Critical configurations of planar robot arms

TL;DR

This paper extends the understanding of critical configurations from cyclic polygons to open polygonal chains (robot arms), characterizing critical points and providing formulas for their Morse index.

Contribution

It introduces the concept of oriented area for open chains, identifies critical configurations as cyclic with antipodal endpoints, and derives the Morse index formula.

Findings

Critical points of open chains are cyclic with antipodal endpoints.

A formula for the Morse index of critical configurations is established.

The results generalize known properties from polygons to robot arms.

Abstract

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.