Linear state feedback control of rigid-link manipulators

•

LINEAR STATE FEEDBACK CONTROL OF RIGID-LINK' MANIPULATORS

by

TECH

iSC!E

HUGE~CH

_{'I ,-' ~}

l 0 lFT

r •. , IIEj( lUC: Tir ~ • I' . ) \" , • , . , • • . . . .

. ' .).f~,·llt ..

Ki. ,I' ~r'~·· 9 1 - eLFT

Da v i d F. Go 11 a

o

6 NO'J. 1979

LINEAR STME FEEDBACK CONTROL OF RIGID-LINK MANIPULATORS

by

Acknowledgement

The au thor wishes to thank Dr. P. C. Hughes for his invaluable comn:ents and his support of this thesis, Dr. S. C. Garg, for his guidance and much appreciated criticaJ. evaJ.uations, and Dr. R. Ravindran for his timely discussion on the· aspects of manipulator control. The author aJ..so wishes to thank Mrs. Laura Quintero for preparing the figures.

Summary

Linear state variable feedback controllers are designed for rigid-link manipulator arms. Two types of feedback schemes are considered, general rigid control (GRC) which allows for both interjoint and intrajoint feedback, and independent joint control (IJC) which allows for only intrajoint feedback.

The design seheme consists of arbitrary closed loop pole placement. Pole assignment algorithms are developed for both GRC and IJC feedback control

sehemes.

The design procedure is augmented by root locus analysis. The configura-tional dependence of the system inertia matrix results in the movement of the

systempoles with changes in the arm joint angles.

Two controllers, IJC and GRC, are designed for a preliminary two-link rigid model of the space shuttle manipulator arm carrying a full payload in planar motion. The two models are simulated and the controllers compared.

2.

3.

4.

5.

6.

7.

8.

CONTENTS Acknaw1edgement Summa.ry Notation INTRODUCfION MATHEMATICAL IDDEIS STABILITY CONrROLLABILITY

POLE ASSIGNMENT ALGORITHM3 INDEPENDENr JOINr CONTROL CONrROLLER DESIGN

SUGGESTIONS FOR FUI'URE RESEARCH

APPENDIX A - EQUATIONS OF MOTION OF A TWO-LINK REMOTE

MANIP~OR WITH PAYLOAD IN PLANAR MOTION

REFERENCES ii iii v 1

3

7 11

13

22

33

49

52

58

r - - - -- - -

---Notation

Upper Case Roman A B C

C

F I J. ~ K

K

Li M MO

-

,

MC 0

,g,

R T, Ti V system matrix input matrix

rate feedback gain.matrix

nondimensional. rate ;feedb.ack gain matrix state feedback matrix

identity matrix

polar moment of 'inertia of the i th link ab out the i th ,-J oint position feedback gain matrix

nondimensional. position feedback gain matrix

length of ith link inertia matrix

portions of inertia matrix (see Appendix A) zero matrix

matrices defined where used kinetiq energy

Lyapunov function, potential. in Lagrange 's formulation

Lower Case Roman

a. ~ c. ~ f, g k. ~ m. ~ m .. ~J P

nondimensional. characteristic equation coefficients

nondimensional. elements of diagonal. rate feedback gain matrix functions defined where used

nondimensional. elements of diagonal. position feedback gain matrix

f . th l'nk mass 0 ~ ~

nondimensional. elements of inertia matrix

x state vector

centre of mass position coordinates

z difference between angular configuration and desired angular

configuration

upper Case Greek

A À I

Lower Case Greek

column veqtor of joint angles (angular configuration)

input vector of des~red joint angles (desired angular

cOnfigura-tion Pi damping ratios À,

~

eigenvalues W . . l. frequencies w_i nondimensional frequencies Script

minor of

M

formed by deleting the ith row and column of

M

'r

1. INTRODUCTION

The digita.l. computer, compared to the human mind, is a very rudimentary device capab1e of dup1icating only the simplest aspects of man' s abi1ity to caJ:.culate and reason. Yet the digita.l. computer is a powerful too1 for it can perform. complex ca.l.culations bi11ions of times faster than man wi th much higher accuracy and re1iability. Simi1ar1y, manipulators, compared to human limbs, are very basic mechanica.l. devices capab1e of dup1ic.ating on1y a tiny fraction of man' s natura.l. dexteri ty. However, 1ike computers, manipulators are capab1e of vast1y outperforming humans in the tasks that they can do. Manipulators can handle llDlch heavier 10ads and can operate in environments that man himse1f could not withstand.

Remote manipulators, or te1eoperators, are manipulators that are separated from their human operators by same form. of barrier. For examp1e, a manipulator handling radioactive fue1 may be separated from its operator by a concrete wa.l.1 severa.l. feet thick. Remote manipulators are usual1y manipulators that operate in an environment that is hosti1e to man. The barrier separating the manipulator and the operator is usual1y a protective

shie1d for man.

Remote manipulators are finding increasing1y important ro1es in

industry and research [Ref. 1]. In recent yeaxs there has been much attention given to the deve10pment of remote manipulators for app1ications in space [Ref. 2]. Of particular importance is the deve10pment of the space shuttle remote manipulator system.

The space shuttle remote manipulator system (SRMS) is a m:!chanical arm that is to be mounted on the side of the space shuttle cargo bay and

used for 10ading and unloading as wel1 as or;i.enting such payloads as low earth orbit satellites. Due to weight limitations the SRMS consists of long slender links which are considerably f1exib1e. The SRMS is illustrated in Fig. 1.

The control of manipulators is perhaps the most important aspect of manipulator design for it is the abi1ity of manipulators to perform. desired tasks that gi yes them their great va.l.ue. There are severa.l. references

avai1able on the design and control of flexible manipulators [Refs.

3, 4, 5,

6] •

One control system design seheme applied to flexib1e manipulators which is cammon to the above references is a feedback controller based on a rigid-link model.

Two distinct types of such feedback control are defined fRefs.

3, 4,

5]:

gener al rigid control (GRC) which allows for both interjoint and intra-joint feedback, and independent intra-joint control (IJC) which allows only for intrajoint feedback. These control schemes are known in control systems literature as centralized and decentralized control, respective1y. Since 1inear feedback controllers based on a rigid-1ink model are sametimes applied to f1exib1e manipulators as we11, there is a definite need to unq.erstand ful1y the control system design methods for rigid-1ink manipulators. The Present Work

The purpose of this technica.l. note is to deve10p 1inear state variable feedback control techniques ·for rigid-1ink manipulators. This is accomplished

for both GRC and IJC control schemes. In addition, a comparison of the two methods is· launched by means of a numerical example. The example used is a preliminary two-link model of the SRMS in restricted (planar) motion carrying its maximum payload.

