Optimal motion planning for a tethered robot: Efficient preprocessing for fast shortest paths queries

We study the problem of planning the shortest path for a polygonal robot anchored to a fixed base point by a finite tether translating among polygonal obstacles in the plane. Specifically, we preprocess the workspace to efficiently answer queries of the following type: Given a source location of the robot and an initial configuration of the tether, compute the shortest path to reach a target location while avoiding obstacles and adhering to the tether's constraints. Our work is an extension of the recent work by Kim et al. [1] who considered the problem for a point robot. Their algorithm relies on a discretization of the workspace and is optimal with respect to this discretization. We first replace their grid-based approach with a visibility-graph based approach. This allows to improve the running time of their algorithm by several orders of magnitude. Specifically, testing on a scenario similar to one presented by Kim et al., the running time is improved by a factor of more than 500. Moreover, our approach, which plans optimal paths, is applicable to polygonal (translating) robots and can be used to plan a shortest path while ensuring a predefined clearance from the obstacles. We report on our experimental results on a variety of scenarios. In all cases the preprocessing time is less than one second on a standard-commodity laptop, and a typical query takes several tens of miliseconds.