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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Abstract. The outstanding locomotor and manipulation characteristics of the octopus have recently inspired the development, by our group, of multi-functional robotic swimmers, featuring both manipulation and locomotion capabilities, which could be of significant engineering interest in underwater applications. During its littlestudied arm-swimming behavior, as opposed to the better known jetting via the siphon, the animal appears to generate considerable propulsive thrust and rapid acceleration, predominantly employing movements of its arms. In this work, we approximate the corresponding complex pattern of arm motion with a sculling-like profile, involving a fast power stroke and a slow recovery stroke. We investigate the propulsive capabilities of a multi-arm robotic system under various swimming gaits, namely patterns of arm coordination, which achieve the generation of forward, as well as backwards, propulsion and of turning. A lumped-element model of the robotic swimmer, which considers arm compliance and the interaction with the aquatic environment, was used to study the characteristics of these gaits, the effect of various kinematic parameters on propulsion and the generation of complex trajectories. This investigation focuses on relatively high-stiffness arms. Experiments with a compliant-body robotic prototype swimmer with eight compliant arms, all made of polyurethane, inside a water tank, successfully demonstrated this novel mode of underwater propulsion. Speeds of up to 0.26 body lengths per second (98.6 mm/s), and propulsive forces of up to 3.5 N were achieved, with a non-dimensional cost of transport of 1.42 with all 8 arms and of 0.9 with only 2 active ones. The experiments confirmed the computational results and verified the multi-arm maneuverability and simultaneous object grasping capability of such systems.