A pole placement approach to multivariable control of manipulators.pdf

A POLE PLACEMENT APPROACH TO MULTIVARIABLE

CONTROL OF MANIPULATORS

Liu de man (Ph.D), Liu zong fu (Prof.)

Department of Automatic Control

Northeast University of Technology, Shenyang, Liaoning

People’s Republic of China

Abstract

atied by the pioneering work of Dubowsky and Desforges

161.

An investigation of methods for multivariable con-

trot systems based upon pole placement by state feed-

back has been carried out by author. The nonlinear

equations of motion for a complete arm have been

treated as linear equations with time varying para-

meters. After these equations have been linearized and

discretized, the system and input matrices have a

structure that permits simplification of the procedure

for finding the block companion form for the system.

Finally we have reduced the determination of the state

feedback gains to the evaluation of explicit expres-

sions requiring only two matrix multiplications and

two matrix subtractions. As in the case of the CO.-

puted torque method, it is necessary to evaluate the

tine varying parameters of system equations.

Our progress in developing computationally efficient

approach for computing the controller gains has led us

to the consideration of on-line pole placement as well

as gain scheduling. Because of the time-varying system

parameters it is desirable to use either a gain sche-

dule based upon a finely divided configuration space

or a parallel processor for on-line evaluation of the

time-varying parameters and controller gains.

Efficient approach for multivariable digital control

of robotic manipulators based upon pole placement by

state feedback are presented as an improved approach

of path control. Implementation of these approaches by

both gain scheduling and on-line computation is dis-

cussed. An example of the computations for a three-

link robot arm and computer simulation experiments are

given.

1. Introduction

Although robot manipulators have been used in in-

dustry for a number of years, their full capabilities

reach far beyond their present-day applications. At

the present time, industrial applications of robot

manipulators are mainly restricted to simple tasks

such as welding, paint-spraying and pick-and-place

operations [TI. In order to utilize robot manipulators

in performance any complex task such as assembling,

object tracking and grappling, it is necessary to al-

leviate some of the basic limitations of present ro-

bots. The basic difficulty in controlling a robot

manipulator arises from the fact that the dynamic

equations describing the manipulator motion are in-

herently nonlinear and highly coupled; because of dy-

namic coupling effects between the joints and varying

effective inertias of the links. The complexity of the

mathematical model of a manipulator makes the robot

control task a difficult and challenging problem. Over

the past few years, the control of robot manipulator

has been a very active area of research and two najor

design categories have emerged. The first category,

originated by the classic works of Paul [9l,Bejezy [I],

and markiewicz [2], covers robot control techniques

based on global Linearization robot dynamics by means

of a nonlinear feedback controller. In these techni-

ques, the nonlinearities in the robot dynamics are

cancelled out by the controller and hence the non-

linear feedback controller is of the same order of

complexity as the robot dynamics itself. The second

category of advanced robot control systems covers me-

thods based on adaptive control theory and was initi-

2. Linearized Model of Robot Dynamics

The dynamics of a multiple-joint robot manipulator

is described by a set of complex differential equa-

tions. In this section, we shalt model the behavior of

the robot for perturbations in the neighborhood of

some nominal operating point.

Consider a robot manipulator having n joints. Using

Euler-Lagrange formulation, the dynamic equations of

motion of the robot joints can be repersented by a set

of coupled nonlinear differential equations of the

general form [1,2]

M< e 1 e +N( 0, 6 )+G( 8 )=T

(1)

Where T(t) is the nX1 vector of joint torques, B(t>,

6 (t), and 6(t) are the nxl vectors of joint angular

positions, velocities and accelerations respectively.

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In equation (I), M( 8) is the nxn inertia matrix,

N( 8, 8 ) is the nX1 Coriolis, centrifugal and fric-

tional torque vector and G( 8 ) is the nX1 gravity

loading vector. The elements of H, N, and G are highly

complex nonlinear functions of the joint configuration

8, the speed of motion 6 and the payload.

