Adaptive tracking control of flexible joint manipulator based on fuzzy model reference approach

C.-W. Park and Y.-W. Cho

Abstract: A direct model reference adaptive control via Takagi-Sugeno fuzzy modelling and parallel distributed compensation (,PDC) is developed for the MIMO plant model to control an uncertain flexible joint manipulator. The proposed control scheme is proposed to provide asymptotic tracking o fa reference signal for the systems with uncertain parameters. From Lyapunov stability analysis and simulation rewlts, the developed control law and adaptive law guarantee the boundedness of all signals in the clmxed-loop system. In addition, the plant state tracks the state of the reference model asymptotically with time for any hounded reference input signal.

1 Introduction

Many of today's robots are driven by actuators with high gear ratios, such as harmonic drivers for high torque and low operation speed. Furthermore, mechanical damage to the robot and the environment can be minimised in an accidental collision involving the arm as the flexible joints and links can absorb a certain amount of impact force due to collision, and this will render joint compliance some- times to he a desirable feature [ I ] .

To this e n 4 much research on the control of flexible joint manipulators has been done, such as model based approaches, which include the feedback linearisation scheme [I ] , the invariant manifold scheme [Z, 31, robust control [4] and adaptive control [5, 61.

However, although the joint flexibility has demonstrated some potential merits, the difficulty with modelling and controlling such a flexible mechanical system with high performance made most robot designers prefes to manu- facture mechanically rigid arms with stiffjoints. Hence, in this paper, we tackle the problem of controlling for flexible joint robots via fuzzy modelling and fuzzy model based controllers, and propose a complete solution to !solving the problem of model uncertainty.

Fuzzy logic controllers are generally considered applic- able to plants that are mathematically poorly understood and where experienced human operators are available. However, fuzzy control has not been regarded as a rigorcas science due to the lack of guaranteed global stability and acceptable performance. To overcome this drawback, since the Takagi- Sugeno (TS) fuzzy model [7], which can expre:is a highly nonlinear functional relation in spite of a small number of fuzzy implication rules, was proposed there has been

0 IEE, 2003 IEE Pmceedrngs online no. 20030017 DOI: 10.1049iip-cta:20~3nn17 Paper first received 19th September 2001 and in revised form 18th October 2002 The authors arc with the ICs Lab., Department of Electrical and Electronic Engineekg, Yonsei University, 134, Shinchan-dong, Seodaemun-ku, Seoul, Korea

198

significant research on the stability analysis and systematic design of fuzzy controllers [S-IO]. In their research, the nonlinear plant is represented by a TS fuzzy model and the control design is carried out based on the fuzzy model via the. so-called parallel distributed compensation (PDC) scheme and linear matrix inequality based optimisation.

To deal with the uncertainties of nonlinear systems, in the fuzzy control system literature, a considerable number of adaptive schemes have been suggested [lO-l5]. An adaptive fuzzy system is a fuzzy logic system equipped with an adaptive law. The major advantage of the adaptive fuzzy controller over the conventional adaptive fuzzy controller is that the adaptive fuzzy controller is capable of incorporating linguistics fuzzy information from human operators. Most of them were based on the feedback linearisation scheme or indirect adaptive approach in which the approximating ability of the fuzzy system was utilised or an online adaptation scheme was usually used to estimate the unknown parameters of the system and an appropriate controller was then designed to control the plant to satisfy a desired performance.

In this paper, to control the flexible joint manipulator, we present an alternative direct adaptive fuzzy controller based on the model reference approach, in which the desired process response to a command signal is specified by means of a parametically defined reference model, for MIMO plants with poorly understood dynamics or plants subjected to parameter uncertainties. Wc utilised the TS fuzzy model for uncertain flexible joint manipulator modelling and PDC. The adaptation law for adjusting the parameters in feedback and feedforward gain of the PDC controller is designed so that the plant output tracks the reference model output.

