on-line tune parameters of premise and consequence parts of fuzzy rules of the fuzzy basis function (FBF) controller. The main part of the fuzzy controller is a fuzzy basis function network to approximate unknown rigid serial-link robot dynamics. Under some mild assumptions, a stability analysis guarantees that both tracking errors and parameter estimate errors are bounded. Moreover, a robust technique is adopted to deal with uncertainties including approximation errors and external disturbances. Simulations of the proposed controller on the PUMA-560 robot arm demonstrate the e ectiveness. c 2000
Elsevier Science B.V. All rights reserved.
1. Introduction
Robot manipulators have highly nonlinear dynam-ics. A strategy, feedback linearization of nonlinear systems, cancels the nonlinearities of robot manipula-tors and imposes a desired linear model so that linear control techniques can be applied [12, 2]. However, the method is based on the exact knowledge of robot dynamics. Without knowing the exact knowledge of robot dynamics, a nonlinear component is required to approximate and cancel the dynamics. Neural net-works and fuzzy systems provide good solutions to this challenging task. In this paper, we design a fuzzy controller for rigid robot manipulators with completely unknown dynamics.
∗Corresponding author. Tel.: 886-2-23635251 ext. 441; fax:
886-2-23671909.
E-mail address: [email protected] (S.-D. Wang)
It has been proved that fuzzy basis function (FBF) expansions can be universal approximators with arbi-trarily small errors [16]. Therefore, a fuzzy basis func-tion network is used to approximate and cancel the unknown dynamics of robot manipulators. As in [15], the control structure and learning rules are derived from a Lyapunov theory extension that guarantee both tracking errors and parameter estimate errors in the closed-loop system are bounded. By taking the uncer-tainties including approximation errors and external disturbances into consideration, such a technique can reject the e ects.
Tuning parameters of fuzzy systems has been an active research area in the past two decades. Most of the approaches can only tune parameters of conse-quence part of fuzzy rules [16–14]. Some approaches can tune parameters of premise and consequence parts of fuzzy rules, however, they do not guarantee global stability and tracking performance [4, 7, 10]. The main topic of this paper is to present an algorithm
0165-0114/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.
to tune all parameters of the fuzzy controller under a perturbation environment. The use of an FBF-based controller in direct closed-loop controllers with the algorithm can guarantee global stability and tracking performance.
The arrangement of the rest of this paper is as follows. In Section 2, the dynamics of the rigid serial-link robot manipulators and an FBF network are introduced. Section 3 presents an algorithm for tuning the FBF-based controller. Based on the Lya-punov synthesis technique, a global stable fuzzy con-troller in a constructive manner too is developed. In Section 4, the FBF controller is used to track desired trajectories for a popular PUMA-560 robot arm suc-cessfully. Finally, Section 5 provides the conclusions. 2. Robot arm dynamics and fuzzy basis function networks
2.1. Robot arm dynamics
Consider a rigid robot manipulator with n serial links described by the equations
M() + Vm(; ˙) ˙ + G() + F ˙ + d = (1)
with vector ∈ Rn being the joint position vector;
M() ∈ Rn×n being a symmetric positive de nite
inertia matrix; Vm(; ˙) ˙ being a vector of Coriolis
and centripetal torques; G() ∈ Rn representing the
gravitational torques; F = K!+ Vf ∈ Rn×n being a
diagonal matrix consisting of the back emf coecient matrix K!and the viscous friction coecients matrix Vf; d ∈ Rn×l being the unmodeled disturbances
vector; and ∈ Rn×lbeing the vector of control input
torques. The structural properties of the robot manip-ulator such as boundedness of M(); Vm(; ˙) and d
and skew-symmetry of matrix ˙M − 2Vmhold for (1).
