Generic Rigidity Research Articles

A sufficient connectivity condition for rigidity and global rigidity of linearly constrained frameworks in R2

We study the bar-and-joint frameworks in $\mathbb{R}^2$ such that some vertices are constrained to lie on some lines. The generic rigidity of such frameworks is characterised by Streinu and Theran (2010). Katoh and Tanigawa (2013) remarked that the corresponding matroid and its rank function can be characterised by using a submodular function. In this paper, we will transfer this characterisation of the rank function to the form of the value of a ``1-thin cover" and obtain a sufficient connectivity condition for rigidity and global rigidity of these frameworks analogous to the results of Lov\'asz and Yemini (1982).

Globally rigid graphs are fully reconstructible

AbstractAd-dimensional framework is a pair$(G,p)$, where$G=(V,E)$is a graph andpis a map fromVto$\mathbb {R}^d$. The length of an edge$uv\in E$in$(G,p)$is the distance between$p(u)$and$p(v)$. The framework is said to be globally rigid in$\mathbb {R}^d$if the graphGand its edge lengths uniquely determine$(G,p)$, up to congruence. A graphGis called globally rigid in$\mathbb {R}^d$if everyd-dimensional generic framework$(G,p)$is globally rigid.In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graphGis strongly reconstructible in$\mathbb {C}^d$if the set of (unlabeled) edge lengths of any generic framework$(G,p)$ind-space, along with the number of vertices ofG, uniquely determine bothGand the association between the edges ofGand the set of edge lengths. It is known that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then it is strongly reconstructible in$\mathbb {C}^d$. We strengthen this result and show that, under the same conditions,Gis in fact fully reconstructible in$\mathbb {C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstructG, without any constraint on the number of vertices (although still under the assumption that the edge lengths come from a generic realization).As a key step in our proof, we also prove that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then thed-dimensional generic rigidity matroid ofGis connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

A necessary condition for generic rigidity of bar‐and‐joint frameworks in d‐space

AbstractA graph is ‐sparse if each subset with induces at most edges in . Maxwell showed in 1864 that a necessary condition for a generic bar‐and‐joint framework with at least vertices to be rigid in is that should have a ‐sparse subgraph with edges. This necessary condition is also sufficient when but not when . Cheng and Sitharam strengthened Maxwell's condition by showing that every maximal ‐sparse subgraph of should have edges when . We extend their result to all .

A note on generic rigidity of graphs in higher dimension

The characterization of rigid graphs in Rd is known only in the low dimensional cases (d=1,2) and is a major open problem in higher dimensions. In this note we consider the other extreme case when d is close to n, the number of vertices of the graph. It turns out that there is a fairly simple characterization as long as n−d is at most four. We also characterize globally rigid graphs in this range.

Which graphs are rigid in ell _p^d?

We present three results which support the conjecture that a graph is minimally rigid in d-dimensional ell _p-space, where pin (1,infty ) and pnot =2, if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from ell _p^d to ell _p^{d+1}. We then prove that every (d, d)-sparse graph with minimum degree at most d+1 and maximum degree at most d+2 is independent in ell _p^d. Finally, we prove that every triangulation of the projective plane is minimally rigid in ell _p^3. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.

Graph Reconstruction from Unlabeled Edge Lengths

A d-dimensional framework is a pair (G, p), where G=(V,E) is a graph and p is a map from V to mathbb {R}^d. The length of an edge uvin E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in mathbb {R}^d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in mathbb {R}^d is globally rigid for some graph G on nge d+2 vertices (where dge 2), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For d=1,2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

Epsilon local rigidity and numerical algebraic geometry

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is [Formula: see text]-locally rigid, a notion we define. A configuration which is [Formula: see text]-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius [Formula: see text] in configuration space. Deciding [Formula: see text]-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.

Rigid Cylindrical Frameworks with Two Coincident Points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity.

Rigidity of Linearly Constrained Frameworks

Abstract We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by Streinu and Theran [14] in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We then point out that results on body–bar frameworks obtained by Katoh and Tanigawa [8] in 2013 can be used to characterise when a graph has a rigid realisation as a $d$-dimensional body–bar framework with a given set of linear constraints.

Algorithm 990

For configurations of point-sets that are pairwise constrained by distance intervals, the EASAL software implements a suite of algorithms that characterize the structure and geometric properties of the configuration space. The algorithms generate, describe, and explore these configuration spaces using generic rigidity properties, classical results for stratification of semi-algebraic sets, and new results for efficient sampling by convex parametrization. The article reviews the key theoretical underpinnings, major algorithms, and their implementation. The article outlines the main applications such as the computation of free energy and kinetics of assembly of supramolecular structures or of clusters in colloidal and soft materials. In addition, the article surveys select experimental results and comparisons.

The maximum likelihood threshold of a graph

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $(n-1)$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold. We conclude by showing that these methods give exact bounds on the number of observations needed for the score matching estimator to exist with probability one.

Rigidity of Frameworks on Expanding Spheres

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterizations are established for the rigidity of generic frameworks for d=1 with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterization of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in one and two dimensions and to some extent between rigidity in two and three dimensions. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.

