A Discrete Topological Complexity of Discrete Motion Planning

TL;DR

This paper introduces a new discrete topological complexity concept for robot motion planning, extending homotopy theory to discrete settings and showing its dependence on the space's homotopy type.

Contribution

It generalizes discrete homotopy to (s, r)-homotopy and defines a discrete topological complexity for robots, linking it to the space's homotopy properties.

Findings

Discrete motion planning exists iff the space is discrete contractible.

Discrete topological complexity depends only on the homotopy type.

The approach reduces the number of motion planning strategies needed.

Abstract

In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning, called discrete topological complexity, required for these robots is reduced. Then we prove some properties of discrete topological complexity; For instance, we show that a discrete motion planning in a metric space X exists if and only if X is a discrete contractible space. Also, we prove that the discrete topological complexity depends only on the strictly discrete homotopy type of spaces.