Robust self-tuning rotated fuzzy basis function controller for robot arms

Rolbust self-tuning rotated fuzzy basis function

controller for

robot arms

C.-K. Lin

S. -D. L'Va n g

Indexing terms: Adoptive fuzzy controllers, Robot arms, Self-tuning fuzzy control

Abstract: An adaptive fuzzy controller is

developed for a serial-link robot arm. The proposed rotated fuzzy basis function (RFBF) controller is a more flexible fuzzy basis function expransion to approximate unknown functions of the robot model. All parameters of RFBF network can be tuned online when the number of rules is determined. In the control design, the unniodelled dynamics are considered. Moreover, the stability analysis shows that the states and tracking errors of the robot arm are uniformly bounded. Simulations of the proposed controller on the PUMA-560 robot arm demonstrate the effelctiveness.

1 Introduction

Fuzzy control has been an active research field in the past decade. Recently, it has also been related to adap- tive control based on fuzzy basis function expansion addressed in [ 1-31. Although adaptive fuzzy controllers in [2, 31 can guarantee the global stability and tracking performance, they can only tune a part of parameters of the fuzzy system. The robustness property of the controller was also not mentioned.

The area of self-tuning fuzzy control has been stud- ied by many researchers, with a few of the notable ones surveyed here. Although self-organising, self-learning and self-tuning fuzzy control are something different, they are all treated as self-tuning fuzzy control here. Procyk and Mamdani [4] proposed a self-organised fuzzy control system which employs a mechanism to

modify or create a fuzzy rule base. Since the pioneer work of Mamdani, considerable research works about self-tuning fuzzy control have appeared. Consequential papers which improve Procyk-Mamdani's controller were proposed in [5, 61. However, such self-organised fuzzy control systems cannot determine the rule number, shapes of membership functions, global stabil- ity and the tracking performance. Many fuzzy-neural paradigms have been proposed recently to confront these deficiencies. For example, Yamaguchi et al. [7] proposed a fuzzy associative memory system called

0 IEE, 1997

ZEE Prmeedings online no. 19971000 Paper received 7th March 1996

The authors are with the Department of Electrical Engineering, National Taiwan IUniversity, 1 Roosevelt Rd., Sec. 4, Taipei 106, Taiwan

1E.F Proc.-Control Theory Appl., Vol. 144, NO. 4, J d y 1997

FAMOUS. In the IF-part of the knowledge pairs of FAMOUS, the membership function is automatically generated from input data by an unsupervised learning algorithm. Another class of adaptive fuzzy systems is based on feedforward five-layer neural networks. Lin and Lee [8] proposed an unsupervised learning neural network-based fuzzy logic control systems employing a reinforcement learning algorithm to tune parameters. This fuzzy controller can be represented by the ad hoc structure of a five-layer neural network which has a direct correspondence from fuzzy control system. Jang [9] also proposed a five-layer self-learning fuzzy control architecture whose learning algorithm is gradient descent. These two five-layer self-tuning fuzzy neural control systems are not capable of guaranteeing stabil- ity and tracking performance.

Fuzzy Basis Function Network is a new fuzzy-neural scheme which integrates fuzzy reasoning systems with radial basis function networks. Nie and Linkens [lo] proposed a fuzzified self-organised radial basis function network employing a simplified fuzzy control algorithm (SFCA) to tune all parameters and construct the net- work. SFCA, which is an unsupervised learning algo- rithm, cannot guarantee the stability and performance. In the past, a fuzzy controller has not been viewed as a rigorous control approach owing to a lack of formal synthesis techniques that guarantee the very basic requirements of global stability and acceptable per- formance. Wang [2, 31 proposed an adaptive fuzzy con- troller which can tune the parameters of consequent part by adaptation law using Lyapunov stability the- ory. This controller can also cope with the unstructured dynamics by adaptive robust techniques.

In this paper, the proposed controller is only based on a general three-layer neural network that implements a fully adaptive fuzzy radial basis function expansion. The architecture of the neural network is basically a multilayer neural network where radial functions are used instead of sigmoid functions in hidden neurons. The hidden-to-output layer interconnection weights correspond to the consequence part of the fuzzy rule base and the input-to-hidden layer interconnection weights can be mapped to the parameters of fuzzy basis functions. The major differences between RFBF networks and three-layer neural networks are the processing elements of neurons and the input-to-hidden layer interconnection weights. The input of the sigmoid (kernel) function of RFBF networks is the square norm of the difference of input vector and input-to-hidden layer weight vector. The use of RFBF controller in direct closed-loop controllers 293

can guarantee the global stability and tracking performance.

2 Preliminaries

2.

