A neuro-fuzzy logic controller for trajectory tracking of uncertain robots

Proceedings of the 1996 IEEE

International Conference on Robotics and Automation Minneapolis, Minnesota

-

April 1996

A

Neuro-Fuzzy Logic Controller for Trajectory Tracking

of Uncertain Robots

Chih-Hsin Tsai

t

Jing-Sin Liu * and

Wei-Song Lin

t

Nankang., Taipei, Taiwan 115,

R.O.C.

University, Taipei, Taiwan

106, R.O.C.

*

_Institute

of

Information Science, Academia Sinica

Department

of

Electrical Engineering, National Taiwan

t

A b s t r a c t This paper presents an adaptive fuzzy computed- troque controller, that enhances fuzzy controllers, with an embedded adjustable two-stages credit assignment and self- learning capability, for uncertain robots, to on-line track a pre- scribed trajectory. A n adaptation law for the parameters of controller is combined with the dead-zone technique Lo guaran- tee a given attenuationregion of trackingerror in the presence of torque disturbance. Siinulations of a two-link robot (carrying a heavy load illustrate the effectiveness and attenuation capability of the controller for on-line trajectory tracking in the presence of ineflial parameters uncertainties and torque disturbances.

1.

Introduction

Motion control of uncertain robots has attracted a. lot of re- searches (see e.g. [I]) and a lot of tracking controller design methods, ranging from traditional computed-torque controller, iudependcnt joint PID controller, robust or variable structures controllers [ 7 ] , [SI, model-based or parameter adaptive con- trollers to more recently intelligent controllers (such as iterative learning controllcrs [Ill, radial basis, multilayered, feedback-

error-learning neural networks [4], [9], [16], [17] or fuzzy logic [4] Lased controllers) have been proposed to treat the robot track- ing problems of varying degree requirements. Advan,:es in the robot controllers design are mainly for developingrnethods that could adapt to variations of robot dynamics and payloads to im- prove performance for tracking the desired trajectory as closely as possible over a wide range of configurations. Previous work has shown that certain physical properties, e.g. passivity prop- erty, linear-in-parameter property, skew-symmetric property of robot dynamics are of use in designing robust and adaptive robot, controllers and in proving the stability. However, in the presence of unstructured uncertainties or torque disturbance, the adaptive approaches can't guarantee the trackin.g perfor- mance or even the stability [6] due to their sensitivity of robot dynamics properties. In addit,ion, to exploit the nonlirtear map- ping capability of neural networks to learn the inverse (dynamics of uncertain robot,s, neural network based robot controllers re- quire a tedious training phase for a given trajectory. Moreover, their generalization capability for different trajectories is not fiilly investigated. Also there are difficulties in choosing appro- priate testing trajectories which contain sufficient information for learning the inverse dynamics. Only local stability around the operating points is ensured for tracking error dynamics.

On the other hand, fuzzy mehtods provide an efficient way to cope with uncertainties. The goal of this paper is thus to con-

struct for uncertain robots a learnable controller-an adaptive fuzzy control system-to guarantee the tracking performance: achieve satisfactory on-line trajectory tracking in the presence of imperfect knowledge by meeting the requirement that the tracking error is attenuated to a prescribed region. Our ap- proach is to employ fuzzy technique which could implement on-line computed-torque control that computes the necessary torque to move along a given desired trajectory, without de- tailed information of robot dynamics.

The organization of the paper is as follows.

In

Section 2, a fuzzy logic implementation of computed-torque control for robot motion tracking is presented. To train the controller network, its parameters are tuned. A modified update law for parame- ters using dead-zone technique is given in Section 3 . The track- ing performance and stability of the closed-loop robot system are investigated in Section 4. Simulations of a two-link robot carrying a heavy payload are given to illustrate the controller performance in Section 5. Section 6 is the conclusion.

2.

Uncertain Robot Dynamics

The dynamics of n-link robots is expressed as [l]

with q E

R"

joint variable, M ( q ) the symmetric positive- def- inite inertia matrix, c ( q , q ) the Coriolis and centrifugal forces, g ( q ) the gravitational force, and U is the input torque. The combined effect. of friction and external torque disturbance is represented by d ( q ,

4,

t ) .

