Parametrized topological complexity of collision-free motion planning in the plane

TL;DR

This paper investigates the parametrized topological complexity of collision-free motion planning for multiple points in the plane, considering unknown obstacle positions, and extends prior work from 3D to 2D using specialized algebraic and topological methods.

Contribution

It analyzes the parameterized motion planning problem in the plane, providing new insights and tools distinct from the spatial case, and advances understanding of obstacle avoidance in 2D.

Findings

Characterization of topological complexity for planar collision-free motion

Development of algebraic and topological tools for 2D case

Extension of prior 3D results to planar scenarios

Abstract

Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety of external conditions, which are viewed as parameters and form part of the input of the underlying motion planning problem. In this paper, we analyze the parameterized motion planning problem for the motion of many distinct points in the plane, moving without collision and avoiding multiple distinct obstacles with a priori unknown positions. This complements our prior work [arXiv:2009.06023], where parameterized motion planning algorithms were introduced, and the obstacle-avoiding collision-free motion planning problem in three-dimensional space was fully investigated. The planar case requires different algebraic and topological tools than its spatial analog.