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Oscillations, SE(2)-snakes and motion control

Concerns motion generation via cyclic variations in selected degrees of freedom (shape variables) in mechanical systems subject to nonholonomic constraints (here, a disk rolling without sliding on a flat surface). It is necessary to take into account certain dynamical phenomena. We consider the group SE(2) of rigid motions in the plane and a concrete mechanical realization, the SE(2)-snake. One version of this is the Roller Racer (a patented toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray (1995), one recognizes in the example 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 law results in certain structures which are used 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 modification of the skateboard that also admits the group SE(2) as a symmetry group, Lewis, Ostrowski, Burdick and Murray (1994) exploited the same type of balance law, but in this 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.