Planning the Initial Motion of a Free Sliding/Rolling Ball

This paper investigates the motion of a free ball with initial velocity and angular velocity on a horizontal plane under friction. The ball will first slide along a parabolic trajectory (with its contact velocity in constant direction) and then roll along a straight trajectory. Curving of the sliding trajectory can be utilized for obstacle avoidance in path planning. A one-to-one correspondence exists between curved trajectories of the ball from one point to another and pairs of sliding and rolling directions that span a cone at the starting point to contain the ending point in the interior. This leads to a compact 2-D trajectory space, in which obstacles induce monotonic constraint curves. Based on the monotonicity, a plane sweep algorithm is described to find all collision-free trajectories. The work has several distinct features: handling of both sliding and rolling; planning in the configuration space of trajectories not that of the ball; computation of not just one collision-free trajectory but all such trajectories; and, finally, combining dynamics with computational geometry. The application of the work to spherical robots is discussed at the end.