Neuro-Fuzzy Dynamic-Inversion-Based Adaptive ... - Semantic Scholar

1342

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007

Neuro-Fuzzy Dynamic-Inversion-Based Adaptive Control for Robotic Manipulators—Discrete Time Case Fuchun Sun, Member, IEEE, Li Li, Han-Xiong Li, Senior Member, IEEE, and Huaping Liu

Abstract—In this paper, we present a stable discrete-time adaptive tracking controller using a neuro-fuzzy (NF) dynamicinversion for a robotic manipulator with its dynamics approximated by a dynamic T-S fuzzy model. The NF dynamic-inversion constructed by a dynamic NF (DNF) system is used to compensate for the robot inverse dynamics for a better tracking performance. By assigning the dynamics of the DNF system, the dynamic performance of a robot control system can be guaranteed at the initial control stage, which is very important for enhancing system stability and adaptive learning. The discrete-time adaptive control composed of the NF dynamic-inversion and NF variable structure control (NF-VSC) is developed to stabilize the closed-loop system and ensure the high-quality tracking. The NF-VSC enhances the stability of the controlled system and improves the system dynamic performance during the NF learning. The system stability and the convergence of tracking errors are guaranteed by the Lyapunov stability theory, and the learning algorithm for the DNF system is obtained thereby. An example is given to show the viability and effectiveness of the proposed control approach. Index Terms—Adaptive control, dynamic-inversion, neurofuzzy (NF) systems, NF variable structure, robotic manipulators.

I. I NTRODUCTION

W

E HAVE witnessed intensive research on the neurofuzzy (NF) adaptive control of nonlinear systems in the past two decades [1]–[11]. Most of these stable adaptive approaches use static neural networks (NNs) [1]–[8] or static adaptive fuzzy systems [9]–[11] to approximate nonlinear function components in the system dynamics and then stabilize the closed-loop system by constructing the inverse controller with some extra input for robustness. However, feedforward NNs or static adaptive fuzzy systems are static mapping and cannot afford to present nonlinear dynamic systems. With the modeling power of the fuzzy logic system and dynamic property of the recurrent NNs, dynamic NF (DNF) systems have important

Manuscript received August 14, 2005; revised February 13, 2006. Abstract published on the Internet January 27, 2007. This work was supported in part by the National Excellent Doctoral Dissertation Foundation under Grant 200041, in part by the National Science Foundation of China under Grants 60625304, 60474025, 60504003, 60621062, and 90405017, in part by the National Key Project for Basic Research of China under Grant G2002cb312205, and in part by the SRFDP of Higher Education under Grant 20050003049. F. Sun, L. Li, and H. Liu are with the Department of Computer Science and Technology, State Key Laboratory of Intelligent Technology and Systems, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. Digital Object Identifier 10.1109/TIE.2007.893056

capabilities not found in feedforward NNs or static adaptive fuzzy systems, such as attractor dynamics and memory property, and are better suitable to present the nonlinear dynamical systems [12]. As one of the DNF systems, dynamic T-S fuzzy models are ones with a rule consequent of linear dynamic equations such as state equations [13], generalized system models [14], and autoregressive with exogenous input form [15], as well as linear singularly perturbed models [16]. A theoretic justification has been proven in [13] that the dynamic T-S fuzzy models are universal approximators for the nonlinear dynamic systems. As a result, advances in modern control theory have made available a large number of powerful design tools for fuzzy controller syntheses based on the dynamic T-S fuzzy models. On one hand, the dynamic T-S fuzzy models are used to develop modelbased control approaches, such as optimal control in continuous time [17] and discrete time [18], robust-optimal fuzzy control using linear matrix inequality (LMI) [13], H∞ control [19], [20], etc. On the other hand, the dynamic T-S fuzzy models are used to approximate the nonlinear dynamic systems. They not only can simulate some dynamic behaviors, such as limit cycles and chaos, but also may provide functions of much larger dynamic NNs (DNNs) due to the modeling power of the fuzzy logic. Lots of adaptive control approaches based on these two types of DNF models are proposed for the identification [21] and control of a class of multi-input multi-output (MIMO) nonlinear systems [22]. A recurrent wavelet-based NF network is proposed in [21] to deal with temporal problems and is more efficient for identifying the nonlinear dynamic systems. The same problem is tackled by Juang [23] using a genetic algorithm with variable-length chromosomes. A recurrent fuzzy NN controller design based on a sliding mode is proposed in [22] to solve the chattering problem through replacing the discontinuous sign term in the sliding mode control (SMC) by the recurrent NF output. However, a reduction of the chattering in the SMC system is still dependent on the choice of network structure and associated parameters, and moreover, the superiority of using the recurrent NF networks to replace the feedforward fuzzy networks [24] in the SMC system is not discussed. Wang and Lee [25] use recurrent NF systems to construct the inverse dynamics control for an autonomous underwater vehicle, and a recursive learning algorithm based on the ordered derivative is employed to fine tune the free parameters of the DNF systems. But, the stability of the closed-loop system is not guaranteed because of gradient learning. Cheng and Chien [26] propose

0278-0046/$25.00 © 2007 IEEE

Authorized licensed use limited to: Tsinghua University Library. Downloaded on November 25, 2008 at 18:41 from IEEE Xplore. Restrictions apply.

