Widening the effect of Lie bracket motion: a semi-global approximation and control for nonholonomic systems using non-power series expansion∗

A nonlinear control system often has complicated input-to-state relationship, mainly due to its controllability structure in nonlinear sense. A difficult but challenging nature of such systems is that they can only be controlled using appropriate combination of periodic inputs, corresponding to the Lie brackets. However, the effect of so-called Lie bracket motion tends to be limited due to small choice of the input amplitudes. In this paper, focusing on a class of nonholonomic systems as typical examples, we propose to approximate the state displacement under periodic signals with larger amplitudes, and utilize the result to design periodic control input. The key of our approach is the use of suitable special function, such as the Bessel functions, for series expansion to predict the state displacement, as well as considering symmetry of the state space.