Category and Topological Complexity of the configuration space $F(G\times \mathbb{R}^n,2)$

TL;DR

This paper computes the Lusternik-Schnirelmann category and topological complexity of the configuration space of two points in a product space, with applications to robot motion planning for rigid bodies in 2D and 3D.

Contribution

It provides explicit calculations of topological invariants for configuration spaces of two points in product spaces, extending their application to motion planning problems.

Findings

Calculated the Lusternik-Schnirelmann category for specific configuration spaces.

Determined the topological complexity for configuration spaces relevant to robot motion.

Applied the results to planar and spatial motion of rigid bodies.

Abstract

The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times\mathbb{R}^n$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.