Practical Distance Functions for Path-Planning in Planar Domains

TL;DR

This paper introduces practical, mathematically sound distance functions for path planning in planar domains, ensuring efficient computation and reliable pathfinding via greedy routing on sampled networks.

Contribution

It develops a method to define coordinate-based distance functions using harmonic measure and f-divergences, applicable to both continuous domains and discrete sampled networks.

Findings

Distance functions guarantee unique minima for path planning.

Greedy routing on sampled networks reliably finds paths between sites.

Networks are often close to planar with dense sampling.

Abstract

Path planning is an important problem in robotics. One way to plan a path between two points $x,y$ within a (not necessarily simply-connected) planar domain $\Omega$, is to define a non-negative distance function $d(x,y)$ on $\Omega\times\Omega$ such that following the (descending) gradient of this distance function traces such a path. This presents two equally important challenges: A mathematical challenge -- to define $d$ such that $d(x,y)$ has a single minimum for any fixed $y$ (and this is when $x=y$), since a local minimum is in effect a "dead end", A computational challenge -- to define $d$ such that it may be computed efficiently. In this paper, given a description of $\Omega$, we show how to assign coordinates to each point of $\Omega$ and define a family of distance functions between points using these coordinates, such that both the mathematical and the computational challenges are met. This is done using the concepts of \emph{harmonic measure} and \emph{$f$-divergences}. In practice, path planning is done on a discrete network defined on a finite set of \emph{sites} sampled from $\Omega$, so any method that works well on the continuous domain must be adapted so that it still works well on the discrete domain. Given a set of sites sampled from $\Omega$, we show how to define a network connecting these sites such that a \emph{greedy routing} algorithm (which is the discrete equivalent of continuous gradient descent) based on the distance function mentioned above is guaranteed to generate a path in the network between any two such sites. In many cases, this network is close to a (desirable) planar graph, especially if the set of sites is dense.