The dynamic modelling of a manipulator is examined in Section 2. The equations of motion developed are a system of coupled, nonlinear, second order differential equations. Linearization is performed in two steps:

rate linearized and totally linearized medels. The three models - linearized, rate linearized and nonlinear - are utilized in the con trol system design procedure for pole placement, root loci plots and digital simulation, respec-tively . .

Sections 3 and 4 discuss stabili ty and controllabili ty of the nonlinear equations of motion. These concepts are of considerable interest since they are the foundation stones of control system design.

Perhaps the chief contribution of this note is the study of pole placement algorithms for both GRC and IJC schemes, in Sections

5

and

6,

respectively. Several pole placement algorithms are defined for the GRC method and their special properties are noted. In addition, relationships between pole placement algorithms are derived which characterize the infinity

of solut~ons to the GRC pole assignment problem. The IJC pole assignment problem is solved for a two-link manipulator, and questions of both existence

and uniqueness are answered. The IJC pole assignment problem for an n-link manipulator still remains an open problem.

The pole assignment algorithms developed here are applied to a numerical example in Section

7.

Both GRC and IJC controllers are designed for a prelim-inary two-link model of the SRMS in planar motion with maximum pay:t"oad. The controller design procedure is augmented by root loci plots utilizing the rate linearized model. Finally, the nonlinear model is simulated with the chosen controllers and the results compared with the linearized case.

2. MATHEMATICAL MODELS

A general rigid-link manipulator may be modelled as a chain of

hinge-connected rigid bodies, with one end of the chain inertially fixed as illustrated in Fig. 2. Since the bodies are hinge-connected rather than point-connected, there is only one degree of freedam for each link. For most manipulators the root joint is actually fixed in inertial space but for the SRMS (Shuttle Remote Manipulator System) this is not the case. The root joint of the SRMS is

attached to the shuttle orbiter which has three rotational and three transla-tional degrees of freedom. However, the orbiter inertias may be considered large in comparison with the inertias of the arm links. Hence, the assumption that the root joint is inertially fixed is a good starting point, at least for small payloads.

The equations of motion for a rigid-link spatial manipulator may be derived using Lagrange' s formulation. Starting at the root and werking towards the tip, the energy components for each link may be found most directly using a link-fixed reference frame. The equations of motion for a two-link arm in planar motion are obtained in Appendix A.

FIG. 2 THREE - LINK SPATlAL MANIPULATOR

WITH END EFFECTOR

- -- - -- - - -- - - -- - - ---.

The eq~tions of motion for an n-link spatial manipulator may be written in matrix form as

where

and

Z

is a column vector of joint angles;

u is a column vector of generalized torques.

(2.1)

In general, u is the sum of control torques and environment al torques. For the fOllowing analysis, however, it will be assumed that environmental torques are negligibly small compared to control torques.

In order to maintain the option of .regulation about any desired configuration within a configuration or task space, the constant vector

lD

is introduced into the equations of motion by the following substitution:

!

=2 - 2D

(2.2)

The equations of motion become

Mz=Nz+u where

and

The vector 1D which nOrmally represents the desired angular configuration may also be regarded ~s an input parameter vector.

The inertia matrix, ~(ZD +

!),

has the following important properties for every possible configuration:

(i)

(ii)

~ is symmetric;

M is positive definite.

Since from. (ii) M is always non-singular, the equations of motion may be written in state-space form as:

x = u

where

Linearization

The desired state vector of the system may be written as , J

(2.4)

(2.5)

(2.6)

That is, it is desired to maintain the arm in a given configuration with no motion at the joints. If it is assumed that the joint angular velocities remain sma1l for all motions of the arm, so that

Il!i

.ill

«

I~

11

then the "rate linearized" state 'space model is obtained by omitting the nonlinear term containing ,!'!:

This is not fu1ly linearized since

(2.7)

(2.8)

is a function of the angular configuration. Since it is desiral:>le to move the arm through large configuration changes, the rate linearized model is required for control system analysis. In particular , the rate linearized model will be used for the construction of root loci.

The fully linearized. ~del is found from the rate linearized model

simply by evaluatingthe inertia matrix at ~ =

Q,

giving the·model:

(2.10)

where

This may be written in the standard form:

.

x=Ax+Bu

whereA and B are constant matrices of appropriate dimensions.

3. STABILITY

stahility is often regarded asthe premiere criterion for control system design in that one must at least stabilize adynamical system before attempting to improve its performance further. For manipulators that are entirely computer controlled, stahilitywill be the premiere design criterion. It should be noted, however, that stahility, even though desirable, is not a

necessi ty for · human operator controlled manipulators. As· illustrated in

Fig. 3, the human operator provides ·an. outer ·visual feedback loop. Even though

the.computer contro~ler ~y be unstahle, the human operator may stahilize the

overall system. Indeed, allowing instahility of the computer controller might

improve other characteristics of system performance tRef.

7].

However, it is

not the intention here to delve into the study of human operator dynamics. The design analysis therefore will only be for a computer controller which will allow for human operator input and which is required to be stahle.

The linear state variable feedback controllers considered in this technical note are of the form

~ =

!

~ = (-!~.

-2,]

~

HUMAN

OPERATOR

INPUT

VISUAL

FEEDBACK

-

T - -,

I

I

I

I

t

t

- I

I

. . . - - - - 1

U

₂

1

I

•

MICRO-

'Yi ,

)'1

... PROCESSOR

CONTROLLER

FIG.

:3

MANIPULATOR WITH AUTOMATIC

AND

VISUAL FEEDBACK LOOPS

where

and

The closed loop linearized model is

I

x

where

The nonlinear system is loeelly asymptotically stable about the

con-figuration 1.D if the eigenvalues of the matrix

(3.4)

where ~

=

~(1.D) lie in the open left complex plane. For any particular

con-troller design, that is choice of K and C, it is desirable that the system

behave satisfactorily for any choiCe of

lP

within the configuration space.

In particular, it will be required that the closed loop eigenvalues lie in

the open left complex plane for every possible

lP.

When this is true, it

implies that the system is locally asymptotically stable about every con-figuration. It does not imply that the nonlinear system is globally asymptotically stable.

Consider the linear state feedback controller

~

= [-!

-Q]!

where

!

and

Q

are positive definite matrices. Define the Lyapunov function

where ~

=

~(1D + 3). Since K and M are always positive definite, therefore the Lyapunov f'unction V is aïso positive definite·. The time derivative of

V is given by

. ·T . V

=

-z

-

C Z

--That this must be so is evident from the interpretation of V asthe total energy. Note that in the mathematical modelling i t is assumed that there is no passive damping at the joints. The time derivative

• T - - ,

[

0 0

]

V .

