The algebraic geometry of motions of bar-and-body frameworks
SIAM Journal on Algebraic and Discrete Methods
Algebraic solution for geometry from dimensional constraints
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
Decomposition plans for geometric constraint systems, part I: performance measures for CAD
Journal of Symbolic Computation
Decomposition plans for geometric constraint problems, part II: new algorithms
Journal of Symbolic Computation
Combinatorial characterization of the Assur graphs from engineering
European Journal of Combinatorics
Geometric properties of Assur graphs
European Journal of Combinatorics
Relative centers of motion, implicit bars and dead-center positions for planar mechanisms
European Journal of Combinatorics
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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 d-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently into minimal pinned isostatic graphs, which we call d-Assur graphs. We also study key properties of motions induced by removing an edge in a d-Assur graph - defining a sharper subclass of strongly d-Assur graphs by the property that all inner vertices go into motion, for each removed edge. The strongly 3-Assur graphs are the central building blocks for kinematic linkages in 3-space and the 3-Assur graphs are components in the analysis of built linkages. The d-Assur graphs 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 modular information for extending the motion induced by inserting one driver in a bottom Assur linkage to the joints of the entire linkage. We also highlight some problems in combinatorial rigidity in higher dimensions (d=3) which cause the distinction between d-Assur and strongly d-Assur which did not occur in the plane.