Elsevier

Neurocomputing

Volume 74, Issues 14–15, July 2011, Pages 2377-2384
Neurocomputing

Neural network-based sliding mode adaptive control for robot manipulators

https://doi.org/10.1016/j.neucom.2011.03.015Get rights and content

Abstract

This paper addresses the robust trajectory tracking problem for a robot manipulator in the presence of uncertainties and disturbances. First, a neural network-based sliding mode adaptive control (NNSMAC), which is a combination of sliding mode technique, neural network (NN) approximation and adaptive technique, is designed to ensure trajectory tracking by the robot manipulator. It is shown using the Lyapunov theory that the tracking error asymptotically converge to zero. However, the assumption on the availability of the robot manipulator dynamics is not always practical. So, an NN-based adaptive observer is designed to estimate the velocities of the links. Next, based on the observer, a neural network-based sliding mode adaptive output feedback control (NNSMAOFC) is designed. Then it is shown by the Lyapunov theory that the trajectory tracking errors, the observer estimation errors asymptotically converge to zero. The effectiveness of the designed NNSMAC, the NN-based adaptive observer and the NNSMAOFC is illustrated by simulations.

Introduction

The design of the motion control for robot manipulators has attracted considerable attention. A number of either model-based or model-free control scheme have been proposed. In the first case, different control schemes have been often used. However, a robot manipulator is a complex nonlinear system, whose dynamic parameters are difficult to forecast precisely. In fact, it is almost impossible to obtain exact dynamic models because of such uncertainties as nonlinear frictions and flexibilities of the joints and links of robot manipulator. To deal with parameter uncertainties, various advanced methods have been proposed, which include robust control techniques [1], [2], sliding-mode control [3], [4], adaptive control [5], [6] and neural network (NN) [7], [8], [9], [10], [11], [12], [13], [14], [15] techniques.

It is well known that the main advantage of using sliding-mode control and adaptive control is strong robustness with respect to system uncertainties and external disturbances. But this type of controller is designed by assuming an exact knowledge of system structure of the robots without including dynamic effects, such as complex nonlinear frictions, elasticity of joints and links, backlash and so on.

NN, one of the most popular intelligent computation approaches, has an inherent learning ability and can approximate a nonlinear continuous function to arbitrary accuracy. To present, NN-based controls have been widely used in trajectory tracking by robot manipulators. While, since the existence of the NN approximation errors, most of the papers get the result that the tracking errors can be uniformly ultimately bounded or can be kept arbitrarily small if some gain parameters are sufficiently large, such as [9], [10], [11], [12], [13], [14], [15]. In [9], [10], and the references therein, NN controllers were developed for a large number of robot manipulator models including rigid link manipulators and flexible joint manipulators. Kim et al. [12], developed an output feedback controller for robot manipulators with online weight adaptation which provided uniformly ultimately bounded (UUB) tracking. Sun et al. [14] presented a discrete NN controller for robots with uncertain dynamics which did not require off-line training; however, this controller also only achieves UUB tracking. In [15], a parametric adaptive controller that adapts for robot dynamic parameters was coupled with a NN to compensate for the unmodeled friction The controller provided asymptotic stability but required the structure of the robots dynamic mode to be known a priori.

The results mentioned above have been developed based on full state measurements. And the joint velocity measurements are obtained by means of tachometers, which are very sensitive to noises. To eliminate the need for tachometers and numerical differentiation but maintain high control accuracy, one option is to use a joint velocity observer. To present, some velocity observers are presented to estimate the joint velocity [12], [16], [17], [18].

In this paper, a neural network-based sliding-mode adaptive control (NNSMAC) is designed for a robot manipulator with modeling uncertainties to do trajectory tracking. The NNSMAC is composed of a sliding-mode control, a RBF NN approximation and an adaptive control. The sliding-mode control and adaptive control terms are used to compensate the approximation errors and the external disturbances. Under the NNSMAC, the tracking errors asymptotically converge to zero. Next, a NN-based observer is designed to estimate the velocity of the links. Finally, based on the NN-based observer, a neural network-based sliding-mode adaptive output feedback control (NNSMAOFC) is designed for the robot manipulator to track desired trajectory. It is shown using the Lyapunov theory that the trajectory tracking errors and the observer estimation errors asymptotically converge to zero.

The paper is organized as follows. In Section 2, the preliminaries about the dynamic model of the robot manipulators and NN approximation of piecewise continuous functions are introduced. In Section 3, the NNSMAC is designed and the stability proof is presented. NN-based adaptive observer and the NNSMAOFC are developed and verified in 4 NN-based adaptive observer design, 5 NNSMAOFC design. In Section 6, an illustrative example is applied to validate controls design. Finally, conclusions are given in Section 7.

Section snippets

Dynamic model

Consider the dynamics of n-link robot manipulatorM(q)q¨+Vm(q,q˙)q˙+Fq˙+fc(q˙)+G(q)+τd=τwith M(q)Rn×n is a symmetric, positive definite inertia matrix, Vm(q,q˙)Rn×n is the centripetal and coriolis matrix, FRn×n denotes the viscous friction coefficients, fc(q˙) is the Coulomb friction coefficients, G(q)Rn×1 is the gravitational vector, τd denotes bounded unknown disturbances including unstructured unmodeled dynamics, τRn×1 is the input torque vector.

