On bar frameworks, stress matrices and semidefinite programming

A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V,E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p in R. Given a framework G(p) in R, a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||q−q || = ||p−p || for all (i, j) ∈ E. This problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems. In this paper, we show that semidefinite programming (SDP) can be effectively used to address the universal rigidity problem. In particular, we use the notion of non-degeneracy of SDP to obtain a sufficient condition for universal rigidity, and to re-derive the known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S. E-mail: [email protected] Research supported by the Natural Sciences and Engineering Research Council of Canada.