Topology of parametrised motion planning algorithms

TL;DR

This paper introduces the concept of parametrised topological complexity, a new invariant for motion planning that accounts for external conditions, and computes it for multi-robot obstacle avoidance in 3D space, revealing higher complexity than traditional measures.

Contribution

It defines and analyzes parametrised topological complexity, providing explicit calculations for multi-robot systems with obstacles, highlighting increased complexity compared to nonparametrised cases.

Findings

Parametrised topological complexity can be significantly higher than standard invariant.

Explicit computation for obstacle-avoiding motion of multiple robots in 3D.

Shows the importance of external conditions in motion planning complexity.

Abstract

In this paper we introduce and study a new concept of parametrised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high degree of universality and flexibility, it can function under a variety of external conditions (such as positions of the obstacles etc). We explicitly compute the parameterised topological complexity of obstacle-avoiding collision-free motion of many particles (robots) in 3-dimensional space. Our results show that the parameterised topological complexity can be significantly higher than the standard (nonparametrised) invariant.