Planning the motion of a sliding and rolling sphere

This paper investigates the free motion of a sphere with initial velocity and angular velocity on a plane under sliding and rolling friction. The sphere will first slide along a parabolic trajectory (with a constant direction of its contact velocity), and then roll along a straight trajectory. Such a curved trajectory can be utilized for obstacle avoidance in path planning. A one-to-one correspondence exists between trajectories connecting two specified locations and pairs of sliding and rolling directions within some 2-dimensional region, called the trajectory space, which is then used for planning. A plane sweep algorithm operating in this space is presented to find all collision-free trajectories in the presence of cylindrical obstacles that are vertically oriented.