=

-3!: .Q. Q

~

is a negative semi-definite f'unction. In order to deduce local asymptotic stability,it is required to show that there is no solution other than x

=

°

such that

V

==

o.

Note that all vectors for whicl1

.

V

=

0

may be written as

Suppose that x(t) == 3!:0 is a solution. Then the nonlinear equation of motion, equation

C3

.2)

becomes :

or

-;t

-M - -....ó() K z

=

°

Since M and K are positive definite, therefore Zo

=

0 is the only constant solutiOn. Hënce a sufficient condition for the-locaï asymptotic stability of the nonlinear system

r

l

_~-~(Q_'!!)

J

x

where !:! = !:!(.zo + !) and.!:! = .!:!(.zo + ~,

!)

is that ~ and Q be positive def'inite. Further, it can be seen that this also implies global asymptotic stallility

since the Lyapunov f'unction, V, is .r.adiaJ..ly unbounded. According to Hahn [Ref'.

8,

p. 109], this is equivalent to saying that V(~) is positive def'inite on all of' Rn' Since !:!(.zo + !) is positive def'inite f'or 'all possible conf'igura-tions, it is clear that

is positive def'inite on all of' Rn if' K is positive def'inite. Theref'ore, a suf'f'icient eondition f'or the global asymptotic stallility of' the nonlinear system is that K and C be positive def'inite.

4.

CONrn.OLLABIL::rrY

Almost two decades ago Kalman introduced the concept of' controllability for dynamic systems [Refs.

9,

10, 11, 12]. The question of' contr.ollabil~ty is a.fundamental one for it is concerned with whether or not a controller e~ists

f'or a given dynamical system. The criteria f'or determining whether a linear dynamic system is contro11able are now we11-known. For nonlinear systems, the groundwork for determining contro11abili ty was done by Markus and Lee [Ref. 13]. The f'ollowing is their def'inition f'or the loc al controllability of' a nonlinear dynamical system.

Def'inition: A system is locally completely contro11able at the origin if' each initial state in some neighborhood of' the or~g~n

can be transferred to the origin in a f'ini te time interval by means of' a measurable control.

The f'ol1owing theorem is due to Markus and Lee and Kalman.

Xheorem:

Def'inition:

A system is locally completely controllable if' its varia-tional equations are completely controllable.

The variational equations f'or the system!

=

!(~, ~) are x=Ax+Bu where

àr.

~ a .. ₌

_dK:"

~J J and

O:r.

b .. ~ ~J =

dü:

J

The nonlinear open loop equations of motion for a manipulator are

[:

I

[

~~,

1

.

x = x + u

-

-_M-:r:N (4.1)

where

.!:!

=

.!:!( r.o

₊ !),

!

= !(ZO +

!'

!),

and

The variational equations evaluated at any desired state

:l. :: .ln' :1.:: :l.n

may be wri tten as

(4.2)

where

B

and

~

exist and are bounded for every choice of .2l> and.in and

These equations are of the well-known form

x=Ax+Bu

The matrix I]

!:.

~] equal to

.

.

Therefore, the equations of motion of a manipulator are locally completely controllable about every possible state and hence are globally completely controllable. This conclusion is valid for the full nonlinear equations.

5.

POLE ASSIGNMENr AffiORITHM3

One of the ~or results of modern control theory is the pole assign-ment theorem. It was shown by Wonham [Ref. 14] that controllabili ty of a multi-input linear dynamical system is equivalent to the ability to assign an arbitrary set of poles to the closed loop systems by using appropriate linear state feedback. The pole assignment theorem leads directly to the pole assignment problem. Namely, given a controllable linear dynamical system and a desired set of closed loop system poles, find the linear state feedback gains that will assign the desired poles to the closed loop system •

. The general pole assignment problem may be solved in many ways [Refs. lq, 15]. A system of second order differential equations such as the linear-ized model of a rigid-link spatial manipulator exhibits special properties and a special form. It is desirable here to solve the pole assignment problem not for general linear dynamical systems but for physical dynamical systems

of second order equations. The problem is stated thus: Gi ven the linear system

where M is symmetric and positive definite, find the linear state variable feedbaCk matrix

!.

=

[-!

-,9,]

where

!

is the posi tion feedback, or stiffness,

matrix and C is the rate feedback, or damping, matrix such that the closed loop system-matrix

(5.2)

has a desired set of eigenvalues.

In general, there are no restrictions on the form of the feedback gain matrices K and C. The solutions developed below are termed GRC (general rigid control)-solutIons IRefs.

3, 4, 5].

The linearized model

may be transf'ormed into an equi vaJ..ent system by the transf'ormation " x=,g~ where M 0 ~

=

o

M The system

o

I

o

.

A 1\ X

=

X + u

o

0 I

is a.n equivaJ.ent system.

All of' the poles of' this system may be arbitrarily assigned by a linear state variable feedback law of the form

where

...

,..

u = F X

"

"

"

F

=

[-lf

-.Q]

(5.4)

(5.5)

The choice of' F is not unique for a given set of desired eigenvaJ..ues. For example, let A " " !~ = [-lf~ -.Q2] (5.6) where 0

-

~

"

0 K

₌

~

-l

(5.7) _2 0 e.";.

ao

a 2 a4 • • • a2n-2 and • •

am-

~

1

"

[I

0 (5.8)

S

= al. as as •

then the 2n eigenvalues of the c10sed loop matrix

l

-~

I

are the roots of the characteristic equation

Note that the c10sed loop matrix (5.9) may be transformed to contro11ab1e companion form [Ref. 16] by a series of e1ementary raw and column switches. The resulting matrix 0 1

~

_

O

l

0 0

r

-

0

_______________

î'--~ -al. -a_{2 . . .} aa!l-l.

must also have equation (5.10) as its characteristic equation. This is readi1y verified by simp1e expansion.

(5.11)

The feedback matrix [equation (5.6)] is the same result as produced by Wolovi ch f s po1e placement algori thm [Ref. 16]. It is obtained here,

however, much IOOre direct1y. In pai:ticular, the equivalence transformation [equation (5.4)] is highly sui ted to the system of second order equations at hand and the role of the inertia matrix is clear.

Another feedback matrix, re1ated in structure to the first, which wil1 assignthe same set of po1es is:

To show that the characteristic equations are the same one can write the characteristic equation as A -I = 0 0 l'. T A+ê T ~l. __ l. where

The equation is identicaJ.. to

Transposing, one obtains as the characteristic equation

1\ "

l!f

+

1:.2J.

+ ~J.

I

= 0 or which is identicaJ.. to A -I = 0 "

Hence, the feedback matrix

!2

will assign the same set of poles.

