Sparse Roadmap Spanners

Asymptotically optimal planners, such as PRM⇤, guarantee that solutions approach optimal as iterations increase. Roadmaps with this property, however, may grow too large. If optimality is relaxed, asymptotically near-optimal solutions produce sparser graphs by not including all edges. The idea stems from graph spanner algorithms, which produce sparse subgraphs that guarantee near-optimal paths. Existing asymptotically optimal and near-optimal planners, however, include all sampled configurations as roadmap nodes. Consequently, only infinite graphs have the desired properties. This work proposes an approach that provides the following asymptotic properties: (a) completeness, (b) near-optimality and (c) the probability of adding nodes to the spanner converges to zero as iterations increase. Thus, the method shows that finite-size data structures can have near-optimality properties. The method brings together ideas from various planners but deviates from existing integrations of PRM⇤ with graph spanners. Simulations for rigid bodies show that the method indeed provides small roadmaps and results in faster query resolution. The rate of node addition is shown to decrease over time and the quality of solutions satisfies the theoretical bounds. Smoothing provides a more favorable comparison against alternatives with regards to path length.