Sliding manipulation of rigid bodies on a controlled 6-DoF plate

We model the full dynamics of a rigid part in three-point frictional sliding contact with a flat rigid 6-degree-offreedom plate. Given a periodic plate motion and the geometric, inertial, and frictional properties of the part, we define an asymptotic twist field mapping each part configuration to a unique part twist (linear and angular velocity). Asymptotic twist vectors in the field approximate the part's cycle-averaged twist at each configuration and are independent of time or the system's initial state. Simulations and experiments show that the trajectory of the part's configuration as it slides on the plate is well described by the field. With the ability to program arbitrary plate motions, part manipulation reduces to finding plate motions that generate asymptotic twist fields to accomplish desired tasks. Several simple fields useful for manipulation tasks (e.g., sensorless part alignment) are verified in simulation and experiment. For the special case of a rigid part with infinitesimal thickness, we show that the part's cycle-averaged twist for any configuration asymptotically converges to a unique asymptotic twist vector.