Abstract
This letter studies the motion of a vertical rolling disk on an arbitrary smooth surface in ${\mathbb {R}}^{3}$ . The disk can roll without slipping about its axis and turn about the surface normal. A global formulation for the dynamics of the rolling disk is proposed without the use of local coordinates, and the model is globally defined on the manifold without singularities or ambiguities. The theoretical results are specialized for two different surfaces; a flat surface and a spherical surface. The proposed motion planning algorithm consists of three phases and each phase is a rest-to-rest maneuver, such that the rolling disk is stationary at both the start and the end of each phase. Simulation results are included that show effectiveness of the motion planning algorithm on the smooth surfaces.
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