The graph realization problem is that of computing the relative locations of a set of vertices placed in Euclidean space, relying only upon some set of inter-vertex distance measurements. This paper is concerned with the closely related problem of determining whether or not a graph has a unique realization. Both these problems are NP-hard, but the proofs rely upon special combinations of edge lengths. If one assumes the vertex locations are unrelated, then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths. This paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension. Efficient sequential and NC algorithms are presented for each condition, although these algorithms have very different flavors in different dimensions.

Conditions for unique graph realizations

This article sets out the rudiments of a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles. The theory rests on ideas of rigid graph theory, some but not all of which are old. The theory, however, has some gaps in it, and their elimination would help in applications. Some of the gaps in the relevant graph theory are as follows. First, there is as yet no analogue for three-dimensional graphs of Laman's theorem, which provides a combinatorial criterion for rigidity in two-dimensional graphs. Second, for three-dimensional graphs there is no analogue of the two-dimensional Henneberg construction for growing or deconstructing minimally rigid graphs although there are conjectures. Third, global rigidity can easily be characterized for two-dimensional graphs, but not for three-dimensional graphs.

Rigid graph control architectures for autonomous formations

This article considers the design of a formation control for multivehicle systems that uses only local information. The control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation. A potential function is obtained from the graph, from which a gradient control is derived. Under this controller the target formation becomes a manifold of equilibria for the multivehicle system. It is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold. A complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well. Finally, the controller is validated experimentally.

Stabilisation of infinitesimally rigid formations of multi-robot networks

This review presents the potential that lattice ~or spring network! models hold for micromechanics applications. The models have their origin in the atomistic representations of matter on one hand, and in the truss-type systems in engineering on the other. The paper evolves by first giving a rather detailed presentation of one-dimensional and planar lattice models for classical continua. This is followed by a section on applications in mechanics of composites and key computational aspects. We then return to planar lattice models made of beams, which are a discrete counterpart of non-classical continua. The final two sections of the paper are devoted to issues of connectivity and rigidity of networks, and lattices of disordered ~rather than periodic! topology. Spring network models offer an attractive alternative to finite element analyses of planar systems ranging from metals, composites, ceramics and polymers to functionally graded and granular materials, whereby a fiber network model of paper is treated in considerable detail. This review article contains 81 references. @DOI: 10.1115/1.1432990#

Lattice models in micromechanics

Abstract Suppose a finite configuration of labeled
points p = (p1,. . . ,pn) in Ed is given along with
certain pairs of those points determined by a graph G such that the
coordinates of the points of p are generic, i.e., algebraically
independent over the integers. If another corresponding configuration
q = (q1,. . . ,qn) in Ed is given such that the
corresponding edges of G for p and q have the same
length, we provide a sufficient condition to ensure that p and
q are congruent in Ed. This condition, together with recent 
results of Jackson and Jordán, give necessary and sufficient conditions for 
a graph being generically globally rigid in the plane.

On Generic Global Rigidity.

We regard a graph G as a set { 1,...,v) together with a nonempty set E of two-element subsets of (1, . . ., v). Letp = (Pi, . . . &) be an element of RIv representing v points in RI. Consider the figure G(p) in RI consisting of the line segments [pi, pj in RI for {i,j) E E. The figure G (p) is said to be rigid in RI if every continuous path in RI', beginning atp and preserving the edge lengths of G(p), terminates at a point q E RIV which is the image (Tp1, . . ., Tpv) of p under an isometry T of RW. Otherwise, G (p) is flexible in RW. Our main result establishes a formula for determining whether G (p) is rigid in RI for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in R3, and other related matters.

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L. Asimow

Papers

The rigidity of graphs