2.2. Fuzzy basis function networks
Assume that there are r rules in a fuzzy rule base and each of which has the following form:
If x1is ˜A1j and x2is ˜A2jand : : : and xn is ˜Anj
then y1is ˜Bj1and y2 is ˜Bj2and : : : ymis ˜Bjm;
where j = 1; 2; : : : ; r; the input vector x = (x1; : : : ; xn)T
contains the input variables to the fuzzy system, yk (k = 1; 2; : : : ; m) are the output variables of the
fuzzy system, and ˜Aijand ˜Bjkare linguistic terms
char-acterized by their corresponding fuzzy membership functions ˜Aij(xi)’s and ˜Bjk(yk); respectively. For an
FBF network, the membership functions ˜Aij(xi)’s are
Gaussian functions. As in [14], we consider the FBF network with singleton fuzzi cation, product infer-ence, and de ning the defuzzi er as a weighted sum of each rule’s output. The scheme of the FBF net-work with n inputs, r rules (hidden units) and m outputs is shown in Fig. 1. Such an FBF network im-plementing the procedures of fuzzi cation, fuzzy in-ference and defuzzi cation performs the m mappings fk : Rn→ R according to fk= r X j=1 wjk (kx − cjk; j); (2)
where x ∈ Rnis the input vector, c
j∈ Rnis the center
vector of the jth rule, j∈ Rn is the width vector
of the fuzzy basis function (·) and hidden-to-output layer interconnections weights are denoted by wjk. The
fuzzy basis function can be represented by j= e−[((x1−c1j)=1j)2+···+((xn−cnj)=nj)2]
or
j= e−[!21j(x1−c1j)2+···+!2nj(xn−cnj)2]: (3)
For ease of notation, we de ne vector c and ! collect-ing all centers and inverse radii of fuzzy basis func-tions as c = [c11 · · · cn1 c12 · · · cn2 · · · c1r · · · cnr]T; ! = [!11 · · · !n1 !12 · · · !n2 · · · !1r · · · !nr]T: (4) The output f of FBF network can be represented in a vector form
f (x(t); c; !; W) = WT (x(t); c; !); (5)
where WT = [w
jk] is an r × m matrix and =
Fig. 1. Network representation of an FBF expansion system.
any given real function f over the input space X , there exists a fuzzy system in the fuzzy basis func-tion expansion form of (5) such that it can uniformly approximate f on the compact set X to arbitrary ac-curacy. Accordingly, let r be the rule number of the FBF network, there exist an ideal matrix W∗; and
ideal vectors !∗and c∗such that
f (x(t)) = W∗T (x(t); c∗; !∗) + ”
r(x(t)): (6)
We employ an FBF network ˆf to approximate f ˆf= ˆWT (x(t); ˆc; ˆ!) (7) with ˆc; ˆ!, and ˆW of the FBF network estimating !∗; c∗and W∗: For notational convenience, we denote
∗= (x(t); c∗; !∗) and ˆ = (x(t); ˆc; ˆ!) as ∗ j=exp{−[!∗21j(x1− c∗1j)2+ · · · +!∗2nj(xn−c∗nj)2]} and ˆ j= exp{−[ ˆ!2 1j(x1− ˆc1j)2+ · · · + ˆ!2nj(xn− ˆcnj)2]}:
In this paper, the rule’s format of an FBF network is represented as [10]
If (c1j; 1j) and (c2j; 2j) and : : : and (cnj; nj)
then (wj1; : : : ; wjm): (8)
By de ning cj= [c1j; : : : ; cnj]Tand j = [1j; : : : ; nj]T
as the center and radius vector of IF part of the jth
rule, (8) can be rewritten in the following simpler form:
If (cj; j) then (wj1; : : : ; wjm):
An alternative representation of the rule’s format is to use the inverse radius vector !j instead of the radius
vector j.
