Sample Complexity of Probabilistic Roadmaps via $\epsilon$-nets

TL;DR

This paper analyzes the sample complexity of probabilistic roadmaps (PRM) in finite regimes, introducing -completeness and leveraging -nets to determine sample bounds for near-optimal motion planning.

Contribution

It introduces -completeness for PRM, characterizes sample bounds using -nets, and proposes an efficient sampling distribution inspired by these concepts.

Findings

Derived bounds on sample size for -completeness.

Proposed a new sampling distribution with fewer samples.

Validated the approach with theoretical guarantees.

Abstract

We study fundamental theoretical aspects of probabilistic roadmaps (PRM) in the finite time (non-asymptotic) regime. In particular, we investigate how completeness and optimality guarantees of the approach are influenced by the underlying deterministic sampling distribution ${\mathcal{X}}$ and connection radius ${r>0}$. We develop the notion of ${(\delta,\epsilon)}$-completeness of the parameters ${\mathcal{X}, r}$, which indicates that for every motion-planning problem of clearance at least ${\delta>0}$, PRM using ${\mathcal{X}, r}$ returns a solution no longer than ${1+\epsilon}$ times the shortest ${\delta}$-clear path. Leveraging the concept of ${\epsilon}$-nets, we characterize in terms of lower and upper bounds the number of samples needed to guarantee ${(\delta,\epsilon)}$-completeness. This is in contrast with previous work which mostly considered the asymptotic regime in which the number of samples tends to infinity. In practice, we propose a sampling distribution inspired by ${\epsilon}$-nets that achieves nearly the same coverage as grids while using significantly fewer samples.