On bar frameworks, stress matrices and semidefinite programming

Abstract

A bar framework G(p) in r-dimensional Euclidean space is a graph G on the vertices 1, 2, . . . , n, where each vertex i is located at point p i in $${\mathbb{R}^r}$$ . Given a framework G(p) in $${\mathbb{R}^r}$$ , 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 i − q j ||2 = ||p i − p j ||2 for each edge (i, j) of G. 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, semidefinite programming (SDP) theory is used to address, in a unified manner, several problems concerning universal rigidity. New results are presented as well as new proofs of previously known theorems. In particular, we use the notion of SDP non-degeneracy to obtain a sufficient condition for universal rigidity, and we show that this condition yields the previously 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.