Generic global rigidity of body–bar frameworks

Abstract

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid inRd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and (d+1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d=2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d⩾3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d⩾1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d⩾1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.