I

Robot arm dynamics

The dynamic equation of an n degree of freedom robot manipulator is given by:

M(e)fi+Vm(@,b)b+G(@)

+F'+Q=Z

( I )

where vector

8

E Rn is the joint position vector; M ( e ) E Rnxn is a symmetric positive definite inertia matrix;

V,

(e, e

) is a vector of Coriolis and centripetal tor- ques; G(8) € R" represents the gravitational torques; F = K,

+

Vf€

Rnx" is a diagonal matrix consisting of the back emf coefficient matrix K , and the viscous friction coefficients matrix K j z d E Rnxl is the unmodelled dis- turbances vector; and 3 E Rnxl is the vector of control input torques. A planar two-link robot arm used for illustration is shown in Fig. 1. The structural pmperties of the robot such as boundedness of V,

(8,

e )

and skew-symmetry of matrix M - V, are well-known and not stated here.

link 6

Fig. 1 A two-link robot manpilator with links 4, 5 and 6fixed

f l

fm

Fig. 2 Network representation of RFBF expansion system

2.2 Rotated fuzzy basis function networks

A three-layer network representation of an RFBF net- work is shown in Fig. 2. The RFBF network can per- form the procedures of fuzzification, fuzzy inference and defuzzification. Let input space X C R" be a com- pact product space. Assume that there are r rules in the rule base and each of which has the following form:

R,:If X I is

AI,

and 5 2 is Az3 and . . . and x, is

A,,

then y1 is B,1 and y2 is B32 and . . . ym is B J m ,

294

wherej = 1, 2, ..., r , the input vector

x

= (xl, x2, ..., xn) E X, x, (i = 1, 2, ..., n) contains the input variables to the fuzzy system, Y k

(k

= 1, 2, ..., m) are the output var- iables of the fuzzy system, and A, and Bjk are linguistic terms characterised by their corresponding fuzzy mem- bership functions pAij(xi) and pBjkCyk), respectively.

As in [2], we consider a subset of fuzzy systems with

singleton fuzzification, product inference, and Gaussian membership functions. Hence, such fuzzy system can be written as

,=1 \ z = 1 /

where f k : X C R"

-

R", pBjk ( f j k ) = 1, and p A g (xJ is the Gaussian membership function of the form

(3)

where

Eo

and ov are real, and vector

E

- =

[Elj

.._

En,]'

is

an n-dimension vector.

From the point of view of geometry, the rule base of an RFBF network can be written as [13]: Rj: If

(Elj,

olj) and 02j) and ... and

(Enj,

onj), then ( c j l ,

oil)

and

(ej,, a&), where

q y

=

[ o ; ~ ,

dj2,

...,

o7J is the radius vector of THEN part of the jth rule, and

q

= [olj, 02j, ..., onjIT and

5 j

=

[E1,,

5.2j,

...,

5.nj]T

are radius and centre vector of I F part of the jth rule, respectively.

Definition 1 : (Fuzzy Basis Function) Define fuzzy basis functions (FBFs) as

z= 1

j = 1 , 2 , . . . , ? -

(4)

08,

...) lioi).

where , U ~ ,

(xi)

are Gaussian membership functions (eqn. 3),

x

= [xl,

x2,

...,

xnIT E X and S, = diag(l/o$ 1/ Motivated by [ l l , 121, the basis functions can be rotated to achieve better performance. We use the rotated fuzzy basis function which is defined as follows.

Definition 2: (Rotated Fuzzy Basis Function) Define rotated fuzzy basis functions (RFBFs) as

j = 1 , 2 , ...,?- ( 5 )

where

8$

is the rotation matrix and zj =

The RFBF expansion can be defined in the following definition.

Definition 3: (Rotated Fuzzy Basis Function Expan- sion) The set of a fuzzy system with rotated fuzzy basis functions consists of all functions of the form

-

5j).

3 = 1 ,=1

where cl are weights of defuzzifier, are rotated fuzzy basis functions.

The concepts of input patterns and rule patterns are introduced in [lo]. The rule pattern of the jth rule is defined by the centre

El,

and radius

4

of the jth rule. From the above definition, the n-dimensional vector

RI $I will be referred to be a rotated input pattern and

the rule pattern is the same as [lo]. In other words, the difference between RFBF and FBF networks is just in that the inputs are multiplied by a rotated matrix for IEE Proc -Contuol Theory A p p l , Vol 144, No 4, July 1997

each ride. Fig. 3 shows the difference of two-dimension elliptic isocontours of FBFs and RFBFs and the latter has more flexibility in structure. In [2, 31, the parame- ters of pAa,(x,) are fixed and the weights c, are adjusta- ble. However, in this paper, the parameters of pA

(x,),

tq

and oq are also adjustable and the para2eter updated law will be stated in a later Section.

