Collision-free spatial motion of rigid bodies via topological complexity

TL;DR

This paper computes the topological complexity of collision-free configurations of multiple rigid bodies in 2D and 3D, and introduces practical, optimal motion planning algorithms for collision avoidance.

Contribution

It provides the first calculation of topological complexity for these configuration spaces and develops implementable algorithms for collision-free motion planning.

Findings

Calculated topological complexity for rigid bodies in 2D and 3D.

Developed optimal, practical motion planning algorithms.

Algorithms are easily implementable in real systems.

Abstract

The Topological complexity a la Farber $\text{TC}(-)$ is a homotopy invariant which have interesting applications in Robotics, specifically, in the robot motion planning problem. In this work we calculate the topological complexity of the configuration space of $k$ distinct rigid bodies without collisions in $\mathbb{R}^d$, for $d=2,3$. Furthermore, we present optimal algorithms which can be used in designing practical systems controlling motion of many rigid bodies moving in space without collisions. The motion planning algorithms we present in this work are easily implementable in practice.