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

A 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 $\text{n(n + 1)}\text{2}$ edge disjoint spanning trees. Some techniques for generating rigid linkages are also discussed.