F

a b

Fig.3

( U ) Isocontours of fuzzy basis functions

(b) Isocontours of rotated fuzzy basis functions

Isocontours of fuzzy basis functions which has two input variables

Another type of rotated fuzzy basis functions we consider has an easily calculated form:

43

=~ e - ( ~ l o + ~ , ~ z ~ + + ~ ) r L z n ) ' + _ZT

3 -Net Z N e t

-

_.

,-Get

( W , g + % %Net

3 = 1 , 2 , . . , T ( 7 )

where o , ~ , o , ~ , ..., are adjustable parameters,

a,

=

[q0,

wjl,

...,

wJnlT and xNel = [l x T T .

It is easy to show that using the definition 2 eqn. 7

can be put into an RFBF form as in eqn. 5. Since the matrix is

y y T

+

a,' a positive-definite matrix, there exists

where A, = diag(4, AI,

&,

.., A,,). The matrix T I can be divided into four blocks:

uJw;+Q31=rp3r3,

9 =i,2,...,r (8)

(9) where

rll

is 1 by 1, T12 is 1 by n,

r21

is n by 1 and

r22

is n by n matrix. Thus equation eqn. 5 can be rewritten

as:

4 .

-

e-xO(rll

+

r12g)ze-(rzl

+rzzdTh: (rZ1+rz2d j = 1 , 2 , . . . , T

where Sj = A; = diag(Al, A2, ..., A,,), $ j =

-r22-1r2,

and

= !Rj := Therefore, eqn. 7 is a rotated fuzzy basis function multiplied by a magnitude modification term

e-h(rll

+

r1&2.

Each fuzzy rule is described by membership functions of linguistic variables. In other words, in fuzzy basis function expansion, the parameters and oij determine the I F part of rule and e, determine THEN part of the rule. From the previous derivation, while cojis deter- mined, we can find

rj

to determine

Eij

and oo by Sj = A/ and TZ2

5,

=

-r21.

Therefore, the I F parts of rules can be reconstructed from rotated fuzzy basis func- tions. The weights c of defuzzifier can be tuned by learning algorithm.

Rub-base initiulisution: To initialise the input pattern and rule pattern of a fuzzy system [lo] is to specify the centre and radius of each FBF. For RFBF, besides the specification of centre

5 j

and radius

q,

a rotated matrix !Rj is needed to be determined. As the rule base is initialised, the initial weight vector w can be deter- mined by the following equations.

(10)

3 -

rZ2

=

xJ

(11)

IEE Pro(:..-Control Theory Appl., Vol. 144, No. 4, .July 1997

Therefore, the initial weight vector from w ~ o = r21TAr21 and

can be calculated

In comparison with general self-organising fuzzy sys- tems as [lo], the size of the RFBF network is smaller. For an RFBF network, there are only ( n

+

1) x r

+

r x

m parameters to be tuned and 2n x r

+

r x m parame- ters for [lo]. The parameters

Si,,

oLi determine the I F part and cI determine the THEN part of the jth rule. Therefore, these parameters are stored and tuned in a compressed form. Another advantage of RFBF is that the experts' experience can be incorporated into the RFBF network by eqns. 11-13. The lack of experts' knowledge is the major difference of Gaussian neural networks to FBF systems. In general FBF systems, only weights c of defuzzification can be tuned by learn- ing algorithm [1, 21. However, the RFBF network can tune both the I F parts and THEN parts of rules. In essential, the RFBF network is a multilayer neural net- works with Gaussian activation functions. The sigmoid functions may result in local errors producing global change for function mapping, however, Gaussian acti- vation functions are capable of eliminating this global interaction [13].

3 Robot RFBF controller design

3.1 RFBF-based controller

The control objective is to design a robust RFBF-based controller so that the movement of robot arms follow the desired trajectory even in the presence of distur- bances. Given a desired robot manipulator trajectory

ut), the tracking error vector is g ( t ) = ut) - fit) and error metric ~ ( t ) can be defined as

s(t)

=

k ( t )

+

A&)

(14)

where A = AT > 0. The RFBF network is used to approximate unknown nonlinear function

- f = M(@)(&+AE)+V,(@,

4)

(@&Ag-A~at(s))+G(@)+F8

The approximation error and unmodelled uncertainties are inevitable. Therefore, two methods are adopted to compensate them: one is the deadzone function and another is a robust term. Deadzones can be incorpo- rated into error metrics by defining continuous func- tion

zA

and &(t) as:

(15)

sA(t)

=

~ ( t )

-

Asut(s(t)/A)

and

s % ( t ) = [ s ? ' . '

Sk

1

(16)

(17)

T

where sat is the saturation function: 1,

z > l

s a t ( z ) =

z,

121

5

1

{

-1, z < - 1

Deadzone functions, which are specified around the zero of their corresponding error metrics, will be used in the adaptation law to tolerate the parameter errors and approximator errors. The architecture of the con- troller is shown in Fig. 4. The control law is given by

-

.(t)

= V R F B F ( t )

+

&t)

+

K s , (18)

where K is a positive definite diagonal matrix, yRFBA!)

