The rigidity of graphs, II
01 Mar 1979-Journal of Mathematical Analysis and Applications (Academic Press)-Vol. 68, Iss: 1, pp 171-190
TL;DR: In this article, a graph G is viewed as a set {1,…, v} together with a nonempty set E of two-element subsets of { 1,.., v}.
About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 1979-03-01 and is currently open access. It has received 319 citations till now.
Citations
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TL;DR: A survey of formation control of multi-agent systems focuses on the sensing capability and the interaction topology of agents, and categorizes the existing results into position-, displacement-, and distance-based control.
1,751 citations
TL;DR: It is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold of the multivehicle system.
Abstract: 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.
511 citations
Cites background from "The rigidity of graphs, II"
...The definitions below are taken from Asimow and Roth (1979)....
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...Theorem 1 (Asimow and Roth 1979): A framework (G, p) is infinitesimally rigid if and only if (G, p) is rigid and p is a regular point....
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...Definition 4 (Asimow and Roth 1979): A framework (G, p) is infinitesimally rigid in the plane if dimðKerJgG ð pÞÞ 1⁄4 3, or equivalently if rankJgG ð pÞ 1⁄4 2n 3:...
[...]
...Definition 4 (Asimow and Roth 1979): A framework (G, p) is infinitesimally rigid in the plane if dimðKerJgG ð pÞÞ ¼ 3, or equivalently if rankJgG ð pÞ ¼ 2n 3: If a framework is infinitesimally rigid, then it is also rigid....
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01 Jan 1990
TL;DR: 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.
Abstract: 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.
357 citations
TL;DR: Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance as mentioned in this paper, which is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidian space realizing those given distances.
Abstract: Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidean space realizing those given distances. We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis on molecular conformation problems.
345 citations
TL;DR: In this paper, the authors give necessary and sufficient conditions for a graph being generically globally rigid in the plane, i.e., algebraically independent over the integers, with respect to a finite configuration of labeled points p = (p1,...,pn) in Ed.
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 Jordan, give necessary and sufficient conditions for a graph being generically globally rigid in the plane.
343 citations
References
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TL;DR: In this paper, the combinatorial properties of rigid plane skeletal structures are investigated, and the properties are found to be adequately described by a class of graph-structured graphs.
Abstract: In this paper the combinatorial properties of rigid plane skeletal structures are investigated. Those properties are found to be adequately described by a class of graphs.
1,117 citations
TL;DR: In this paper, the authors considered the problem of determining whether a graph G(p) is rigid in RIV with respect to the line segments of a continuous path in RI.
Abstract: 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.
566 citations
01 Jan 1975
314 citations
TL;DR: In this article, the authors give a counterexample of a closed polyhedral surface (topologically a sphere), embedded in three-space, which flexes, where the singular points look like two dihedral surfaces that intersect at just one point in their edges.
Abstract: Are closed surfaces rigid? The conjecture that in fact they all are r i g i d a t least for polyhedra has been with us a long time. We propose here to give a counterexample. This is a closed polyhedral surface (topologically a sphere), embedded in three-space, which flexes. (See Gluck [5] for definitions and some references for the history of the problem.) Certain ambiguities arising from definition io of the eleventh book of Euclid's Elements have led many to conjecture the rigidity of closed surfaces. In 1813 Cauchy [2] proved that strictly convex surfaces were rigid, and this result is the basic tool for many other rigidity theorems. Recently Gluck [5] has shown that almost all simply connected closed surfaces are rigid. On the other hand we have shown [3] that there are immersed surfaces which flex. The ideas in [3] are part of the motivation behind the example described here. The first step is to find an example of an immersed flexible sphere that is not only immersed but has just two singular points in its image. Locally the singular points look like two dihedral surfaces that intersect at just one point in their edges. The next step is to alter the polyhedron only in the neighborhood of these singular points in such a way that the dihedral surfaces flex as before, but one dihedral surface is " crinkled " such that near the intersection point it is pushed in. When this is done the resulting polyhedron still flexes, but the singular points have been erased; no new ones have been created, so it is embedded.
150 citations
01 Jan 1969
TL;DR: In this paper, two relatively simple proofs for the famous theorem of Steinitz on the characterization of graphs of 3-polytopes are presented, together with some new results on the structure of 3connected graphs.
Abstract: : Two relatively simple proofs for the famous theorem of Steinitz on the characterization of graphs of 3-polytopes are presented, together with some new results on the structure of 3-connected graphs. (Author)
93 citations