# Topological complexity of the work map

## TL;DR

This paper introduces the topological complexity of the work map in robotics, a homotopy invariant that measures the complexity of algorithms controlling both the robot's configuration space and its task.

## Contribution

It generalizes the classical topological complexity by defining a new invariant for maps, providing a broader framework for analyzing robotic control complexity.

## Key findings

- Defines the topological complexity of the work map
- Generalizes classical topological complexity to maps
- Provides a homotopy invariant for robot control analysis

## Abstract

We introduce the topological complexity of the work map associated to a robot system. In broad terms, this measures the complexity of any algorithm controlling, not just the motion of the configuration space of the given system, but the task for which the system has been designed. From a purely topological point of view, this is a homotopy invariant of a map which generalizes the classical topological complexity of a space.