Assur decompositions of direction-length frameworks

Nixon, Anthony (2020) Assur decompositions of direction-length frameworks. In: Graphs and Combinatorial Optimization : from Theory to Applications : CTW2020 Proceedings. AIRO Springer . Springer, Cham, pp. 131-143. ISBN 9783030630713

Text (NixonAssurDirectionLength)
NixonAssurDirectionLength.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial.
Download (146kB)

Abstract

A bar-joint framework is a realisation of a graph consisting of stiff bars linked by universal joints. The framework is rigid if the only bar-length preserving continuous motions of the joints arise from isometries. A rigid framework is isostatic if deleting any single edge results in a flexible framework. Generically, rigidity depends only on the graph and we say an Assur graph is a pinned isostatic graph with no proper pinned isostatic subgraphs. Any pinned isostatic graph can be decomposed into Assur components which may be of use for mechanical engineers in decomposing mechanisms for simpler analysis and synthesis. A direction-length framework is a generalisation of bar-joint framework where some distance constraints are replaced by direction constraints. We initiate a theory of Assur graphs and Assur decompositions for direction-length frameworks using graph orientations and spanning trees and then analyse choices of pinning set.