The closed loop eigenvalues may aJ..so be assigned by the li~ear state variable feedback matrix

(5.13) where

!3~ ~:

~

• :2 L -_ _ _ _ .:::...Wn and

o

(5.15)

o

.

L -_ _ _ _ --=::,..2

SnWn

The 2n eigenvalues of the closed loop matrix

are the roots of the n characteristic equations

(i = 1, n) (5.16)

This pole placement routine is favourable to the analyst because i t retains the form of a system of n second order differential equations rather than the form of a single differential equation of order 2n. This does not imply, however, that the performance of the system with this choice of feedback gains is necessarily better • The form of the feedback

/ \

matrix, Fs, may be regarded as a coalescence of the mathematical and

physicaJ.-natures of the control problem. As such i t has intuiti ve appeal. In addition, the feedback matrix decouples the system of equations. Note, however, that it is the equivalent system that is decoupled, not the

original system.

The final step in the pole placement process is returning to the original system. The feedback matrix F which will assign the same set of poles to the original system as the' matrix! does to the equivalent system is found by

~

A

I __

Mo

_!!o

1

F=!~=F

l

(5.17)

For the linear dynamical system

(5.18)

with linear state variable feedback

u=Fx

the following three choices of F are sufficient for assigning the corresponding closed loop characteristic equations:

(i) Let 1\ 1\ F

=

_{[-!J. ~}

_-.21

_~] (5.20) where 0 -1 0 1\ 0 0 -1 K

=

0

.

~

-1

(5.21) ao _{a 2} a4 ••• a2I1-2 and "-C

=

o

(5.22)

Then the 2n po1es of the closed loop system are the roots of the characteristic equation (ii) Let where 1\ " T

!2

=

lh

(5.25) and (5.26)

Then the 2n poles of the closed loop system are the roots of the same char-acteristic equation

(iii) Let 1\ " F =

[-!s!:! -Es !:!]

where

"-!s =

and 2 ~l.Wl.

₀

"

Es

= 2

S

2»2

.

(5.30)

0

·2snWn

Then the 2n poles of the closed loop system are the roots of the n character-istic equations

(i

=

1, n)

(5.31)

The first method has an interesting property regarding the rate feed-back, or daJIq?ing, matrix.

"

E

=

El.

!:!

has the form

Therefore, if all of the joint angular rates are measured, then it is only necessary to feed the rate information to one joint to be able to arbi trarily assign the poles!

The pole assignment problem may also be solved by choosing the feed-back matrix,

!"

wi th the form

rather than the form:

f\

The elosed loop system becomes

• - I

-[

0

:

I

J

~

=

----

i

-

--

-

~ (5.33)

which is the identical form as the equivalent closedloop system used above (e.g., equation

(5.9)].

Clearly, then, the three choices of

!

givenpreviously are also sufficient to assign the system poles where!

=

M F. Hence,

"-" F

₌

_[-~ !J. -M CJ.] (5.34)

-

-"- A F

₌

_[-l:!

_!2 ... ~ .Q2] ( 5 .35) and _I'

"

F

₌

[-M Ks -M Cs] (5.36)

-

-

-

-are three additional pole assignment methods.

The second method has an interesting property regarding the rate feedback, or damping, matrix.

has the form

x

x

C =

°

x

Therefore, if rate information is fed to all of the joints, then it is only necessary to measure the angular rate of one joint to be able to arbitrarily assign the poles~

The third feedback matrix is the pole assignment method chosen by Whi tney [Refs.

3

.

, 4,

5].

In addi tion to retaining the form of a system of n second order differential equations rather than a single differential equation of order 2n, this method is also the decoupled solution.

The decoupled solution is a unique solution which is characterized by a one-to-one relationship between joint response and input. For example,

,,. 'f

!. "

a change in the desired angle of the second joint of a two-joint manipulator with decoupled control Will cause only the second joint to rotate. The first joint will maintain i ts original angle.

The decoupled solution, however, is in a sense "structurally unstable". That is, for small changes in the system parameters, namely the inertia matrix elements and the feedback gains, the solution will no longer be decoupled [Ref. 17]. Mathematically, this may be seen by the form of the feedback matrix [equation

(5.36)].

" A

F = [-M Ks -M Cs]

-

-

--The inertia matrix must be reconstructed exactly so that it "cancels" the actual system inverse inertia, ~-J..

There are two sources of variation in the parameters of an actual manipulator that will cause deviation from the decoupled solution. The first is the inability of the designer to exactly reconstruct the inertia matrix. This, however, is not the major source of uncertainty. The second

source is much more significant. The system inertia matr~x depends on the configuration • The feedback gains are constant and are calculated for only one reference configuration. When the configuration changes, the system is no longer decoupled. A manipulator that is required to regulate about any configuration within a large configuration space is, thus, only decoupled near the reference configuration • Note, however, that by using nonlinear feedback one may compute the inertia matrix as a function of the configura-tion, thus making decoupled performance, on the whole, viable.

In all, six solutions to the GRC pèl:J!e assignment problem are given .above. In addition to the special properties of sorne of these solu-tions there are also special relasolu-tionships between the solusolu-tions. These are useful in characterizing the inf~tely many pole assignment solutions.

Let

!

=

[-!

i

-Q]

(5.37)

be s olution to the pole assignment problem. Then the following feedback matrices also assign the same poles:

( i)

E:.

=

[-!?

_QT]

(5.38)

(ii) F =

I-M-J. !!:!

I

_!:!-l.

_Q

.!i]

_(5.39)

I

I

(iii)

_K

=:= [-~!

!:!-J.

:

_~.Q

!:!-l.]

(5.40)

(iv) F =

[_!:-J. .!i-J.

!

~

.!i

I I

_.!:-J.

~-J. Q

.!:

!:!]

(5.41)

(v) (5.42)

where P is any. nonsingular matrix. Cases (iv) and (v) are generaJ.izations of caSës (ii) and (iii), respectively. Note that if the given solution [equation

(5.37)]

is the decoupled solution, then cases (i) and (ii) are identical.

6.

INDEPENDENr JOINT CONrROL

Independent joint control (IJC) is a restricted form of linear state feedback controL It is characterized by the constraint that the angular position and rate measured at any joint may only be fed back to the input for the same joint. This is equivalent to the constraint that the position and rate feedback matrices, K and C, be diagonal. Independent jointcontrol is a special case of what is-known -in the literature as decentralized control [Refs. 18, 19, 20, 21].

