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 six-degree-of-freedom (6-DoF) plate. When the plate moves periodically, we show the part's dynamics are well approximated by a first-order system represented by an asymptotic velocity field that maps part configurations in SE(2) to unique velocities (linear and angular) in 2. The form of the asymptotic velocity field depends on the plate's motion, the location and friction coefficient of each contact point, and the inertial properties of the part. Asymptotic velocity vectors in the field approximate the part's cycle-averaged velocity at each configuration and are independent of time or the system's initial state. For the special case of a rigid part with infinitesimal thickness, we prove that asymptotic velocities are always unique and well defined. With the ability to program arbitrary periodic plate motions, part manipulation reduces to finding plate motions that generate asymptotic velocity fields to accomplish desired tasks. Several fields useful for manipulation tasks (e.g. sensorless part alignment) are verified in simulation and experiment.