On the Targeted Complexity of a Map

TL;DR

This paper investigates the topological complexity of work maps in robotics, aiming to optimize motion planning by analyzing properties and inequalities of targeted complexity related to configuration and workspace spaces.

Contribution

It introduces the concept of targeted complexity for work maps, explores its properties, and compares it with existing notions like relative topological complexity, providing new insights for motion planning optimization.

Findings

Targeted complexity is homotopically invariant.

Inequalities help reduce the number of motion planners.

Relative topological complexity is a special case of targeted complexity.

Abstract

We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion planners for work maps. In this regard, we focus on the useful set of works. We check some basic properties of the targeted complexity of maps, such as homotopical invariance, reduction, the product of maps, and so on. Then we compare these targeted complexities, and we find some inequalities in reducing the number of motion planners. We show that the relative topological complexity of pair of spaces defined by Short is a special case of the targeted complexity of work maps.