Continuing the work presented in part I, ‡ we consider the problem of moving a point robot through a two dimensional workspace containing polygonal obstacles moving on unknown trajectories. We propose to use sensor information to predict the trajectories of the obstacles, and interleave path planning and execution. We define a locally minimum velocity path as an optimal robot trajectory, given only local information about obstacle trajectories. We show that the complexity of a path planning problem can be characterized by how frequently the robot must change directions to approximate the locally minimum velocity path. Our results apply to both robots with and without maximum velocity limits.