Design of Stabilizing Feedback Controllers for High-Order Nonholonomic Systems

This letter presents a novel stabilizing control design strategy for driftless control-affine systems with an arbitrary degree of nonholonomy. The proposed approach combines a time-varying control component that generates motion in the direction of prescribed Lie brackets with a state-dependent component, ensuring the stability of the equilibrium. The coefficients of the state-dependent component are derived in such a way that the trajectories of the resulting closed-loop system approximate the gradient flow of a Lyapunov-like function. In the case of a quadratic Lyapunov function, this guarantees the exponential stability of the equilibrium. The usability of this approach is demonstrated on general two-input systems having the fourth degree of nonholonomy. The proposed stabilization scheme is illustrated with several examples.