Abstract
A new discontinuous robust control law for a rigid manipulator with bounded parameter uncertainties to track a desired trajectory, is presented. Global exponential stability is proved by the use of a natural Lyapunov function based on a transformation of the manipulator's differential equation due to Slotine and Li [2]. Convergence is to a sliding mode along which the tracking error is reduced at an arbitrary exponential rate. It is also shown how adaptation of bounds on uncertainties, and parameter adaptation, can be incorporated. For a continuous approximation of the discontinuous control law,practical stability (essentially, global uniform ultimate boundedness) of the tracking error is proved. Simulation results for the continuous control law exhibit excellent robust tracking.
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Abdallah, C., Dawson D., Dorato P., and Jamshidi M., “Techniques for the robust control of rigid robots,” inControl and Dynamic Systems, vol. 53, pp. 387–426, 1992.
Slotine, J. J. E. and Li, W., “On the adaptive control of robot manipulators,”Int. J. of Robotics Research, vol. 6, pp. 49–59, 1987.
Gutman, S., “Synthesis of min-max strategies,”JOTA, vol. 46, pp. 515–523, 1985.
Utkin V. I., “Variable structure systems with sliding modes,”IEEE Trans. on Automatic Control, vol. AC-22, pp. 212–222, 1977.
De Carlo, R. A., Zak S. H., and Mathews G. P., “Variable structure control of nonlinear multivariable systems: A tutorial,”Proc. IEEE, vol. 76, pp. 212–232, 1988.
Corless, M. J. and Leitmann, G., “Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems,”IEEE Trans. Automatic Control, vol. AC-26, pp. 1139–1144, 1981.
Spong, M. W. and Vidyasagar, M.,Robot Dynamics and Control, John Wiley & Sons Inc., 1989.
Ortega, R. and Spong, M. W., “Adaptive motion control of rigid robots: A tutorial,”Automatica, vol. 25, pp. 877–888, 1989.
Middleton, R. H., “Hybrid adaptive control for robot manipulators,”Proceedings of the 27th Conference on Decision and Control, 1988.
Chen, Y. H. and Leitmann, G., “Robustness of uncertain systems in the absence of matching conditions,”Int. J. Control, vol. 45, pp. 1527–1542, 1987.
Corless, M. and Leitmann, G., “Deterministic control of uncertain systems: A Lyapunov theory approach,” Ch. 11 inDeterministic Control of Uncertain Systems, A. S. I. Zinober, ed., Peter Pevegrinus Ltd., London, 1990(IEE Control Eng. Series # 40).
Yoo, D. S. and Chung, M. J., “A variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties,”IEEE Trans. on Aut. Control, vol. 37, pp. 860–864, 1992.
Najson, F. and Kreindler, E., “Tracking control of rigid manipulators: A Lyapunov approach,” Dept. of Elec. Eng., Technion-Israel Institute of Technology, EE Pub. No. 817, January 1992.
Najson, F. and Kreindler, E., “On compensation for neglected actuator dynamics,”Automatica, vol. 30, pp. 1501–1502, 1994.
Aubin, J. P. and Cellina, A., “Differential Inclusions,” Springer-Verlag, Berlin, 1984.
Ryan, E. P., “Subspace attractivity and invariance: Ultimate attainment of prescribed dynamic behaviour,” Ch. 4 inDeterministic Control of Uncertain Systems, A. S. I. Zinober, ed., Peter Pevegrinus Ltd., London, 1990 (IEE Control Eng. Series # 40).
Najson, F., “A Lyapunov Approach to Robust Stability of Nonlinear Systems: Applications to Manipulators,” MSc Thesis, Dept. of Elec. Eng., Technion-Israel Institute of Technology, April 1992.
Dawson, D. M., Qu, Z., Lewis, F. L. and Dorsey, V. F., “Robust control for the tracking of robot motion,”Int. J. Control vol. 52, No. 3, pp. 581–595, 1990.
Wijesoma, S. W. and Richards, R. J., “Robust trajectory following of robots using computed torque structure with VSS,”Int. J. Control, vol. 52, pp. 935–962, 1990.
Dolphus, R. M. and Schmitendorf, W. E., “Robot trajectory control: Robust outer loop design,”Dyn. and Control, vol. 1, pp. 109–126, 1991.
Slotine, J. J., E. and Li, W.,Applied Nonlinear Control, Prentice-Hall Inc., pp. 398–403, 1991.
Wen, J. T., Kreutz-Delgado, K. and Bayard, D. S., “Lyapunov function-based control laws for revolute robot arms: Tracking control, robustness, and adaptive control,”IEEE Trans. on Automatic Control, vol. AC-37, pp. 231–237, February 1992.
Zhu, W. H., Chen, H. T., and Zhang, Z. J., “A variable structure robot control algorithm with an observer,”IEEE Trans. on Robotics and Automation, vol. 8, pp. 486–492, 1992.
Sotirov, Z., “A nonlinear robust controller for robot manipulators,”Proc. of the 1992 IEEE Int. Conf. on Robotics and Automation, Nice, France, pp. 2134–2139, May 1992.
Chang, T. H. and Hurmuzlu, Y., “Trajectory tracking in robotic systems using variable structure control without a reaching phase,”Proc. 1992 ACC, pp. 1505–1509.
Tianpei, L., Qijie, Z. and Chunyi, S., “The variable-structure control of robot manipulators with adaptive feed forward compensation and its implementation,”Chinese J. of Automation, vol. 3, pp. 322–327, 1991.
Spong, M. W., “On the robust control of robot manipulators,”IEEE Trans. on Automatic Control, vol. 37, pp. 1782–1786, Nov. 1992.
LaSalle, J. and Lefschetz, S., “Stability by Liapunov's direct method, with applications,” Academic Press, 1961.
Lakshmikantham, V., Leela, S. and Martynyuk, A. A., “Practical Stability of Nonlinear Systems,” World Scientific Publishing Co., 1990.
Chen, Y H. and Chen, J. S., “Adaptive robust control of uncertain systems,” in “Control and Dynamic Systems,” Ed. Leondes, C. T., Academic Press, Inc. vol. 50, 1992.
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Najson, F., Kreindler, E. Robot robust path tracking. Dynamics and Control 6, 333–359 (1996). https://doi.org/10.1007/BF02169812
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DOI: https://doi.org/10.1007/BF02169812