Directed graphs, decompositions, and spatial linkages

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 dd-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 dd-Assur graphs. We also study key properties of motions induced by removing an edge in a dd-Assur graph — defining a sharper subclass of strongly dd-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 dd-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≥3d≥3) which cause the distinction between dd-Assur and strongly dd-Assur which did not occur in the plane.