Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then be used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using the “chained form” and cast that work into a coordinate-free setting.

[1]  S. Chern,et al.  Exterior Differential Systems , 1990 .

[2]  Ole Jakob Sørdalen,et al.  Conversion of the kinematics of a car with n trailers into a chained form , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[3]  R. Gardner,et al.  The GS algorithm for exact linearization to Brunovsky normal form , 1992 .

[4]  Wei-Liang Chow.  Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung , 1940 .

[5]  Henry Hermes,et al.  Distributions and the Lie algebras their bases can generate , 1989 .

[6]  R. Brockett.  Control Theory and Singular Riemannian Geometry , 1982 .

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

[8]  Richard M. Murray,et al.  A motion planner for nonholonomic mobile robots , 1994, IEEE Trans. Robotics Autom..

[9]  G. Jacob.  MOTION PLANNING BY PIECEWISE CONSTANT OR POLYNOMIAL INPUTS , 1992 .

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

[11]  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.

[12]  Anthony M. Bloch,et al.  Control of mechanical systems with classical nonholonomic constraints , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

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

[14]  G. Foussereau,et al.  Comptes rendus des séances de l'Académie des Sciences et annales de chimie et de physique; 1892 , 1893 .

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

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

[17]  Richard M. Murray,et al.  Steering nonholonomic systems in chained form , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[18]  V. Rich.  Personal communication , 1989, Nature.