# Proximity Queries for Absolutely Continuous Parametric Curves

## TL;DR

This paper introduces efficient methods for proximity queries of parametric curves in motion planning, enabling fast collision detection and distance computation crucial for autonomous robot navigation.

## Contribution

It provides a novel approach to compute bounds on obstacle proximity for a broad class of curves using convex bounding techniques and closed-form arc length bounds.

## Key findings

- Methods are computationally efficient and accurate.
- Applicable to curves with trigonometric or polynomial bases.
- Demonstrated effectiveness through numerical simulations.

## Abstract

In motion planning problems for autonomous robots, such as self-driving cars, the robot must ensure that its planned path is not in close proximity to obstacles in the environment. However, the problem of evaluating the proximity is generally non-convex and serves as a significant computational bottleneck for motion planning algorithms. In this paper, we present methods for a general class of absolutely continuous parametric curves to compute: (i) the minimum separating distance, (ii) tolerance verification, and (iii) collision detection. Our methods efficiently compute bounds on obstacle proximity by bounding the curve in a convex region. This bound is based on an upper bound on the curve arc length that can be expressed in closed form for a useful class of parametric curves including curves with trigonometric or polynomial bases. We demonstrate the computational efficiency and accuracy of our approach through numerical simulations of several proximity problems.