Motion Planning and Feedback Control of Rolling Bodies

This paper examines the problem of planning and stabilizing the trajectory of one smooth body rolling on the surface of another. The two control inputs are the angular velocity of the moving body about two orthogonal axes in the contact tangent plane; spinning about the contact normal is not allowed. To achieve robustness and computational efficiency, our approach to trajectory planning is based on solving a series of optimization problems of increasing complexity. To stabilize the trajectory in the face of perturbations, we use a linear quadratic regulator. We apply the approach to examples of a sphere rolling on a sphere and an ellipsoid rolling on an ellipsoid. Finally, we explore the robustness and performance of the motion planner. Although the planner is based on non-convex optimization, in practice the planner finds solutions to nearly all randomly-generated tasks, and the solution trajectories are smoother and shorter than those found in previous work in the literature.