Rigidity Of Frameworks Research Articles

Rigidity, Tensegrity, and Reconstruction of Polytopes Under Metric Constraints

Abstract We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset {\mathbb {R}}^{d}$ and $Q\subset {\mathbb {R}}^{e}$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack these questions and we verify them in three relevant special cases: $P$ and $Q$ are centrally symmetric, $Q$ is a slight perturbation of $P$, and $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q:V(G_{P})\to {\mathbb {R}}^{e}$ of the edge-graph $G_{P}$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.

When is a Planar Rod Configuration Infinitesimally Rigid?

AbstractWe investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

Flexibility and rigidity of frameworks consisting of triangles and parallelograms

A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons.

Angle Rigidity for Multiagent Formations in 3-D

This paper establishes the notion and properties of angle rigidity for 3D multi-point frameworks with angle constraints, and then designs direction-only control laws to stabilize angle rigid formations of mobile agents in 3D. Angles are defined using the interior angles of triangles within the given framework, which are independent of the choice of coordinate frames and can be conveniently measured using monocular cameras and direction-finding arrays. We show that 3D angle rigidity is a local property, which is in contrast to the 3D bearing rigidity as has been proved to be a global property in the literature. We demonstrate that such angle rigid and globally angle rigid frameworks can be constructed through adding repeatedly new points to the original small angle rigid framework with carefully chosen angle constraints. We also investigate how to merge two 3D angle rigid frameworks by connecting three points of one angle rigid framework simultaneously to the other. When angle constraints are given only in the surface of a framework, angle rigidity of convex polyhedra is studied, in which the cases of triangular face and triangulated face are considered, respectively. The proposed 3D angle rigidity theory is then utilized to design decentralized formation control strategies using local direction measurements for teams of mobile agents. Simulation examples are provided to validate the convergence of the formations.

Cooperative Manipulation via Internal Force Regulation: A Rigidity Theory Perspective

This paper considers the integration of rigid cooperative manipulation with rigidity theory. Motivated by rigid models of cooperative manipulation systems, i.e., where the grasping contacts are rigid, we introduce first the notion of bearing and distance rigidity for graph frameworks in <inline-formula><tex-math notation="LaTeX">$\mathsf {SE}(3)$</tex-math></inline-formula>. Next, we associate the nodes of these frameworks to the robotic agents of rigid cooperative manipulation schemes and we express the object-agent internal forces by using the graph rigidity matrix, which encodes the infinitesimal rigid body motions of the system. Moreover, we show that the associated cooperative manipulation grasp matrix is related to the rigidity matrix via a range-nullspace relation, based on which we provide novel results on the relation between the arising interaction and internal forces and consequently on the energy-optimal force distribution on a cooperative manipulation system. Finally, simulation results enhance the validity of the theoretical findings.

Equilibrium stresses and rigidity for infinite tensegrities and frameworks

Asymptotic equilibrium stresses are defined for countably infinite tensegrities and generalisations of the Roth-Whiteley characterisation of first-order rigidity are obtained. Generalisations of prestress stability and second order rigidity are given for countably infinite bar-joint frameworks and are shown to give sufficient conditions for continuous rigidity relative to certain prescribed motions. The proofs are based on a new short proof for finite frameworks that prestress stability ensures continuous rigidity.

DNA framework-engineered chimeras platform enables selectively targeted protein degradation

A challenge in developing proteolysis targeting chimeras (PROTACs) is the establishment of a universal platform applicable in multiple scenarios for precise degradation of proteins of interest (POIs). Inspired by the addressability, programmability, and rigidity of DNA frameworks, we develop covalent DNA framework-based PROTACs (DbTACs), which can be synthesized in high-throughput via facile bioorthogonal chemistry and self-assembly. DNA tetrahedra are employed as templates and the spatial position of each atom is defined. Thus, by precisely locating ligands of POI and E3 ligase on the templates, ligand spacings can be controllably manipulated from 8 Å to 57 Å. We show that DbTACs with the optimal linker length between ligands achieve higher degradation rates and enhanced binding affinity. Bispecific DbTACs (bis-DbTACs) with trivalent ligand assembly enable multi-target depletion while maintaining highly selective degradation of protein subtypes. When employing various types of warheads (small molecules, antibodies, and DNA motifs), DbTACs exhibit robust efficacy in degrading diverse targets, including protein kinases and transcription factors located in different cellular compartments. Overall, utilizing modular DNA frameworks to conjugate substrates offers a universal platform that not only provides insight into general degrader design principles but also presents a promising strategy for guiding drug discovery.

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).

Global rigidity of 2-dimensional direction-length frameworks with connected rigidity matroids

A 2-dimensional direction-length framework ( G , p ) consists of a multigraph G = ( V ; D , L ) with realisation p : V → R 2 . The edges of G represent geometric constraints: edges in L have fixed length, and edges in D have fixed gradient. A direction-length framework ( G , p ) is globally rigid if every ( G , q ) which satisfies the same geometric constraints as ( G , p ) can be obtained from ( G , p ) by an isometry of the plane. We characterise global rigidity for the class of 2-dimensional direction-length frameworks ( G , p ) for which p is generic and the rigidity matroid of G is connected. Specifically, we show that such frameworks are globally rigid if and only if both D and L are non-empty, and every 2-separation of G is direction-balanced. This extends previous work by Jackson and Jordán (2010).

Sufficient connectivity conditions for rigidity of symmetric frameworks

It is a famous result of Lovász and Yemini that 6-connected graphs are rigid in the plane (Lovász and Yemini, 1982). This was recently improved by Jackson and Jordán (2009) who showed that 6-mixed connectivity is also sufficient for rigidity. Here we give sufficient graph connectivity conditions for both ‘forced symmetric’ and ‘incidentally symmetric’ infinitesimal rigidity in the plane.

Infinitesimal rigidity and prestress stability for frameworks in normed spaces

A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid – i.e. whether every other suitably close framework with the same distance constraints is an isometric copy – is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces.

Coincident Rigidity of 2-Dimensional Frameworks

Fekete, Jord\'an and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characterisation to an arbitrary set T of vertices and verify our conjecture when |T| = 3.

The Theme of Fashion

Abstract: The aim of this paper is to study the domain of World Wide Web site development and also put forward a methodology to assist with this process. These methodologies have their proselytizers and those who decry the constraints and rigidity of prescriptive frameworks. The methodology presented here is unintended to be a panacea for the problems of web development; rather it is hoped it will provide a functional framework for guiding the process. The main study points out that the demographic profile of customers, the online seller of the product, the type of products to be purchased, and the characteristics of online shopping websites had an optimistic impact on the intention and online shopping behavior of the customers. The paper would help the marketers to implement ways and means to pull online shoppers. It will be helpful for the customer to understand the advantages and disadvantages of online shopping. Keywords: Proselytizers ,rigidity ,unintended ,demographic ,optimistic .

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

A bar-joint framework (G,p) in a (non-Euclidean) real normed plane X is the combination of a finite, simple graph G and a placement p of the vertices in X. A framework (G,p) is globally rigid in X if every other framework (G,q) in X with the same edge lengths as (G,p) arises from an isometry of X. The weaker property of local rigidity in normed planes (where only (G,q) within a neighbourhood of (G,p) are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for ℓp-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in X, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in X.

High Protonic Conductivity of Three Highly Stable Nanoscale Hafnium(IV) Metal-Organic Frameworks and Their Imidazole-Loaded Products.

Attracted by the exceptional structural rigidity and inherent porous structures of the Hf-based metal-organic frameworks (MOFs), we adopted a rapid synthesis approach to preparing three nanoscale MOFs, Hf-UiO-66 (1), Hf-UiO-66-(OH)2 (2), and Hf-UiO-66-NH2 (3), and systematically explored the water-assisted proton conductivities of the original ones and the post-modified products. Interestingly, the proton conductivities (σ) of all three MOFs exhibit significant temperature and humidity dependence. At 98% RH and 100 °C, their optimal σ values can reach up to 10-3 S·cm-1. Consequently, imidazole units are loaded into 1-3 to obtain related MOFs, Im@1, Im@2, and Im@3, and the σ values of the imidazole-loaded products are boosted to 10-2 S·cm-1. Note that these modifications not only do not change the frameworks of the pristine MOFs but also do not affect their high chemical and water stability. The proton-conductive mechanisms of these MOFs before and after modification have been thoroughly discussed based on structural analyses, N2 and H2O vapor adsorptions, and activation energy values. The excellent structural stability as well as the durability and stability of their proton conduction ability indicate that these MOFs can be used in the field of fuel cells and so on.

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension d>2. We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in {mathbb {R}}^d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in {mathbb {R}}^{2d} to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

Global rigidity of (quasi-)injective frameworks on the line

A realization of a graph G is a pair (G,p) where p maps the vertices of G into Euclidean space Rd. The realization is injective if p is injective and quasi-injective if for each edge of G, p maps the endpoints of the edge to different points in space. The realization is globally rigid if any realization (G,q) in Rd with the same edge lengths is congruent to (G,p).In this paper we characterize graphs that have an injective (quasi-injective, respectively) non-globally rigid realization in R1, and we show that the problem of recognizing these graphs is NP-complete in both the injective and the quasi-injective cases. Our characterizations are based on the notion of NAC-colorings, which have been used previously to investigate similar problems in the plane. We also give an overview of related results and open problems in rigidity theory.

Almost‐Rigidity of Frameworks

AbstractWe extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities. © 2020 Wiley Periodicals LLC.

Rigidity of symmetric frameworks in normed spaces

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all ℓp spaces with p≠2). Complete combinatorial characterisations are obtained for half-turn rotation in the ℓ1 and ℓ∞-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight gain graphs.

Global rigidity of direction-length frameworks

A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair (G,p), where G=(V;D,L) is a ‘mixed’ graph and p:V→R2 is a point configuration for V. It is globally rigid if every direction-length framework (G,q) which satisfies the same constraints can be obtained from (G,p) by a translation or a rotation by 180°. We characterise the mixed graphs G with the property that every generic framework (G,p) is globally rigid.