A Decomposition-Based Approach to Reasoning about Free Space Path-Connectivity for Rigid Objects in 2D

TL;DR

This paper introduces a decomposition-based method to efficiently analyze the free space for rigid objects in 2D, enabling collision detection, caging verification, and free space characterization with fast computation.

Contribution

It presents a novel approach that decomposes the configuration space into 2D slices and uses alpha-complexes for efficient approximation of free space connectivity.

Findings

Achieves collision and caging analysis in under 2 seconds

Provides conservative approximations of free space components

Effectively characterizes narrow passages and free space volume

Abstract

In this paper, we compute a conservative approximation of the path-connected components of the free space of a rigid object in a 2D workspace in order to solve two closely related problems: to determine whether there exists a collision-free path between two given configurations, and to verify whether an object can escape arbitrarily far from its initial configuration -- i.e., whether the object is caged. Furthermore, we consider two quantitative characteristics of the free space: the volume of path-connected components and the width of narrow passages. To address these problems, we decompose the configuration space into a set of two-dimensional slices, approximate them as two-dimensional alpha-complexes, and then study the relations between them. This significantly reduces the computational complexity compared to a direct approximation of the free space. We implement our algorithm and run experiments in a three-dimensional configuration space of a simple object showing runtime of less than 2 seconds.