Related papers: A Unified Dissertation on Bearing Rigidity Theory
A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the…
Rigidity regulates the integrity and function of many physical and biological systems. This is the first of two papers on the origin of rigidity, wherein we propose that "energetic rigidity," in which all non-trivial deformations raise the…
Rigidity is an emergent property of materials - it is not a feature of individual components that comprise the structure, but instead arises from interactions between many constituent parts. Recently, it has been recognized that…
The theory of disordered elastic systems is one of the most powerful frameworks to assess the physics of multiple systems that span from ferromagnets to migrating biological cells. In this formalism, one assumes that the system can be…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
The modelling of heterogeneous and architected materials poses a significant challenge, demanding advanced homogenisation techniques. However, the complexity of this task can be considerably simplified through the application of micropolar…
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 characterisations are established…
We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding…
This paper studies robustness of multivariable systems with parametric uncertainties, and establishes a multivariable version of Edge Theorem. An illustrative example is presented.
A new type of elasticity of random (multifractal) structures is suggested. A closed system of constitutive equations is obtained on the basis of two proposed phenomenological laws of reversible deformations of multifractal structures. The…
This work presents a novel approach for \textit{bearing rigidity} analysis and control in multi-robot networks with sensing constraints and dynamic topology. By decomposing the system's framework into \textit{subframeworks}, we express…
Respecting deformational constraints and predeformations poses a substantial challenge in the description of nonlinear elasticity. We here outline how group theory can play a beneficial role to overcome this challenge. Specifically, group…
This work considers the problem of estimating the unscaled relative positions of a multi-robot team in a common reference frame from bearing-only measurements. Each robot has access to a relative bearing measurement taken from the local…
Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction…
In this paper we present a geometrical framework to study the uniformity of a composite material by means of double groupoid theory. The notions of vertical and horizontal uniformity are introduced, as well as other weaker ones that allows…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
This paper establishes a unified element-based framework for formation control by introducing the concept of the deformation gradient from continuum mechanics. Unlike traditional methods that rely on geometric constraints defined on graph…
We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class.…
This Thesis is devoted to the analysis of entanglement in relevant physical systems. Entanglement is the conducting theme of this research, though I do not dedicate to a single topic, but consider a wide scope of physical situations. I have…
The modern theory of elasticity and the first law of thermodynamics are cornerstones of engineering science that share the concept of reversibility. Engineering researchers have known for four decades that the modern theory violates the…