Does a robot path have clearance c?

TL;DR

This paper develops a data structure for efficiently verifying if a given polygonal path maintains a specified clearance from polygonal obstacles, optimizing query times especially for paths with few segments.

Contribution

It introduces a novel preprocessing method to quickly determine obstacle proximity to a path segment, improving efficiency for small-segment paths in polygonal obstacle environments.

Findings

Achieves $O(t ext{ log } n)$ preprocessing time and $O((n / \sqrt{t}) ext{ log }^{7/2} n)$ query time.

Provides a scalable solution for multiple obstacle proximity queries with adjustable parameter t.

Significantly improves query efficiency for paths with few segments compared to existing methods.

Abstract

Most path planning problems among polygonal obstacles ask to find a path that avoids the obstacles and is optimal with respect to some measure or a combination of measures, for example an $u$-to-$v$ shortest path of clearance at least $c$, where $u$ and $v$ are points in the free space and $c$ is a positive constant. In practical applications, such as emergency interventions/evacuations and medical treatment planning, a number of $u$-to-$v$ paths are suggested by experts and the question is whether such paths satisfy specific requirements, such as a given clearance from the obstacles. We address the following path query problem: Given a set $S$ of $m$ disjoint simple polygons in the plane, with a total of $n$ vertices, preprocess them so that for a query consisting of a positive constant $c$ and a simple polygonal path $\pi$ with $k$ vertices, from a point $u$ to a point $v$ in free space, where $k$ is much smaller than $n$, one can quickly decide whether $\pi$ has clearance at least $c$ (that is, there is no polygonal obstacle within distance $c$ of $\pi$). To do so, we show how to solve the following related problem: Given a set $S$ of $m$ simple polygons in $\Re^{2}$, preprocess $S$ into a data structure so that the polygon in $S$ closest to a query line segment $s$ can be reported quickly. We present an $O(t \log n)$ time, $O(t)$ space preprocessing, $O((n / \sqrt{t}) \log ^{7/2} n)$ query time solution for this problem, for any $n ^{1 + \epsilon} \leq t \leq n^{2}$. For a path with $k$ segments, this results in $O((n k / \sqrt{t}) \log ^{7/2} n)$ query time, which is a significant improvement over algorithms that can be derived from existing computational geometry methods when $k$ is small.