(1) can be put in the form

i

= f(z)

+

G ( z ) u

+

v(z,t) ( 2 )

w i t h z T

=

( q T , ~ T ) , f = M - l ( - c - g ) , G = l l / l - l , v = M - l d .

Given a desired trajectory q d ( t ) E R n , denote the it11 joint tracking error state as e , = [ q 2 - q 2 d ,

it

- q ; d l T and let e

=

[e:, .

. .

,

.TIT.

The robot motion tracking problem treated here is to design an adaptive fuzzy controller for uncertain robots snch that e is attenuated to a given region in the pres- ence of lumped uncertainties and bounded torque disturbances. For this purpose, this paper presents an adaptive fuzzy system for on-line learning the computed-torque control to guarantee the tracking performance. This is presented in Fig.1. Details about the design are described in the following section.

I'

Figure 1

3.

A Fuzzy Controller for Trajectory

Tracking

As a preliminary, we introducebasic fuzzy concepts

[lo],

[12]. Let s

=

(SI,.

.

.

s z n ) represents the input (vector e or x) of the

fuzzy system. Denote

RS,%

the fuzzy if-then rules for the ith joint of the robots. In general, the j t h rule in the ith knowledge rule base, either Rz,c for robot position and velocity or R e , , for robot tracking error, of the fuzzy system is defined by a set of linguistic rules of the following form:

where AS,k reference antecedent fuzzy set of S k , and BI refer- ence consequent fuzzy set of the outputs of the fuzzy system. This set of fuzzy if-then rules forms a control rule base whose antecedent parts are related to the measurement and whose con- sequent parts determine the control action. The quality of the control action is inferred by a fuzzy inference engine and is eval- uated by the credit assignments mechanism. Fig. 2 shows the concepts of the novel approximate reasoning fuzzy system em- bedded with adjustable two-stages credit assignment for fuzzy if-then rules ( 3 ) .

3.1 Two-Stages Credit Assignment

The basic idea of rule credit assignment is to reward good rules by increasing their certainty of the consequent fuzzy set while punish bad ones by decreasing their certainty. There are two rule credit assignment stages presented in the fuzzy system of Fig. 2 . First, at stage I, we reshape the consequent fuzzy set BZ of the original fuzzy rule base. This paper uses LR

parametrization [12] as the consequent membership functions. Thus after stage I rule credit assignment, those membership functions become

BZ.

By fuzzy implication inference, the corresponding output ac- tion (recommendation) of each rule is defined a s [9], [13] where A : ( s ) is an arbitrary fuzzy set input to the fuzzy system,

Ai(.) := AS,l (SI)*.. .*AZ,2n(s2n) denotes thematchingdegree,

*

is the T-norm [12], I the implication function and B ~ ( u , ) is a reshaped B : ( u % ) in original rule base ( 3 ) as a consequence of stage

I

credit assignment. On the other hand, the stage

I1

credit assignment is imposed on the fuzzy output where we have determined the corresponding output action of each rule. Here, we refine them by giving a credit assignment, wz,, to the j t h rule. Then the output fuzzy set becomes:

Figure 2

w:, .AI o

R : ( S , U )

(5)

where "

."

is the multiplication operation.

3 . 2

A

fuzzy computed-torque controller

Similar to the center-average defuzzification method [9], the defuzzification of a multi-input multi-output fuzzy system with credit assignment is defined as

(6)

u ( t )

=

where

Et,,

denotes

E;,t E the centroid of t h e set{u :

Bl(u)

2

A ; ( s ) } _{( 7 )}

(using the

LMOM

method [5]), U:, represents the credit as-

signed to

R:,,

for joint

i,

while w : ~ , i

#

k , is used to counteract the dynamic interactions between the robot joints

i

and

k.,

Fig. 3 shows four different types of processing nodes in the network. Each corresponds, respectively, to a substage shown in Fig. 2:

Input Layer: The two inputs to the fuzzy system are e , ,

the tracking error of zth joint, and x, the position and velocity of the robot.

