Asymptotically Near-Optimal Planning With Probabilistic Roadmap Spanners

Abstract

Asymptotically optimal motion planners guarantee that solutions approach optimal as more iterations are performed. A recently proposed roadmap-based method, i.e., the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{\tt PRM}^{*}$ </tex></formula> approach, provides this desirable property and 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, especially for relatively high-dimensional instances. In graph theory, there are algorithms that produce sparse subgraphs, which are known as graph spanners, that guarantee near-optimal paths. This paper proposes different alternatives for interleaving graph spanners with the asymptotically optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{\tt PRM}^{*}$</tex></formula> algorithm. The first alternative follows a sequential approach, where a graph spanner algorithm is applied to the output roadmap of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\hbox{\tt PRM}^{*}$</tex></formula> . The second one is an incremental method, where certain edges are not considered during the construction of the roadmap as they are not necessary for a roadmap spanner. The result in both cases 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 through roadmap spanners.