Conditions For Rigidity Research Articles

Rigidity for the perimeter inequality under Schwarz symmetrization

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

Widths and rigidity

We consider the Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions in $L_2$ is rigid, that is, it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$ for $p<2$ the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We present some sufficient conditions for rigidity. We prove that the independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$, the metric that corresponds to convergence in measure. In the case of $L_p$ for $1<p<2$ the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, for example, that the first $N$ trigonometric functions can be approximated by very low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$ for ${p<1}$. Bibliography: 34 titles.

Rigidity of terminal simplices in persistent homology

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on varepsilon >0 implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through varepsilon -perturbations of filtration function. The condition for a homology class or a bar in dimension n depends only on the barcodes in dimensions n and n+1.

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).

Calculation of the boring bar design for static rigidity and strength with simultaneous boring of a stepped hole with further optimization

A method for simultaneous processing of stepped holes in the bedding of a submersible pump and the boring bar design is proposed. The results of the study of existing methods for processing stepped holes and the design of boring tools are presented. It was revealed that the considered methods and designs of tools cannot be used for processing stepped holes in the submersible pump frame. The calculation of the developed design of the boring bar for static stiffness and strength, as well as the calculation of the stress-strain state (SSS) of the cutting plates by the finite element method using the ANSYS program, was carried out. The purpose of calculating the boring bar for static stiffness is a finite element study of the radial displacement of the boring bar cutter plates in the process of machining a stepped hole. As a result, the value of deformation in the axial and radial directions, which occur at the top of the cutting plate, is established. Also, the values of the cantilever overhang of the boring cutters are established, under which the conditions for the static rigidity of the boring bar are met. Taking into account the obtained results, the boring bar design was optimized.

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.

Transverse rigidity is prestress stability

Recently, V. Alexandrov proposed an intriguing sufficient condition for rigidity, which we will call “transverse rigidity”. We show that transverse rigidity is actually equivalent to the known sufficient condition for rigidity called “prestress stability”. Indeed this leads to a novel interpretation of the prestress condition.

Gradient-based bearing-only formation control: An elevation angle approach

This paper proposes an elevation angle rigidity theory in both 2D and 3D spaces, and applies it to solve multi-agent formation control with only inter-agent bearing/direction measurements in agents’ local coordinate frames. Motivated by the sensor technology in measuring elevation angle and angular diameter, we develop elevation angle rigidity by attaching each agent in a multi-agent framework with a rod in 2D and a ball in 3D, respectively. By defining the elevation angle rigidity matrix, conditions for infinitesimal elevation angle rigidity are derived. Compared to previously developed angle rigidity-based and bearing rigidity-based formation control laws, the proposed elevation angle rigidity-based control law can maintain the gradient-based control property. Compared to distance-based formation control laws, less sensor measurements are required. The formation maneuvering with desired translation and rotation is also realized by using only local bearing measurements. Simulation examples illustrate the advantages and effectiveness of the proposed approach.

Globally rigid powers of graphs

The characterization of rigid graphs in Rd for d≥3 is a major open problem in rigidity theory. The same holds for globally rigid graphs. In this paper our goal is to give necessary and/or sufficient conditions for the (global) rigidity of the square G2 (and more generally, the power Gk) of a graph G in Rd, for some values of k,d. Our work is motivated by some results and conjectures of M. Cheung and W. Whiteley from 2008, the Molecular Theorem of N. Katoh and S. Tanigawa from 2011, which settled the case of rigidity for k=2,d=3, and the potential applications in molecular conformation and sensor network localization.We first consider the case when k=d and characterize those graphs G for which Gd is globally rigid in Rd, for all d≥1, and then focus on the case when k=d−1. We provide a new, direct proof for the 3-dimensional bar-and-joint version of the Molecular Theorem (d=3) and a necessary condition for the rigidity of Gd−1 in Rd, for all d≥3. We conjecture that this condition is also sufficient.The global rigidity of square graphs in R3 is still an open problem. We formulate a Molecular Global Rigidity Conjecture, which proposes a combinatorial characterization of globally rigid square graphs in terms of vertex partitions and edge count conditions. We prove that the condition is necessary. For the general case we give a best possible connectivity based sufficient condition by showing that if G is 3-edge-connected then Gd−1 is globally rigid in Rd, for all d≥3.Our results imply affirmative answers to the conjectures of M. Cheung and W. Whiteley in two special cases.

