Complexity of the forward kinematic map

TL;DR

This paper introduces a new method to analyze the complexity of robot arm designs by quantifying inherent discontinuities in the forward kinematic map, aiding in understanding motion planning challenges near singularities.

Contribution

It defines the complexity of the forward kinematic map, explores its properties, and provides explicit computations for common robot configurations, advancing qualitative analysis of robot arm design.

Findings

Complexity measures inherent discontinuities in robot kinematic maps.

Explicit computations for singularities in common robot configurations.

Insights into motion planning instabilities near singular points.

Abstract

The main objective of this paper is to introduce a new method for qualitative analysis of various designs of robot arms. To this end we define the complexity of a map, examine its main properties and develop some methods of computation. In particular, when applied to a forward kinematic map associated to some robot arm structure, the complexity measures the inherent discontinuities that arise when one attempts to solve the motion planning problem for any set of input data. In the second part of the paper, we consider instabilities of motion planning in the proximity of singular points, and present explicit computations for several common robot arm configurations.