Elsevier

Automatica

Volume 41, Issue 11, November 2005, Pages 1957-1964
Automatica

Brief paper
Continuous finite-time control for robotic manipulators with terminal sliding mode

https://doi.org/10.1016/j.automatica.2005.07.001Get rights and content

Abstract

A continuous finite-time control scheme for rigid robotic manipulators is proposed using a new form of terminal sliding modes. The robustness of the controller is established using the Lyapunov stability theory. Theoretical analysis and simulation results show that faster and high-precision tracking performance is obtained compared with the conventional continuous sliding mode control method.

Introduction

The asymptotic stability of robot manipulators can be achieved by computed torque control or inverse-dynamics control (Sage, De Mathelin, & Ostertag, 1999). Asymptotic stability implies that the system trajectories converge to the equilibrium as time goes to infinity. It is now known that finite-time stabilization of dynamical systems may give rise to a high-precision performance besides finite-time convergence to the equilibrium. This can be achieved by some continuous nonsmooth feedback controllers (Haimo, 1986, Hong et al., 2001). Recently this approach has been applied to control robot manipulators (Hong, Xu, & Huang, 2002). However, the robustness issue has not been fully addressed. Another approach is discontinuous terminal sliding mode (TSM) control with robustness for matched disturbances and parametric uncertainties with known bounds (Yu and Man, 1998, Yu and Man, 2002).

Discontinuous TSM control has been widely applied to robotic manipulators for finite-time stability. However, the negative fractional powers existing in the TSM control may cause the singularity problem around the equilibrium (Barambones and Etxebarria, 2001, Barambones and Etxebarria, 2002; Parra-Vega and Hirzinger, 2001, Tang, 1998). Recently, a discontinuous nonsingular TSM control scheme has been developed to avoid this problem (Feng, Yu, & Man, 2002). In order to reduce the chattering of the discontinuous control, the boundary layer approach was usually adopted in these works. However, the finite time stability was lost because of the asymptotic stability in the boundary layer, even for a nominal system.

In this paper, we first propose an improved version of TSM, which is then applied in both the reaching phase and the sliding phase of sliding mode control system, resulting in a new continuous TSM control for robotic manipulators with global finite-time stability. The robustness issue is ad-dressed using the Lyapunov stability theory, and the analytical results show that the proposed approach enables faster and higher-precision tracking performance compared with the conventional TSM control with boundary layers.

The rest of this paper is organized as follows. The new forms of TSM are proposed and some relevant properties are analyzed in the next section. In Section 3, a continuous finite-time TSM controller is developed and analyzed for the trajectory tracking of robotic manipulators. Simulation examples are given to demonstrate the effectiveness of the proposed control algorithms in Section 4. Finally, we end this paper with some conclusions and suggestions for further research.

Section snippets

New forms of TSM

In this section, the new forms of TSM are proposed and some of their properties are analyzed for their application in control of robotic manipulators.

Definition 1

The TSM and fast TSM can be described by the following first-order nonlinear differential equationss=x˙+β|x|γsign(x)=0,s=x˙+αx+β|x|γsign(x)=0,respectively, where xR,α,β>0,0<γ<1.

Remark 1

Expression (1) is slightly different from the previously reported TSM and fast TSM (Yu and Man, 1998, Yu and Man, 2002) which are expressed ass=x˙+βxq/p=0,s=x˙+αx+βxq/p=0,

Continuous finite-time control of robotic manipulators

In this section we will develop a continuous finite time control for trajectory tracking of robotic manipulators with the new forms of TSM introduced in Section 2. As a result, the high precision tracking can be acquired with faster convergent speed compared with conventional continuous sliding mode control.