= C4 and _d(t) is the robust term. There are three main reasons for using rotated fuzzy basis function expan- sion system as the basic component of an adaptive fuzzy controller. First, the rotated fuzzy basis function expansion can be represented by a three-layer neural network as shown in Fig. 3 rather than five-layer neu- ral network as in [2, 31. The general learning algorithm can be used to train the network. Secondly, the experi- ences of human experts can be incorporated into the controller by the linguistic IF-THEN rules. The third is that the fuzzy system is a universal approximator. It

has been proven in [7] that for any given real function1 over X , there exists a fuzzy system in the fuzzy basis function expansion form of (eqn. 16) such that it can uniformly approximate f on the compact set

X

to arbi- trary accuracy. Accorangly, we have the following assumption:

I I I

Fig.4 Diagram of clvsed-loop system

Astumption I : There exist matrices Q* and

c'

such over

a

that

f

approximate -

f

with arbitrary accuracy compact set

X,

i.e.

3 ~ * and C* s.t. I j ( g ( t ) ) - - -

j ( ~ ~ , ~ * , : ( t ) ) /

5

Hence, eqn. 1 can be rewritten as:

M g t ) =

-

(V,

+

K ) s a ( t )

+

- j ( O * , C*,:(t))

-

K R F B F ( t ) -

a(t)

+

s(t)

+

?d (19)

where the disturbance

d(t)

=j(s(t))

-

-f(Q*,

-

C ,

~ ( t ) ) sat- isfies lai(t)l 5 E~

The architecture of RFBF controller as an approxi- mator is shown in Fig. 2. Denote the approximator having (n

+

I) inputs, r rules and m outputs as follows:

, . A

- ?(:Net

,

fi,

(3

= V R F B F ( t ) = C 4 J ( f 1 2 : N e t ) ( 2 0 )

where E Rrx(n+l) and C E Rmxr are RFBF network weight matrix and defuzzification weight matrix, respectively, and RFBF vector is

4 q f i X N e t ) =

[

;

]

a n d f i =

["I

(21) For simplicity, we define

d

=

@(axNet)

and gj* = H Q ' z N e t ) . Thus, eqn. 19 becomes

4 ( & T X N e t )

4 ( & : X N e t )

w,

- T

M i ( t ) = -(Vm+K)s,(t) + C * $ * - f i $ - ! $ ( t )

-

- + J ( t ) + z d

( 2 2 )

The RFBF can be expressed by Taylor series expansion [31 as

-

-$* =

6

+

$'fi2:Net

+

O((figNet)2) or

-

4

= $'&.Net

+

o((fizNet)2) (23) (24) where

4

=

d*

-

6,

= Q* - Si and

4

= diag(d1,

...,

I&). Substituting esn. 23 into eqn. 22, we can get M S ( t ) =

-(Vm

+

K)sa(t)

+

64

-

+

(?$'figNet

+

d -

2

( 2 5 )

where C = C" - d a n d _d =

Cq$'QzNet

+

c " O ( ( a ~ ~ ~ , ) ~ )

+

- a(t)

+

zd.

In the following, the norm of vector or matrix,

ll*ll,

is

Frobenius norm [14]. The upper bound of the norms of

0((a~,,,)~)

is given by:

(26) where

114

11

s ~1 and 11@11 s K ~ . And, the upper bound of

the norm of _d is as follows:

1, 2 2 0

-1, 2 < 0

sgn(x) =

The parameters

6

Si

and

d,

are updated by the fol- lowing adaptation law which is similar to the back- propagation algorithm:

where K,, KQ and K, are positive symmetric constant matrices determining the adaptation rate.

3.2 Stability analysis

A stability theorem is presented for the tuning algo- rithm eqns. 28-30.

Theorem 1: Consider the dynamic eqn. 1 with the control law eqn. 18 and weight tuning algorithm eqns. 28-30. All states in eqn. 31 will remain bounded and the tracking errors will approach zero.