In this paper, the parameters of ˜Aij(xi)’s (cij and
ij) and wjk are all adjustable and learning rules will
be stated in a later section. 3. Robot FBF controller design 3.1. FBF-based controller
In practical robotic systems, the load may vary while performing di erent tasks, the friction coef- cients may change in di erent con gurations and some neglected nolinearities as backlash may appear as disturbances at control inputs, that is, the robot manipulator may receive unpredictable interference from the environment where it resides [2]. Therefore, the control objective is to design a robust FBF-based controller so that the movement of robot arms follow the desired trajectory and all signals in the closed-loop system are bounded even when exogenous and endogenous perturbations are present. Denote the tracking error vector e(t) and error metric s(t) as
e(t) = d(t) − (t); s(t) = ˙e(t) + e(t); (9)
where d(t); is the desired robot manipulator trajectory
vector and = T¿ 0: Therefore, di erentiating s(t)
and using (9), the dynamics of robot arms can be rewritten as
M ˙s = −Vms + f + d − (10)
where the unknown nonlinear function f as f = M()( d + ˙e) + Vm(; ˙)(d+ e)
+G() + F ˙: (11)
De ne the control law as
= Ks + ˆf + ˆd (12)
where K = KT¿ 0; the output vector of fuzzy basis
function networks ˆf estimates f and ˆd is the robus-tifying term to attenuate exogenous and endogenous disturbances. The architecture of the closed-loop sys-tem is shown in Fig. 2. Using the control in (12), we get closed-loop dynamics as
M ˙s = −(K + Vm)s + ˜f + d − ˆd (13)
where the approximation error ˜f is denoted as ˜f = f − ˆf = W∗T (x(t); c∗; !∗)
− ˆWT (x(t); ˆc; ˆ!) + ”r: (14)
For simplicity of discussion, we de ne ∗= (x(t); c∗; !∗), ˆ = (x(t); ˆc; ˆ!) and ˜ = ∗− ˆ to
obtain a rewritten form of (14) ˜f = W∗T˜ + ˜WTˆ + ”
r; (15)
where ˜W = W∗− ˆW: In this paper, a method is
pro-posed to guarantee the closed-loop stability and the tracking performance, and on-line tune centers and radii of fuzzy basis functions. For achieving the goal, linearization technique is employed to transform the nonlinear fuzzy basis functions into partially linear form so that Lyapunov theorem extension can be ap-plied. Therefore, take the expansion of ˜ in a Taylor series to obtain ˜ = ˜ 1 ... ˜ r = @ 1 @! ... @ r @! != ˆ! ˜! + @ 1 @c ... @ r @c c= ˆc ˜c + h or ˜ = AT˜! + BT˜c + h; (16) where ˜! = !∗− ˆ!; ˜c = c∗− ˆc; A = @ 1 @! · · · @ r @! != ˆ!; B = @ 1 @c · · · @ r @c c= ˆc;
h is a vector of higher-order terms and @ j @! and @ j @c are de ned as @ j @! T = 0 · · · 0| {z } (j−1)×n @ j @!1j · · · @ j @!nj 0 · · · 0| {z } (r−j)×n @ j @c T = 0 · · · 0| {z } (j−1)×n @ j @c1j · · · @ j @cnj 0 · · · 0| {z } (r−j)×n : (17) Substituting (16) into (15) yields
˜f + d = ˜WTˆ + ˆWT˜ + ˜WT˜ + ”r+ d
= ˜WTˆ + ˆWTAT˜! + BT˜c+ d; (18)
where d = ˜WT˜ + ˆWTh + ”r+ d: By substituting
(18) into (13), the closed-loop system dynamics can be rewritten as
M ˙s = −(K + Vm)s + ˜WT ˆ
Fig. 2. The diagram of a closed-loop system.
where ˜d = d − ˆd: Before de ning the robustifying term ˆd, there are some assumptions required to hold in the following discussion.
Assumption 1. The norms of optimal weights, kW∗k,
k!∗k and kc∗k; are bounded by known positive real
values, i.e., kW∗k6W
m; k!∗k6!m and kc∗k6cm
with some known Wm; !mand cm(the norm of a vector
or matrix in this paper, k • k, is the Frobenius norm [17]).
