Control of non holonomic or under-actuated mechanical systems: The examples of the unicycle robot and the slider

¹ MINES ParisTech-CAOR, PSL, 60 Bvd St-Michel 75006 Paris, France.
brigitte.dandrea-novel@mines-paristech.fr
² SAGEM, 100 avenue de Paris 91344 Massy Cedex, France.
sylvain.thorel@sagem.com

Received: 6 June 2016
Accepted: 7 June 2016

Abstract

This paper is dedicated to the control of some classes of mechanical systems, not stabilizable by means of at least continuous state feedback laws. This is the case for non holonomic mechanical systems, an example being the unicycle robot, or for under-actuated mechanical systems, an example being the slider. The similarity between these two kinds of dynamical systems will be analyzed, with respect to controllability, stabilizability and flatness properties. The control design for slider stabilization could be viewed, to some extent, as an extension of the control law used for unicycle systems, but it is not so obvious if one wants to stabilize the slider at a chosen reference position, with a chosen fixed orientation. In fact, a new fixed point stabilization result will be given for the slider, and therefore, a switched control strategy will be proposed for both systems, based on differential flatness property to track moving non singular reference trajectories, and switching to periodic time-varying feedback laws to ensure final stabilization at rest equilibrium points. The method will be illustrated with success through simulation results and also first experimental results in the case of a terrestrial quadrotor tracking non singular reference trajectories.