A Ray Intersection Algorithm for Fast Growth Distance Computation Between Convex Sets

TL;DR

This paper introduces a fast, efficient algorithm for computing the growth distance between convex sets, with applications in robotics, using a novel ray intersection approach on Minkowski differences.

Contribution

The paper presents a new algorithm that reduces growth distance computation to ray intersection problems and demonstrates state-of-the-art performance with open-source code.

Findings

Algorithm achieves state-of-the-art performance in benchmarks.

Open-source implementation demonstrates practical efficiency.

Applicable to motion planning and rigid-body simulation in robotics.

Abstract

In this paper, we discuss an efficient algorithm for computing the growth distance between two compact convex sets with representable support functions. The growth distance between two sets is the minimum scaling factor such that the sets intersect when scaled about some center points. Unlike the minimum distance between sets, the growth distance provides a unified measure for set intersection and separation. We first reduce the growth distance problem to an equivalent ray intersection problem on the Minkowski difference set. Then, we propose an algorithm to solve the ray intersection problem by iteratively constructing inner and outer polyhedral approximations of the Minkowski difference set. We show that our algorithm satisfies several key properties, such as primal and dual feasibility and monotone convergence. We provide extensive benchmark results for our algorithm and show that our open-source implementation achieves state-of-the-art performance across a wide variety of convex sets. Finally, we demonstrate robotics applications of our algorithm in motion planning and rigid-body simulation.