For simplicity, we de ne an optimal matrix ∗ in-cluding all optimal weights and an estimating matrix
ˆ as ∗= W ∗ 0 0 0 !∗ 0 0 0 c∗ and ˆ = W0ˆ 0ˆ! 00 0 0 ˆc : From the above assumption, ∗ is bounded by a
known positive real value m(k∗k6m):
Obvi-ously, W∗; !∗, and c∗ are bounded by kW∗k6m;
k!∗k6mand kc∗k6m; respectively.
Assumption 2. The approximation errors and distur-bances are bounded, i.e., speci ed band b satisfying
k”rk6band k dk6b ; respectively.
Assumption 3. The vector of higher order terms in (16), h; is bounded by khk6kd:
Assumption 4. Since the values of fuzzy basis func-tions are positive and not greater than one, ˜ is bounded by k ˜ k61: Therefore, ˜WT˜ is bounded by
k ˜WT˜ k6k ˜WTkk ˜ k = k ˜Wk6k ˜k:
Based on the above assumptions, we can nd the bound of sTd as
ksTdk 6kskk ˜WT˜ + ˆWTh + ”r+ dk
6kskk ˜k + kskk ˆWkkhk + ksk(b + b):
(20) The robustifying term ˆd eliminating the partial bound of d is denoted as
ˆd = kdk ˆWks (21)
where kd¿khk is assumed to be satis ed. The
para-meters are updated by the following learning rules: ˙ˆ
W = KW ˆ sT− KWksk ˆW;
˙ˆ! = K!A ˆWs − K!ksk ˆ!;
˙ˆc = KcB ˆWs − Kcksk ˆc;
(22) where KW; K!; and Kc are diagonal positive square
matrices and is a positive real value. The rst terms of (22) are similar to the modi ed back-propagation algorithm that can tune weights and gains of nodes (neurons) [5]. As to the last terms of (22), they are similar to the e-modi cation of adaptive control theory [9]. The stability proof will be stated later.
3.2. Stability analysis
In adaptive control, the phenomenon of the possible unboundedness of weight estimates will occur when the persistency of excitation (PE) condition fails to hold. There are some techniques as -modi cation and e-modi cation can overcome this problem [9]. In [6], a weight tuning rule for neural networks is proposed to guarantee the boundedness of weight estimates even though PE does not hold. A proof being similar to the proof of [6] is to show that the control scheme with learning rules (22) can guarantee the boundedness of all signals generated in the closed-loop system without making any assumptions of PE conditions.
Theorem 1. Suppose that the vector d(t) is bounded
and Assumptions 1 and 2 hold. Consider the dynamic equations (1) with the control law (12) and learning rules (22). Make no assumptions of any sort of PE conditions on ˆ ; A ˆW and B ˆW. Then
(1) the error metric s(t) and weights ˆc; ˆ! and ˆW (or ˆ) will remain uniformly ultimately bounded (UUB) and
(2) the tracking errors will be kept as small as de-sired by increasing K:
Proof. Let the Lyapunov-like function candidate be V (t) = 1
2(sTMs + tr( ˜WTKW−1W)˜
+tr( ˜!TK−1
! ˜!) + tr( ˜cTKc−1˜c)): (23)
By the property of skew-symmetry of ˙M − 2Vm and
(18), the time derivative of V (t) along the trajectories of learning rules (19) and (22) is evaluated by
˙V = −sTKs +1 2sT( ˙M − 2Vm)s + sT( ˜f + d) −sTˆd + tr( ˜WTK−1 W W)˙˜ +tr( ˜!TK−1 ! ˙˜!) + tr( ˜cTKc−1˙˜c) = −sTKs + sTW˜Tˆ + sTWˆTAT ˜! + BT˜c +sTd − sTˆd − tr( ˜WT( ˆ sT− ksk ˆW)) −tr( ˜!T(A ˆWs − ksk ˆ!)) −tr( ˜cT(B ˆWs − ksk ˆc)): (24)
Using the facts tr(ATB)6kAkkBk and tr( ˜T(∗−
˜
))6k ˜kk∗k − k ˜k2 [2], and applying (20) and
(21) results in ˙V 6 − sTKs + ksktr( ˜T(∗− ˜)) +kskk ˜k + kskk ˆWkkhk + ksk(b + b) − sTˆd 6 − sTKs + ksk(k ˜kk∗k − k ˜k2) +kskk ˜k + kskk ˆWkkhk +ksk(b + b) − kdk ˆWkksk 6 − ksk{Kminksk + (k ˜k − c)2− D}; (25)
where Kminis the minimum singular value of K, c=
m=2 + 1=2 and D = b + b + c2: Therefore, if ksk ¿ sor k ˜k ¿ , where s= KD min and = c+ D ; (26)
then ˙V60. This implies that the Lyapunov derivative ˙V is negative outside the compact set (ksk ¡ s or
k ˜k ¡ ). In other words, outside the compact set
given by (26) the tracking errors and parameter errors will decrease. As for inside the compact region around the origin, the tracking errors and parameter errors are bounded. Therefore, according to a standard Lyapunov theorem extension [6], we can prove that s(t) and ˜ are UUB. Since ∗ is bounded (Assumption 1), ˆ
is also UUB. The explicit bound of s(t) is derived in (26) and the bound can be kept as small as desired by increasing Kmin.