Экспериментальное исследование жесткости технологической системы на базе промышленного робота KUKA KR 300 R2500 ULTRA

In modern machine-building production, there are many parts that have both significant overall dimensions and complex hard-to-reach surfaces for processing. The manufacture of such parts is carried out on specialized equipment, which has the same significant overall dimensions. However, the complexity of the design of a number of parts, such as the body of a rocket engine, large volume tanks, non-separable metal structures, etc., as well as their inaccessibility, level the technological capabilities of this equipment. Currently, industrial robots equipped with tool spindles are used to process such parts. The presence of a large number of degrees of freedom of this equipment, the relatively small overall dimensions and the vastness of the working area allow them to be effectively used in this field of mechanical engineering. However, the insufficiently studied rigidity of this equipment, the lack of specific recommendations for their use significantly limit the scope of their application. Thus, the issues related to the mutual influence of the rigidity of the technological system, cutting forces and cutting conditions in the manufacture of large parts using industrial robots are relevant and require a comprehensive study. In order to study dynamic processes in the manufacture of large-sized parts using an industrial robot, experiments were carried out to determine the amplitude-frequency characteristics of vibrations that occur during processing. An experimental study made it possible to establish that the rigidity of the technological system varies over a wide range. At the same time, the rigidity of the part under an impulse load difIn modern machine-building production, there are many parts that have both significant overall dimensions and complex hard-to-reach surfaces for processing. The manufacture of such parts is carried out on specialized equipment, which has the same significant overall dimensions. However, the complexity of the design of a number of parts, such as the body of a rocket engine, large volume tanks, non-separable metal structures, etc., as well as their inaccessibility, level the technological capabilities of this equipment. Currently, industrial robots equipped with tool spindles are used to process such parts. The presence of a large number of degrees of freedom of this equipment, the relatively small overall dimensions and the vastness of the working area allow them to be effectively used in this field of mechanical engineering. However, the insufficiently studied rigidity of this equipment, the lack of specific recommendations for their use significantly limit the scope of their application. Thus, the issues related to the mutual influence of the rigidity of the technological system, cutting forces and cutting conditions in the manufacture of large parts using industrial robots are relevant and require a comprehensive study. In order to study dynamic processes in the manufacture of large-sized parts using an industrial robot, experiments were carried out to determine the amplitude-frequency characteristics of vibrations that occur during processing. An experimental study made it possible to establish that the rigidity of the technological system varies over a wide range. At the same time, the rigidity of the part under an impulse load differs from the rigidity during milling by at least an order of magnitude, and the rigidity of the spindle during milling differs from the rigidity of the part by at least 2 orders of magnitude. Based on this, in the process of designing the machining of large-sized non-rigid parts using industrial robots, it is necessary to take into account the worst conditions for rigidity. It is advisable to base further recommendations on the appointment of cutting conditions under changing processing conditions on the basis of mathematical modeling of the rigidity of the elements of the technological system. Also, to automate the process of removing the parameters of the rigidity of the technological system, an urgent task is to determine by test determination of vibrations on the spindle of an industrial robot and indirect assessment of microdisplacements on the part

A Unified Dissertation on Bearing Rigidity Theory

This work focuses on bearing rigidity theory, namely the branch of knowledge investigating the structural properties necessary for multi-element systems to preserve the inter-unit bearings under deformations. The contributions of this work are twofold. The first one consists in the development of a general framework for the statement of the principal definitions and properties of bearing rigidity. We show that this approach encompasses results existing in the literature, and also provides a systematic approach for studying bearing rigidity on any differential manifold in SE(3)^n, where n is the number of agents. The second contribution is the derivation of a general form of the rigidity matrix, a central construct in the study of rigidity theory. We provide a necessary and sufficient condition for the infinitesimal rigidity of a bearing framework as a property of the rank of the rigidity matrix. Finally, we present two examples of multi-agent systems not encountered in the literature and we study their rigidity properties using the developed methods.

