Extremal Configurations of Hinge Structures

TL;DR

This paper investigates the geometric properties of hinge structures in various dimensions, characterizing maximal configurations and their applications in robotics and molecular modeling.

Contribution

It provides a novel analysis of body-and-hinge chains, identifying conditions for maximal configurations and their geometric simplicity in arbitrary dimensions.

Findings

Maximal configurations correspond to critical points of a Morse-Bott function.

In 3D, these configurations have applications in robotics and molecular structures.

The squared distance function serves as a Morse-Bott function on the configuration space.

Abstract

We study body-and-hinge and panel-and-hinge chains in R^d, with two marked points: one on the first body, the other on the last. For a general chain, the squared distance between the marked points gives a Morse-Bott function on a torus configuration space. Maximal configurations, when the distance between the two marked points reaches a global maximum, have particularly simple geometrical characterizations. The three-dimensional case is relevant for applications to robotics and molecular structures.