Rigidity of multi-graphs. I. Linking rigid bodies in n-space

AbstractA linkage of rigid bodies in n-space consists of a set of n-dimensional rigid bodies in n-space, certain pairs of which are linked together by one or more rigid bars using universal (ball) joints. Such a structure is canonically associated (bodies to vertices, bars to edges) with a multi-graph. We show that a multi-graph can be realized as a rigid linkage of rigid bodies in n-space if and only if it contains n(n + 1)2 edge disjoint spanning trees. Some techniques for generating rigid linkages are also discussed