Asymptotically near-optimal is good enough for motion planning

Asymptotically optimal motion planners guarantee that solutions approach optimal as more iterations are performed. There is a recently proposed roadmap-based method that provides this desirable property, the PRM* approach, which minimizes the computational cost of generating the roadmap. Even for this method, however, the roadmap can be slow to construct and quickly grows too large for storage or fast online query resolution. From graph theory, there are many algorithms that produce sparse subgraphs, known as spanners, which can guarantee near optimal paths. In this work, a method for interleaving an incremental graph spanner algorithm with the asymptotically optimal PRM* algorithm is described. The result is an asymptotically near-optimal motion planning solution. Theoretical analysis and experiments performed on typical, geometric motion planning instances show that large reductions in construction time, roadmap density, and online query resolution time can be achieved with a small sacrifice of path quality. If smoothing is applied, the results are even more favorable for the near-optimal solution.