Rule Matchzng Layer: Each node here obtains the rule matching degree $ ( s ) = A1) ( s ) . To make (6) a form of computed-torque control, two types of antecedent mem- bership functions are used. For s = x , A i , L is the Gaussian membership function defined by

For s = e , we choose a special class of bell-shape member- ship functions that satisfy

C36: ( e t ) = 1 (9)

where the summation is over the number of Re,% rules for ith joint.

Fuzzy Implication Layer: Each node in this layer obtains a singleton implication fuzzy set and compute its location

",,

by

Figure

3

where c",; is the center of A i ,

pi,i

the stage

I

credit of

BI

ai

=

-

a&,,, the difference of left and right spread of fuzzy membership function A i . On the other hand, the Rd rules are chosen to be of Takagi-Sugeno type. Its consequelit membership function

BI

is a sing1.eton with support represented as the form of synthesis input

.T

c3 a71 .

=

iji,d

-

a: ei, (11) where CL: r:

( k i i , k i i ) T .

Suppose the credits of

all

these rules are assigned to be 1, then we have

De,furziJication. Layer: This layer contains n nodes which compute the on-line computed-torque control according to Let CL:

=

a , for all

j

in (11) and for convenience, we choose

pft =

l / w i I such that the credits of all rules a t joinit

i

are as- signed simultaneously in stages

I

and

I1

and the number of adjust.able parameters could be reduced. Accordingly, using

(lo),

(11) and (12), the equation(6) resolvesinto thecomputed- torgue form u ( t ) = D-l

(z,d"))

(F(z,dCa))

+

h ( e , t ) )

=

(6). (12) where

! I

W l I G l ( I )

+

c11

...

WlnGl(I)

-t

C 1 n w n l G n ( ~ )

+

cnl

...

W m G l ( I ) -k cnn

*p

~ Gld

-

aTei fn qnd

-

aKen

b

D (1,8(")) =

F ( z , d C a ) )

=

[

-

;

f 1

]

,h(e,

t )

=

[

;

]

with

d W )

= (8!W),...,8p))T,

Si"'

=

( W % I , . . . , W , ~ ) ~ , Wtl

=

( W t , ,

. . .

, w t , ) na T

,

j i

= ($

, - . .

,6?)*,

d c a )

=

@a), ' ' '

,

& q T , @a'

=

( W t i C t , .

. .

, W i t m cu m a t ,

~. .

, a y ,

f,"

=

( ~ ~ , . . . , ~ ~ , ~ ~ , . . . , ~ ~ ) ~ ,

with ,f: =

-G:,

and

.fl

=

G i

((ii)-'

-

1)1/2 where m is the number of R z , z rules. In the

above, C := ( c ; ~ ) is a bias matrix.

R e m a r k : As a funct,ion approximator, the fuzzy system uses the term w,,Gt

+

cil to learn the ( i , j ) element of matrix G,

and the term 8$ca'T f, ~ to learn

f,

in vector

f.

It c : a n there-

fore be easily seen that the fuzzy system tries to on-line learn

a computed-torque control in the presence of uiicertainties in robot dynamics.

4.

Learning algorithm

Referring to the controller network in Fig.3, the connections linking from the second layer to the output layer and the pa- rameters in the nodes of the third layer (i.e., wZr, c',,, and a : ) are adjustable. Define 8,

=

( B ! " " ) , 8 ~ w ) ) . Let the parameter

8,

be updated according to

= 8 0 ,

+

A@z(t) (13) where 8,, is the initial parameter value.

4.1 A d a p t a t i o n law

Plugging (13) into (2), subtracting

D u ( t )

and adding its identity

( F ( Z , ~ ( ' ~ ) )

+

h(e, t ) ) to the right hand side of ( 2 ) we have

ijc

= ijrd-aaet+

ft

-

Q ! c Q ) ~ ~ ~ )

+ ~ 7 = ,

( g t j - W : G ~

-

c13) '11,

(14)

(15)

(

Substituting (3) into (4) and after some manipulations, the tracking error equation for ith joint can be written as

6 ,

=

A l e ,

-

b,wTA%,

+

b z c ,

where

In this paper,an adaptation law for tuning the parameters of fuzzy controller for each joint is given by

A%,

=

R,'w,b~P,e, (17)

where P,

=

PT

>

0 is the solution of the following Riccati-like equation (18) 1 P I A t

+

ATPz

+

- P l b t b T P ,

+

Ql

=

0 P2 with Ql

>

0 , p

>

0 and

R ,

=

Block diag

(RIC),

RS"',

R(") 1 )

R!W)

=

Block dzag

(R:'),

. . .

