The controller neglects the changes of the parameters and the loadings. The result is reduced speed of operation and limited precision of the end effector. It is evident that to substantially increase the manipulator capabilities, it is essential to apply advanced control concepts to robot manipulators. One of the basic control schemes is the computed torque technique 2-4 based on the Lagrange-Euler (L-E) or the Newton-Euler (N-E) equations of motion. Basically the computed torque technique has a feedforward and a feedback component. A PID and command matching controller based on the linearized model of the robot dynamics is presented in Ref. 5. A command matching controller consists of feedforward and feedback components. The feedforward component ensures regulation and the feedback component allows desired stabilization. Due to the physical configuration and unknown (imprecise) robot parameters, a decentralized robust controller is proposed here and is applied to a 5-axis robot manipulator model. Decentralized robust control is particularly well suited to robot manipulator applications, because modeling them is highly complex and they contain unknown parameters. The structure of this paper is as follows: In Section 2, the mathematical model of an n-link robot, and its perturbed linear state equation is given. Two classes of control schemes of PD and PID are reviewed in

1. I N T R O D U C T I O N In the 1960s, George Devol demonstrated what was to become the first U N I M A T E industrial robot, a device combining the articulated linkage of the teleoperator with the servoed axes of the numerically controlled milling machine. Since then, the use of the industrial robot, along with computer-aided design (CAD) systems, and computer-aided manufacturing (CAM) systems, characterize the latest trends in the automation of the manufacturing processes. By and large, the study of the mechanics and control of manipulators is not a new science, but merely a collection of topics taken from "classical" fields. Mechanical engineering contributes methodologies for the study of machines in static and dynamic situations. Mathematics supplies tools for describing spatial motions and other attributes of manipulators. Control theory provides tools for designing and evaluating algorithms to realize desired motions or force applications. Electrical engineering techniques are brought to bear in the design of sensors, circuits, and interfaces for industrial robots and computer science contributes a basis for programming these devices to perform a desired task. The problem of dynamic control of multi-link robot manipulators has been a very active area of research in recent years and several solutions have evolved. Basically, the dynamic control problem can be stated as follows: What torques must be exerted by the robot joint motors at every instant of sampling time in order to move the manipulator links from their present configuration to a new desired configuration along a specified trajectory? Although the control problem can be stated in such a simple manner, its solution is quite complicated because of the highly interactive

* AT & T Professor of Manufacturing Engineering and Director of CAD Laboratory.

309

310

R o b o t i c s & C o m p u t e r - l n t c g r a t e d M a n u f a c t u r i n g • V o l u m e 6, N u m b e r 4, 1989

Section 3. A robust decentralized control scheme is proposed in Section 4 for a 5-axis robot manipulator and its simulation results are given. Conclusions will be made on the proposed scheme in Section 5. 2. D Y N A M I C M O D E L O F AN n-LINK R O B O T The mathematical model of a robot manipulator having n degrees of freedom can be obtained from the well-known Lagrange-Euler (L E) equations of motion in the following form: (1)

M(q(t))~l(t) + n(q(t)), dl(t)) + ,q(q(t)) = z(t)

where q e R" x ~ is an angle or a distance for revolute or prismatic joint; q, q • R n x ~ are the velocity and acceleration of q, respectively; M ( q ) • R ~ × 1 is the symmetric positive-definite inertia matrix; n(q. ?1) • R" × ~ is the Coriolis and centrifugal force vector, y(q) • R ~x ~ is the gravitational force vector; and z ( t ) • R "× 1 is an applied torque/force vector. The elements of M(q), n(q, q), and y(q) are highly coupled, nonlinear, and slowly time-varying functions of q and q. After this point, the time argument t will be dropped for convenience. Now, in order to derive a linearized perturbed model around a nominal trajectory, suppose that the control torque vector is perturbed slightly by At, i.e. r = "~ + Az, and let the resulting perturbation in the joint angle vector be Aq, i.e. q = q + Aq, where denotes the nominal trajectory. Then from the nonlinear model (!), we have ^

M ( q + aq)(q + a~) + n(~ + a q , ~ + a~)

+ 9(q + Aq) = f + At.

