Exact Pareto-Optimal Coordination of Two Translating Polygonal Robots on a Cyclic Roadmap

We consider planning optimal collision-free motions of two polygonal robots under translation. Each robot has a reference point that must lie on a given graph, called a roadmap, which is embedded in the plane. The initial and the goal are given for each robot. Rather than impose an a priori cost scalarization for choosing the best combined motion, we consider finding motions whose cost vectors are Pareto-optimal. Pareto-optimal coordination strategies are the ones for which there exists no strategy that would be better for both robots. Our problem translates into shortest path problems in the coordination space which is the Cartesian product of the roadmap, as a cell complex, with itself. Our algorithm computes an upper bound on the cost of each motion in any Pareto-optimal coordination. Therefore, only a finite number of homotopy classes of paths in the coordination space need to be considered. Our algorithm computes all Pareto-optimal coordinations in time O(2mn log(mn)), in which m is the number of edges in the roadmap, n is the number of coordination space obstacle vertices, and α = 1 + d(5` + r)/be where ` is total length of the roadmap and r is total length of coordination space obstacle boundary and b is the length of the shortest edge in the roadmap.