An approximation algorithm for $d_1$-optimal motion of a rod robot with fixed rotations

Given a translating and rotating rod robot in a plane in the presence of polygonal obstacles with the initial and final placements of the rod known, the d1-optimal motion planning problem is defined as finding a collision-free motion of the rod such that the orbit length of a fixed but arbitrary point F on the rod is minimized. In this paper we study a special case of this problem in which the rod can translate freely, but can only rotate by some pre-specified given angles around F. We first characterize the d1-optimal motion of the robot under the given conditions and then present a (1 + e)-approximation algorithm for finding the optimal path. The running time of the algorithm is bounded by a polynomial in terms of some parameters related to the problem input.