Controllability and stabilizability of a class of systems with higher-order nonholonomic constraints

A theoretical framework is established for the control of higher-order nonholonomic systems, defined as systems that satisfy higher-order nonintegrable constraints. A model for such systems is developed in terms of differential–algebraic equations defined on a higher-order tangent bundle. A number of control-theoretic properties such as nonintegrability, controllability, and stabilizability are presented. Higher-order nonholonomic systems are shown to be strongly accessible and, under certain conditions, small time locally controllable at any equilibrium. The applicability of the theoretical development is illustrated through a third-order nonholonomic system example: a planar PPR robot manipulator subject to a jerk constraint. In particular, it is shown that although the system is not asymptotically stabilizable to a given equilibrium configuration using a time-invariant continuous feedback, it is strongly accessible and small-time locally controllable at any equilibrium.