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.

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 .

Spectral conditions for graph rigidity in the Euclidean plane

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G,p) of graph G together with a map p of the vertices of G into a Euclidean space. We view the edges of (G,p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lovász and Yemini discovered a new characterization and proved that every 6-connected graph is rigid. Combined with a combinatorial characterization of global rigidity given by Jackson and Jordán in 2009, it is actually proved that every 6-connected graph is globally rigid. Consequently, if the algebraic connectivity of a graph is greater than 5, then it is globally rigid. In this paper, we improve this bound and show that for a graph G with minimum degree δ≥6, if its algebraic connectivity is greater than 2+1δ−1, then G is rigid and if its algebraic connectivity is greater than 2+2δ−1, then G is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least 8 is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint spanning 2-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees.

On the global rigidity of tensegrity graphs

A tensegrity graph is a graph with edges labeled as bars, cables and struts. A realization of a tensegrity graph T is a pair (T,p), where p maps the vertices of T into Rd for some d≥1. The realization is globally rigid if any realization (T,q) in Rd in which the bars have the same length and the cables and struts are not longer and not shorter, respectively, is an isometric image of (T,p). A tensegrity graph is weakly globally rigid in Rd if it has a generic globally rigid realization in Rd, and strongly globally rigid in Rd if every generic realization in Rd is globally rigid.In this paper we give a necessary condition for weak global rigidity in Rd and prove that in the d=1 case the same condition is also sufficient. In particular, our results imply that a tensegrity graph has a generic globally rigid realization in R1 if and only if it has a generic universally rigid realization in R1. We also show that recognizing strongly globally rigid tensegrity graphs in Rd is co-NP-hard for all d≥1.

Sufficient Conditions for 2-Dimensional Global Rigidity

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 23 October 2020Accepted: 07 July 2021Published online: 19 October 2021Keywordsrigid graph, redundant rigidity, global rigidity, essential connectivityAMS Subject Headings05C40, 05C75Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

AbstractLet $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

Angle Rigidity and Its Usage to Stabilize Multiagent Formations in 2-D

Motivated by the challenging formation stabilization problem for mobile robotic teams wherein no distance or relative position measurements are available but each robot can only measure some of relative angles with respect to its neighbors in its local coordinate frame, we develop the notion of “angle rigidity” for a multipoint framework, named “angularity”, consisting of a set of nodes embedded in a Euclidean space, and a set of angle constraints among them. Different from bearings or angles defined in a global frame, the angles we use do not rely on the knowledge of a global frame, and are signed according to the counter-clockwise direction. Here, angle rigidity refers to the property specifying that under proper angle constraints, the angularity can only translate, rotate, or scale as a whole when one or more of its nodes are perturbed locally. We first demonstrate that this angle rigidity property, in sharp comparison to bearing rigidity or other reported rigidity related to angles of frameworks in the literature, is not a global property since an angle rigid angularity may allow flex ambiguity. We then construct necessary and sufficient conditions for infinitesimal angle rigidity by checking the rank of an angularity's rigidity matrix. We develop a combinatorial necessary condition for infinitesimal minimal angle rigidity. Using the developed theories, a formation stabilization algorithm is designed for a robotic team to achieve an angle rigid formation, in which only angle measurements are needed. Simulation examples demonstrate the advantages of the proposed angle-only formation control approach.

Rigidity for perimeter inequality under spherical symmetrisation

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.