Motion Control for Nonholonomic Systems on Matrix Lie Groups

Abstract : In this dissertation we study the control of nonholonomic systems defined by invariant vector fields on matrix Lie groups. We make use of canonical constructions of coordinates and other mathematical tools provided by the Lie group setting. An approximate tracking control law is derived for so-called chained form systems which arise as local representations of systems on a certain nilpotent matrix group. After studying the technique of nilpotentization in the setting of systems on matrix Lie groups we show how motion control laws derived for nilpotent systems can be extended to nilpotentizable systems using feedback and state transformations. The proposed control laws exhibit highly oscillatory components both for tracking and feedback stabilization of local representations of nonholonomic systems on Lie groups. Applications to the control and analysis of the kinematics of mechanical systems are discussed and numerical simulations are presented.

[1]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[2]  E. Norman,et al.  ON GLOBAL REPRESENTATIONS OF THE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS AS A PRODUCT OF EXPONENTIALS , 1964 .

[3]  F. Wilson.  The structure of the level surfaces of a Lyapunov function , 1967 .

[4]  H. Hermes,et al.  Nonlinear Controllability via Lie Theory , 1970 .

[5]  R. Brockett.  System Theory on Group Manifolds and Coset Spaces , 1972 .

[6]  J. Wood.  Power conversion in electrical networks , 1974 .

[7]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[8]  H. Hermes,et al.  Nilpotent bases for distributions and control systems , 1984 .

[9]  Jean-Paul Laumond,et al.  Feasible Trajectories for Mobile Robots with Kinematic and Environment Constraints , 1986, IAS.

[10]  Anatoliĭ Timofeevich Fomenko,et al.  Recursion formulas for the Lie integral , 1991 .

[11]  Gerardo Lafferriere,et al.  Motion planning for controllable systems without drift , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[12]  H. Sussmann,et al.  Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[13]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[14]  Jean-Michel Coron,et al.  Global asymptotic stabilization for controllable systems without drift , 1992, Math. Control. Signals Syst..

[15]  Rui Yang.  Nonholonomic Geometry, Mechanics and Control , 1992 .

[16]  Richard M. Murray,et al.  Nonholonomic control systems: from steering to stabilization with sinusoids , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[17]  Jean-Baptiste Pomet.  Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift , 1992 .

[18]  P. Krishnaprasad.  Optimal Control and Poisson Reduction , 1993 .

[19]  Zexiang Li,et al.  Smooth Time-Periodic Feedback Solutions for Nonholonomic Motion Planning , 1993 .

[20]  L. Dai,et al.  Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability , 1993 .

[21]  Linda Bushnell,et al.  Steering Three-Input Chained Form Nonholonomic Systems Using Sinusoids: The Fire Truck Example , 1993 .

[22]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[23]  Naomi Ehrich Leonard.  Averaging and Motion Control of Systems on Lie Groups , 1994 .

[24]  Richard M. Murray,et al.  Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems , 1994, Math. Control. Signals Syst..

[25]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[26]  V. Arnold,et al.  Dynamical Systems VII , 1994 .

[27]  Perinkulam S. Krishnaprasad,et al.  On the Geometry and Dynamics of Floating Four-Bar Linkages , 1994 .

[28]  S. Sastry,et al.  Optimal path planning on matrix Lie groups , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[29]  C. Samson.  Control of chained systems application to path following and time-varying point-stabilization of mobile robots , 1995, IEEE Trans. Autom. Control..

[30]  Dimitrios P. Tsakiris,et al.  Motion Control and Planning for Nonholonomic Kinematic Chains , 1995 .

[31]  Naomi Ehrich Leonard,et al.  Motion control of drift-free, left-invariant systems on Lie groups , 1995, IEEE Trans. Autom. Control..

[32]  Herschel Rabitz,et al.  Control of molecular motion , 1996, Proc. IEEE.

[33]  R. Murray,et al.  Exponential stabilization of driftless nonlinear control systems using homogeneous feedback , 1997, IEEE Trans. Autom. Control..

[34]  Perinkulam S. Krishnaprasad,et al.  On approximate inversion and feedback stabilization for systems on matrix Lie groups , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[35]  Pascal Morin,et al.  Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop , 1999, SIAM J. Control. Optim..