Motions and stability of a piecewise holonomic system: the discrete Chaplygin sleigh

We discuss the dynamics of a piecewise holonomic mechanical system: a discrete sister to the classical non-holonomically constrained Chaplygin sleigh. A slotted rigid body moves in the plane subject to a sequence of pegs intermittently placed and sliding freely along the slot; motions are smooth and holonomic except at instants of peg insertion. We derive a return map and analyze stability of constant-speed straight-line motions: they are asymptotically stable if the mass center is in front of the center of the slot, and unstable if it lies behind the slot; if it lies between center and rear of the slot, stability depends subtly on slot length and radius of gyration. As slot length vanishes, the system inherits the eigenvalues of the Chaplygin sleigh while remaining piecewise holonomic. We compare the dynamics of both systems, and observe that the discrete skate exhibits a richer range of behaviors, including coexistence of stable forward and backward motions.