Walter Whiteley

A Theory of Network Localization

In this paper, we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly

Rigidity, computation, and randomization in network localization

We provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

The union of matroids and the rigidity of frameworks

From the pattern of its rigidity matrix, we show that a k-frame on a graph (or multigraph) has the matroid structure of the union of k copies of the cycle matroid of the graph. This matrix pattern is applied to three central results about the rigidity of frameworks. An immediate corollary of this matroid union is a characterization of rigid bar and body frameworks in n-space (Tay’s Theorem). This is further specialized to characterize the independence and the rigidity of body and hinge structures in n-space (a new theorem). The two-frame, or union of two copies of the graphic matroid, is truncated to produce plane bar and joint frameworks giving a characterization of minimal infinitesimally rigid bar and joint frameworks in the plane (Laman’s Theorem). Finally, these techniques are used to characterize the graphs of infinitesimally rigid frameworks on other surfaces, such as the flat torus, the cylinder, cones, etc., using matroid unions of cycle and bicycle matroids of the graph.

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

THE ALGEBRAIC GEOMETRY OF STRESSES IN FRAMEWORKS

A bar-and-joint framework, with rigid bars and flexible joints, is said to be generically isostatic if it has just enough bars to be infinitesimally rigid in some realization in Euclidean n-space. We determine the equation that must be satisfied by the coordinates of the joints in a given realization in order to have a nonzero stress, and hence an infinitesimal motion, in the framework. This equation, called the pure condition, is expressed in terms of certain determinants, called brackets. The pure condition is obtained by choosing a way to tie down the framework to eliminate the Euclidean motions, computing a bracket expression by a method due to Rosenberg and then factoring out part of the expression related to the tie-down. A major portion of this paper is devoted to proving that the resulting pure condition is independent of the tie-down chosen. We then catalog a number of small examples and their pure conditions, along with the geometric conditions for the existence of a stress which are equivalent ...

Graphical properties of easily localizable sensor networks

The sensor network localization problem is one of determining the Euclidean positions of all sensors in a network given knowledge of the Euclidean positions of some, and knowledge of a number of inter-sensor distances. This paper identifies graphical properties which can ensure unique localizability, and further sets of properties which can ensure not only unique localizability but also provide guarantees on the associated computational complexity, which can even be linear in the number of sensors on occasions. Sensor networks with minimal connectedness properties in which sensor transmit powers can be increased to increase the sensing radius lend themselves to the acquiring of the needed graphical properties. Results are presented for networks in both two and three dimensions.

The algebraic geometry of motions of bar-and-body frameworks

This paper generalizes and extends previous results on bar-and-joint frameworks to bar-and-body frameworks: structures formed by rigid bodies in space linked by rigid bars and universal joints. For a multi-graph which can form an isostatic (minimal infinitesimally rigid) bar-and-body framework, a single polynomial—the pure condition—is found which describes those bad positions of the bars for which infinitesimal rigidity fails. (The proof is much shorter than the previous derivation for bar-and-joint frameworks and the condition is linear in the variables.) The pure condition is used to describe the infinitesimal motions of a 1-underbraced framework in terms of the screw centers of motion of the bodies. The factoring of the polynomial condition is given by the lattice of isostatic blocks in the framework, with at most one irreducible factor for each block. For frameworks realized at generic points of an irreducible factor the infinitesimal motions and the static stresses are also given by the factoring an...

Sensor and network topologies of formations with direction, bearing, and angle information between agents

Sensor and network topologies of formations of autonomous agents are considered. The aim of the paper is to suggest an approach for such topologies for formations with direction, bearing and angle information between agents in the plane and in 3-space. A number of results are translated from prior work in this field and in the study of constraints in CAD programming, in rigidity theory, in structural engineering and in discrete mathematics. Some new results are presented both for the plane and for 3-space. A number of unsolved problems are also mentioned.

Information structures to secure control of rigid formations with leader-follower architecture

This paper is concerned with rigid formations of mobile autonomous agents that have leader-follower architecture. In a previous paper, Baillieul and Suri gave a proposition as a necessary condition for stable rigidity. They also gave a separate theorem as a sufficient condition for stable rigidity. This paper suggests an approach to analyze rigid formations that have leader-follower architecture. It proves that the third condition in the proposition given by Baillieul and Suri is redundant, and it proves that this proposition is a necessary and sufficient condition for stable rigidity. Simulation results are also presented to illustrate rigidity.

Constraining Plane Configurations in Computer-Aided Design: Combinatorics of Directions and Lengths

Configurations of points in the plane constrained by directions only or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph with doubled edges to describe the combinatorial properties of direction-length designs.