(2)

Expanding M, n, and 0 about the nominal trajectory Q using the Taylor series, ignoring the second- and higher-order terms in Aq and Aq, and also assuming that M ( q + Aq) ~c M(c~), from (2) we can have M(q)(~ + Aq) + n(q, ~) + F t A q + EAdl + ,c1(~) + F 2 A q = f + Az

F1

F2 =

=

3q O"

M(q)Aq + EAq + (F1 + F2)At~ = Az.

Az = K p e + K ~

(4)

Let F = (F t + F2), D = M(t~), then (4) can be rewritten as DA~t + EA?t + F A q = Az.

(5)

The above equation can be put into a state equation of the following form: =

-D-"F

-O-'EJLA4 j +

[on] ,

AT

(6a) = ""

22

+ A1

A2

.¢c = A x + Bu

x2

u

3.1. PD control

The control structure of PD control is very simple, and is described by:

With the above relations, (3) can be written as

dt Aq

3. REVIEW O F R O B O T C O N T R O L M E T H O D S Compared with many control problems, manipulator control is quite complicated. The dynamic equation for the manipulator is inherently nonlinear. This means that linear control theory cannot be applied directly. Although there are many proposed solutions to the nonlinear problem, one ends up using methods from linear control theory. To solve this control problem, a linearization procedure can be performed first. When nonlinearities are not severe, local linearization may be used to derive linear models which are approximations of the nonlinear equations in the neighborhood of an operating point. If the system does not operate over a small range in its variables, then as the system moves, the operating point can be moved along with it. At each new operating point, a new linearization is performed. The result of this sort of moving linearization is a linear but time-varying system. Although this quasi-static linearization of the original system is useful in some analysis and design techniques, it are not used in the present work. After obtaining a linear time-invariant model, two of the present industrial robot control systems will be introduced. The relationship between performance and expense of a robot is a kind of trade-off. For economic reasons, manufacturers of industrial robots design control systems with simple control laws. The most popular control structures for modern industrial robots are PD and P I D controls. Here, some properties of these two types of control systems are considered.

(3)

where F t • R n×", E • R" ×" and F2 • R ~×" are given as E =

where A • R 2n × 2n and B • R 2n × n are system and control matrices; x ~ [Aq, Aq] r and u = Ar are state and input vectors, respectively. System (6) is completely controllable.

(6b)

B2

(6c)

(7)

where e = Aqa - Aq, and Aqe is the desired trajectory of Aq. Since the matrices, B2, At, and A 2 in (6b) are arbitrary, we simply choose B 2 = I , A 1 -----0 , and A 2 = 0, then rewrite (6b) as:

,IF q+ [~]A,. [~]=[0 0 0JLAqJ

(8)

Since (8) is a decoupled system, we can separate it into five 2 x 2 subsystems, and design the controller for each subsystem, respectively. Each subsystem has the same matrix form which can be described by (8). Considcring the PD control structure in Fig. l and applying (7) to subsystem (8) with B2 = l, A , = A 2 = 0 being now scalars, the resultant closed-loop system

Robust decentralized control of 5-axis robot • M. JAMSHIDIand C.-S. TSENG

311

loop system becomes: Aq

1

y=

Fig. I. Structure ofa PD controller for a 5-axis robot with one reference input.

C

BE. f:

[1

o

S

1

I

I

T,me(see)

000F.Of: OIOE,O

I

1

I

where A~a is the desired trajectory of A~. N o w arbitrarily choosing K n = 25, K a --- 10, Aq = I r(t) (unit ramp input), and A~d = l u(t), (unit step input), and solving (I 1), we have the results given in Fig. 4.

I

0 50E,0~

±qd

Aq d

g

000E OIOEI., I~_.

I

,0~

I

1

I

I

Ttme (see)

I

I

0 50E*OI

Fig. 3. Structure of a PD controller for a 5-axis robot with two reference inputs.

0 50E*01~-

I

1

I

I

I

'

Time (see)

[

I

L

Kd

~

.

000E+O0

0 50E~OI

Time (sec)

o 5,E.I -

can be described by:

-Kp

-

0 50E+OI

Fig. 2. Responses of PD control for a S-axis robot with one reference input.

