Controllability of Single Input Rolling Manipulation

This paper investigates controllability of underactuated rolling systems consisting of a smooth object rolling on a moving smooth surface. Our system consists of a spherical ball which rolls on the inside of an ellipsoidal bowl. The bowl has a single translational degree of freedom not aligned with any of its principal axes. The single control input is the bowl's acceleration in this direction. The object and contact motions are governed by a nonlinear system of equations derived from the kinematics and dynamics of rolling. Using existing results on small time local accessibility (STLA) and weakly positive Poisson stable (WPPS) vector elds, and assuming that the ball stays in the bowl, we show that the ball is globally controllable on its ve-dimensional space of con gurations relative to the bowl. We are currently working on motion planning algorithms with our experimental setup to control the equilibrium con guration of the ball.