The concept of fixed modes was introduced by Wang and Davison [Ref. 18]. The fixed modes of a system correspond to the poles of the closed loop transfer matrix that remain invariant with arbitrary decentralized linear state feedback. The concept of fixed modes is a generalization of uncon-trollable modes. It is shown by Davison [Ref.21] that a necessary and sufficient condition for the existence of a solution to the decentralized pole assignment problem (with added dynamics allowed) is that there be no fixed modes.

It will not be shown that the linearized model

(6.1)

has no fixed modes.

,

The matrix

(6.2)

has 2n eigenvalues at the origin of the complex plane.

The closed loop system matrix with arbitrary decentrali~ed feedback may be wri tten as

.

'

where

1S

and

Q

are diagonal.

The characteristic equation of this matrix is given by

A -I

=

0

(6.4)

(where

!:

=

À

l).

Since A is nonsingular, this is identical to

(6.5)

The constant term in the characteristic equation is simply

(6.6)

By choosing K with no zeros on the diagonal, the constant term will be non-zero and hence the closed loop system matrix will have no eigenvalues at the origin of the complex plane. Therefore the system

has no fixed modes.

Since the system has no fixed modes, there exists a decentralized controller that will assign any arbitrary set of poles. I t should be emphasized, however, that the controller will, in general, have added

dynamics • Wang and Davison [Ref. 18] give an algorithm for solving the pole assignment problem by adding integrators at each control station. In the fOllowing analysis, the use of added dynamics is not allowed in the solution. It may be stated then that gi ven the addi tional constraint that there be no added dynamics, then the absence of fixed modes is only a necessary condition for the existence of a decentralized controller that will assign any arbitrary set of poles. The decentralized linear state feedback pole assignment problem may be stated thus:

Given the system

find a feedback matrix

I

!

=

[-!

l

-Q]

(6.8)

where K and C are diagonal such that the closed loop system matrix

has a desired set of eigenvalues.

Since K and C are diagonal, there are 2n unknowns to be solved for. There are also 2n equations. The equations, however, are nonlinear. The

decentralized pole assignment problem is solved below for a two degree-of-freedom system.

Two-Link Manipulators Consider the system

(6.10)

where

(6.11)

is symmetric and positive definite. Let

(6.12)

and

and let the desired characteristic equation be written as

The equations may be nondimensionalized by choosing M22 as the characteristic inertia and Wh as the characteristic frequency.

The nondimension~ closed loop equations are

.

x :::

[

--~

--M K x where

[

M,.,/M:.,

M,~M"

_J

[~U

~:'l

M::: ::: M2J./M22. ID2J.

K

:::

[

K,/~"M"

K,/~"M"j

:::

[

:'

:

,l

and

ë

=

[

c,/:M"

C,/~M',J

:::

[

:'

:,

J

The desired characteristi'c equation becames

À4 _{+ aJ.Às + a2À2 + asÀ + 1} ::: ₀

(6.14 )

(6.15)

(6.16)

(6.17)

(6.18)

(6.19)

Calculating the cnaracteristic equation of the c10sed loop matrix, treating kJ., k2,cJ. and c2 as unknowns, and mJ.J. and mJ.2 (::: m2J.) as constants, and equating the re sult to the desired characteristic equation, yie1ds four algebraic equations in the four unknowns. The solution of these equations yields the feedback gains necessary to produce a given set of coefficients (aJ.' a2' as), which in turn correspond to a given set of c1osed-1oop eigen-values. ..

The four equations are:

(6.20)

( 6.21)

(6.22)

(6.23)

For any desired set of poles, ,the coeffi'cients (aJ.' a2' a3) can be found. Given these and the inertia matrix, one can solve for the feedback gains (kJ., k 2 , cJ.' c2). If this solution exi~ts, then the arbitrary po1e

assignmentprob1em has a solution. '

Solution for Feedback Gains

Equations

6.20

and

6.22

may be written as

Assuming

cJ. and C2 can be aalculated by

(a3 - aJ.k2) c 2

=

[ia

1 {kJ. - IDiJ.k2)

Now the produce CJ.C2 is wri tten

(aJ.kJ. - IDJ.J.a3)(a3 - aJ.k2) cJ. c 2

=

fM

12 -(kJ. - IDnk2) 2 (6.24)

(6.25)

(6.26) (6.27)

(6.28)

Assuming k₂

f

0,

- k2(a~k~k2 - ~~~ask2) (as - a~k2)

C~C2

=

/M

12

-(k~k2 - m~~k22) 2

Using equation (6.

ç

3), this becomes

k 2( a~

fM

1 - mna sk2)( as - a~k2)

fMI2---( fM

1 - m~~k22) 2

=

-Define ~l 0) then c~ C2

=

=

-Equation (6.21) becomes (6.29) (6~30) (6.31) (6.32)

Finally, di viding by mu and rearranging the terms, one obtains

p4k 22(aJ.P2 - ask2) (as - al.k2) f'(k2)

=

k 22 - aaP~2 + p2 + - - - -

=

0

(p2 _ k22) 2

(6.35)

It is important to note ~ere that the parameter p is the only parameter representing the, system inertias in' this equation. The parameter

( 6.36)

may be regarded as a me~sure of' the singularity of' the system. The va1ue of' p is such that

O<p~l

As p approaches 0, the inertia matrix becomes near1y singular • When p equals 1, the inertia matrix is diagonal.

. _{Any real root k 2 of' the above f'unction, f'(k2), such that k 2}

f

0 and k2

f

p nl1 yie1d a solution to the po1e assignment problem. The existence of' a solution may be f'ound by the ;t?fop~rties ,_{of' the f'unction, f'(k 2):}

(6.37)

(6.38)

Let

(6.39)

(6.40)

Hence f'or as

f

pal.

Therefore a solution almost always exists for the dec~ntralized pole assignment problem. The set of coefficients for which a solution may not exist lies on

a plane in the coefficient space, (a~, a2 , as), given by

as = pa:~

(6.4-2)

It will now be shown that a solution always exists. Setting as

=

pa~ in equation

(6.35)

one obtains

(6.43)

Note that the discontinuity at k2

=

P is removable. Evaluating the limit

one obtains

,

( 6 .44)

From this equation and equations

(6.37)

and

(6.38),

it can be seen that a sufficient condition for the existençe of a solution to the decentralized pole assignment problem is

4(aaP - 2)

a~2

< _ _ _ _

_

(6.45)

or

It is important to note that the assumption [equation

(6.25)}

made in the development of the solution to this point is not necessary for all possible solutions. The original equations [equations

(6.20) - (6.23)}

with as

=

pa~ are:

c~ + IIl~~C2

=

a~

_/MI

_·

( 6.46)

k~ + m~~k2 + C~C2

=

a 2

/M I

(6.47)

k2c~ + k~C2

=

pa;!.

