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 impacts. Impacts are modeled using the Routh two-dimensional impact model, which incorporates the effects of spin and friction. Using this model, we derive an explicit mapping between preimpact and postimpact velocities of the puck as a function of the velocities of the impacting objects and the coefficients of restitution and friction. We will see that this mapping depends fundamentally on whether or not relative sliding between the objects occurs throughout the duration of the impact event, or terminates during the impact. 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 preimpact and postimpact velocities. Our results confirm the intuitive notion that a large coefficient of friction between the mallet and puck, and between the puck and table walls, enhances the ability to control puck motion through impacts.