Control of Planar Rigid Body Sliding with Impacts and Friction

In this paper we continue our investigations into robotic air hockey by studying the problem of controlling the trajectory of a puck subject to intermittent impacts. Impacts are modeled using the Routh two-dimensional impact model which incorporates spin and friction. Using this model, we derive an explicit mapping between pre-impact and post-impact velocities of the puck. We will see that this mapping is discontinuous and separates the velocity space of the puck into regions determined by whether or not relative sliding ends during impact. This discontinuity depends in a fundamental way on the relative velocity of the objects at impact and the coe cients of restitution and friction. We then present results on planning of puck trajectories and control of the puck through impacts with a mallet or striker, which is essentially the problem of inverting the mapping between pre-impact and post-impact velocities. Our results con rm the intuitive notion that a large coe cient of friction between the mallet and puck, and between the puck and table walls, enhances the ability to control puck motion through impacts.