ρ-Exponential Stabilization for Caplygin Systems: Simplified Controllability Condition and Examples

The paper investigates the global ρ-exponential stabilizability for nonholonomic Caplygin systems that are linear in certain state variables. A simple and easily verified controllability condition is proposed to guarantee the global ρ-exponential stabilizability. For an important subclass of Caplygin systems, the controllability condition is further reduced to some conditions relating to the degree and non-zero property of the lowest nonzero polynomials in the Taylor expansion of constraint function. A new feature in this paper is that all parameters can be explicitly determined from the constraint function. Moreover, a novel coordinate transformation between Caplygin systems can be used to enhance the proposed criterion so that it can be applied to various situations. Several interesting examples, including of the rolling wheel system and a set-point problem of hopping robot, are provided to validate the effectiveness of the proposed results.