Oscillations , SE ( 2 ) { Snakes and Motion

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 at surface). In earlier work, we identiied 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 kine-matic 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. Speciically 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, Mars-den and Murray, one recognizes in the example of this paper a balance law called the momentum equation, which is a direct consequence of the interaction of the SE(2){symmetry of the problem with the constraints. The systematic use of this type of balance law results in certain structures in the example of this paper. We exploit these structures to demonstrate that the single shape freedom in this problem can be cyclically varied to produce a rich variety of motions of the SE(2){ snake. In their study of the Snakeboard, a patented modii-cation of the skateboard that also admits the group SE(2) as a symmetry group, Lewis, Ostrowski, Bur-dick and Murray, exploited the same type of balance law as the one discussed here to generate motions. A key diierence however is that, in the present paper, we have only one control variable and thus controllability considerations become somewhat more delicate. In the present paper, we give a self{contained treatment of the geometry, mechanics and motion control of the Roller Racer.