Oscillations, SE(2)-snakes and motion control: A study of the Roller Racer

Abstract

This report 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 identi ed 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. Speci cally we concern ourselves with the group SE(2) of rigid motions in the plane and a concrete mechanical realization dubbed the 2{node, 1{module SE(2){snake. In a restricted version, it is also known as the Roller Racer (a patented ride/toy).Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this report 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 report. We exploit these structures to demonstrate that the single shape freedom in this problem can becyclically varied to produce a rich variety of motions of the SE(2){snake.In their study of the snakeboard, a patented modi cation of the skateboard that also admits the group SE(2) as a symmetry group, Lewis, Ostrowski, Burdick and Murray, exploited the same type of balance law as the one discussed here to generate motions. A key di erence however is that, in the present report, we have only one control variable and thus controllability considerations become somewhat more delicate.In the present report, we g i v e a self{contained treatment of the geometry, mechanics and motion control of the Roller Racer.3

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