Collision free motion planning on a wedge of circles

TL;DR

This paper presents an optimal collision-free motion planning algorithm for two robots on a wedge-shaped track of three circles, demonstrating that the topological complexity of the configuration space is 3.

Contribution

It introduces a continuous motion planning algorithm that is proven to be optimal with a topological complexity of 3 for this specific problem.

Findings

Topological complexity of the configuration space is 3.

The algorithm requires exactly 3 continuous instructions.

The motion planning method is optimal for the given problem.

Abstract

We exhibit an algorithm with continuous instructions for two robots moving without collisions on a track shaped as a wedge of three circles. We show that the topological complexity of the configuration space associated with this problem is 3. The topological complexity is a homotopy invariant that can be thought of as the minimum number of continuous instructions required to describe the movement of the robots between any initial configuration to any final one without collisions. The algorithm presented is optimal in the sense that it requires exactly 3 continuous instructions.