SUN et al.: NF DYNAMIC-INVERSION-BASED ADAPTIVE CONTROL FOR ROBOTIC MANIPULATORS

an adaptive SMC approach with the T-S fuzzy models for a class of MIMO perturbed nonlinear systems in order to solve tracking problems. The parameters of controller are adjusted according to the tracking error, and the stability of the closedloop system is guaranteed by the Lyapunov theory. However, it requires the upper bound of input uncertainty, which is difficult to be determined sometimes in practical plant. Feng et al. [27] discuss the fuzzy model reference adaptive control for a class of nonlinear systems, where the model reference adaptive control law is first designed in each local region and, then, constructed in a global domain. However, the proposed control approach may become too complicated if more subsystems are required to model the plant, and too many parameters used in the controller not only consume much computing time but also are hard to be tuned. Besides, the aforementioned DNF-based adaptive control approaches are almost developed in a continuous time and require the system states to be within a compact set for the well-defined approximation. Although Sun et al. [28] deal with this problem using DNNs, there are few research papers in this field that are using the DNF systems. Furthermore, it is very hard to guarantee the system performance at the initial control stage with existing NF adaptive control approaches, which is very important for system stability and adaptive learning. This paper is concerned with the DNF adaptive control for a robotic manipulator by a dynamic T-S fuzzy model that is derived from a given robot dynamic equation by a local sector nonlinearity approach [13] with an exact model construction. An n-dimensional state (2) is proposed and used as the rule consequents to simplify the mathematical operations and avoid the multiplication operations in two sides of the state equation by variables in each rule consequent. NF dynamicinversion is introduced in the control law to make the state of the DNF system strictly to be the interpolation between the initial state and the desired trajectory such that the system dynamic performance can be improved at the initial control stage through assigning parameters of the DNF system. An inversion compensation control is constructed by the partly known robot dynamics in the NF dynamic-inversion such that the DNF adaptive controller is only used to compensate for the remaining inversion dynamics other than the whole inversion dynamics of a robot, and as a result, a better control precision can be obtained. The NF variable structure control (NF-VSC) proposed by Sun and co-workers [4], [28] is introduced in the control structure to further improve the DNF learning convergence and the system stability. This dynamic feature of the NFVSC distinguishes this paper from other work on the static VSC that only uses a simple switching. The proposed NF adaptive controller is compared with the adaptive control using DNNs [6] to show its effectiveness in the simulation of a two-link manipulator. This paper is organized as follows. In Section II, fundamentals of the robot model, the DNF system, and the dynamical T-S fuzzy modeling are briefly reviewed. In Section III, the DNF adaptive controller design based on a dynamic-inversion is developed together with a complete control structure and the learning algorithms for parameter adaptation. Section IV shows a simulated example. Finally, the features of the proposed DNF adaptive controller are summarized in Section V.

1343

II. P ROBLEM S TATEMENT A. Discrete-Time Dynamics Equation of Robotic Manipulators The discrete-time model of an n-link rigid robot manipulator [28], [29] can be represented as ˙ + 1) − q(k)) ˙ ˙ M (q(k)) (q(k = F (q(k), q(k)) + u(k)

(1)

˙ ˙ where F (q(k), q(k)) = (D(q(k)) − D(q(k)))q(k) − f (q(k), ˙ ˙ q(k)), M (q(k)) = D(q(k))/δ, q(k) ∼ = q(k) + a(k)δ q(k), with a(k) denoting the slope change of robot joint trajectories at any discrete instant [29] and δ as the sampling interval. D(q(k)) = D T (q(k)) ∈ Rn×n is the inertia matrix, αm ≤ M −1 (q(k)) ≤ αM with αm , αM > 0 are known constants. ˙ F (q(k), q(k)) represents centrifugal, Coriolis, and gravitational torques, and u(k) ∈ Rn is the piecewise constant generalized force input. B. DNF Systems The robot discrete-time dynamics equation can be represented by the following dynamic T-S fuzzy model, which is described by fuzzy IF-THEN rules as Rule i : If z1 (k) is Fi1 ,

z2 (k) is Fi2 , . . . , zp (k) is Fip

˙ + 1) = Ai x(k) + bi + u(k), Then M i q(k i = 1, . . . , m

(2)

where Fij (i = 1, 2, . . . , m, j = 1, 2, . . . , p) are fuzzy sets described by membership functions Nij , x(k) = (q T (k), q˙ T (k))T ∈ R2n is the state vector, and u(k) ∈ Rn is the input vector. Besides, Ai ∈ Rn×n , bi ∈ Rn are appropriate constant matrices and vectors, respectively. m is the number of fuzzy rules, and z1 (k) ∼ zp (k) are some measurable variables or nonlinear functions as defined in Section II-C, i.e., the premise variables. C. DNF System Modeling Using Sector Nonlinearity Approach Suppose that there are p different nonlinear functions in ˙ of the robot dynamics (1), and M (q(k)) and F (q(k), q(k)) these nonlinear items are denoted as zi (x) (i = 1, . . . , p). If the maximum and minimum values of zi (x) are denoted as zimax and zimin , then we have zi (x) = Ei1 zimax + Ei2 zimin ,

i = 1, . . . , p

(3)

where Ej1 and Ej2 (j = 1, . . . , p) are the membership functions satisfying Ej1 + Ej2 = 1. Thus, the membership functions can be calculated as Ei1 =

zi − zimin zimax − zimin

Ei2 =

zimax − zi , max zi − zimin


i = 1, . . . , p.

