Control and stabilization of nonholonomic Caplygin dynamic systems

A theoretical framework is established for the control of nonholonomic Caplygin dynamic systems, i.e., nonholonomic systems with certain symmetry properties which can be expressed by the fact that the Lagrangian and constraints are cyclic in certain of the variables. A model is presented in terms of differential-algebraic equations defined on a phase space. A nonlinear control system in a normal form is then introduced to completely describe the dynamics. The structure of the normal form equations allows identification of a base spaces, on which a set of decoupled controllable dynamics is defined. The emphasis of the present work is on control of the normal form equations on the complete phase space. The control system, linearized at an equilibrium, always has uncontrollable eigenvalues with zero real part, their number being equal to the number of nonintegrable constraints. However, the equilibrium is shown to be strongly accessible and small time locally controllable. Conditions for smooth asymptotic stabilization to an m -dimensional equilibrium manifold are presented