AO =

(ll)

3.2. PID control P I D control has almost the same structure as P D control except for an additional integration term in

it

000E,CCkl

0]IAA: ]

Ag/ + Kp

(9)

and y=[l

0]

A4 "

(to)

Since Kp and Kd are arbitrary constants, we simply let Kp = 25, Kd = 10 (poles locate at --5, --5), and Aq = l u(t) (unit step input), then the d y n a m i c robot model (10) is simulated with the plot the response of each state, input, and output in Fig. 2 using the T I M D O M / P C package. 6 If one desires to control not only the trajectory of Aq, but also the trajectory of A4, the control structure of Fig. 1 can be modified to that of Fig. 3. Then applying (7) to the subsystem (9), the resultant closed-

.... i .... ; .... 71

000E¢O0

000E,O0

fl

050E~Oi

Time {sec)

I

I

I

I

I

Time (see)

I

L

I

I

0 50EtOI

Fig. 4. Responses of PD control for a S-axis robot with two reference inputs.

Robotics & Computer-Integrated Manufacturing • Volume 6, Number 4, 1989

312

the P I D control law. The structure of the P I D control law can be described by: AT = Kpe + K i f e d t

0 47E,01

(12)

+ KdE

where e = Aqn - Aq. Using the P I D control structure in Fig. 5, and applying (12) to subsystem (9), we have:

.... : .... 7-:

- 59E-0-a 0 0 0 E *00

........

,..........

Time (sec)

0.50E ,01

1

i 0 50E ¢01

0 'OE,OI

A0 = A/~

0 -K,

+

0 -Kp

1 -K~

[o] [o] 0

Az~ +

Ki

0

Aq A¢~ c

Aqd

Kp 000E-OC

y=[0

1 0]

Aq

(13)

0 IC,E*C~

M where Az

A0

g

= fAq dr, and Az. = ~Aq~dt.

=

0

A0

0

-- K i

+

1

-- Kp - Kd

0

Az~+

Kp

1 0]

I

I

0 Kp

Aq .

/ I

~ , i T i m e (sec)

I 0 5OF.0~

A0~

~q

(14) D i i~ :LI "~.,/

Using Kp = 25. Ki = 1. K, = 10. A~, = lu(t) (unit step input). Aq, = Ir(t) (unit ramp input), and Az, = lp(t) (unit parabolic input), we have the results given in Fig. 8. 4. D E C E N T R A L I Z E D R O B U S T C O N T R O L In this section, a decentralized robust control is introduced, and is applied to a 5-axis robot. The study of decentralized systems becomes important when one attempts to design the controllers for large-scale systems. One of the most basic problems which arises in

~q

Aqd

1

Fig. 6. Responses of PID control for a 5-axis robot with one reference input.

A0

?

I

T~me (sec)

0 00E.OC

All

Aq,+

I

I

Aq

[o] iDol [o] 1 K i

y=[0

O

1

[; (? ~:).~:;,f

•

'~qd

÷

N53.:

d-

Fig. 7. Structure of a PID controller for a 5-axis robot with two reference inputs.

this class of systems is the 'decentralized robust servomechanism problem', i.e. to find a decentralized robust controller so that the outputs of the system asymptotically tracks the given reference inputs independent of any external disturbances which affect the system, and independent of any variations in the plant parameters, and gains of the system. In Section 4.1, we focus our attention to the decentralized robust control of a known plant, while in Section 4.2 a decentralized robust controller for a 5-axis robot is designed.

D

Fig. 5. Structure of a PID controller for a 5-axis robot with one reference input.

4.1. Decentralized robust control of a known plant First, let us consider the decentralized stabilization problem. A linear time-invariant multivariable system

Robust decentralized control of 5-axis robot • M. JAMSHIDIand C.-S. TSENG O ,3E*02

O00E.O0 0 5IE~01~

T,me (seE)

000E.O0 0 54E+01 "

T,me (see)

