# A Homotopy Invariant Based on Convex Dissection Topology and a Distance Optimal Path Planning Algorithm

## TL;DR

This paper introduces a new homotopy invariant based on convex dissection topology for 2D path planning, enabling efficient encoding of path classes and an algorithm that actively explores and finds optimal paths within these classes.

## Contribution

It develops a novel homotopy invariant for 2D spaces and proposes the CDT-RRT* algorithm that effectively explores homotopy classes for optimal path planning.

## Key findings

- The proposed algorithm performs comparably to state-of-the-art methods.
- The homotopy invariant efficiently encodes all path classes.
- CDT-RRT* actively explores unknown homotopy classes.

## Abstract

The concept of path homotopy has received widely attention in the field of path planning in recent years. In this article, a homotopy invariant based on convex dissection for a two-dimensional bounded Euclidean space is developed, which can efficiently encode all homotopy path classes between any two points. Thereafter, the optimal path planning task consists of two steps: (i) search for the homotopy path class that may contain the optimal path, and (ii) obtain the shortest homotopy path in this class. Furthermore, an optimal path planning algorithm called CDT-RRT* (Rapidly-exploring Random Tree Star based on Convex Division Topology) is proposed. We designed an efficient sampling formula for CDT-RRT*, which gives it a tendency to actively explore unknown homotopy classes, and incorporated the principles of the Elastic Band algorithm to obtain the shortest path in each class. Through a series of experiments, it was determined that the performance of the proposed algorithm is comparable with state-of-the-art path planning algorithms. Hence, the application significance of the developed homotopy invariant in the field of path planning was verified.