(4)

1344


By substituting (3) into the robot dynamics (1), we can obtain the DNF model of the robot shown in (2), and the membership function N ji in the ith fuzzy rule can be chosen as Ej1 or Ej2 . III. DNF A DAPTIVE C ONTROL B ASED ON D YNAMIC I NVERSION In this section, we first derive the dynamics of tracking error metric of a robot manipulator. Then, a DNF system is used to approximate the robot dynamics, by which the dynamicinversion is derived. Finally, a hybrid control, composed of the dynamic-inversion and the NF-VSC terms, is obtained. A. Dynamics of Tracking Error Metric for a Robotic Manipulator The following tracking error metric is defined for the DNF adaptive controller design: ˙ S(k) = C (x(k) − xd (k)) = Λq(k) + q(k) − Cxd (k)

(5)

where S(k) = [s1 (k), . . . , sn (k)]T , x(k) = [q T (k), q˙ T (k)]T , T ˙T xd (k) = [q T d (k)] is the desired state trajectory to be d (k), q tracked, and C = [Λ, I] ∈ Rn×2n , Λ = ΛT > 0. Usually, Λ is chosen as Λ = diag(λ1 , . . . , λn ) > 0. Besides, by using center-average defuzzifier, product inference, and singleton fuzzifier, the DNF system (2) can be written as m

˙ = θi (z(k)) M i q(k+1)

i=1

m

θi (z(k)) (Ai x(k)+bi )+u(k)

i=1

(6)

m p

p

where θi (z(k)) = j=1 Nij / i=1 j=1 Nij (i = 1, . . . , m). Since the DNF model (2) is the equivalent expression of mthe robot dynamics (1), it is evident that M (q(k)) = ˙ + 1) = S(k + i=1 θi (z(k))M i . Besides, substituting q(k 1) − Λq(k + 1) + Cxd (k + 1) into (6) yields ˜ + 1) = M (q(k)) S(k

m

Rule i : if z1 (k) is Fi1 ,

z2 (k) is Fi2 , . . . , zp (k) is Fip

ˆ i (k)q ˆ i (k)ˆ ˆ˙ (k + 1) = A Then M x(k) î (k) + u(k) − u(k), +b

i = 1, . . . , m

(9)

where Fij (i = 1, . . . , m; j = 1, . . . , p) are fuzzy sets deˆ (k) = scribed by the membership functions Nij and x T T T 2n ˙ ˆ (k)) ∈ R is the state of the DNF system. u(k) is (ˆ q (k), q ˆ (k) → xd (k), u(k) → 0, a robust control component, and as x then the DNF system constructed is equivalent to the robot ˆ i (k), system in terms of state and control input. Besides, M ˆ ˆ i (k), and bi (k)(i = 1, . . . , m) denote the estimates of M i , A Ai , and bi at the instant k, and can be represented as ˆ i (k) = M 0 + ∆M ˆ i (k) M i ˆ i (k) = A0 + ∆A ˆ i (k) A i î (k) = b0 + ∆b î (k), b i

i = 1, . . . , m

(10)

where M 0i , A0i , and b0i are the known consequent parameters which are determined by the sector nonlinear approach î are the ˆ i , and ∆b ˆ i (k), ∆A described in Section II-C; ∆M 0 estimate errors. If no a priori knowledge about M i , A0i , and b0i is used, we have M 0i = 0 A0i = 0

b0i = 0.

(11)

The tracking error metric for the DNF system (9) can be defined as ˆ˙ (k) − Cxd (k). x(k) − xd (k)) = Λˆ q (k) + q S 0 (k) = C (ˆ (12)

θi (z(k))

i=1

× (M i h(k) + Ai x(k) + bi ) + u(k) ˙ where q(k + 1) = q(k) ≈ q(k) + a(k)δ q(k), S(k + 1) − rS(k)

poor approximation to the robot dynamics because of parameter uncertainty and disturbance. For accurate trajectory tracking of robotic manipulators, a stable DNF adaptive control approach will be developed to compensate for the robot remaining dynamics. The following DNF system with the same antecedents as that in (2) is constructed to approximate the robot dynamics shown in (1).

(7)

˜ + 1) = S(k

˙ h(k) = (a(k)Λδ−I) q(k)+rS(k)+C (xd (k)−xd (k+1)) (8) and r = r T > 0 is a design parameter used to assign the dynamics of S(k), which can be chosen by the equation in [4, eq. (12a)], and r = I − r. B. Dynamics of the Tracking Error Metric for a DNF System The error dynamics (7) in the whole fuzzy space has been derived for robot trajectory tracking under the assumption that the robot dynamics can be represented by a dynamic T-S fuzzy model. Usually, the dynamic T-S fuzzy model constructed by the local sector nonlinear approach given in Section II-C is a

By the same operation as that of (7), S 0 (k) dynamics in the whole state space can be written as ˆ (k)S ˜ 0 (k+1) = M

m

ˆ ˆ i h(k) θi (z(k)) M

i=1 m

+

î (k) ˆ i (k)ˆ θi (z(k)) A x(k) + b

i=1

+ u(k) − u(k) m ˆ ˆ (k) + b0i θi (z(k)) M 0i h(k) + A0i x = i=1 m

+

θi (z(k))

i=1

ˆ î (k) ˆ i (k)h(k)+∆ ˆ i (k)ˆ × ∆M A x(k)+∆b + u(k) − u(k)


(13)


ˆ (k)= where M r 0 S 0 (k)

m

ˆ

i=1 θi (z(k))M i ,

˜ 0 (k + 1) = S 0 (k + 1)− S

ˆ ˆ˙ (k)+r 0 S 0 (k)+C (xd (k)−xd (k+1)) h(k) = (a(k)Λδ−I) q (14) and r 0 > 0, will be designed to assign the dynamics of S 0 (k) and r 0 = I − r 0 .

