Oscillations, SE(2)-Snakes and Motion Control

This paper is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a flat surface). In earlier work, we identified an interesting class of such problems arising in the setting of Lie groups, and investigated these under a hypothesis on constraints, that naturally led to a purely kinematic approach. In the present work, the hypothesis on constraints does not hold, and as a consequence, it is necessary to take into account certain dynamical phenomena. Specifically we concern ourselves with the group SE(2) of rigid motions in the plane and a concrete mechanical realization dubbed the SE(2)--snake. In a restricted version, it is also known as the Roller Racer (a patented toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this paper a ..