Control of Nonholonomic Systems

When the generalized velocity of a mechanical system satis es an equality condition that cannot be written as an equivalent condition on the generalized position, the system is called a nonholonomic system [1, 2]. Nonholonomic condition may arise from constraints such as pure rolling of a wheel or from physical conservation laws such as the conservation of angular momentum of a free oating body. Nonholonomic systems pose a particular challenge from the control point of view, as any one who has tried to parallel park a car in a tight space can attest. The basic problem involves nding a path that connects an initial con guration to the nal con guration and satis es all the holonomic and nonholonomic conditions for the system. Both open loop and closed loop solutions are of interest: open loop solution is useful for o -line path generation, closed loop solution is needed for the real time control. Nonholonomic systems typically arise in the following classes of systems:

[1]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

[2]  J. Zabczyk.  Some comments on stabilizability , 1989 .

[3]  R. A. DeCarlo.  Linear Systems: A State Variable Approach With Numerical Implementation , 1989 .

[4]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[5]  P. Crouch,et al.  Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models , 1984 .

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

[7]  A. Isidori.  Nonlinear Control Systems , 1985 .

[8]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[9]  Naomi Ehrich Leonard,et al.  Averaging for attitude control and motion planning , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[10]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[11]  H. Sussmann,et al.  A continuation method for nonholonomic path-finding problems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[12]  John T. Wen,et al.  Trajectory tracking control of a car-trailer system , 1997, IEEE Trans. Control. Syst. Technol..

[13]  S. Sastry,et al.  Trajectory generation for the N-trailer problem using Goursat normal form , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[14]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[15]  J. Latombe,et al.  On nonholonomic mobile robots and optimal maneuvering , 1989, Proceedings. IEEE International Symposium on Intelligent Control 1989.

[16]  Karim Ait-Abderrahim,et al.  Feedback stabilization of a nonholonomic wheeled mobile robot , 1991, Proceedings IROS '91:IEEE/RSJ International Workshop on Intelligent Robots and Systems '91.

[17]  Leo Dorst,et al.  The geometrical representation of path planning problems , 1991, Robotics Auton. Syst..

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

[19]  Wei-Liang Chow.  Über Systeme von linearen partiellen Differential-gleichungen erster Ordnung , 1941 .

[20]  Steven Dubowsky,et al.  Coordinated manipulator/spacecraft motion control for space robotic systems , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[21]  Yuandan Lin,et al.  A universal formula for stabilization with bounded controls , 1991 .

[22]  A. Bloch,et al.  Control and stabilization of nonholonomic dynamic systems , 1992 .

[23]  Zexiang Li,et al.  Motion of two rigid bodies with rolling constraint , 1990, IEEE Trans. Robotics Autom..

[24]  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.

[25]  O. J. Sordalen,et al.  Exponential stabilization of mobile robots with nonholonomic constraints , 1992 .

[26]  Steven Dubowsky,et al.  On the dynamics of manipulators in space using the virtual manipulator approach , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[27]  Eduardo Sontag.  Gradient techniques for systems with no drift: a classical idea revisited , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[28]  Giuseppe Oriolo,et al.  Free-joint manipulators: motion control under second-order nonholonomic constraints , 1991, Proceedings IROS '91:IEEE/RSJ International Workshop on Intelligent Robots and Systems '91.

[29]  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.

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

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

[32]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[33]  C. Fernandes,et al.  Near-optimal nonholonomic motion planning for a system of coupled rigid bodies , 1994, IEEE Trans. Autom. Control..

[34]  J. Laumond.  Controllability of a multibody mobile robot , 1991 .

[35]  Michel Taïx,et al.  Efficient motion planners for nonholonomic mobile robots , 1991, Proceedings IROS '91:IEEE/RSJ International Workshop on Intelligent Robots and Systems '91.

[36]  Jean-Paul Laumond,et al.  Finding Collision-Free Smooth Trajectories for a Non-Holonomic Mobile Robot , 1987, IJCAI.

[37]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[38]  John T. Wen,et al.  Nonholonomic path planning with inequality constraints , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[39]  R. Murray,et al.  Convergence Rates for Nonholonomic Systems in Power Form , 1993, 1993 American Control Conference.