1345

mine r 0 , and the inverse design method will be discussed in another paper. D. Adaptive Controller Design The following control law is considered for the robot trajectory tracking: u(k) = uI (k) + u(k) = uI (k) + up (k) + un (k)

C. NF Dynamic Inversion Dynamic inversion used in the controller design is defined as the inverse model of the DNF system shown in (9), with a state specified by the desired dynamics, which is similar to the stable inversion proposed by Zhao and Chen [30]. Stable inversion is the inverse model of the multivariable system with a desired state trajectory as its model input, which guarantees the inverse model to be bounded. For a DNF system (9), if the desired dynamics is chosen as S 0 (k + 1) = r 0 S 0 (k)

(15)

then the dynamic-inversion of the DNF system is obtained from (13) as uI (k) = u(k) − u(k) = uIc (k) + uIa (k) with uIc = −

m

0ˆ i=1 θi (z(k))(M i h(k)

(16)

ˆ (k) + b0i ) + A0i x

m î (k) ˆ ˆ i h(k)+∆ ˆ i (k)ˆ uIa (k) = − θi (z(k)) ∆M x(k)+∆b A i=1

where the NF dynamic-inversion is composed of the inversion compensation control uIc (k) and the DNF adaptive control uIa (k). uIc (k) is constructed by the partly known robot dynamics and used to compensate for the known part of the robot inversion dynamics, while uIa (k) is only used to compensate for the remaining inversion dynamics of a robot. ˆ (k) will converge to the desired trajectory specified Since x by dynamics (15), the primary difference between the dynamicinversion and the stable inversion lies at the initial stage of the control process. The advantage of the dynamic-inversion is ˆ (k) at the initial control stage can be that the state trajectory x designed in advance such that the good dynamic performance of the closed-loop system is guaranteed. Besides, the dynamics relation (15) ensures the state of the DNF system to be in the compact set strictly such that the dynamic-inversion always exists. It will be proven that the DNF system (9) will approximate the robot dynamics (2) using an appropriate control law composed of dynamic-inversion and the DNF learning algorithm, and thus, the robot state will track the desired trajectory xd . As a result, the dynamic-inversion (16) will finally approximate the inverse dynamics of the robot. Remark: There are two methods for determining the design parameter r 0 . One is through trial and error until a good control performance is obtained at the initial control stage; the other is through an inverse system design that involves the determination of both the feedback controller and the reference input. In this paper, the trial-and-error method is used to deter-

(17)

where u(k) is composed of the dynamic-inversion uI (k) and the robust control component u(k). Dynamic inversion uI (k) acting as a feedforward controller is used to approximate the robot inverse dynamics. up (k) is employed to compensate for the model uncertainty [see (20)] resulting from the replacement of robot state by the DNF state, while un (k) in the feedback loop is used to enhance the stability and robustness of the robot control system. up (k) and un (k) will be defined later in this section. Define the state deflection metric of the robot from the DNF system as ˜ (k) S e (k) = C x

(18)

T ˙T ˜ (k) = x(k) − x ˆ (k) = (q T with x e (k), q e (k)) and C defined as before. Subtracting (13) from (7) and using (15) and (17), we have

˜ e (k+1) M (q(k)) S m ˆ i (k) h(k) θi (z(k)) ∆M i−∆M = i=1

+

î (k) ˆ i (k) x(k)+∆bi −∆b + ∆Ai −∆A

m

ˆ i (k) ((a(k)Λδ−I) q˙ e (k)+rC x ˜ (k)) θi (z(k)) M

i=1

ˆ i (k)˜ +A x(k) +up (k)+un (k)

(19)

where ∆M i = M i − M 0i , ∆Ai = Ai − A0i , ∆bi = bi − b0i , ˜ e (k + 1) = S e (k + 1) − rS e (k), h(k) = h(k) + (r − and S r 0 )S 0 (k), the compensation term is assumed to be m ˆ i (k) ((a(k)Λδ−I) up (k) = − θi (z(k)) M i=1

ˆ i (k)˜ ˜ (k))+ A × q˙ e (k)+rC x x(k) .

(20)

Substituting (20) into (19) leads to ˜ e (k + 1) M (k)S m θi (z(k)) = i=1

×

ˆ i (k) h(k) + ∆Ai − ∆A ˆ i (k) ∆M i − ∆M î (k) + un (k) × x(k) + ∆bi − ∆b

˜ (k)Y (k) + un (k) =W


(21)

1346


˜ (k) = W − W ˆ (k) = (w ˜ ij (k)) ∈ Rn×nr where W

then, the robot state deflection metric will enter the sector defined as

W = (∆M 1 , . . . , ∆M m , ∆A1 , . . . , ∆Am ,

n

Ω = ∪ sei (k)| sle (k) ≤ Al (k)

∆b1 , . . . , ∆bm ) ∈ R

n×nr

ˆ m (k), ∆A ˆ 1 (k), . . . , ∆A ˆ m (k), ˆ (k) = ∆M ˆ 1 (k), . . . , ∆M W ˆ1 (k), . . . , ∆b ˆm (k) ∈ Rn×nr ∆b T Y (k) = θ (z(k)) ⊗ h (k), θ (z(k)) ⊗ xT (k), θ (z(k))

T

IV. A PPLICATION E XAMPLE nr = (3n+1)m.

(22)

Finally, the nonlinear control component is defined as un (k) = −pa (k)Gsat (S eg (k))