Proof: Consider the Lyapunov function candidate

Evaluating the time derivative of V(t) along the trajectories of the tuning law eqns. 28-30, then

V

= 0, when Is;I

5 A,

i = 1 , 2

When lsil > A for each i, by the property of skew- symmetry of

M-

2 ~ , , WG can get

(32)

Therefore, if

aA,

all Zy and all

LjY

are bounded at initial time t = 0, they will remain bounded for all time t > 0. If

g(0) is bounded, then j ( t ) is

also bounded for all time t , and since y,Xt) is bounded specified, J ( t ) is bounded as well. Next, WG will show that sA

-

0 as t

-

W . It is easy to show by Barbalat's lemma:

.i'

[ V ( r )

+ s z K ~ ~

1

d r with _{( 3 3 )} V I ( t ) = V ( t ) -

(34) T

Vl(t) = -S*KSa

Thus, we have shown that every term in eqn. 31 is bounded; hence

sa

is bounded and

s

is bounded as well. This implies that

V ( t )

is a uniformly continuous func- tion of time. Since VI is bounded by 0, and

v

5 0 for

all time t , Barbalat's lemma can be applied to prove that + 0 and hence

aA

+ 0 as t + w. Q.E.D.

4 Simulation results

In this section, PUMA-560 is taken as the robot manipulator to be controlled. As in [15], the fourth, fifth and sixth links of PUMA-560 were fixed, and the angles of the second and third links were considered to be O1 and

e,,

respectively. The numerical values of the robot model can refer to [15]. The desired trajectories for 8, and 0, were chosen as:

O d l = 0.5

+

0.2(sin

t

+

sin

at)

Oda = 1.3 - O.l(sin

t

+

sin

at)

( r a d ) for 191 and ( r a d ) for 0 2 The proposed RFBF controller was compared with the well-known Slotine-Li's adaptive controller. The Slotine-Li's method [ 121 estimated nine parameters; the proposed method approximated nonlinear functions eqn. 15. The derivative gains of Slotine-Li's method were ,KO = diag(250, 250). For the RFBF network, there were 20 rules in the rule base and the parameters of RFBF network were given by eqns. 28-30. The adaptation rates were specified as Kc = 100.01,,,, KQ = 50.01,,t~+11x~n+11 and K, = 0.0515,, ( I p x p is a p x p identity matrix). Figs. 5 and 6 show the desired trajectories and trajectories obtained from RFBF and Slotine-Li's controller. The maximum tracking errors of 01 and O2 after the first two seconds of movement of the robot arm using the Slotine-Li's method were 0.72" and 0.60". Using the proposed RFBF controller, the errors were found to be 0.39" and O.OX", respectively. This comparison shows the proposed controller can obtain more accurate tracking performance due to the good approximation capability of the RFBF network as shown in Figs. 8-11 show the simulation results with IEE Proc -Control Theory A p p l , Vol 144, No 4 July 1997

bounded disturbances

s&

These results imply robustness of the proposed RFBF controller.

the 01 I I 0 2 4 I time,s Fig. 5 ~ desired trajectory .... RFBFN controller

Simulations,for 8, ( t ) using Slotine-Li's and RFBFN controller

Slotine-Li's controller 0 2 4 6 time,s Fig. 6 ~ desired trajectory ... RFBFN controller

Simulations for &(t) using Slotine-Li's and RFBFN controller

- - Sloth-Li's controller

time, s

Fig.7 Input torques: q ( t ) and t,(t)

01 I I

0 2 4

time,s

Fig. 8 Function approximation of%(t)

~. desired trajectory .. RFBFN controller

5 Conclusions

A self-tuning rotated fuzzy basis function network has been proposed. The controller design is based on the proposed RFBF which is capable of incorporating experts’ experience into a controller. The self-tuning rotated fuzzy basis function control system used a smaller network size than other neural fuzzy systems to achieve the desired performance. The controller is

flexible because all parameters of the RFBF network can be tuned by adaption law once the rule number is

determined. By combining the deadzone functions and robustness techniques, we can show that the controller is robust. Sound tracking results could be obtained by the proposed controller. Simulation shows that the RFBF controller is more accurate than the well-known Slotine-Li’s controller. The stability analysis shows that the control system can be guaranteed to be asymptotically stable.

6 References

timeas

Fig.9 Function approximation o f h i t )

~ desired trajectory

.... RFBFN controller

01 I I I

0 2 4 6

time, s

Fig. 10 , Simulationsfor 0,jt) using RFBFN with disturbances

~ desired trajectory ...’ RFBFN controller ‘“I 0 2 4 6 time, s Fig. 11 ~ desired trajectory RFBFN controller

Simulationsfor @ i t ) using RFBFN with disturbances

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