Densification strategies for anytime motion planning over large dense roadmaps

We consider the problem of computing shortest paths in a dense motion-planning roadmap G. We assume that n, the number of vertices of G, is very large. Thus, using any path-planning algorithm that directly searches G, running in O(VlogV + E) ≈ O(n²) time, becomes unacceptably expensive. We are therefore interested in anytime search to obtain successively shorter feasible paths and converge to the shortest path in G. Our key insight is to provide existing path-planning algorithms with a sequence of increasingly dense subgraphs of G. We study the space of all (r-disk) subgraphs of G. We then formulate and present two densification strategies for traversing this space which exhibit complementary properties with respect to problem difficulty. This inspires a third, hybrid strategy which has favourable properties regardless of problem difficulty. This general approach is then demonstrated and analyzed using the specific case where a low-dispersion deterministic sequence is used to generate the samples used for G. Finally we empirically evaluate the performance of our strategies for random scenarios in ℝ² and ℝ⁴ and on manipulation planning problems for a 7 DOF robot arm, and validate our analysis.