0 50E,OI

',

0 50E,OI

313

where w • R q is a disturbance vector which may or may not be measurable, ei is the error in the system which is the difference between the output y~ and the specified reference input y[cf and the matrices A, Bi, E, C~, D~ and F~ have appropriate dimensions. It is further assumed that the plant matrices (A, B, C) are subject to variations leading to [(A + 6A), ( B + 6B), ( C + 6C)], where 6 A , fib and 6 C represent plant variations which may be, for the robot manipulator, due to changes of the gripper load. The following theorem, due to Davison, 7 sets forth existence conditions and robust strategy for the system (18)-(20). Theorem 1

Necessary and sufficient conditions for the existence of a robust linear time-invariant, decentralized controller for (18)-(20) such that lim e(t) = 0 for all measurable t~co

000E,O0

~

Time (secl

i i

o 50E,OI

or unmeasurable disturbances co described by ~o = ~ l z t, 21 = P l a t , with ((~l, P I ) observable and zl(0) may or may not be available, for all specified Yrcf described by Yref G p - G ~ z z 2, z2 = P 2 z 2 , where (t~2, P2) is observable and z2(O) is known, and such that the controlled system is stabilizable would be: =

1.

Fig. 8. Responses of PID control for a 5-axis robot with two reference inputs.

2. with p local control stations is described by:

(Cm, A, B) has no unstable fixed modes, i.e. none of the closed-loop poles unaffected by decentralized control lie on the right-half s-plane. The closed-loop poles unaffected by decentralized control for the system

P

2 = Ax + ~

[: ,.J [ lt

(15)

Biu i

i=1

(16)

Yi = C i x , i = ! . . . . . p

where x • R" is the state, u i • R m' and y~ • R" are the input and output, respectively, of the ith local control station, i = 1. . . . , p. The 'decentralized stabilization problem' is to find p local output feedback control laws with dynamic compensation to stabilize the resultant closed-loop system. Suppose that the feedback control laws are described by:

i = I..... p

(17)

where z~ • R"' is the state of the ith feedback controller, v~• R m' is the ith local external input, and S;, R~, Q~, K~ are real constant matrices with proper dimensions. An important case of the dynamic system is when both the plant and the output measurement are subject to disturbance, i.e. consider a linear time-invariant system, P

Yc = A x + ~, Biu i + E w

Then, any decentralized robust controller which regulates (18)-(20) must have the following structure: ul

i

K o ( i + Kith, i = 1. . . . , p

where (i is the state vector of a 'decentralized stabilizing c o m p e n s a t o r ' described by = a°¢, + 6:7, +

Zl = Sizl + Riyi Ui = Q i z i + K i y i + vl,

j = ! . . . . . q not contain 2j, respectively, where Cm = (C r C r . . . . . C r ) represents the output measurement matrix.

(18)

i=1

Yl = c i x + Diul + F i w

(19)

el = Yi - y[cf

(20)

with Yl = Yi - Diu~ and Ais are matrices of appropriate dimensions and rh is the output vector of a 'decentralized general servocompensator' which satisfies rh = F i r h + Fiei

where (F~, V,.)are arbitrary controllable pairs and ei is defined by (20). The following algorithm provides a simplified procedure to fine a robust controller with polynomialtype disturbance and a known plant. Algorithm 1

Robust controller with known plant, a

314

Robotics & Computer-Integrated Manufacturing • Volume 6, Number 4, 1989

Step 1: The first augmented states are formed from x and ~l=e

If we choose Kt and K z to be diagonal matrices, then we can decouple the system into five subsystems. Each system can be described by:

u = ~q Ktll with K = [Ta(1, 1)] ÷, where ' + ' represents the pseudo-inverse. /"1(I, 1) is the steady-state tracking gain matrix for first controller, defined, in general as,

~Ci(J.l - A)-IBI q- Di i =j T~(i,j) = [ C i ( 2 I A)-~Bi i #j" D~ is a matrix relating control u~ the output, i.e. y~ = C~x + D~u~ and scalar cq > 0 is obtained by a one-dimensional search such that the eigenvalues of the closed-loop matrix

Acl =

~IDK

are at appropriate locations on the lefthalf s-plane for ' m a x i m u m speed of response', i.e. desired settling time, overshoot, etc.