Suppose that the nominal operating point of the

faniputator is repersented by the joint torque vector

T, and the joint poshition, velocity and acceleration

vector 8, 6, and 6. Now suppose that the joint tor-

qup vector is perturbed slightly by T(t>, that is T(t)

=T+ T(t); and let the resulting perturbations in the

joint position, velocity and acceleration vectors be

6(t), b<t), and 6'(t),* i.e., B(t)= 8 + 6(t), Q(t)

=6+b(t) and 8(1)=8+b(t). Assuming M(6+6)=

H( 6 1 for small 6 [g] ;xppnding N and G about the

operating point P= [T, 8,8 ] using Taylor series and

ignoring second and higher order terms in 6 and 8,

equation (1) can be Linearized as [3]

of motion. The corresponding state variable repre-

sentation is

X=G X+H U

1; H=[ :-]

(4)

Where

[bll)]; b=I

Ut)=

8<t) -A-'C -A-'B

The system is discretized by using Euler approximation

with a sampling interval of time T to obtain

X(k+ 1)=F X(k)+I utk)

(5)

Where 6=I+GT and LHT; ICR'"'"

We assuae that the velocity and position are measured

at each joint and can be used for state feedback. The

control taw is given by

Aii(t)+B6(t)+CG(t)=T(t)

(2)

Where the constant nxnmatrices A, B, and C are de-

fined by

Where K and F are nX2n matrices,and X,(k) is composed

of the increments of the desired path X., and the de-

sired velocity Xaw, i.e., X.(k)= [X,, Xa,]'

4. Transformation to Block Companion Form

and the matrix A is always symmetric positive-definite,

and hence nonsingular [I]. Equation (2) gives a set of

coupled Linear time-invariant differential equations

which describe the incremental behavior of the robot

dynamics for perturbations in the neighborhood of the

nominal operating point P.

Two alternative repersentations of the linearized

model of the robot are now possible; namely, the state

-space model in the time-domain and the transfer-func-

tion model in the frequency-domain.

The state-space model of the robot

defining the elements of 6 and 8 as

-variables and rewriting equation (2)

format

The computations for finding gains corresponding to

the desired pole assignment are reduced when the sys-

tem equations are transformed into block companion

fora.

In block companion fora the state equations are

given by

X, (k+ 1) = G. X. (k) + H, u( k)

(7)

is obtained by

the system state

in the standard

Where X,(k) is the state vector expressed in the

transformed coordinates and T. is the transformation

matrix defined by

(3)

This aodet is of the order 2n with the 2nx

] ,the nxl input vector T(t) and

state

H, = T,I= [o 11

We construct the required similarity transformation by

using a simplified version of the method of Shieh and

Tsay [61.

=ITcL

6(t)

vector/

the nX

8Ct)

1 output vectors 6(t), all in the time-domain.

Tc=[:::]

T,,T:

and

T., = [O I] [fi GI]-'

3. Discretized Equations of Motion

For digital control we must discretize the equations

For the robotic manipulator the system and input ma-

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trices are

6. Direct Computation of the Gain Matrices

-A-'CT I-A-'BT

The structure of the system, input and transforma-

tion aatrices attows us to compute the gains corre-

sponding to a desired pole assignment directly. Let

Where A-'T is an nxnnonsingular matrix [I], thus we

obtain

To, = [AT" 01

T,, = [AT-' AT-']

K= [K, K,] ; F= [F, F,]

K, = [I + A, A. -(A, + A )]AT-' - C

K. = [21- ( A, + A, )]AT-' -B

(15)

(16)

F, = A.AT-'

(17)

' 1

F. = A,AT-'

(18)

Therefore

Given the matrices A, B, and C of system parameters

and the matrices of desired poles, the computation of

the controller gains only requires two matrix multi-

p 1 i cat ions and two matrix subtractions.

Tc=[ AT-'

AT-' AT-'

It can be easily verified that

A-' T'

T;' =

7. Experiment simulation

-A-'T

In the section, we show that the gains can be computed

directly from a simple expression.

The purpose of this experiment is to demonstrate the

computation of the gain matrices for multivariable

control and to demonstrate the performance that can be

expected through the use of a centralized controtter.

1) Model Description. Control of the three-joint tink-

age shown in Fig.1 was studied using computer simu-

tation.

5. Pole Placement

The block companion form simplifies the determina-

tion of the gains corresponding to the pole assignment.

Substituting equation (6) into equation (?), yields,

Where F = F. T., X. (k) = T,X, (k)

First, the gain matrix K, (= [K,, Kc,]) is found such

that

det( z I - (G. - H. K, 1) = det( z I - A )

(10)

Where A is a Znx2n diagonal matrix with the desired

poles as entries on the main diagonal. The above equa-

tion can be written as,

1 )=det(zI-[ :' Ay]) (11)

Fig.1 Model of a robot arm

det(z1-

2) Equations of Motion. The kinetic and potential

energy of the system are

Ge, -&, 6. -&.

Therefore

G.,-K.,=-A,A.

G..

