A rank condition for ρ-exponential stabilization of dynamic Caplygin systems

The paper investigates the global ρ-exponential stability of dynamic nonholonomic Caplygin systems, which is composed of double integrator cascading nonholonomic constraints. A novel decomposition of state is assumed first so that the constraints are linear in certain state variables. A simple and easily verified rank condition for the global ρ-exponential stability of Caplygin systems is derived. A feature of our design is that all parameters can be explicitly determined from the constraint function or an important class of Caplygin systems in which one of the decomposed states is scalar, the rank condition can be explicitly represented as the conditions relating to the degree and non-zero property of the lowest degree polynomials of the Taylor series expansion of the constraint function at the origin. Moreover, an alternative form of Caplygin system in different coordinates is presented so that the proposed coordinate-dependent criterion can be applied. Examples such as extended power form, the rolling wheel and hopping robot systems are shown to belong to this class of Caplygin systems and thus their ρ-exponential stability can be checked by the proposed test.