Motion Planning for Nonholonomic Dynamic Systems

Abstract

The problem of motion planning for nonholonomic dynamic systems is studied. A model for nonholonomic dynamic systems is first presented in terms of differential- algebraic equations defined on a phase space. A nonlinear control system in a normal form is introduced to completely describe the dynamics. The assumptions guarantee that the resulting normal form equations necessarily contain a nontrival drift vector field. We show that the linearized control system always has uncontrollable eigenvalues at the origin. However, any equilibrium is shown to be strongly accessible and small time locally controllable. A motion planning approach using holonomy is developed for nonholonomic Caplygin dynamical systems, i.e. nonholonomic systems with certain symmetry properties which can be expressed by the fact that the constraints are cyclic in certain of the variables. The theoretical development is applied to physical examples of systems that we have studied in detail elsewhere: the control of motion of a knife edge moving on a plane surface and the control of motion of a wheel rolling without slipping on a plane surface. The results of the paper are also applied to the control of motion of a planar multibody system using angular momentum preserving joint torques. Results of simulations are included to illustrate the effectiveness of the proposed motion planning approach.