(23)

where S eg (k) = (s1eg (k), . . . , sneg (k))T , sleg (k) = sle (k)/Al (k) with sle (k) being the ith component of S e (k) and Al (k) being the size of a sector; sat(S eg (k)) = (sat(s1eg (k), . . . , sat(sneg (k)))T , G = diag(g 1 , . . . , g n ) > 0 is a constant matrix for enhancing the system robustness, and sat(·) is defined by |y| ≤ 1 ⇒ sat(y) = y

|y| > 1 ⇒ sat(y) = sgn(y)

pa (k) = 

In this section, the above developed control approach is employed in the position control of a two-link manipulator. The dynamics equation and parameters of a two-link manipulator are the same as those of Sun et al. [4]. The desired joint angle trajectory for the robot to follow is q1d (t) = 0.5(sin t + sin 2t)

Also, the initial simulation condition for a robot motion is q1 (0) = 1.0

q˙1 (0) = −0.5

q2 (0) = 1.0

q˙2 (0) = −2.0 (28)

 |yj (k)|

Al (k) = αM (p + g l )2 pa (k)/2rl g l

j=1

(24) with p = maxi,j,k |wij (k)|. The following theorem gives a stable adaptive control law and learning algorithm for the DNF system. Theorem: If the system (2) is digitally controlled by the control law (17) and the DNF system has the following adaptive learning algorithm:

A. Derivation of the Robot T-S Fuzzy Model The discrete-time dynamics equation for a two-link manipulator can be represented by (29), as shown at the bottom of the page, where Mij (φ(k)) = Dij (φ(k))/δ(i, j = 1, 2), Dij , F12 , q1 , and q2 are defined in [4]. There are five nonlinear terms founded in the discrete-time dynamics equation, which are

ˆ (k) ˆ (k) = ηr S e (k)Y T (k) + σ W 0 − W (25a) ∆W and

z1 = m2 r1 r2 cos (φ(k)) ˙ z2 = m2 r1 r2 sin (φ(k)) φ(k) ˙ z3 = m2 r1 r2 sin (φ(k)) θ(k) z4 = −m2 r2 cos (θ(k) + φ(k))

2 2 ∆M (k) ≤ 1 − rM /αM + rm ηm Y (k)

q2d (t) = 0.5(cos 3t + cos 4t). (27)

where qi , qid (i = 1, 2) are the ith robot joint angle and the corresponding desired trajectory, respectively.

and nr

(26)

where η = diag(η1 , . . . , ηn ) > 0 is the learning rate matrix, σ > 0 and W 0 = (w0,1 , . . . , w0,nr ) are the design parameters, rm and ηm are defined as the minimum eigenvalues of r and η, respectively, and rM denotes the maximum eigenvalue of r. Proof: See Appendix I.

∈ Rnr

= (y1 (k), . . . , ynr (k)) ,



i=1

M11 φ(k)

M12 φ(k)

(25b)

˙ + 1) q1 (θ(k), φ(k)) g u1 M12 φ(k) θ(k ˙ + 1) = q2 (θ(k), φ(k)) g + u2 φ(k M22

˙ 2F12 (φ(k)) φ(k) + M11 φ(k) +

˙ −F12 (φ(k)) θ(k) + M12 φ(k)

z5 = −(m1 + m2 )r cos (θ(k)) .

˙ ˙ F12 (φ(k)) φ(k) + M12 φ(k) θ(k) ˙ φ(k) M 22


(30)

(29)


1347

With zimax and zimin (i = 1, . . . , 5) determined by (30), the following dynamic T-S fuzzy model can be obtained by substituting (30) into (29) as Rule i : If z1 (k) is Fi1 , z2 (k) is Fi2 . . . , z5 (k) is Fi5 ˙ + 1) = Ai x(k) + bi + u(k), Then M i q(k i = 1, . . . , 32 (31) where Fij (j = 1, . . . , 5) are fuzzy sets described by the membership function Nij , and M i , Ai , bi (i = 1, . . . , 32) are the model parameters. To illustrate the performance of the DNF-based adaptive control approach under model uncertainty, 0%–50% random perturbations are considered for the model parameters M i , Ai , bi (i = 1, . . . , 32), and are used as their estimates denoted by M 0i , A0i , b0i (i = 1, . . . , 32). Some of them are shown in Appendix II. B. DNF Adaptive Controller Design The basis function of the DNF system can be chosen by (22). There are 224 parameters required to be adjusted; the initial weights are chosen as zero vectors (matrix) or as random numbers. Parameters p and h are determined by the equation in [4, eq. (37)] with α0 = 0.999, pmin = 0.02, pmax = 2.5, hmin = 2.5, and hmax = 16.5. Simulations are done using a fourth-order Runge–Kutta algorithm with an integral step of 0.001 s and a controller sampling interval δ = 0.02 s. The design parameters are chosen as C = [Λ I]

Λ = diag(23

25)

αM = 0.01. (32)

In the following, the performance of the proposed DNF adaptive control approach will be illustrated in comparison with the DNN adaptive one [6]. These two kinds of controllers are simulated in the same initial conditions, and the whole control software for the DNN adaptive control is from [6] directly. The learning rates for the DNF adaptive control are chosen as η1 = 15

η2 = 40

σ = 2.5 × 10−6

(33)

and the learning rates for the DNN adaptive controller are chosen as ΓDk = 13

ΓCk = 12

ΓGk = 25.