The robot manipulator dynamics given in (1)

NNSMAC design

This paper considers the trajectory tracking problem of the robot manipulator described above. A trajectory tracking control objective is: on the basis of q, given a desired qd, design a controller for the mobile robot such that for any q(0)Rn, q(t)qd(t)0 as t.

In this section, a NNSMAC composed of a sliding-mode control, a RBF NN identifier and an adaptive control is designed for the robot manipulator to do trajectory tracking, which is shown in Fig. 1.

In the following, some variables are

NN-based adaptive observer design

In some cases, only the link position can be measured, and the velocity of the link should be estimated. In this section, a NN observer is designed to estimate the link position and velocity.

From (1), we getq˙=pp˙=H(q,p)M1(q)τd+M1(q)τwhere H(q,p)=M1(q)(Vm(q,p)p+Fp+fc(p)+G(q)).

According to the approximation property of NN described in Section 2.2, the function H(q,p) can be expressed asH(q,p)=W0Tϕ0(q,p)+ε0(q,p),ε0(q,p)ε0Nwhere W0 and ε0(q,p) are the optimal weight of the RBF NN and

NNSMAOFC design

In this section, the information provided by the NN-based adaptive observer is used to designed the NNSMAOFC for the robot manipulator to do trajectory tracking. The NNSMAOFC design procedure is shown in Fig. 2.

In the following, some variables are defined ase^=q^qde^˙=q^˙q˙d=p^+l1q˜q˙ds^=e^˙+le^e=qqd=q^qd+(qq^)=e^+q˜where q^ is the observer of q, e^ is the error between the observer and the desired trajectory.

Based on the estimation of q and p, the estimation of f(X) defined in (12) can

Simulation results

In order to illustrate the effectiveness of the designed NNSMAC, the NN-based adaptive observer and the NNSMAOFC, a two-link robot manipulator described in Fig. 3 is considered. The dynamic model of the two-link robot manipulator is described in (1), where M(q)=M11M12M21M22,Vm=Vm11Vm12Vm21Vm22M11=m1l23+4m2l23+m2l2cosq2,M12=M21=m2l23+m2l22cosq2,M22=m2l23Vm11=m2l22sin(q2)q˙2,Vm12m2l22sin(q2)(q˙1+q˙2),Vm21=m2l22sin(q2)q˙1,Vm22=0G(q)=m1gl2cosq1+m2gl2cos(q1+q2)+m2glcosq1m2gl2cos(q1+q2)Fq˙+fc(q˙)=fd

Conclusion

This paper addressed the robust trajectory tracking problem for a robot manipulator in the presence of uncertainties and disturbances. First, under the assumption that the full dynamics of the robot manipulator can be measured, we designed a NNSMAC, which is a combination of sliding mode technique, neural network (NN) approximation and adaptive technique, to ensure desired trajectory tracking by the robot manipulator. It was shown using the Lyapunov theory that the tracking error asymptotically

Acknowledgments

The authors gratefully acknowledge the reviews for their constructive and insightful comments for further improve the quality of this work. This work is supported in part by the National Science Foundation of China (nos. 60574004, 60736024).

Tairen Sun was born in 1985. He received his M.S. degree in Operations Research and Cybernetics from Zhongshan University, Guangzhou, PR China, 2008. He is currently pursuing the Ph.D. degree at School of Automation, South China University of Technology, Guangzhou, PR China. The main research interests include nonlinear systems, neural network control, etc.

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Tairen Sun was born in 1985. He received his M.S. degree in Operations Research and Cybernetics from Zhongshan University, Guangzhou, PR China, 2008. He is currently pursuing the Ph.D. degree at School of Automation, South China University of Technology, Guangzhou, PR China. The main research interests include nonlinear systems, neural network control, etc.

Hailong Pei was born in 1965. He received his Ph.D. degree in Automatic Control from South China University of Technology, Guangzhou, PR China, 1992. From 1997 to 1998, he did postdoctoral research in The Chinese University of Hong Kong. He is now a professor at School of Automation, South China University of Technology, Guangzhou, PR China. His research interests include robot control, neural networks, nonlinear control, etc.

Yongping Pan was born in 1982. He received his M.S. degree in Control Theory and Control Engineering from Guangdong University of Technology , Guangzhou, PR China, 2007. He is currently pursuing the Ph.D. degree at School of Automation, South China University of Technology, Guangzhou, PR China. The main research interests include neural network control, fuzzy control, etc.

Hongbo Zhou was born in 1980. He received his M.S. degree from Yanshan University, Qinhuangdao, PR China, 2007. He is currently pursuing the Ph.D. degree at School of Automation, South China University of Technology, Guangzhou, PR China. His research interests include robot control, intelligent control, etc.

Caihong Zhang was born in 1981. She received her M.S. degree from Dalian University of Technology, Dalian, PR China, 2005. Since September 2005, she had been with Qingdao University of Science and Technology. She is currently pursuing the Ph.D. degree at School of Automation, South China University of Technology, Guangzhou, PR China. Her research interests include robust adaptive control, intelligent control, etc.

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