Geometric Properties of Assur Graphs

TL;DR

This paper investigates the geometric characteristics of Assur graphs, focusing on singular configurations with self-stresses, and introduces a new geometric characterization linked to dead-end positions in mechanical linkages.

Contribution

It offers a novel geometric characterization of Assur graphs based on singular realizations, enhancing understanding of their static and kinematic properties.

Findings

Identifies special singular realizations with self-stresses

Relates singular positions to dead-end mechanism configurations

Provides a new geometric perspective on Assur graphs

Abstract

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks - Assur graphs - which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam.