Sequential motion planning of non-colliding particles in Euclidean spaces

TL;DR

This paper characterizes the topological complexity of sequential motion planning for multiple non-colliding particles in Euclidean spaces, showing it matches the complexity with stationary obstacles, revealing fundamental insights into collision avoidance.

Contribution

It provides a complete topological description of motion planning complexity for non-colliding particles amidst moving obstacles, extending previous static obstacle results.

Findings

Complexity matches that of static obstacle problems.

Topological instabilities are fully characterized.

The Isotopy Extension Theorem plays a key role in the analysis.

Abstract

In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system consisting of non-colliding autonomous entities performing tasks in space whilst avoiding collisions with several moving obstacles. The Isotopy Extension Theorem from manifold topology implies, somewhat surprisingly, that the complexity of this problem coincides with the complexity of the corresponding problem in which the obstacles are stationary.