Step: 2:

Let u = ~t2Kq2, ~2 = rh, and apply u to the augmented closed-loop system of Step 1 and find ~tz through the onedimensional search to find desirable eigenvalues for the second augmented closed-loop matrix

[ A ~t*BK ~2BK ] Acl = C ct*DK ctzDK 0 1 0 where ~t~' represents the desired value of cq fixed at the end of Step I.

Step k: In a similar fashion let u

=

otkKqk, ~k =

qk-~ and apply u to the (k - 1)th closedloop system such that ctk is obtained for a set of desired eigenvalues of Ac~_ ,. The overall control is thus given by

M"

=

y=[1

[o k,

0 F< k2JLA~J +

0]

[o] [o] AT+

?q A0

'

MATRIX A 0.000E + 00 0.100E + 01 .200E + 02 -.500E + 01 MATRIX B (1) 0.000E + 00 0.100E + 01 M A T R I X C (1) 0.100E + 01 0.000E + 00 M A T R I X D (1) 0.000E + O0 M A T R I X K (1) 0.100E + 01 NO. O F R O O T S O F D E L T A ( S ) L = 1 R O O T S O F D E L T A ( S ) F O R X (1) = 0 T(1, 1 , 1 ) = . 0 5 R A N K O F K (1) = 1 SPECIFIED Lambda L = 1 T U N I N G O R D E R (SI . . $ 2 ) < = (1 . . N C I ) s (1) = l R A N K O F T(I, 1, 1) IS 1 T H E R E EXISTS D E C E N T R A L I Z E D R O B U S T CONTROL DISTURBANCE W = 1 E(1, !) = 0 E(2, 1 ) = 1 F(I) = 0 Y(id) = ! THE DESIRED SPEED FOR STABILITY A L P H A IS - 5

j=!

y = [l

ol

[

u3

where w is the disturbance vector and AQ represents joint position vector for all 5 axes.

(21)

To design the decentralized robust controller for the subsystem (21), we first check the conditions in T h e o r e m 1 and if the conditions all hold, then we apply Algorithm 1 to carry out the decentralized robust control. A p r o g r a m which realizes the algorithm was written for an IBM P C / X T and the results, using k~ = - 2 0 , and k 2 = - 5 arbitrarily, are shown below:

k . = K Z

4.2. Decentralized robust control of a 5-axis robot F r o m (13), the dynamic equations of general model can be described by:

w

ALPHA EV(I) = EV(2) = EV(3) =

= .5 -2.213712 +j 3.544638 - 2 . 2 1 3 7 1 2 + j - 3.544638 -.5725746 +j 0

ALPHA EV(I) = EV(2) = EV(3) =

= .7 - 2.075059 + j 3.488124 - 2.075059 + j - 3.488124 -.8498812 + j 0

315

Robust decentralized control of 5-axis robot • M. JAMSHIDIand C.-S. TSENG at ALPHA EV(I) = EV(2) = EV(3) =

,

~

~,

-!i~7~(o~~" ~

= .9 -!.921322 +j 3.44401 - 1.921322 + j - 3.a.4401 - 1.157356 + j 0

y

^

;! VV-I_ :k. aQd

ALPHA EV(I) = EV(2) = EV(3) =

= -

1.1 1.755317 + j 3.419099 1.755317 + j - 3.419099 1.489367 + j 0

ALPHA EV(1) = EV(2) = EV(3) =

= -

1.325 1.563087 + j 3.420369 1.563087 + j - 3.420369 1.873841 + j 0

Fig. 9. Structure of a decentralized robust controller for a 5-axis robot.

0

12Eo01

t""\ / It

ot

•

- 72E.00 ~'" ~ O00E,O0

Time (sec)

050E.O,

0 IOE ¢01

ALPHA EV(I) = EV(2) = EV(3) =

= 1.375 - 1.520903 + j 3.424966 - 1.520903 + j - 3.424966 -1.958193 +j 0

i 000E.O0 0,0E.O,~

~

ALPHA EV(I)= EV(2)= EV(3) =

= !.425 - 1.479242+j 3.431042 - 1.479242 + j - 3 . 4 3 1 0 4 2 -2.041518 + j 0

ALPHA EV(1) = EV(2) = EV(3) =

= 1.475 - 1.438214 + j 3.438492 - 1.438214 + j - 3.438492 -2.12357 +j 0

O00E.O0

A t = ~1T1(1, 1)- It/l where /'1 (I, 1) - t = 0.05. Thus, the resultant c l o s e d - l o o p system b e c o m e s :

Aq,

[ =

0 ji q]

1 720

O5

r/l T1 (1 , 1) - I

I

t

l

I

I

T,me (sec)

,

T,me (sec}

I

I

I

I

I0 50[E.O~

0.50E.OI

Fig. 10. Responses of a decentralized robust control for the 1st axis of a 5-axis robot without disturbance.

