Symmetry Adapted Assur Decompositions

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.

Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

“…Further applications of Theorem 4.4 and other operations are recently discussed in [ 14 , 15 , 22 ].…”

“…The main contributions of this paper are (i) to develop a concise approach to analyze the rigidity of symmetric frameworks based on inductive constructions and (ii) to give the first combinatorial characterization for frameworks with non-cyclic symmetry, which is far more complicated than the cyclic case. After the publication of the technical report [ 8 ] of this paper, our formulation and results on inductive constructions were used for analyzing the infinitesimal rigidity of symmetric frameworks [ 15 , 22 ] and the symmetric-forced rigidity of symmetric frameworks on surfaces [ 14 ]. Also the matroid construction given in Sect.…”

Assur Decompositions of Direction-Length Frameworks

“…Proof. Since the equivalence of (1), (2) and (3) can be proven by adapting the technique used in [11,Theorem 3] Then take the bottom component with its edges to the pinned vertices. In R pin DL (G, p) apply a permutation of rows and a permutation of column vertices to place these rows and columns at the top left of the matrix.…”