,

Rln))

(19) (20)

4.2

A

dead-zone modification of a d a p t a t i o n law

A

continuous version of (7) could be expressed as

e,

= R ~ l w , b ~ P , e , (21) In practice, bounds on the components of the desired param- eters vector are generally known, so that unboundedness of

vector 8 , ( t ) can be avoided by suitably modifying the adap- tation policy. Furthermore, to counteract the modeling error and the parameter estimation errors, a deadzone of size do is employed in the adaptation law. Let 6 %= 8, ~ f ( 8 , ( . Define

P

=

B l o c k

dzag

PI,...,^,), and B

=

Block

d r a g (bl,...,b,) . Suppose B , ( t ) is required to be inside a set M Q ,

,

then the mod- ified adaptationlaw [13] is :

6,

=

0, for e T P b b T P e

<

dg, oth-

erwise we have

where the distance measure d ~ ( 8 , )

=

0 for 8T(R;'w,bTP,e,)

5

0,

otherwise

d ~ ( 6 ' ~ ) = mzn[l,dist(6',,Me,)/~*] , E *

>

0 . ( 2 3 )

4 . 3 Adaptation law for symmetric G

First, note that the lower component of

8,

is 8 j w ) ( t ) = 0 for

eTPbbTPe

5

d g , otherwise we have

4,

= ( I - d r y r ( 8 , ) 8 1 1 8 T ; ) R ( , W ) - 1 w j W ' b T P ~ e 1 ( 2 4 ) Since G is symmetric, we let w Z J

=

w J r in the network. In

this case, we modify the continuous adaptation law as follows: The adaptat,ion algorithm for symmetric G replaces

it"'

of (27) by another $ ! w ) , where

)

(25)

1

2

4;"'

:= it/l T O W O f

-

(d")

+

8 ( w ) T . 4.4 Selection of adaptation gains

It is interesting to observe that the matrices

(R!C),

Rr"'),

R;")

represent the inverse of adaptation gains of

8

!

'

"

)

,

8:"':

respectively. Since for robot dynamics the variation of the com- ponents of G is usually smaller than those o f f , the adaptation gains of 8:"' (and thus

4;"))

could be smaller than that of 6:""'.

This provides us a guideline to choose controller parameters.

5.

Guaranteed tracking performance

Assume t,hat there exist parameters

e;,

. . .

,QA

such that sat- isfactory approximation accuracy of on-line computed-torque input can be achieved, or

Define the parameters estimation error as 8, = 8,

-

e,*.

Then the error equation (5) can be rewritten as

where the coupling effects between robot joints and the com- bined effect of approximation error and disturbance are

5 . 1 Feasibility of controller

Denote

8")

= 8(")

-

d W ) * .

In the case there exist e and

6

small enough such that c g

5

6 and l l ! 8 w ) 1 1 2

5

6, we can show that

D

(z,

d W J )

exists, which in turn, guarantees the feasibility of computed-torque controller (12). Let \iG(z)112

2

g

>

0 in robot dynamics. Since

llD (z,

d " ) )

-

G(z)112

5

6

+

E

if

6

and E are small enough. Then, using matrix properties [2] we have

I,

+

G-'(z)(D(z,B(")*)G(z!)

is invertible which im- plies that G(z)(l,+G-' ( z ) ( D ( z , S ( " ' )

)

- G ( z ) ) ) = D ( z ,

d")'

)

is invertible.

5 . 2 Initial parameters for better tracking performance In practice, nominal knowledge about the manipulator dy- namics is available. Hence, an "approximation" to G ( z ) , de- noted by Go (xlnominal robot parameters), is known a priori.

