[elearnica.ir]-adaptive ann control of robot arm using structure of lagrange equation

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Proceedingsof the American Control Conference

San Diego,

California June

1999

Adaptive

ANN

Control

of

Robot Arm

Using Structure

of

Lagrange Equation

Masaki YAMAKITA and Takashi SATOH

Department of Control and Systems Engineering

Tokyo Inst i tute

of

Technology

Abstract

Several adaptive control algorithms for robot manipu-

lators using artificial neural network ANN) have been pro-

posed and an algorithm using physical structure of robot

manipula tors also has been proposed. But th e struct ure of

ANN should b e modified according to t he change of the de-

gree of freedom of the system. In this p aper, we propose the

algorithm in which structure of ANN is independent of the

degree of freedom and compare it to some other algorithms

by simulation and experiment.

1 Introduction

Adaptive control algorithms for robot manipulators have

been studied for a decade, e.g.,

[2]

and they usually use a

property that the dynamic equation of th e system can be

represented by

a

multiplication of

a

known regressor matri x

and an unknown dynamic parameter vector. I t is not easy,

however, to o btain t he regressor matrix if the kinematic pa-

rameters ar e unknown. Several adaptive con trol algorithms

using Artificial Neural Network ANN) have been also pro-

posed to overcome the problem. Th e explicit calculation of

the regressor matrix is not required in these algorithms

[l]

[3].

Th e learning c apability of the network was very low be-

cause the str uctur e of the dy namic equations was not tak en

into account.

In

[4]

an adaptive control algorithm which considers the

struc ture of the pr operty has been proposed, however, since

the allocation of the neural elements was fixed, the stru cture

of the neural network should be modified according to the

degree of the system and neural elements may not be used

effectively.

In this paper

a

new adaptive

A N N

controller which over-

comes the problem is proposed. It uses a structure of dy-

namic energy of the system. Th e efficiency

of

the proposed

method is compared to other c onstruc tions of the n eural net-

works by the numerical simulations and experiments.

2

Structure

of

ANN

For comp utat iona l efficiency we use RBF Radia1 Basis

Function ) neural network which is known that it can ap-

proximate any cont inuous non-linear function with any ac-

curacy for a bounded set of the d0main.A neural network

4~

which has

n

input, one output and p functions can be

represented by

where B

E

R p , < E R p , p ; E R . Of course, multi output

functions can be represented by a combination of the net-

works. We introduc e the following assumption as in

[3].

[Assumption]

Let 4 ~ ) e a continuous function to be estimated and as-

sume that there exists a known matrix function Y, satisfying

in th e domain where

11 11

stands for

a

matrix induced norm or

the Frobenious norm and

&

is an unknown constant matrix.

Please notice th at this condition is satisfied by th e R BF neu-

ral network and also by three layers neural networks if we

assume boun ds of weighting matrices as in

[3].

3 Design of Controller

We consider the system whose dynamic equation is repre-

sented by

M(q) i

+

(C(q7

4 +

D ) i

+G(q)

=

5 )

where q is

a

generalized coor dina te vector, is a general-

ized force vector and M ( q ) ,C q, ,

D,

G(q)are inertia, Col-

ioris/centrifugal and da mping, grav ity terms, respectively.

Let

q d ,

q d ,

i [ d

E L , be desired trajectories and

e

:= q - q d ,

s :=

i + A e (A

>

0), r := qd h e . If we use a control inpu t

defined by

where

B = -r eYT(q,

4 1

r)s, re

> 0

8)

= raYallSll, r > 0, 9)

and

8

is estimated function of

8 ,

then

q ,

converges to the

desired signal

q d , q d .

The convergence of the error can be

shown as in

[l] 3]

under the assumption. Th at is proved in

appendix.

0-7803-4990-6199 10.000 999

AACC

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as

..........GONTlL0LLE-R;

: r - -

Figure

1:

Control System

Th e remained problem of t he control design is how to con-

struct 4 ~ .n the following 4 alternative methods are com-

pared and Method 3 and Method 4 a re methods proposed in

this paper.

Method1

[l]

4~ is constructed by a single

N N

with the knowledge

of passivity structure of

4,

and it

is

constructed by

and

d ,

G can be derived as in Method 2. Please notice

that the allocation of the neural elements is not fixed

in this Method.

k s constructed by a multiplication of estimated ma-

trices and neural elements and the potential function

U

s constructed by the another

N N

as

Met ho d4 proposed)

n

M

= xkits

i l

O = eG


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network depends on th e allcation of the cente r of each basis

function,

p i .

Further study shoud consider adaptive alloca-

tion of the center, and also another type of neural network,

e.g., thr ee layered neural network shold be compared to the

RBF neural network.

APPENDIX

A.1 The Proof of Stability

where

Choose a candidat e of Lyapunov function V

=

Vi

+

%,

where = 8

,

= & .

Th e time derivative

of

VI an be derived as

2

= s T M i + - s T M s

2

=

s T M ( i

&)

+

-sTMs

1

2

=

sT(r C D ) q G

M i r ) +

- sTMs

S T ( - & &

6+ b)qr

sT(-Y D S+ v KdS)

-sTDs TKds+

STV

STY T I

=

=

=

D S

+

v

KdS)

[3] Wei-Der Chang, et. al: ADAPTIVE ROBUST

NEURAL-NETWORK BASED CONTROL FOR SISO

SYSTEM IFAC,13th Triennial World Congress, San Fran-

cisco,USA, 1996)

[4] Shuzhi

S

Ge: ROBUST ADAPTIVE CONTROL

WARKS IFAC,13th Triennial World Congress, San Fran-

cisco,USA,1996.

OF ROBOTS BASED ON STATIC NEURAL NET-

Simulation results

Th e respances for desired angle Figure 2 - 5 )

5

5

- s T D s TKds+ sTv

Ty

+

~

-sTDs

TKds

+

sTv TY

+

BTYalls/l.

Figure 4: Method3

Figure

5: Method4

Th e respances for desired anglar velocity Figure 6

-

9)

he t ime derivative of V2 can be also calculated as

r2 = i T r g - f B + T r - l a

- 8 T r e - 1 j T r a - l i .

~~~~~

,

?ad

sec

Q

ad

SCE

am m tm >m &m

Im

am

om

Im ,m >m uo

xm

m

method

4

10

Therefore the time derivative of

V

is

am

v = V i + ,a

a m a m

-sTDs STKds+ sTv Tra-iii+ YfiIIsll 1 . 4 I 4

=

-sTDs TKds+

sTv GTre-ii

+&Y&lls11

Figure 6: Method1 Figure 7: Method2

+tiT

Yn 1311

= -sTDs TKds

:R;R

d rad

SCC

In

a

a m

to t he transfer function from s t o

e.

Furthemore it can be om

nm

zm >m

[elearnica.ir]-adaptive ann control of robot arm using structure of lagrange equation

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