-Kc, =A, +A.

(12)

(13)

and

K,, =G,, + A, A.

K,,=G,,-(A,+A.>

Secondly, the gain matrix F, (= [F,, F..] is choosed

as I

1 1.

+-SI,( 2 1: 6: +'-<

4 e , + e, )* + 1. i,(cos e, ) e a( e .t e, )

+-%os*( 4 e. + e. )e: + i. i. COS( e. + e, )COS( e.) e:

+ I: COS*( e ~ ) 6: )

Where the components of the inertia tensors are with

respect to the center of mass and

1 1

(14)

HcFc=[

] =[

Therefore,

v = -'sine 2 .m.g-('sin( 2 e + e, )+ 1, sine, 1m.g

Where g is the gravity coefficient and equals approxi-

mately 9.8062m/sa. The dynamical equations are

F.,=O

;

F..=A,

The gains are then transformed back to the original

coordinates by post multiplying K. by T., i.e.,

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By combining sections 3, 4, and 5, we obtain

The above equations are obtained based upon tagrange's

equations which given as

All the elements of matrix A are only functions of

9. and e,, and all elenents of matrix B are linear

combinations of the products 0. and functions of 9.

and 9,. By dividing the workspace of 9. and 9. into

N regions of equal size, the gains FA and Fa of equa-

tions (17) and (18) can then be stored in a took-up

table for each region, because these gains involve

only position dependent terms. Also, the position-de-

pendent and constant terms of K, and K. can be stored.

At each sampling instant the elements of K, and K, can

be readily computed using the measured angular velo-

city. The number of time regions N determines the

trade-off between expected performance and memory re-

quirement. Better performance is achieved for larger N

at the expense of more computer memory [lo]

The Lagrangian is given by

Y=3-v

Where 9, is the ith generalized coordinate and T, is

the i th generalized noncoservative forces.

Then, we have for example

9. On-line Gain Computation

Let the nonlinear robot dynamics (equation (1)) be

approximated in the time interval t, < t< t, + At by

the 1 inear t ime-invariant nul t ivariab le mode L.

A' 6 (t)+ B' 6 (t)+ C' 8 ( t) = T ( t)

Where 9(t) and T(t) are incremental variatbes and

the model matrices [A', B', C'] are evaluated at the

operating point P' = [T(t, 1, 9 (t, 1,

equations

6 (t, )] using the

A'=[M],ij B'=[-],i ; C'=[- ae l.i

In this approach, the simplicity of expressions (15)-

(18) is utilized in computing the controller gains on-

tine during the operation of the robot. For a robot

under computer control with the sampling period T,,

the controller gains should be updated ideally every

T. seconds and then used to generate the control

action (6) for the subsequent interval. In practice,

however, it is sufficient to update the gains every

At (>>T,) seconds, where At depends on the speed of

robot motion and required accuracy. The value of At

determines the trade-off between dynamics performance

and on-line computational burden. Better performance

is achieved for smaller At, i.e. more frequent up-

dating, at the expense of more on-line computations.

a(N+ G)

B= - -

aN.

aN, aN,

I ab, ad.

___-

O P

in which

a N,

monstrate the performance of the control system in

following a specified trajectory for both position and

velocity of the tip of the arm. In order to exercise

control of the arm in terms of joint variables we

a 9,

a (N, + G, )

a (N. + 6, )

.c= I

a 9,

a 8.

a (N. + G, )

a (N, + G, 1

solved for the e,'s at points along the path speci-

a8,

a 8.

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cording to an algorithm given by Lin, Chang, and Luh

[Ill. In the experinents the arm starts fro. rest,

accelerates to a desired speed, and then returns to

rest.

-48

8. 2

9.4

8.6

1.1

Torque 3

Fig.4 Control effort for circular tracking <T=lOms)

Horizontal X-Y Plane

Fig.2 Tracking a circular path tT=lOms)

Tracking error

Horizontal X-Y Plane

Fig.5 Tracking a circular path (T=5ms)

9.8

8. 2

0.4

8.6

8. I

8.0

8.2

1.4

8.6

8.1

Tracking error

Velocity

Fig.3 Performance of circular tracking (T=10ns)

til.)

#.I 8. I

0.4

0.6

9.8

-69

8. I

0.4

0.6

8.8

Torque 1

Veloci ty

Fig.6 Performance of circular tracking (T=5ms)

-6.

8. 2

8.4

8.6

8.1

-68

E.? 8.4

9.6

0.:

Torque 2

Torque 1

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