# Configuration Spaces and Robot Motion Planning Algorithms

## TL;DR

This paper surveys topological challenges in robot motion planning, analyzes configuration space complexity, and presents optimal algorithms for collision-free multi-particle navigation, including detailed topology of configuration spaces on trees.

## Contribution

It introduces new results on the topology of configuration spaces and develops optimal motion planning algorithms with minimal discontinuities.

## Key findings

- Analysis of topological complexity of configuration spaces
- Explicit optimal motion planning algorithms for particles
- Topology of configuration spaces on trees and cohomology classes

## Abstract

The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible for discontinuities in algorithms for robot navigation. Then we present explicit motion planning algorithms for coordinated collision free control of many particles moving in Euclidean spaces or on graphs. These algorithms are optimal in the sense that they have minimal number of regions of continuity. Moreover, we describe in full detail the topology of configuration spaces of two particles on a tree and use it to construct some top-dimensional cohomology classes in configuration spaces of n particles on a tree.