2 Takagi-Sugeno fuzzy model based control

Consider the continuous-time nonlinear system described by the Takagi-Sugeno fuzzy model. The ith rule of a continuous-time TS model is of the following form:

R': If x i ( f ) is M; a n d . . .and x,(t) is ML

then x(t) = A,x(t) + B,u(t) (1)

I€€ Proc.-Conna; Themy Appl., Yul. 150, No. 2, M u c h 2003

where

XT(0 = [XI(O. x2(t), . . . 1 x,(t)l

U'(t) = [ill ( t )% U Z ( f ) , . . . 3 U,&)]

Given a pair of inputs (x(t), u(t)) , the final output of the fuzzy system is inferred as follows:

where iv,(t) = n;= lq:(x,(l)) and q : ( x j ( f ) ) is the grade of membership of xj(t) in Mj.

To design fuzzy controllers to stabilise fuzzy system (2), we utilise the concept of PDC. The PDC controller shares the same hzzy sets with fuzzy model (2) to construct its premise part. That is, the PDC controller is of the following form:

R': If xl(t) is M ; and. . . and x,(f) is ML

then ~ ( t ) = -K;x(t) (3) T where x ( t )= [~rl(t), xZ(t), . . . , ~ , ~ ( t ) ] and i = I , . . . , 1.

PDC controller (3) is inferred as follows: Given a state feedback x(t) , the final output of the fuzzy

where w,(t) = n;= IM;(.,(f)). By substituting the controller (4) into the model (2), we

can construct the closed-loop fuzzy control system as follows:

A sufficient condition for ensuring the stability of the closed-loop fuzzy system ( 5 ) is given in Theorem 1, which was derived in [2]. Theorem I : The equilibrium of a fuzzy control system ( 5 ) is asymptotically stable in the large if there cxists a common positive definite matrix P such that

CgP + PG, = -Q, (6 )

for all i , j = 1, 2, . . . , I , where G , = A ; - BjKj and Q, is a positive definite matrix.

Thc design problem of model based fuzzy control is to select K; ( j = I, 2,. . . , 0 which satisfy the stability condi- tions (6). In [9], the common, P problem was solved efficiently via convex optimisation techniques for LMIs (linear matrix inequalities). However, the fuzzy control (4) does not guarantee the stability of the system in the presence of parameter uncertainty. Moreover, the design of the control parameters is not possible for the systems whose parameters are unknown. To overcome these draw- backs, in this research, an adaptive control scheme is developed for the plant models whose parameters are unknown.

3 Adaptive fuzzy control based on model reference approach

In this section, an adaptive fuzzy model reference control scheme for the MlMO TS fuzzy system is developed. Consider again the nonlinear plant represented by the TS model ( I ) or (2), whcre state X E R" is available for measurement, A; ER" X n , B; ER" x y (i= I , . . . , l ) are

IEE rmc.~coniroi neOv A,$ 6'01. I S U , N". z. n40rnrCll zuu3

unknown constant matrices and ( A j , B;) are controllable. The control objective is to choose the input vector U E Rq such that all signals in the closed-loop plant are bounded and the plant state x follows the state x, E R " of a reference model specified by the system

where ( A , ) , E R " ~ " (i= I , . . .,[) satisfy the stability condition of the fuzzy system given in Theorem 1, ( B , J j j t R n , X Y , and r c R q is a bounded reference input vector. The reference model and input r are chosen so that x,(t) represents a desired trajectory that x has to follow.

3.1 Control law design If the matrices A ; , B; were known, we could apply the control law

where pi(x) = y,(x), and obtain the closed-loop plant

(9) . E;=, Ej=l X ' ; W P , ( ~ { ( A ; - B , K ; k + BjLYrJ

X = xi=, E;=, W ' ( X ) P j ( X )

Hence, if K/* E R4 '' and LF E RY the algebraic equations

are chosen to satisfy

Ai - B,K; = (A"!), B;L; = (Bm)o ( I O )

then the transfer matrix of the closed-loop plant is the same as that of the reference model and x(t) + x,,,(t) exponen- tially fast for any bounded reference input signal r( t ) . However, the design of the control parameters is not possible for the systems whose parameters are unknown. To overcome this drawback, in this research, the following controller is developed for the plant models of which parameters are unknown.

Let us assume that 4. L,# in ( I O ) exist, i.e. that there is sufficient structural flexibility to meet the control objective, and propose the control law

(11) / + ) ( - K j ( f h +L;(r)r)

U = z;=i P j N

where K,(t), L,.(1) are the estimates of K,?, L;*, respectively, to be generated by an appropriate adaptive law.

