Decidability of Robot Manipulation Planning: Three Disks in the Plane

This paper considers the problem of planning collision-free motion of three disks in the plane. One of the three disks, the robot, can autonomously translate in the plane, the other two move only when in contact with the robot. This represents the abstract formulation of a manipulation planning problem. Despite the simplicity of the formulation, the decidability of the problem had remained unproven so far. We prove that the problem is decidable, i.e., there exists an exact algorithm that decides whether a solution exists in finite time.