Abstract
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.
Highlights
This paper examines the problem of planning and stabilizing the trajectory of one smooth body rolling on the surface of another
This is relevant for systems such as a ball-type mobile robot rolling over smooth terrain (Figure 1(a)) or a robot hand planning multi-finger rolling motions to reorient an object (Figure 1(b))
To stabilize a planned trajectory to small perturbations, we use linear feedback control based on a linear quadratic regulator. For this to be successful, the linearized trajectory must be controllable, so we examine the controllability of rolling trajectories and provide examples of uncontrollable trajectories
Summary
This paper examines the problem of planning and stabilizing the trajectory of one smooth body rolling on the surface of another. In this work we present a method to generate motion plans and stabilizing feedback controllers for general, smooth, three-dimensional objects in rolling contact. We first solve a convex problem that uses the two rolling velocity inputs to drive two of the five configuration variables directly to their desired values This motion serves as the initial trajectory guess for direct-collocation constrained. MOTION PLANNING FOR ROLLING SYSTEMS First- and second-order roll-slide kinematics, as discussed above, allow sliding at the contact, but we focus on planning for first-order systems in pure-rolling contact This is a well-known driftless nonholonomic system, where the rolling constraints do not necessarily translate to constraints on the achievable relative configuration. Li and Canny derive the first-order contact equations for rolling objects parameterized by orthogonal coordinate systems, analyze the controllability properties, and provide a geometric motion planning algorithm for a sphere on a plane [14]. Feedback stabilization of rolling is addressed by Walsh et al, who present a control law to exponentially stabilize linearized trajectories [21]; Sarkar et al who demonstrate the use of feedback linearization to control dynamic rolling motions for two planes in contact with a sphere [5]; and Choudhury and Lynch, who stabilize the orientation of a ball rolling in an ellipsoidal dish actuated along a single degree of freedom [22]
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