3.2 Adaptive law design By adding and subtracting the desired input term, namely,

E;=, P~(x){-B,(K;x - L;r))

E:=] rl,(x)

in the plant equation and using (IO), we obtain

199

Furthermore, by adding and subtracting the estimated input term multiplied by E;=,wiBi/EI= Iwi, that is,

By using (12) and (13), we ca," express the equal.ion ofthe tracking error defined as e(t)=x(r) - x,(t), i.e.

where Ej = &(/) - K,? and ij = Lj(/) ~ L,?. In the dynamic equation (14) of tracking error, Bi are

unknown. We assnine that L,? are either positive <definite or negative definite and define r;' = L,? sgn(!J, where 1.- .i 1 if L,? is positive definite and b = - I if L,? is negative definite. Then Bi = (B,JtiLT-' and (14) becomes

i EL E,=l W , ( A P j ( m " , ) o

C L C;=l w;(x)Pj(x) e = e

Now, by using the tracking error dynamics (15), we derive the adaptive law for updating the desired control para- meters K,?, L,? so that the closed-loop plant model (12) follows the reference model (7) . We assume that the adaptive law has the general structure

k.(t) I = F,(x.x,,e, r). Lj = Gj(x,xnx,e,i~) (16)

(i = I , . . . , 1) are functions ,sf known where Fj and signals that are to be chosen so that the equilibrium

&.e =I(*, Lie = L;. e, = 0 (17)

of (15), (16) has some desired stability properties. We propose the following Lyapunov function 'candidate:

( I 8)

where P=PT > 0 is a common positive definite matrix of the Lyapunov equations- (A,)~P+P(A,),< - Q, for all Q u = Q $ > O (i, j = I , . . . , r) whose existence is guaranteed by the stability assumption for A,. Then, after some straightforward mathematical manipulations,

i V(e , kj. ij) = eTPe + E tr(KTrjKj + tTrjLi)

j= I

200

we obtain the time derivative V of V along the trajectory of (15) , (16)as

x Per' + i;rjij} J= I

In the last two terms of (19), if we let

f: I;;Tr,& j= I

(20b)

we can make V negative, i.e.

Hence, the obvious choice for an adaptive law to make k negative is

Theorem 2: Consider the plant model (2) and the reference model (7) with the control law ( 1 I ) and adaptive law (22). Assume that the reference input r and the state x, of the reference model are uniformly hounded. Then the control law ( 1 1) and the adaptive law (22) guarantee that:

( i ) K ( f ) , L(t) , e(/) are hounded (ii) e ( / ) + 0 as /+ cu.

Proof: From (18) and (21), it directly follows that V is a Lyapunov function for the system (15), (16), which implies that the equilibriumgiven hy (1:) is uniformly stable, which, in turn, implies that the trajectory K(/) , L(/) , e(t) is hounded for all / > O . Because e = x -x,, and x , E ~ ' ~ , we have that

IEE Proc.~Conrrol Theon. Appl.. Yo/. 150, No. 2. March 2003

x E 9,. From (1 I ) and r E 9,, we also have that U E 9,; therefore, all signals in the closed loop are hounded.

From (IS) and (21) we conclude that because V is houndcd from below and is nonincreasing with time, it has a limit, i.e.

Now, let us show that

It, Iim ~ ( e ( t ) , k,(t), i,(t)) = V, < 00 (231

From (21) and (23), it follows that

which implies that e E p 2 . Because e, E,, E,, r E Y , , it follows from (15) that b~ ie,, which, together with

0 e E 2'2, implies that e(t) + 0 as f + 00.

4 Tracking control of an uncertain flexible joint manipulator

In this section, the validity and effectiveness of the proposed controller are examined through the simulation of tracking control for a flexible joint manipulator.

The control objective is to follow a given trajectory qd( f ) and to produce a torque vector U such that the trajectory error approaches 0 as I + 0. In the simulation, we examine the effects of parametric variation on behaviours of the closed-loop systems with the proposed T-S model based adaptive control scheme.

4.7 joint manipulator To apply the suggested AFC, we need a T-S fuzzy model representation of the manipulator.

