Control and stabilization of nonholonomic dynamic systems.

Abstract

A theoretical framework is established for the control of nonholonomic dynamic systems, i.e. dynamic systems with nonintegrable constraints. In particular, we emphasize control properties for nonholonomic systems that have no counterpart in holonomic systems. A model for nonholonomic dynamic systems is first presented in terms of differential-algebraic equations defined on a phase space. A reduction procedure is carried out to obtain reduced order state equations. Feedback is then used to obtain a control system in a normal form. The assumptions guarantee that the resulting normal form equations necessarily contain a nontrival drift vector field. Conditions for smooth ($C\sp{\infty}$) asymptotic stabilization to an m-dimensional equilibrium manifold are presented; we also demonstrate that a single equilibrium solution cannot be asymptotically stabilized using continuous static or dynamic state feedback. However, any equilibrium is shown to be strongly accessible and small time locally controllable. An approach using geometric phases is developed as a basis for the control of 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 variables. The theoretical development is applied to physical examples of systems that we have studied in detail elsewhere; the control of a knife edge moving on a plane surface and the control of a wheel rolling without slipping on a plane surface. The results are also applied to the reorientation of planar multibody systems using joint torque inputs and to the reorientation of a rigid spacecraft using momentum wheel actuators, since in these examples conservation of angular momentum gives rise to nonintegrable motion invariants.