A Topologist's View of Kinematic Maps and Manipulation Complexity

TL;DR

This paper explores the topological properties of kinematic maps in robotics, analyzing how geometry influences manipulation complexity and providing a framework to measure the difficulty of planning robust manipulations.

Contribution

It introduces a topological framework for understanding kinematic maps and defines a complexity measure related to manipulation difficulty in robotic devices.

Findings

Topological obstructions affect the existence of continuous inverse kinematic solutions.

The complexity measure quantifies the difficulty of finding robust manipulation plans.

Relations between topological properties and manipulation stability are established.

Abstract

In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss mechanical devices that consist of rigid parts connected by joints and show how the geometry of the joints determines the forward kinematic map that relates the configuration of joints with the pose of the end-effector of the device. We explain how to compute the dimension of the joint space and describe topological obstructions for a kinematic map to be a fibration or to admit a continuous section. In the second part of the paper we define the complexity of a continuous map and show how the concept can be viewed as a measure of the difficulty to find a robust manipulation plan for a given mechanical device. We also derive some basic estimates for the complexity and relate it to the degree of instability of a manipulation plan.