This knowledge could be employed to satisfy the accuracy re- quirement of 6 small enough, by selecting a suitable initial pa- rameters 0:'. This is done here by using the least square (LS) algorithm [14]. Our approach is based on the element- by-element minimization of the approximation error through a sufficient number of training data

{x'~)}:

E ( w o t J ) = C k

I

(Go ( ~ ( ~ ) l n o m i n a l robot pnra?neters) - D ( z ( ~ ) , 8 ( w ) ) ) z 3

l2

(28) where the set {z('j'> contains either the sampled feature points on or the points close to the desired trajectory. Furthermore, the same technique can also be used to satisfy the approxima- tion accuracy requirement for f .

In addition, since the inertiamatrix (thus G matrix) is lower and upper bounded [l], the bias matrix C call Le select,ed with

an aim to meet the bounds requirement of G in the process of

LS minimization.

5 . 3 Tracking performance

The following shows the tracking performance of adaptive fuzzy controller for uncertain robot manipulators.

Theorem. Consider the robots ( 2 ) with unknown but bounded

f , ( z ) , v,(z!t) a n d g , ( z )

#

0,i

=

l,...,n

.

Assume that t,hecon- troller (12) is adopted with the adaptation policy ( 2 4 ) . Then in the bounded state space z E R

=

{z

:

llzll

5

r},

we have

8;

and the control input are bounded. Let

Q = B l o c k d i a g

(Ql,

' . .

,

Q n ) and assume that there exists

<

= up^,,^,^ Zt[l<t(8t, z , t ) ( j 2 , then e converges t,o the residual set {e : e T Q e

5

p 2 [ or e T P b b T P e

5

d i } . Moreover, for e t and

e9 small enough such that

E

5

d o / 2 p 2 , then e converges to the deadzone { e : e T P b b T P e

5

d i } .

Proof: Let Ve be a positive-definite function of the form

= ( E l , . .

.

,

1 2

v,

=

-cy=loe,Te,

VQ

= ET=.=,

O F e t

5

0. Therefore, we can guarantee the bound- edness of

O i .

Thus the arguments 8%,z in (12) are hoimded, in view of the assumption of

z

E 0 . S o we have U bounded.

Next, for the whole error system, define the Lyapunov func-

tion candidate as V = VI

+

V,

+

. . .

+

V, where for each joint

we deihe the Lyapunov function

( 2 9 )

V, = I d ;

+

$aTR,6,,

if e T P b b T P e

5

d z

=

$eTP,e,

+

+ i ? ~ , i , ,

otherwise

Taking the time derivative of Vi, we readily obtain V, = 0 for

In the case d M ( 8 , )

#

0,Q, will be in the exterioi of

Met

and

O T ( R ~ ' w , b ~ P , e , )

>

0 which implies Q ~ l ( R ~ 1 w l b f P 8 e t )

> 0.

Suppose

Mot

is appropriately selected such that 8: is in the interior of

M e , ,

then we have

so we have

. 1 1

V

5

--eTQe 2

+

- p 2 ( 2 ( 3 2 )

Thus e converges t,o the residual set { e : eTQe

5

p 2 c or e T P b b T P e

5

d g } .

Finally, consider the case t.hat ~f and E~ are small enough siich that

5

rlo/2p2 for eTPbbTPe

>

d g . Thus

V

=

--eTQe I

-

-p2eTPbbTPe 1

+

cbTPe

2 2

5

-;eT&, 1

-

(

:p21/eTPbll

-

()

llbTPell

( 3 3 )

1

5

- - e T Q e

2

Then, following the arguments of [16] we can conclude that e

conveqes to the deadzone { e : e T P b b T P e

5

d i } .

C).

E . D .

Corollary. Consider t.he robot (2) with a symmetric G matrix. Then all properties described in Theorem also hold for the robot corit,roller (1 a)-( 2 5 ) .

Proof: Not,e that. t,he derivative of Lyapunov function (29) can be decomposed as

i:

= e?P&

+

E.;,Ryi.,,

+

q R p i L ;

+

8 $ " ) ' R y p

( 3 4 )

v,

=

e T P , e , '

+

E ~ , R : C ' ~ ~ ,

+

C T R ! ~ ) ~ ~

+

B ~ W ) ' R { W ) ~ ! W ) ( 3 5 )

I n view of t,lw properties: (i)Gt3

=

CJ,,, due to lwi3

=

u3ir

(ii)R:3)

=

RI",

one can verify that Substituting ( 2 5 ) into (4), we have

6.