Remark 1. Without the last terms of (22), the ˆ , A ˆW and B ˆW should be persistently exciting signals. In other words, positive numbers Ti; i; i (i = 1; 2; 3)
exist such that given t¿t0; there exists ti ∈ [t; t + Ti]
such that [ti; ti+ i] ⊂[t; t + Ti] and
1 Ti Z ti+i ti +i( )+i( )Td ¿” iI ∀t¿t0; where +1= ˆ ; +2= A ˆW and +3= B ˆW:
Remark 2. It can be found that an implicit parameter in (26) determines the magnitudes of ksk and k ˜k: A smaller will result in a smaller ksk and a larger
Fig. 3. A two-link robot manipulator with links 4, 5 and 6 xed.
4. Simulation results
Computer simulations were conducted on the PUMA-560 robot manipulator to verify the availabil-ity and performance of the proposed controller. Fig. 3 depicts a 6-link planar robot arm with the fourth, fth and sixth links xed to be a two-link robot manipula-tor. Therefore, the angles of the second and third links were considered to be 1 and 2, respectively. The
numerical values of parameters of the robot model were speci ed as that in [3]. For demonstrating the tracking performance of our proposed controller, the desired trajectories for 1and 2were set as
d1= 0:5 + 0:2(sin t + sin 2t) (rad) for 1
and
d2= 1:3 − 0:1(sin t + sin 2t) (rad) for 2;
respectively.
kd = 100:0: The initial values of centers ˆc(0) and
ˆ
W(0) were set to be small random numbers and the inverse radii ˆ!(0) was speci ed to be 1. Fig. 4 shows the desired trajectories and trajectories obtained from FBF and Slotine-Li’s controller. The maximum track-ing errors of 1 and 2 after the rst two seconds of
movement of the robot arm using the Slotine-Li’s method were 0.72◦ and 0.60◦. Using the proposed
FBF controller, the maximum errors were found to be 0.42◦and 0.06◦, respectively. This comparison shows
that the proposed controller can obtain more accurate tracking performance due to the good approximation capability of the FBF network as shown in Fig. 5. Fig. 6 shows control inputs with smooth curves. Fig. 7 shows the process of tuning centers and inverse radii of some FBFs. After the tuning process, we found that all 20 rules are located in a reasonable input range with suitable radii. Take one rule as an example: If (c1= (0:2570120; 0:1702516; 0:2400224;
−0:1067455); !1= (0:7214551; 0:7807048;
0:9510864; 0:9238370))
then (w11; w12) = (0:02314668; 0:0031): (27)
Finally, Fig. 8 shows the simulation results with bounded disturbances d that are 2 Hz square waves
with 10 Nm magnitudes. The errors are only slightly larger than that without disturbances. These results imply the robustness of the proposed FBF controller. All these simulations were carried out using C pro-grams on pentium-120 PC and the running time is about 3 min.
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