Combinatorial characterization of the Assur graphs from engineering

AbstractWe introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This 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 the 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

One-Dimensional Synthesis of Graphs as Tensegrity Frameworks 1 Abstract

The edge set of a graph G is partitioned into two subsets EC∪ES. A tensegrity framework with underlying graph G and with cables for EC and struts for ES is proved to be rigidly embedable into a 1-dimensional line if and only if G is 2-edge-connected and every 2-vertex-connected component of G intersects both EC and ES. Polynomial algorithms are given to find an embedding of such graphs and to check the rigidity of a given 1-dimensional embedding.

Combinatorial synthesis approach employing graph networks

The paper proposes a methodology to assist the designer at the initial stages of the design synthesis process by enabling him/her to employ knowledge and algorithms existing in graph network theory. The proposed method comprises three main stages: transforming the synthesis problem into a graph theoretic problem; devising the topology possessing special engineering properties corresponding to the system requirements; finding the geometric configuration of that topology that will possess the desired properties. To clarify the idea and to demonstrate its generality, the approach is presented through three synthesis case studies from different engineering domains: electrical networks, statics and kinematics. As is highlighted in the paper, the approach of employing graph theory in the synthesis process offers several unique advantages. Among these advantages are: gaining a general perspective on different synthesis problems from different engineering domains by transforming them into the same graph problem; employing the same graph algorithms for different synthesis problems; establishing the existence of configurations with special properties solely from the topology of the system; transferring knowledge and methods between different engineering disciplines for both the topology and the geometry generation steps