Sample Complexity of Probabilistic Roadmaps via ϵ-nets

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 (δ,)-completeness of the parameters \mathcal{X},r, which indicates that for every motionplanning problem of clearance at least δ > 0, PRM using \mathcal{X},r returns a solution no longer than 1+ϵ times the shortest δ-clear path. Leveraging the concept of ϵ -nets, we characterize in terms of lower and upper bounds the number of samples needed to guarantee (δ, ϵ)-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 -nets that achieves nearly the same coverage as grids while using fewer samples.