(34)

Two sets of simulation experiments are set up to illustrate the robustness of the closed-loop system using the DNF and DNN adaptive controllers, respectively. Experiment 1 has disturbance control torques with magnitudes of 250.0 (N · m) for joint one and 125.0 (N · m) for joint two exerted during time interval t ∈ [20, 25]; experiment 2 has the same disturbance control torques with a payload variation from m2 = 6.25 kg to m2 = 10.25 kg. r 0 = diag(0.091, 0.091) is chosen for the DNF adaptive control approach. Figs. 1 and 2 present the angle tracking errors for the two joints during the first 40 s and last 10 s of operations in experiment 1, respectively. Figs. 3 and 4

Fig. 1. Angle tracking errors during first 40-s operation in experiment 1. (a) Joint one. (b) Joint two.

present the angle tracking errors for two joints during the first 40 s and last 10 s of operations in experiment 2, respectively. It has been shown that the DNF adaptive controller results in a better control performance than the DNN adaptive one in terms of convergence of the tracking errors and stable precision for robot trajectory tracking under the influence of disturbances and model variations. The DNF adaptive controller shows a less initial response time than the DNN adaptive one in [6]. Moreover, when the disturbance control torques and the payload variation happen, the DNF adaptive controller gives a stronger robustness than the DNN adaptive one does. The reason for these is two folds. One is attributed to the modeling capability of the DNF system, where good NN structure and initial weights are determined for the DNF system, while the initial weights in the DNN adaptive controller are totally unknown. The other is an introduction of the inversion compensation control in the dynamic-inversion control (16) constructed by the partly known robot dynamics such that the DNF adaptive controller only is used to compensate for the remaining inversion


1348


Fig. 2. Angle tracking errors during last 10-s operation in experiment 1. (a) Joint one. (b) Joint two.

dynamics of the robot, while the DNN adaptive controller is required to compensate for the whole robot inversion dynamics.

Fig. 3. Angle tracking errors during first 40-s operation in experiment 2. (a) Joint one. (b) Joint two.

A PPENDIX I P ROOF OF T HEOREM

V. C ONCLUSION A stable DNF adaptive control approach integrating the merits of the DNN approach and the modeling power of the fuzzy logic system has been developed for the trajectory tracking of a robotic manipulator with poorly known dynamics. With the modeling power of fuzzy logic, the robot DNF model can be derived from the given robot dynamic equation by the local sector nonlinearity approach, which guarantees an exact model construction. The control law developed contains a dynamicinversion constructed by the DNF system, an adaptive compensation, and an NF-VSC component. The NF-VSC can guarantee the system stability and improve the system performance. Using the Lyapunov stability theory, the complete system stability and the tracking error convergence can be proven, and the learning algorithm is obtained thereby for the DNF system. Simulations for a two-link manipulator show the superiority of the proposed DNF adaptive control approach to that by using the DNNs.

Let the Lyapunov function be defined by V (k) =

1 1 T S (k)M (k − 1)S e (k) + S T (k)S 0 (k) 2 e 2 0 1 ˜T ˜ (k − 1) + tr W (k − 1)η −1 W 2

(A.1)

whose first-order forward difference is given by ∆V (k + 1) = V (k + 1) − V (k) ˜ = ST e (k)rM (k)S e (k + 1) 1 (k)(I + r)M (k)(I − r)S e (k) − ST 2 e 1 ˜T ˜ e (k + 1) + S (k + 1)M (k)S 2 e



1349

2 S 0 (k)2 − 1−r0M ˆ −σS T e (k)rηr W 0 − W (k) Y (k) T 1 2 ˆ ˆ − σ tr W 0 − W (k) rηr W 0 − W (k) . 2 (A.3) Since 1 ˆ (k) − rm σ W 0 − W F 2 T 1 ˆ (k) rηr W 0 − W ˆ (k) W0 − W − σ 2 tr 2 2 1 ˆ (k) (A.4) ≤ − rm σ(1 + rm σηm ) W 0 − W F 2 ˆ − σS T e (k)rηr W 0 − W (k) Y (k) 2 ˆ (k) ≤ σrM ηM S e (k) Y (k) W 0 − W . F

(A.5) Substituting (A.4) and (A.5) into (A.3) yields 2

1 ˜ 2 ∆V (k+1) ≤ − rm σ W S 0 (k)2 (k) − 1−r0M F 2 1

2 2 1−rM /αM +rm − ηm Y (k)2 2 −∆M (k) S e (k)2 2 1 ˆ (k) − rm σ(1+rm σηm ) W 0 − W F 2 1 + rM σW −W 0 2F 2 2 ˆ (k) + σrM ηM S e (k) Y (k) W 0 − W .

Fig. 4. Angle tracking errors during last 10-s operation in experiment 2. (a) Joint one. (b) Joint two.

1 + ST (k)∆M (k)S e (k) 2 e

1 2 − ST 0 (k) I − r 0 S 0 (k) 2 ˆ T (k)η −1 W ˜ (k) − tr ∆W 1 ˆT ˆ (k) . (k)η −1 ∆W − tr ∆W 2

F

(A.6) Since − (A.2)

First, the outside of the sector is considered. With (21) and the NF learning algorithm (25a), it can be derived along the lines of theorem in [28] that 2 1 2 1 ˜ ˆ (k) ∆V (k+1) ≤ − rm σ W (k) − rm σ W 0 − W F F 2 2 1 + rM σW −W 0 2F 2 1

2 2 1−rM /αM +rm − ηm Y (k)2 2 −∆M (k) S e (k)2

2 1

2 ˆ (k) σrm + ηm σ 2 rm W 0 − W F 2 2 2 ˆ (k) + rM ηM σ S(k) Y (k) W 0 − W

F

1

2 = − σrm + ηm σ 2 rm 2 ˆ (k) × W 0 − W F

2

2 2 − ηM rM S(k) Y (k)/ rm + ηm σrm

1 4 2 2 + rM ηM σ S(k)2 Y (k)2 / rm + ηm σrm 2

1 4 2 2 ≤ rM ηM σ S(k)2 Y (k)2 / rm + ηm σrm . 2 (A.7)


