Combinatorial Characterization of the Assur Graphs from Engineering

TL;DR

This paper bridges kinematic linkage concepts with rigidity theory by characterizing Assur graphs combinatorially, confirming existing conjectures, and providing new algorithms for analyzing minimal linkages.

Contribution

It translates Assur graphs into the language of rigidity theory, offers mathematical characterizations, and confirms several conjectures with new analytical techniques.

Findings

Assur graphs are characterized by properties in rigidity theory.

Confirmed multiple conjectures regarding Assur graphs.

Developed algorithms for analyzing minimal linkages.

Abstract

We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. The paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from minimal linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work.