After the T-S fuzzy model was proposed there have been efforts to construct an efficient T-S fuzzy model for a given nonlinear system. If the T-S fuzzy model does not exactly model the nonlinear system, the designed controller may not he able to guarantee the control performance and the stability of the closed-loop control system. To develop a systematic procedure, a T-S fuzzy model-

ling method, exact T-S fuzzy modelling, has recently heen developed. The basic idea of exact T-S fuzzy modelling for nonlinear systems was first discussed in [16]. Here, the word 'exact' means that the defuzzified output of the

T-S fuzzy modelling for flexible

IEE Proc.-ConIml Theoq Appl.. El. 150, No. 2, March 2003

constructed T-S fuzzy model is mathematically identical to that of the original nonlinear system.

Consider the single link flexible joint manipulator shown in Fig. 1 whose dynamics can he written as

x, =x, ~. MgL . k

x2 = --smxI I - i ( ~ , -x3) x3 = x,

(26) K 1

J 4, = + X I - x z ) + - u

where I, J are, respectively, the link and the rotor inertia moments, M is the link mass, k is the joint elastic constant, L is the distance from the axis of rotation to the link centre of mass and g is the gravitational acceleration, respectively.

The system (26) has a nonlinear term, sin(xl(t)). If this nonlinear term can he represented as a weighted linear sum of some linear functions, then the T-S fuzzy model of (26) can he Constructed.

For this purpose, we first need the following theorem.

Theorem 3 [17]: Consider the following nonlinear term:

= x I x z . . . x n , where xi E [a', f&] It can be exactly represented by a linear weighted sum of the form

where giIi ,... = q, pj2; ,... = n;=2F{, in which r{, is positive semi-definite for all xi€ [a, Q,], defined as follows:

Proof: Theorem 3 can he proved using inductive reasoning. If n = 1, then Theorem 3 is obviously true. When n = 2, the nonlinear equation isf2 =xIxz, which can be represented as the weighted sum of linear functions of x t as follows:

Fig. 1 Flexible joinr manipuluior configuration

201

where and the membership functions RI and R2 are, respectively,

Assuming that Theorem 3 holds when n = k , then the nonlinear function fx+ I =xIx2...xk+ I can be represented by a weighted linear sum of linear functions of XI in the following form:

(r:+lnk+l I + 2 rk+lnk+l 2 )),TI

Hence, Theorem 3 holds for all n

Corollary I [17]: Assume thatx(t) E [Q, Rz]. The nonlinear term

f ( x ( t ) ) = sin(x(f)) (3 1) can he represented by a linear weighted sum of linear functions of the form

.fM)) = (i P ; ~ , ( ~ w ) x ( t ) (32)

wheregl(x(f))= I, g2(x(f))=a and

11, = rl. r , = , r2=--

p2 = r2 sin(.r(f)) - m(f)

(1 - ' * )X( t )

x(t) - sinl:~y(t)) (1 - a)z-(t)

for x(f) # 0

r , = I , r2 = 0 for x(t) = 0

a = sin-'(max(R, n,)) Proof It follows directly from Theorem 3. Using Corollary I , an exact T-S fuzzy model of (26) can be represented as follows [17].

Plant rules:

Rule I : IF xl(f) = RI THEN ~ ( t ) = A,x(t) + B , u ( f )

and

Rule2: IF x,(t) E THEN ~ ( t ) = A,x(t) + B2a(t) ( 3 3 )

where

0 1 0 0

; i :] A z = [ - < - - - I -_ :! B I = ~ = [ ; \ -

0 -k

- 0 - J J

aMgl k 0 1 0 0 0

k 0

J J J (341

202

for xl(f) # 0 (35)

r , = I, r2 = 0 for xl(f) = 0

where a = sin-' (max(RI Q,)) and Ti is positive definite for all x l ( f ) E [Q, Q,]. In the simulation, [a, Q2] was chosen as [ -2 .85 2.851.

Although the exact fuzzy model of the flexible joint manipulator does not have any modelling uncertainties since the defizzified output of the T-S fuzzy model is exactly the same as that of the original nonlinear flexible joint manipulator (26), the exact modelling scheme may have some demerit. If the nonlinearities in the system model have very complicated form or the number of them is very large, the methodology presented in Theorem 3 cannot he applied easily.

An alternatice T-S fuzzy modelling technique, the lineari- sation method is often utilised to construct a T-S fuzzy model for a nonlinear system. The linearisation based T-S fuzzy modelling technique is the most popular as it is simple and the consequent rule base becomes intuitive although the modelling error inevitably exists.

