Optimal shape and motion planning for dynamic planar manipulation

Abstract

This paper presents a framework for optimizing both the shape and the motion of a planar rigid end-effector to satisfy a desired manipulation task. Both shape and motion play key roles in determining contact interaction, and while both are commonly seen as design/control freedoms, their synergies are rarely formally explored. In this paper we study quasistatic problems like cam design, or dynamic problems like ball throwing, where both the shape and motion of the surfaces at contact are relevant. We frame this design problem as a nonlinear optimization program, where shape and motion are decision variables represented as splines. The task is represented as a series of constraints, along with a fitness cost, which force the solution to be compatible with the dynamics of frictional hard contact while satisfying the task. We illustrate the approach with the example problem of moving a disk along a desired path or trajectory, and we verify it by applying it to three classical design problems: the rolling brachistochrone, the design of teeth of involute gears, and the pitch curve of rolling cams. We conclude with a case study involving the optimization and real implementation of the shape and motion of a dynamic throwing arm.