/M I

(6.48)

Let

(6.50)

From equation

(6.49)

one has

k~ ;:: pm~~

_(6.51)

Equations

(6.46) - (6.48)

become:

(6.52)

(6.53)

(6.54)

It is c1ear from equations

(6.52)

and

(6.54)

that if as ;:: pa~ and k_{2 ;::} p, then equation

(6.25)

is not a required assumption. In this case, a solution may be obtained from equations

(6.52)

and

(6.53).

FrQm equation

(6.53),

the product C~C2 is written

(6.55)

Multip1ying equation

(6.52)

by c_{2 ,} assuming C2

f

0,

yie1ds:

(6.56)

SUbstituting equation

(6.55)

and rearranging the terms gives

(6.57)

Dividing by m~~

f

0 yie1ds

(6.58)

A sufficient condition for the existence of real roots to this equation is

or

4(aaP - 2) a:2

> ____

_

l. _ (6.60)

This result, along with inequality (6.45), guarantees the existence of a solution to the decentralized pole assignment problem for all points on the plane in the coefficient space (al., a:2' as) defined by as :;:: pai. Therefore a solution tb the decentralized pole assignment problem always exists.

The solution is almost never unique. It is only unique on a curve on the plane as :;:: pal. defined by

4(aaP - 2) al.:2:;::

--....---ps In this case the unique solution is given by

aJ.P:2 Cl. :;:: C:2 :;:: -2-(6.61) (6.62) (6.63) (6.64 )

On the surface defined by as :;:: pal. and al.2 _{> 4(aaP-2)/ps there are always}

two solutions to the decentralized pole assignment problem. The solutions

in this c~se are found from the real roots of the polynomial

(6.65)

The corresponding values of cl. are then found from either equation (6.52)

or equation (6.55). In this case, as above, the values of k2 and kJ. are

given by equations (6.62) and (6.63), respectively.

Everywhere else in the coefficient space there are always at least two solutions and at most six solutions. This may be seen as follows.

The roots of f(k_{2 )} (equation (6.35)] will also be the roots of

when as

f

pal. or when as:;:: pal. and al.2

_<

_{4(aaP-2)/ps. This is simply "a}

(6.67)

The roots of this po1ynomial such that k₂

f

0 and k₂

f

p Y1-e1d solutions

. to the decentralized linear state feedback po1e as~ignment llrob1em. The

'corresponding values of k_{1 ,} c~ and c₂ are then found .from equations (6 •. 23},

(6

:

26)

and

(6.27),

respective1y.

The third method of solution given above (equation

(6.67)]

i.s

almost always ·required. In particular, it 'Wi11 be emp10yed for the f0110Wing cases. Let a desired characteristic -equation be given by

(6.68)

Let

and

The nondimensional desired characteristic equation is

which when expanded becomes

'where the identity

W1W2

= 1 has been used. cases arise when either ~l

=

S2

or wl

=

w2.

-

=

1

(6.69)

(6.70)

Two special po1e location In these cases

Since the parameter p is always strictly less than 1 for manipulators, the set of coefficients, (a~, a2 , as), does not lie on the plane as

=

pa~

and hence the real roots of equation

(6.67)

yield the solutions to the decentralized pole assignment problem. Note that when the parameter p equals 1, the inertia matrix is diagonal. In this case, the solutions are trivially found.

7 • CONTROLLER DESIGN

Controller design procedure depends on the characteristics and performance requirements of the manipulator. For example, an industrial robot manipulator wi th virtually no payload may be required to regulate about essentially one configuration. In this case, a controller may be designed by assigning a suitable set of poles to the closed loop system" with the inertia matrix evaluated at the desired configuration. A remote manipulator, such as the SRMS, however, is required to carry large pa,yloads. Moreover, it must regulate satisfactorily about any configuration within a large range. Mathematically, a manipulator carrying a large payload

corresponds to a nearly singular inertia matrix.

While it is true that the inertia matrix for any real physical system is never singular, it may be nearly singular. The degree of sin-gularity may be measured, for the two degree-of-freedom case, by the parameter

O<p,:::l

The closer p is to zero the more nearly singular is the inertia matrix. A manipulator which carries a large payload at the end of slender links is an example where one encounters a nearly singular inertia matrix.

The design procedure for such a manipulator, especially one with a nearly singular inertia matrix, is augmented by root locus analysis. A,;set of closed loop poles is assigned using a pole placement algorithm

and then a root locus plot is made for changing configurations • The

initial poles se1ected may be varied until suitable root loci with respect to configuration changes are achieved. Since the root loci in gener al are calculated numerically, the procedure is best illustrated wi th a numerical example.

A Numerical Example

The equations of motion of a two-link remote manipulator with payload in planar motion are derived in Appendix A. The rate linearized model is

where

o c

=!:! +!:!

COS'/'2

(7.3)

The closed loop rate linearized model is

(7.4)

wÀere K and C are the position and rate feedback matrices which are chosen on the basis of-a pole assignment algorithm. Since the inertia matrix is only dependent on one parameter, '/'2' the root loci are much simplified.

The preliminary values of lf and MC for the SRMS carrying its full payload. [Ref.

22]

are listed in Table 1. The corresponding nonéümensional values are listed in Table 2. The reference design position for the pole placement routines is chosen to be '/'2 = 0, i.e., a straight arm.

Table

1.

Preliminarl SRMS Inertia Matrix Elements

Element Value Dimension

0 M_n

1.9692

x

10

6 slug-ft 2

~2

1.5296

x

10

6 slug-ft 2 0 M22

1.5296

x

10

6 slug-ft 2 c M~~

1.5266

x

10

6 slug-ft 2 c M~2

7.6332

x

10

5 slug-ft 2 c M22

0

slug-ft 2

Talüe 2. Preliminary Nondimensional SRMS Inertia Matrix Elements

Element Value 0

1.2874

m~~ 0

1.0000

m~2 0

1.0000

m22 c

0.99804

m~~ c

0.49903

m~2 c

0

m22

De Controller

. utilizing the pole assignment algorithm developed in Section 6, the

fol~~Wing feedback matrices are calculated that will assign the poles for

W1

=

W2

= 1 and

S1

=

S2

= 0.707: K₁ =

lOo~59

000:217]

ë

1 =

[ -00

9

_:72

00:794]

(7.5) and (7.6)

[

002:06

0

l

-

[

1.

0

9

6

-00:320

l

K

2 :::: C2 :::: 0.1820 0

These equations are derived from the roots of equation (6.67). For the example at hand, there are two real roots: 0.09217 and 0.1820. Equations (7.5) and (7.6) are two distinct solutions to the decentralized pole assign-ment problem. These solutions are termed IJCl and IJC2, respectively. Note that both solutions have one negative rate feedback gaine Since the rate feedback matrices are not posi ti ve ... p.efini te, the system may not be globally asymptotically stable. This does not imply that i t will not be.