By applying the Lyapunov linearisation method [17] at operating points xI = - x , 0, x , we obtain the T-S fuzzy model for the robot manipulator as follows:

R u l e / : I F x l - - x T H E N i = A , x + B I u Rule2: I F r l Z O T H E N i = A 2 x + B Z n Rule 3: IF x, THEN i = A 3 x + B 3 u

r o 1 0 0 1

k J

0 1 0 0

MgI k 0 - 0 k

0 0 0 1 k J J

I

0 -- k O

and

0

Bl = B2 = B, = [ The whole state space formed by the state vector of the original nonlinear equations is partitioned into three differ- ent fuzzy subspaces whose centre is located at the centre of the corresponding membership functions shown in Fig. 2.

4.2 Control results To apply the proposed adaptive fuzzy control scheme, the reference model for the plant state x to follow should be specified. In this simulation, the closed-loop eigenvalues for each subsystem are chosen to he the same, which in

I€€ Proc.-Corrml T h m q Appl.. Yol. I50 No. 2. Mcarch 2003

MF3 (about nj 351 MF, (about -4 MF2 (about 0)

30-

25-

ol 20.: -

iii 15- XI (0 - 0

10-

5

Fig. -x w 2 Memhership/imctions

:.

;.

...

'., k.: The PDC controller shares the same fuzzy sets as the fuzzy model to construct its premise part. That is, the PDC controller is of the following form:

...... reference model 12-

R': if xi is MF;,

then ~ ( t ) = -Ki[.rl xz xi x4]' + L,r(t) (38)

The feedback control gains Ki and L; of each fuzzy state feedback controller are updated by adaptive law so that the closed-loop plant follows the reference model (37).

Now, by using (22), we derive the adaptive law for updating the elements of K j and Lj so that the closed- loop plant follows the reference model:

.

.'.. ...... LFM based . . . .

where B:, = [0 0 0 11

I - LFM based reference model

Fig. 4 Regulation based on LFM

The parameters of the nominal plant model used in this simulation are as follows:

M = 0.2687kg J = 0.03 kgm' L = I m,

k = 3 1 N / m J = 0 . 0 0 4 k g m 2 and g = 9 . 8 m / s 2

(40)

To test the adaptation abilities of the proposed scheme, the mass oflink is vaned with time as m = 0.2687 + 0.15 sin 3nt and the initial value for state xI is assumed to he xI = n/6.

The designed adaptive fuzzy controller was applied to the original nonlinear model of flexible joint maipnlator (26) in the simulation. Figs. 3-5 show the simulation results of regulation of joint angle with exact fuzzy model (EFM) and linearisation based fuzzy model (LFM). From these figures, it is shown that the regulation problem can be solved under parametric uncertainties. Figs. 6-8 show the tracking control results with both EFM and LEM. In both cases, the tracking can be accom- plished successfully. The response characteristics of EFM based control such as response time is better than that of LFM based control. This is due to the fuzzy modelling ability of LFM. If more fuzzy rules, that is, linearisation at more operating points can be possible, the difference between the models can be reduced.

EFM based . . - . reference model

18-

16- :. . .

-2~

4- ~

20 0 5 10 15 20

time, 5 time, s

Fig. 6 Tmcking based on EFM Fig. 8 Packing error

5 Conclusions

In this paper, we have developed an altemative T-S fuzzy model based adaptive control scheme via a model reference approach for flexible joint manipulators with parameter uncertainty in their model. We have used an exact fuzzy modelling method and a linearisation based modelling method to represent the flexible joint manip- ulator. The adaptation law adjusts the contmller para- meters on-line so that the plant output tracks the reference model output. The developed adaptive law guarantees the boundedness of all signals in the closed-loop system and ensures that the plant state tracks the state of the reference model asymptotically with time for any bounded reference input signal. The proposed adaptive fuzzy control scheme was ;applied to the tracking control of a single link flexible joint manipulator to verify the validity and effectiveness of the control scheme. From the simulation results, we conclude that the suggested scheme can (effectively

LFM based relerence model

- , 5 1 - - - 0 5 10 15 20

time, s

Fig. 7 Tracking based on LFM

LFM based ,..... EFM based

achieve the trajectory tracking in spite of parameter perturbation.

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