N o w we have the following c o n t r o l law for the first axis:

aql ]

I

Aql

T h e structure of d e c e n t r a l i z e d r o b u s t c o n t r o l in (22) is s h o w n in Fig. 9. S i m p l y c h o o s i n g ct, = 1.325, disturbance w = 0, a n d solving (21), we have results in Fig. 10. If the system has a d i s t u r b a n c e w = 1, the results are s h o w n in Fig. 11. Clearly, c o m p a r i n g the response of the c l o s e d - l o o p system with or w i t h o u t d i s t u r b a n c e , the o u t p u t always t r a c k s the reference i n p u t independently of the disturbance.

L r/~

-[10] Aqt ] y=l-I

0 01 Aql rh

.

(22)

5. C O N C L U S I O N S C o n t r o l of r o b o t m a n i p u l a t o r s has received a great deal of a t t e n t i o n in recent years. V a r i o u s classes of c o n t r o l a l g o r i t h m s are a p p l i e d for that purpose. In this paper, a n u m b e r of m u l t i v a r i a b l e c o n t r o l m e t h o d s such as P D a n d P I D are reviewed first a n d then a r o b u s t decentralized c o n t r o l is designed for a 5-axis robot. The n u m e r i c a l s i m u l a t i o n indicates that the

316

Robotics & Computer-Integrated Manufacturing • Volume 6, Number 4, 1989 self-tuned r o b u s t c o n t r o l l e r p r o p o s e d can h a n d l e disturbances in r o b o t dynamics.

0 12E.01

REFERENCES

N\ -

\

\

I ~"'1~ "7 5E *0 0 0 O0 (.00

...~1

~ I Time (see)

I

I

I

I 0 50E~OI

0 fOE~Ol

O00E*O0

Time (sec)

0 50E,OI

/

°I/, 000E*O0

I

I Time

I

I (see)

I

I

I 0 50E,OI

Fig. 1I. Responses of a decentralized robust control for the 1st axis of a 5-axis robot with disturbance w = 1.0.

1. Luh, J. Y. S.: Conventional controller design for industrial robots--a tutorial. IEEE Trans. Systems, Man Cybernetics SMC-13:298 316, 1983. 2. Paul, R. P.: Modelling, trajectory calculation and servoing of a computer-controlled arm. Stanford Artificial Intelligence Laboratory, A.I. Memo 177, 1972. 3. Bejczy, A. K.: Robot arm dynamics and control. Technical Memorandum 33 669, Jet Propulsion Laboratory, 1974. 4. Markiewicz, B. R.: Analysis of the computed torque drive method and comparison with conventional position servo for a computer-controlled manipulators. Technical Memorandum 33-601, Jet Propulsion Laboratory, 1973. 5. Seraji, H., Jamshidi, M., Kim, Y. T., Shahinpoor, Mo: Linear multivariable control of two-link robots. J. Robot Syst. 3 (4): 349-365, 1986. 6. Jamshidi, M.: T I M D O M / P C -A muitivariable control software package for IBM/PC. CAD Lab. Systems/Robotics, Report LCAD 85-04, EECE Department, University of New Mexico, Albuquerque, NM, July 1985. 7. Davison, E. J.: Decentralized robust control of unknown systems using tuning regulators. IEEE Trans. Contr. AC23, 35-47, 1976. 8. Jamshidi, M.: Larqe-scale Systerns .Modelin,q and Control. New York, Elsevier, 1983. 9. Tseng, C.-S.: On computer-aided decentralized robust control of a five-axis robot. M. S. Thesis, CAD Lab. Systems/Robotics, EECE Department, University of New Mexico, Albuquerque, NM, May 1987.