Closed-loop root loci for the two solutions as 72 varies from 0 to 1800 _{are shown in Fig s.}

4

_and

5,

_{re specti vely • For the example at hand, i t}

is found that both solutions are in fact highly sensitive to 12. In both cases, the discrepancy between the initial pole locations (12

=

0) and the design pole locations is due to the rounding error in feedback gains, which were kept to only four significant figures. Also, in both cases the root loci cross into the right half complex plane.

Although a negative rate feedback gain does not necessarily imply increased sensitivity, it is still undesirable. Clearly, if the joint with the positive rate feedback gain were to be locked, then the remaining system would surely be unstable.

It has been found numerically for the given example that for

S1

=

S2

the values of

W1

which yield all posi tive feedback gains are gi ven by

W1

>

7 .8 ( 7 .7)

In view of this condition, let us return to the pole placement algorithm~

The fOllowing values of

w

and

S

are assigned: '

W1

=

10

._-~.-...

Im

Re

.

Im

The resulting feedback matrices are given by _ "[3.85

6

Ks =

o

_ . [0.2243 Cs

=

o

Only the solution with the larger vaJ..ue for kl. is given.

(lJC3)

The root loci (Y2

=

0° to 180°) are illustrated in Fig. 6. Note that one root is virtuaJ..ly constant.

The minimum damping ratio is increased slightly by increasing the initiaJ.. damping ratio

h.

'Xbe solution for

WJ.

=

10

~

=

0.1

~l.

=

0.866

is

The root loci are shown in Fig.

7.

This is the finaJ.. Independent Joint Control design.

GRC Controller

(IJc4)

The GRC pole placement algörithm used in the following controller design is the initially decoupled sOlution given by equation (5.36). As in the IJC design, the initiaJ.. pole locations are given by

The resulting feedback matrices are

_ [2.285 Kl.

=

1.499 1.499

J

1.000 [ 3.232 ëJ.

=

2.120 2.120

J

1.414 ( GRC1)

Im

Re

FIG. 6

IJC3

ROOT

LOCI

Im

FIG.

7 IJC4 ROOT LOCI

The root loci are illustrated in Fig. 8. The po1e locations given by

-

i'SJ.

=

2

~l.

=

~2

=

0.707 and

are next assigneq,. The corresponding feedback matrices are

[ 9.142

0.3750

J

_

[ 6.463

_1.060

J

K

2

=

C2

=

5.996 0.2500 4.239 0.7070 (GRC2) and (GRC3)

[ 228.5

0.01499

]

_[32.32

0.2

1

20

J

Ks

=

Cs

149.9 0.01000 21.20 0.1414

respective1y. The r<;>ot loci are i11ustrated in Figs. 9 and 10. Note that the increase in

wJ.

does not significant1y change the nature of the root loci.

The solution GRC2 is improved slightly by increasing

S2

and decreasing

SJ..

The initiaJ. poles given by

WJ.

=

2

S2

=

0.866 are assigned by the f'ollowing feedback matl'ices:

Im

Re

FIG.

8 GRCI ROOT LOCI

Im

Re

FIG. 9 GRC2 ROOT LOCI

---,:::--Im

Re

FIG.

10 GRC3 ROOT LOCI

[ 9.142 5.996 0. 3750] 0.2500 1.299 ] 0.8660 (GRc4)

The root loci for this case are shown in Fig. 11. This is the final General

Rigid Control solution. The greater sensitivity of roots to /2 should be

noted, in comparison with Fig.

7.

Simulation

Simulation is usually the last stage of the design procedure. The nonlinear terms which were previously ignored are included in the analysis. The system with each of the IJC4 and the GRC4 controllers is numerically

integrated for a unit step input at /2' The simulation results are shown

in Figs. 12 and 13. Qualitative differences in the two responses should be noted.

Conclusions

Before attempting to compare the two controllers, IJC4 and GRC4, it is instructive to outline the design difficulties encountered by both

algo-rithms. Since the configuration space includes all possible values of /2'

the inertia matrix elements vary (]Ver a wide range. In addition, the

near-singularity of the example inertia matrix indicates much larger changes in the inertia matrix elements than, say, for a manipulator with a smaller payload. The large variation of the inertia matrix elements with respect

to changes in /2 is evident in all of the root locus plots. In particular ,

the damping ratios of the roots undergo much change. The roots tend to become either underdamped or overdamped.

As illustrated in Figs. 8-11, the GRC solutions all contain

signi-ficant variation in damping ratios. As /2 increases, the smallest root

(in magnitude) tends to become underdamped, while the largest root tend.s to become overdamped. This phenomenon is not changed significantly by the initial

pole locations assigned. The IJC solutions (Figs. 6 and 7) manage to partially

overcome this difficulty. As illustrated in Fig.

7,

the root with the lowest

magnitude is virtually constant. The other toot, however, tends to become underdamped.

Perhaps the most severe limitation on the IJC controller is the maximum bandwidth obtainable, a measure of which is the maximum value of

W2' It is found that it is necessary to decrease W2 by one order of

magni-tude in order to avoid solutions with negative rate feedback gains. The GRC solution has no such constraint and therefore may be designed with

maximum bandwidth 6ijl.

=

W2

=

1). The magnitudes of the roots, however,

also vary with changes in /2' As illustrated in Fig. 11, the bandwidth of

the GRc4.'~solution for /2

=

1800 _{is approximately the same as the bandwidth}

for the IJC4 $olution even though initially (/2

=

0) it is five times

greater.

Comparing the simulation results (Figs. 12 and 13), it can be seen that the motion at the fir st joint due solely to the input at the second

R

_e

FIG.

11

GRC4

ROOT

.

LOCI

46

c

-C> ah c C> In C

.

20.00

1l0.00

60.00

BO.OO

100.00

T

0 · ·1 C>

-C 1 c C N c c

....

c o~~ _ _

-+

______ +-____ -+

______ +-____

~ C>

o

00

20.00

1l0.00

60.00

BO.OO

100.00

Cl o

....

1 Cl o N 1

T

FIG.

12 IJC4

SIMULATION WITH

STEP

INPUT

AT Y2

.

Cl Cl

hf.

'

O::O~----~.;:;rO~~~I1!O .~O~O ---:6~0~. O~O ---:8~0-:. O;-O

.

-~l

00.00

In Cl o I o

-Cl ,. I o 'g .

-o

T

o~ _ _ ~~ _ _ _ _ ~+-

____

-+

______

+-~

__

~

.

c

o

00 20.00 110.00 60.00 80.00 100.00 Cl o

-

I c· o

.

