Optimal Shape and Motion Planning for Dynamic Planar Manipulation

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. 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 metric,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.