1350


Substituting (A.7) into (A.6) yields

A PPENDIX II Part model parameter estimates (M 0i , A0i , b0i ) are listed as

∆V (k + 1) 2

1 ˜ 2 (k) − 1 − r0M ≤ − rm σ W S 0 (k)2 F 2 1

2 1 − rM /αM + ηc Y (k)2 − ∆M (k) − 2 1 × S e (k)2 + rM σW − W 0 2F 2

(A.8)

with

2 2 4 2 ηc = ηm rm Y (k)2 . − σηM rM / rm + ηm σrm

(A.9)

Let 2 1 1 T ˜ V (k) = V (k) + rm σ W (k − 1) ≤ λ1 X (k)X(k) F 2 2 (A.10) T T ˜ where X(k) = (S T λ1 = e (k), S 0 (k), W (k − 1)F ) , −1 −1 max (αm , 1, rm σ + ηm ). With (A.10), (A.8) can be written as

2 1 1 ˜ ∆V (k + 1) ≤ − β1 S e (k)2 − rm σ W (k − 1) 2 2

2 S 0 (k)2 + γ(k) − 1 − r0M ≤ −λV (k) + γ(k)

(A.11)

2 with β1 = (1 − rM )/αM + ηc Y (k)2 − ∆M (k)

161.55 M 10 = 57.8

49 b10 = 98

214.59 M 010 = 83.56

53.7 0 b10 = 103.45

1 2 min β1 , 1 − r0M , σrm , /λ1 2

1 γ(k) = rM σ W − W 0 2F . 2

71.9 86.49

0 A10 = 0

A010 =

0 141.55 0 47.8

0 0 0 0

0 0 A20 = 0 0

A020 =

0 0 0 0

47.8 57.8

171.43 63.58

81.55 17.8

108.15 24.68

63.9 85.47

17.8 57.8

18.74 57.83

··· ··· ···

61.55 7.8 0 0 41.55 −2.2 M 30 = A30 = 7.8 57.8 0 0 17.8 57.8

−147 b30 = −98

65.74 8.85 0 0 56.19 −2.84 M 030 = A030 = 10.21 64.18 0 0 26.2 83.24

−202.1 b030 = −116.6 ···

··· R EFERENCES

(A.12)

If γ(k) ≤ γc , where γc > 0 is a constant, it follows from (A.11) that V (k) ≤ (1 − λ)k V (0) + γc 1 − (1 − λ)k /λ.

··· ··· ···

61.55 7.8 M 20 = 7.8 57.8

−49 b20 = −98

75.54 8.9 0 M 20 = 11.33 59.7

−62.27 b020 = −98.34

··· λ=

57.8 57.8

(A.13)

It is shown from (A.13) that the tracking error metric of the robot will be driven into a small neighborhood of zero o(γc ).

[1] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 1, pp. 4–27, Mar. 1990. [2] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [3] R. J. Wai and P. C. Chen, “Robust neural-fuzzy-network control for robot manipulator including actuator dynamics,” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1328–1349, Jun. 2006. [4] F. C. Sun, Z. Q. Sun, and P. Y. Woo, “Stable neural network-based adaptive control for sampled-data nonlinear systems,” IEEE Trans. Neural Netw., vol. 9, no. 5, pp. 956–968, Sep. 1998. [5] F. L. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems. London, U.K.: Taylor & Francis, 1999. [6] S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. Singapore: World Scientific, 1998.



