Abstract
A design methodology is presented for tracking control of a class of second-order nonholonomic systems. The method consists of three steps. In the first step we transform the system into an extended chained-form system. This extended chained-form system is in cascade form and we apply a linear feedback to the first subsystem. In the second step, the second subsystem is exponentially stabilized by applying a backstepping procedure. In the third and final step it is shown that the closed-loop tracking dynamics of the extended chained-form system are globally exponentially stable under a persistence of excitation condition on the reference trajectory. The control design methodology is applied to an underactuated planar manipulator with two translational and one rotational joint (PPR). The simulation results show a good performance of the tracking controller.
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References
Kolmanovsky I., McClamroch N.H. (1995) Developments in Nonholonomic Control Problems. IEEE Control Systems Magazine 15, 20–36
Brockett R.W. (1983) Asymptotic Stability and Feedback Stabilization. Differential Geometric Control Theory (Brockett R. W., Milman R. S., Sussmann H. J. (Eds.)) 181–191
Reyhanoglu M., van der Schaft A.J., McClamroch N.H., Kolmanovsky I. (1999) Dynamics and Control of a Class of Underactuated Mechanical Systems. IEEE Transactions on Automatic Control 44, 1663–1671
Pantely E., Lefeber E., Loría A., Nijmeijer H. (1998) Exponential Tracking Control of a Mobile Car using a Cascade Approach. In: IFAC Workshop on Motion Control, France, September, 221–226
Jiang Z. P., Nijmeijer H. (1999) A Recursive Technique for Tracking Control of Nonholonomic Systems in Chained Form. IEEE Transactions on Automatic Control 44, 265–279
Reyhanoglu M., van der Schaft A.J., McClamroch N.H. (1996) Nonlinear Control of a Class of Underactuated Systems. In: 35th IEEE Conference on Decision and Control, Japan, December, 2, 1682–1687
Panteley E. and Loría A., On Global Uniform Asymptotic Stability of Nonlinear Time-Varying Systems in Cascade. Systems and Control Letters 33, 131–138
Imura J., Kobayashi K., Yoshikawa T. (1996) Nonholonomic Control of a 3 Link Planar Manipulator with a Free Joint. In: 35th IEEE Conference on Decision and Control, Japan, December, 2, 1435–1436
Khalil H.K. (1996) Nonlinear Systems, 2nd edn. Prentice Hall, New York
Oriolo G., Nakamura Y. (1991) Free-Joint Manipulators: Motion Control under Second-Order Nonholonomic Constraints. In: IEEE/RSJ International Workshop on Intelligent Robots and Systems’ 91. Intelligence for Mechanical Systems, Japan, November, 3, 1248–1253
Oriolo G., Nakamura Y. (1991) Control of Mechanical Systems with Second-Order Nonholonomic Constraints: Underactuated Manipulators. In: 30th Conference on Decision and Control, England, December, 3, 2398–2403
Spong M.W. (1995) The Swing Up Control Problem for the Acrobot. IEEE Control Systems Magazine 15, 49–55
Arai H., Tanie K., Shiroma N. (1998) Nonholonomic Control of a Three-DOF Planar Underactuated Manipulator. IEEE Transactions on Robotics and Automation 14, 681–695
Arai H. (1996) Controllability of a 3-DOF Manipulator with a Passive Joint under a Nonholonomic Constraint. In: IEEE International Conference on Robotics and Automation, USA, April, 4, 3707–3713
Arai H., Tanie K., Shiroma N. (1997) Feedback Control of a 3-DOF Planar Underactuated Manipulator. In: IEEE International Conference on Robotics and Automation 1, 703–709
De Luca A., Mattone R., Oriolo G. (1995) Dynamic Mobility of Redundant Robots under End-Effector Commands. In: Nonlinear Control and Robotics Preprints, Universitá di Roma “La Sapienza”, Italy
Lefeber E., Robertsson A., Nijmeijer H. (1999) Linear Controllers for Tracking Chained-Form Systems. Lecture Notes in Control and Information Sciences, Stability and Stabilization of Nonlinear Systems (Aeyels D., Lamnabhi-Lagarrigue F., van der Schaft A.J. (Eds.)), Springer, Heidelberg, 246, 183–199
Lefeber E., Robertsson A., Nijmeijer H. (1999) Output Feedback Tracking of Nonholonomic Systems in Chained Form. In: 5th European Control Conference, Germany, September, cd-rom, Paper 772 (AP-4)
Rugh W.J. (1993) Linear System Theory (Kailath T. (Ed.)), Information and System Sciences Series, 1st edn. Prentice-Hall, New Jersey
van Beek B., de Jager B. (1999) An Experimental Facility for Nonlinear Robot Control. In: IEEE International Conference on Control Applications, Hawai'i, August, 1, 668–673
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Aneke, N.P.I., Nijmeijer, H., de Jager, A.G. (2001). Trajectory tracking by cascaded backstepping control for a second-order nonholonomic mechanical system. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the Year 2000. Lecture Notes in Control and Information Sciences, vol 258. Springer, London. https://doi.org/10.1007/BFb0110206
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DOI: https://doi.org/10.1007/BFb0110206
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