16 research outputs found
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
Distance constraints solved geometrically
International Symposium on Advances in Robot Kinematics (ARK), 2004, Sestri Levante (Italia)Most geometric constraint problems can be reduced to give coordinates to a set of points from a subset of their pairwise distances. By exploiting this fact, this paper presents an algorithm that solves distance constraint systems by iteratively reducing and expanding the dimension of the problem. In general, these projection/backprojection iterations permit tightening the ranges for the possible solutions but, if at a given point no progress is made, the algorithm bisects the search space and proceeds recursively for both subproblems. This branch-and-prune strategy is shown to converge to all solutions.Peer Reviewe