[7] C. Y. Lee and J. J. Lee, “Multiple neuro-adaptive control of robot manipulators using visual cues,” IEEE Trans. Ind. Electron., vol. 52, no. 1, pp. 320–326, Feb. 2005. [8] H. Hu and P. Y. Woo, “Fuzzy supervisory sliding-mode and neuralnetwork control for robotic manipulators,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 929–940, Jun. 2006. [9] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems-Neural and Fuzzy Approximator Techniques. New York: Wiley, 2002. [10] S. Jagannathan, “Adaptive fuzzy logic control of feedback linearizable discrete-time dynamical systems under persistence of excitation,” Automatica, vol. 34, no. 11, pp. 1295–1310, 1998. [11] R. J. Wai and K. H. Su, “Adaptive enhanced fuzzy sliding-mode control for electrical servo drive,” IEEE Trans. Ind. Electron., vol. 53, no. 2, pp. 569–580, Apr. 2006. [12] C. Venkateswarlu and K. V. Rao, “Dynamic recurrent radial basis function network model predictive control of unstable nonlinear processes,” Chem. Eng. Sci., vol. 60, no. 23, pp. 6718–6732, 2005. [13] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and AnalysisLinear Matrix Inequality Approach. New York: Wiley, 2001. [14] T. Taniguchi, K. Tanaka, and H. O. Wang, “Fuzzy descriptor systems and nonlinear model following control,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 442–452, Aug. 2000. [15] J. Zhang, “Modeling and optimal control of batch processes using recurrent neuro-fuzzy networks,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 417–427, Aug. 2005. [16] H. P. Liu, F. C. Sun, and Z. Q. Sun, “Stability analysis and synthesis of fuzzy singularly perturbed systems,” IEEE Trans. Fuzzy Syst., vol. 13, no. 2, pp. 273–284, Apr. 2005. [17] S. J. Wu and C. T. Lin, “Optimal fuzzy controller design: Local concept,” IEEE Trans. Fuzzy Syst., vol. 8, no. 2, pp. 171–185, Apr. 2000. [18] ——, “Discrete-time optimal fuzzy controller design: Global concept approach,” IEEE Trans. Fuzzy Syst., vol. 10, no. 1, pp. 21–38, Feb. 2002. [19] S. G. Cao, N. W. Rees, and G. Feng, “H∞ Control of uncertain fuzzy continuous-time systems,” Fuzzy Sets Syst., vol. 115, no. 2, pp. 171–190, 2000. [20] A. Wudhichai and S. K. Nguang, “Fuzzy H infinity output feedback control design for singularly perturbed systems with pole placement constraints: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 14, no. 3, pp. 361–371, Jun. 2006. [21] C. J. Lin and C. C. Chin, “Recurrent wavelet-based neuro fuzzy networks for dynamic system identification,” Math. Comput. Model., vol. 41, no. 2, pp. 227–239, 2005. [22] F. J. Lin, C. H. Lin, and P. K. Huang, “Recurrent fuzzy neural network controller design using sliding mode control for linear synchronous motor drive,” Proc. Inst. Electr. Eng.—Control Theory Appl., vol. 151, no. 4, pp. 407–416, Jul. 2004. [23] C. F. Juang, “Temporal problems solved by dynamic fuzzy network based on genetic algorithm with variable-length chromosomes,” Fuzzy Sets Syst., vol. 142, no. 2, pp. 199–219, Mar. 2004. [24] F. Da, “Decentralized sliding mode adaptive controller design based on fuzzy neural networks for interconnected uncertain nonlinear systems,” IEEE Trans. Neural Netw., vol. 11, no. 6, pp. 1471–1480, Dec. 2002. [25] J. S. Wang and C. S. Lee, “Self-adaptive recurrent neuro-fuzzy control of an autonomous underwater vehicle,” IEEE Trans. Robot. Autom., vol. 19, no. 2, pp. 283–295, Apr. 2003. [26] C. C. Cheng and S. H. Chien, “Adaptive sliding mode controller design based on T-S fuzzy system models,” Automatica, vol. 42, no. 6, pp. 1005–1010, 2006. [27] G. Feng, S. G. Cao, and N. W. Rees, “Stable adaptive control of fuzzy dynamic systems,” Fuzzy Sets Syst., vol. 131, no. 2, pp. 217–224, 2002. [28] F. C. Sun, H. X. Li, and L. Li, “Robot discrete adaptive control based on dynamic-inversion using dynamical neural networks,” Automatica, vol. 38, no. 11, pp. 1977–1983, 2002. [29] A. Zagorianos, S. G. Tzafestas, and G. S. Stavrakakis, “On line discretetime control of industrial robots,” Robot. Auton. Syst., vol. 14, no. 4, pp. 289–299, Jun. 1995. [30] H. C. Zhao and D. G. Chen, “Tip trajectory tracking for multilink flexible manipulators using stable inversion,” J. Guid. Control Dyn., vol. 21, no. 2, pp. 314–320, 1998.

1351

Fuchun Sun (S’94–A’96–M’02) was born in Jiangsu Province, China, in 1964. He received the B.S. and M.S. degrees from the Naval Aeronautical Engineering Academy, Yantai, China, in 1986 and 1989, respectively, and the Ph.D. degree from the Department of Computer Science and Technology, Tsinghua University, Beijing, China, in 1998. He was with the Department of Automatic Control, Naval Aeronautical Engineering Academy, for more than four years. From 1998 to 2000, he was a Postdoctoral Fellow with the Department of Automation, Tsinghua University, where he is currently a Professor with the Department of Computer Science and Technology. His research interests include intelligent control, networked control system and management, neural networks, fuzzy systems, nonlinear systems, and robotics. He has authored or coauthored two books and over 100 papers which have appeared in various journals and conference proceedings. Dr. Sun was the recipient of the Excellent Doctoral Dissertation Prize of China in 2000 and the Choon-Gang Academic Award by Korea in 2003, and was recognized as a Distinguished Young Scholar in 2006 by the National Science Foundation of China. He has been a member of the Technical Committee on Intelligent Control of the IEEE Control Systems Society since 2006. He serves as a member of the Editorial Board of the International Journal of Soft Computing: A Fusion of Foundations, Methodologies and Applications.

Li Li was born in Shandong Province, China, in 1978. She received the B.S. and M.S. degrees from Shandong Architecture Engineering College, Jinan, China, and Northwest Polytechnical University, Xi’an, China, in 2000 and 2003, respectively. She is currently engaged in doctoral research in the Department of Computer Science and Technology, Tsinghua University, Beijing, China. Her research interests include intelligent control, fuzzy systems, and adaptive control.

Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree from the National University of Defence Technology, Hunan, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997. Currently, he is an Associate Professor with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. He is a “Lotus Scholar” with the Central South University, Changsha, China—an honorary professorship endowed by the Ministry of Hunan Province. Over the last 20 years, he has had opportunities to work in different fields, including military service, industry, and academia. His research interests include fuzzy and intelligent control and industrial process control with special interest in electronic packaging. Dr. Li was recognized as a Distinguished Young Scholar in 2004 by the China National Science Foundation. He serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN , AND CYBERNETICS—PART B: CYBERNETICS.

Huaping Liu was born in Sichuan Province, China, in 1976. He received the Ph.D. degree from the Department of Computer Science and Technology, Tsinghua University, Beijing, China, in 2004. From 2004 to 2005, he was a Postdoctoral Fellow with the Department of Automation, Tsinghua University, where he is currently an Assistant Professor with the Department of Computer Science and Technology. His research interests include intelligent control and robotics.
