Mathematics – Combinatorics
Scientific paper
2010-10-27
Mathematics
Combinatorics
37 pages, 21 figures
Scientific paper
The decomposition of a system of constraints into small basic components is an important tool of design and analysis. Specifically, the decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned 3-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs) which we call 3-Assur graphs. This analysis extends the Assur decompositions of plane linkages previously studied in the mathematical and the mechanical engineering literature. These 3-Assur graphs are the central building blocks for all kinematic linkages in 3-space. They share a number of key combinatorial and geometric properties with the 2-Assur graphs, including an associated lower block-triangular decomposition of the pinned rigidity matrix which provides a format for extending the motion induced by inserting one driver in a bottom Assur linkage to the joints of the entire linkage in a systemic, modular way. This analysis is the initial paper in a series of papers that present new results in an ongoing investigation of (i) fast algorithms for these decompositions, and (ii) inductive constructions of the basic 3-Assur graphs, as analogs of known results in the plane. This analysis also connects the decompositions of linkages to other combinatorial decompositions of CAD constraints and for modularized analysis of larger engineering systems.
Shai Offer
Sljoka Adnan
Whiteley Walter
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