N I

T

FIG .

.

13 GRC4 SIMULATION WITH

,

STEP

.

INPUT

point to see, however, is that the IJC solution as weil as the GRC so~tion

is capable of nearly decoupled performance, even though it is not possible to mathematically decouple the system with independent joint control.

Finally, the two controllers may be compared on the basis of tl1e magni tudes of the feedback ga.Ï!ns. The feedback gains calculated for the GRC controllers are in all cases much larger than the gains for the IJC

controllers even wh en the assigned poles are the same. It does not follow that the IJC controller will require a smaller total control effort aathough the possibility certainly warrants further investigation.

Summarizing the above, i t may be said th at the IJC controller (at

least in this. case) shows same advantages:

(i) lower sensitivity of second pole, and

(ii) lower gains.

Of course it has same disadvantages as weil, such as a reduced bandwidth.

Both solutions (IJC ,and GRC) are far from ideal. The reason for this is that

both solutions are constrained to be constant linear state feedback solutions. The large variation of the inertia matrix elements, combined with the maximum

range for the second jJoint angle, tends to nullify the advantages and aggravate

the disadvantages of both controllers, which are designed on the basis of onlY one initial configuration.

8 .

SUGGESTIONS FOR FUXURE RESEARCH

General Rigid Control

As shown in Section

5,

there ~e infinitely many ways of assigning

the same set of closed loop poles to a linear mechanical system. It would be of great value to compare the performance of these different schemes with respect to criteria other than pole locations. For example, i t would be interesting to compare the decoupled solution

A

"-F

₌

_{(-~!3 -~ .23]} with the solution

/\ _/\

F

₌

_{(-!3 ~ -.23 M]}

It would be beneficial to know the advantages and disadvantages of the different methods. For example, i t has been shown that one need only

measure the angular rate of one joint and still be able to assign the system poles.

Natura.].ly, the question of optimali ty arises. An alg~ri thm has be.en

developed by Klein (Ref. 23] 'for a gener al linear dynamical system that will

simultaneously assign the systempoles and minimize the sensitivityof the.

system poles to small changes in the system parameters. It would be of great interest to determine the algorithm for systems of second order differ-ential equations. In particular, the role of the inertia matrix in Klein's

Independent Joint Control

Perhaps themost challenging undertaking is the solution to the

decentralized, linear .state feedback pole assignment problem for a

manipu-lator wi th any number of links. The method of solution of the four nonlinear

equations developed in Section 6 for a two::-link 'manipulator does not extend

easily to even a three-link manipulator. For example, i t is found that for a two-link manipulator there are two equations which are linear in the rate

feedback unknowns c~ _{and c2 • The six nonlinear equations for a three-link}

manipulator are:

(8.1)

I (m2~3 + m3sk2) c~ + (m~~k3 + m3sk~) c 2 (8.3)

(8.4)

(8.5)

(8.6)

where Mi is the minor of

M

formed by deleting the i th. row andcolumn of

M

and evaluating the determinant. The first equation and the fifth equation are linear in the tbree rate feedback variables cJ.' c 2, and c3. The third

equation, however, contains the term C~C2C3.

Practically speaking, it is not necessary to solve the problem analytically. If one suspects that a solution might exist, then one can rely on a computer to search for the solution numerically. Numerical

solutions, however, will not yield any insights into controller design.

Sensitivity Analysis

Perhaps the largest problem affecting both IJC and GRC designs is the sensitivity of the closed loop poles, particularly with respect to changes in the arms configuration. It would be of great value to campare different pole assignment methods on the basis of the sensitivity of the closed loop poles. In addition, it would be of interest to determne practical limits, if any, that might be imposëd on pole assignment solutions based on the

resulting closed loop pole sensitivity. For example, it has been found that the IJC solutions with negative rate feedback gains (IJCl and IJC2) have'highly sensitive closed loop poles.

Nonlinear Feedback Control

Nonlinear state feedback control is a logical means for overcoming the difficulties that arise due to the configurational dependence of the inertia matrix. In this method, the inertia matrix is reconstructed as a function of the configuration • Then, utilizing any of the GRC pole place-ment algorithms, it is possible to assign a constant set of poles to the

closed loop system. The feedback control is then a nonlinear function of

7... Some authors also refer to this scheme as adaptive control. In general,

independent joint control precludes the possibility of effective adaptive control because information about all joints is not available at a particu-lar joint. This does not imply that decentralized nonlinear state feedback control cannot improve upon linear IJC solutions. However, the improvement, if any, must be limited due to the above consideration.

APPENDIX A

EQUATIONS OF MJrION OF A TWO-LINK REIDTE MANIPULATOR WITH PAYLOAD IN PLAN.AR IDrION

The equations of' motion of' a two-1±nk remote manipulator with pay1oad. in p1anar .motion -are derived be10w using Lagrange IS f'ormulation. These

equations are essentially the rigid body motion portion of' the equations derived by Hughes [Ref'. 24]. The appropriate ref'erence f'ra,mes are i11us-trated in Fig. 14.

Let r1 be theposition vector to an e1emental mass partiele in link

1.

;11

_{1 is the velocity of' the partic1e as seen in f'rame 1, the inertial f'rame}

.

~Ü

=

_i111

=

@

C:F)

Let T1 be the total kinetic energy of' link 1. Then

~.r:-;-'

T1

=

~

111

h

2(X2

+

y2)dm

1;i.nk 1

FIG. 14 REFERENCE FRAMES FOR A TWO-LINK

MANIPULATOR WITH PAYLOAD IN PLAN AR

MOTION

C

y

COSY2 - x

.

=@

V₂

=

_{r 2}1 -y SinY2 + X 7

.,

~ 0

C

COSY2 - X

=®

-y sinY2 +X 0 sinY2 ) COSY2 SinY2 ) COSY2 COSY2 - x SinY2 + x

o

;2 +@

(

x

,

-y

COSY2 - x sinY2 ) COSY2 -yo Sin~2 + L1

(),J. + ;2) +

®

(

L:hJ

Let T2 be the total kinetic energy of link 2 • Then

T2

=

~

111

_V2 T Vz1m link 2

=

~

(J 2 + LJ.

2m

2 +

2L~m~2COSY2

-

2L~m~2sinY2)

1J.2 1. • 2

+"2

J2 Y2 <,;' '.Yl.

where

is the position vector to the centre of ma~s of link 2. For simplicity it is assumed here that

Y2

=

o.

Let ra be the position vector to an elemental mass particle in the payload

~3 ~0)

(

n

'-=a

{

:

L

j

Clearly,

Let Ta be the total kinetic energy of the payload. Then

Ta

=

~

111

:ta

T1adJn payload