Simulation

+

m z T l r z s i n ( q z ) d z

+

m z h c o ~ ( q ~

+

4 2 ) = u2

+

d2

where the combinedeffects of friction and the external torque disturbance are

d l

=

2.0sin(1jl)

+

2.5sin(&)

+

0 . 5 s i n ( t ) dz

=

5.0sin(d.1)

+

4.0sin((iz)

+

0 . 4 s i n ( t )

and 11

=

2, 12

=

1.6, r1

=

0.511, r2 = 0.512, 5 1 = 5 2

=

5 ,

mi = 0.5, m z = 6.35. The excessive ratio between ml and mg is to emphasize the payload effect.

For each joint position or velocity, only four linguistic labels { N B ,

N S ,

E'S,

P B }

are used with membership functions

NB

: f ( r )

=

l / ( l + e z p ( 5 ( r + l ) ) ) , N S :

f ( ~ )

=

e z p ( - ( ~ + 0 . 6 ) ~ ) , P S : f ( r )

=

e ~ p ( - ( r - 0 . 6 ) ~ ) , P B : f ( ~ )

=

i / ( l + e q ( - 5 ( r - l ) ) ) .

where T is the joint position or velocity. Moreover, we let a i

=

a;

=

0 for all j , i.e., the left and right spreads of the consequent membership function are equal in the adaptation period. We consider there are no fuzzy control rules. We set bias cIJ

=

1

for all

i , j .

The initial parameters cul (0) and cu2 (0) are chosen randomlyin the interval ( - 5 0 , 5 0 ) . To obtain a set of appropriate initial parameters

QP),

in ( 2 8 ) we use 32 testing points on the desired trajectories over one period [0,8]. The following nominal inertia parameters are used in LS minimization:

5; =

4.8,

=

5.1,

m(:

=

0.48,

mi =

6.30.

The other controller parameters are chosen as &1

=

Q 2

=

1012, E*

=

0.05, C Y ~ J

=

L Y Z J

=

100, a 1 , 2

=

0 1 2 , ~

=

65 and

R1 =

Block d i a g (0.0051256,3000I25@, 20001256),

R2

=

Block d i a g ( 0 . 0 0 2 5 1 2 5 6 , 2 0 0 0 I 2 5 6 , 3 O O 0 1 2 5 6 )

.

Two cases with different p values are simulated for

q x d ( t )

=

~ / 1 2 s i n ( 0 . 5 ~ t )

92d(t)

=

2 . 5 ~ / 1 2 ~ 0 ~ ( 0 . 5 ~ t )

4-

2.5~/24co~(0.25~t). with the initial state q l ( 0 )

=

-1.5, q z ( 0 )

=

-1.2, q l ( 0 )

=

i z ( 0 )

=

0. They are compared, with the responses shown in Fig. 4((a) position error (b) velocity error (c) control input) where solid line and dotted line denote p1

=

p2

=

0.01 and p1

=

p2

=

0.5, respectively. The tracking errors for the last joint are relatively small, whereas larger tracking error are ob- served in the first joint. In both simulation cases, the tracking errors are gradually within a tolerable accuracy region as the learning progresses.

7.

Conclusion

For trajectory tracking of uncertain robotic manipulators, we have proposed a learnable controller for on-line learning of

the compuled-torque control to adapt to changing environment characterized by imperfect knowledge. The controller consists of a fuzzy system embedded with adjustable credit assignments and supervised by the desired trajectory. A combination of coli- troller parameter adaptation law with a dead-zone technique is used to guarantee the stability and the attenuation of tracking error to a prescribed region. Simulations of a two-link robot manipulator have demonstrated that the controller is efficient in achieving satisfactory tracking accuracy in the presence of significant uncertainties. Simulations also show how a priori

knowledge about the robot dynamics can be applied in an aux- iliary manner to speed up the learning process and to improve the tracking accuracy.

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_ _ _ ~

... ,-- ... ... ... ..~

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4