Geometric Nonlinear Analysis of Plane Frames With Generically Nonuniform Shear-deformable Members

This paper employs the Direct Stiffness Method (DSM) to carry out geometric nonlinear analysis of plane frames with nonuniform physical-geometric characteristics. At the element level, a flexibility system of equations based on the principle of virtual forces (PVF) is established to calculate the tangent stiffness matrix and the equivalent nodal loads. The formulation allows for the easy modeling of shear-deformable frame elements with generic rigidity variation along their axes. In addition, Green's theorem is considered to express all the necessary section properties in terms of boundary integrals. This considerably simplifies the modeling of complex cross sections of arbitrary shapes. A “boundary-element” mesh is then used to model the geometric description of the cross sections. At the structure level, to determine the nonlinear equilibrium trajectories for the frame, we apply a co-rotational updated Lagrangian formulation along with an incremental-iterative full Newton-Raphson process. Large displacements and internal member forces are accurately reconstituted. Frameworks having elements with geometrically complex cross-sections varying along their axes are analyzed to validate the strategy proposed.

Boundary-integral-based process for calculating stiffness matrices of space frame elements with axially varying cross section

This paper presents a strategy to directly compute the stiffness matrix of 3D (space) frame elements having arbitrary cross sections and generic rigidity variation along their axes. All the necessary section properties are determined by means of formulations based purely on boundary integrals. To determine the torsional constant and the torsion center, this strategy applies the Boundary Element Method (BEM). To model thin-walled cross-sections, the strategy calls for activating integration algorithms devised specifically to deal with the nearly singular integrals involved. To express all other section properties (i.e. area, first and second moments of area, and the shear form factors) in terms of boundary integrals, the strategy employs Green's theorem. The existing boundary-element meshes, used to determine the torsion constants, are employed to evaluate the corresponding boundary integrals. In applying the proposed strategy – the pure boundary-integral-based process (PBIP) – we consider space frame elements with geometrically complex cross-sections varying along their axes.

WITHDRAWN: Boundary-integral-based process for calculating stiffness matrices of space frame elements with axially varying cross section

Abstract This paper presents a strategy to directly compute the stiffness matrix of 3D (space) frame elements having arbitrary cross sections and generic rigidity variation along their axes. All the necessary section properties are determined by means of formulations based purely on boundary integrals. To determine the torsional constant and the torsion center, this strategy applies the Boundary Element Method (BEM). To model thin-walled cross-sections, the strategy calls for activating integration algorithms devised specifically to deal with the nearly singular integrals involved. To express all other section properties (i.e. area, first and second moments of area, and the shear form factors) in terms of boundary integrals, the strategy employs Green's theorem. The existing boundary-element meshes, used to determine the torsion constants, are employed to evaluate the corresponding boundary integrals. In applying the proposed strategy – the pure boundary-integral-based process (PBIP) – we consider space frame elements with geometrically complex cross-sections varying along their axes.

The generic rigidity of triangulated spheres with blocks and holes

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal 3-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).

Graph invariants for unique localizability in cooperative localization of wireless sensor networks: Rigidity index and redundancy index

Rigidity theory enables us to specify the conditions of unique localizability in the cooperative localization problem of wireless sensor networks. This paper presents a combinatorial rigidity approach to measure (i) generic rigidity and (ii) generalized redundant rigidity properties of graph structures through graph invariants for the localization problem in wireless sensor networks. We define the rigidity index as a graph invariant based on independent set of edges. It has a value between 0 and 1, and it indicates how close we are to rigidity. Redundant rigidity is required for global rigidity, which is associated with unique realization of graphs. Moreover, redundant rigidity also provides rigidity robustness in networked systems against structural changes, such as link losses. Here, we give a broader definition of redundant edge that we call the “generalized redundant edge.” This definition of redundancy is valid for both rigid and non-rigid graphs. Next, we define the redundancy index as a graph invariant based on generalized redundant edges. It also has a value between 0 and 1, and it indicates the percentage of redundancy in a graph. These two indices allow us to explore the transition from non-rigidity to rigidity and the transition from rigidity to redundant rigidity. Examples on graphs are provided to demonstrate this approach. From a sensor network point of view, these two indices enable us to evaluate the effects of sensing radii of sensors on the rigidity properties of networks, which in turn, allow us to examine the localizability of sensor networks. We evaluate the required changes in sensing radii for localizability by means of the rigidity index and the redundancy index using random geometric graphs and clustered graphs in simulations.

A characterisation of the generic rigidity of 2-dimensional point–line frameworks

A 2-dimensional point–line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point–line and point–point distances. It is rigid if every continuous motion of the points and lines which preserves the constraints results in a point–line framework which can be obtained from the initial framework by a translation or a rotation. We characterise when a generic point–line framework is rigid. Our characterisation gives rise to a polynomial algorithm for solving this decision problem.

Periodic Rigidity on a Variable Torus Using Inductive Constructions

In this paper we prove a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice. We follow the approach of modelling periodic frameworks as frameworks on a torus and use the language of gain graphs for the finite counterpart of a periodic graph. In this setting we employ variants of the Henneberg operations used frequently in rigidity theory.