Abstract

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.

Highlights

  • Assur decompositions of mechanisms date back to the work of the engineer, Leonid Assur [1], as a tool to simplify the analysis and synthesis of mechanisms

  • We introduce the basic terminology for symmetric frameworks and summarise the key results concerning “symmetry-forced” rigidity of frameworks

  • For a symmetry group Cn which acts freely on the inner vertices of a pinned Cn -symmetric graph Ĝ, but not freely on the pinned vertices of Ĝ, we obtain the necessary condition that every subgraph H 0 = (I 0, P 0 ; E 0 ) of the quotient Cn -gain graph Ĥ, which contains the pinned vertex that is fixed by Cn, must satisfy |E 0 | ≤ 2|I 0 | − 1

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Summary

Introduction

Assur decompositions of mechanisms date back to the work of the engineer, Leonid Assur [1], as a tool to simplify the analysis and synthesis of mechanisms. The graph of this extended framework, or symmetric scheme, is generically redundant, but in the symmetry analysis of the associated orbit matrix, this is minimally rigid for symmetric motions, or S-isostatic. In those cases, only the S-Assur decomposition is possible, and this new decomposition of the underlying graph provides additional insight for analysis and synthesis. In the final section, we outline some extensions of our work based on the philosophy of applying the decomposition techniques to constraint systems that generate square matrices (Section 7)

Pinned Frameworks
Assur Decompositions
Symmetric Graphs
Symmetric Frameworks and Orbit Matrices
Basic Definitions
Counting Conditions for Pinned S-isostatic Graphs
Decomposing Pinned S-Isostatic Graphs
Pinned Symmetric Frameworks
S-Drivers and Strongly S-Assur Graphs
S-Assur Graphs Which Are Pinned-Isostatic
Mapping between Gain Graphs and Covering Graphs
Subgroup Assur Decompositions
S-Assur Graphs Which Are Not Pinned Isostatic at S-Regular Configurations
S-Assur Graphs Which Are Redundant and Rigid at S-Regular Realisations
S-Isostatic Graphs with Combined Components
Extensions to “Anti-Symmetric” Orbit Matrices
Extensions to Matrices for Other Constraint Systems
Inductive Constructions

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