The dynamics of an n-link rigid robotic manipulator can be written asM(q)q¨+C(q,q˙)q˙+G(q)=τ+τd,where q,q˙,q¨Rn are the vectors of joint angular position, velocity and acceleration,

Simulations

Consider the simulation example of the two-link rigid robotic manipulator in Feng et al. (2002):a11(q2)a12(q2)a12(q2)a22q¨1q¨2+-b12(q2)q˙12-2b12(q2)q˙1q˙2b12(q2)q˙22+c1(q1,q2)gc2(q1,q2)g=τ1τ2+τd1τd2,wherea11(q2)=(m1+m2)r12+m2r22+2m2r1r2cos(q2)+J1,a12(q2)=m2r22+m2r1r2cos(q2),a22=m2r22+J2,b12(q2)=m2r1r2sin(q2),c1(q1,q2)=(m1+m2)r1cos(q2)+m2r2cos(q1+q2),c2(q1,q2)=m2r2cos(q1+q2).The parameter values are r1=1m, r2=0.8m, J1=5kgm, J2=5kgm, m1=0.5kg and m2=1.5kg. The reference signals are given by qd1=

Conclusions

We have developed a new class of continuous TSM controllers for the trajectory tracking of robotic manipulators with the finite-time convergent property. The new form of TSM can be used to design not only the sliding mode with finite-time convergence to the equilibrium, but also the continuous TSM control laws to drive system states to reach TSM in finite time. By properly choosing the fractional powers, the proposed TSM controllers can enjoy benefits of both high precision and chattering

Acknowledgements

This work was supported in part by the Australian Research Council (ARC) Discovery and LIEF grants and in part by Dalian Maritime University Research Fund.

Shuanghe Yu received a BE in automatic control from Beijing Jiaotong University, China, and ME in control theory and applications and a Ph.D. degree in navigation, guidance and control from Harbin Institute of Technology, China, in 1990, 1996 and 2001, respectively. From 2001 to 2003, he was a postdoctoral research fellow at Central Queensland University, Australia. From 2003 to 2004, he was a research fellow at Monash University, Australia. Since the autumn of 2004, he has been a Professor at

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Shuanghe Yu received a BE in automatic control from Beijing Jiaotong University, China, and ME in control theory and applications and a Ph.D. degree in navigation, guidance and control from Harbin Institute of Technology, China, in 1990, 1996 and 2001, respectively. From 2001 to 2003, he was a postdoctoral research fellow at Central Queensland University, Australia. From 2003 to 2004, he was a research fellow at Monash University, Australia. Since the autumn of 2004, he has been a Professor at Dalian Maritime University, China. Dr. Yu's research interests include nonlinear control, intelligent control, robust and adaptive control, and their applications in industrial process, robot and micro–nano manipulation systems. He has published over 30 refereed papers in technical journals, books and conference proceedings.

Xinghuo Yu received BE and ME degrees from University of Science and Technology of China in 1982 and 1984, respectively, and a Ph.D. degree from South-East University, China in 1987. Prof. Yu is currently with the Royal Melbourne Institute of Technology, Australia, where he is Professor of Information Engineering and the Associate Dean for Research and Innovation of the Science, Engineering and Technology Portfolio. Professor Yu's research interests include variable structure control, chaos and chaos control and soft computing and applications. He has published over 300 refereed papers in technical journals, books and conference proceedings as well as co-edited 10 research books. Prof. Yu has served as an Associate Editor of IEEE Trans. Circuits and Systems Part I (2001–2004) and IEEE Trans. Industrial Informatics (2005-Present), respectively, and several other scholarly journals. Prof. Yu was the sole recipient of the 1995 Central Queensland University (CQU) Vice Chancellor's Award for Research. He is a Fellow of Institution of Engineers Australia and a Senior Member of the IEEE. Professor Yu was made an Emeritus Professor of CQU in 2002 in recognition of his outstanding services to the university.

Bijan Shirinzadeh received BE and ME degrees from University of Michigan, and a Ph.D. degree from University of Western Australia. He is an Associate Professor, and the Director of Robotics and Mechatronics Research Laboratory at Monash University, Australia. His research interests include laser-based measurement, sensory-based control, micro–nano manipulation systems and mechanisms, modeling, planning/simulation in virtual reality, medical robotics, and automated fabrication and manufacturing.

Zhihong Man received a BE from Shanghai Jiaotong University, China in 1982, and MS from Chinese Academy of Sciences in 1986, and a Ph.D. degree from University of Melbourne, Australia in 1993. He is an Associate Professor of Nanyang Technological University, Singapore. He received NTU 2004 Excellence in Teaching Award. His research interests include robotics, fuzzy systems, neural networks, signal processing, and nonlinear control. He has published over 120 journal and conference papers in these areas.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Yong-Yan Cao under the direction of Editor Mituhiko Araki.

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