Rigidity percolation and geometric information in floppy origami

Rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted Erdös-Renyi graphs, with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n(F) of uncanceled degrees of freedom and gamma(r) of redundant bonds. Upon crossing the coexistence line, gamma(r) and n(F) are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely, that the density of uncanceled degrees of freedom is a "free energy" for rigidity percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1) core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma(c), Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1) rigid core, i.e., the average coordination of these subgraphs is exactly 2g, although gamma(c), the average coordination of the whole system, is smaller than 2g. gamma(c) is found to converge to 2g for large g, i.e., in this limit Maxwell counting is exact globally as well.

A fast computer algorithm, the pebble game, has been used successfully to analyze the rigidity of two-dimensional (2D) elastic networks, as well as of a special class of 3D networks, the bond-bending networks, and enabled significant progress in studies of rigidity percolation on such networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on bcc and fcc lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first-order rigidity percolation transition, which is in contrast to the second-order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed previously for Bethe lattices and networks with "chemical order," here it is in a regular randomly diluted network. In the case of site dilution, the transition is also first order for bcc lattices, but results for fcc lattices suggest a second-order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent nonuniversality, make this phase transition highly unusual.

Origami-inspired mechanisms have revolutionized robotic design by enabling lightweight, compact, and highly adaptable structures capable of complex motions and transformations. However, the internal presence of multiple closed-loop structures within origami mechanisms significantly complicates mobility analysis, making the calculation of their degrees of freedom particularly challenging. Common analytical methods, such as the Jacobian matrix approach, are often labor-intensive and require specialized expertise, limiting their practical application. To address this challenge, we propose a novel framework that leverages large language models (LLMs), specifically ChatGPT, to simplify and automate the mobility analysis of origami-inspired mechanisms. By training ChatGPT to understand and apply the Jacobian matrix method, the framework enables the automated generation of code for analyzing the mobility of various origami structures. The effectiveness of the proposed approach is demonstrated through three typical origami mechanisms: Miura, Waterbomb, and Kresling patterns. Results show that the framework achieves high reliability, with a success rate exceeding 95% under optimal conditions. This study not only pioneers the application of LLMs in the mobility analysis of origami mechanisms but also opens a new perspective for integrating AI-driven methods in solving complex mechanical problems.

An Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game

A dual polarization `eggbox'-inspired frequency selective surface with cross-conductive elements that can change the frequency response of two orthogonal linear polarizations is demonstrated for the first time in this article. The additional degree of freedom from the dual-mode metamaterial in combination with the `eggbox' origami structure which exploits the symmetrical planes of the metamaterial leads to an unprecedented re-configurability performance compared to traditional Miura-Ori based structures which have fewer degrees of control.

The continuum robot is a soft robot with infinite degrees of freedom. Origami has a high capacity for spatial deployment. This paper proposes a flexible continuum robot based on origami and mortise-tenon structure (FCRBOM). The robot consists of some flexible hinges based on origami and mortise-tenon structure (FHBOM). The design process of the FCRBOM is given. The compliance of the FCRBOM is analyzed by the compliance matrix method. The Finite element analysis (FEA) is used to simulate and analyze the FCRBOM, and the correctness of the theoretical analysis is verified. Then a spatial FCRBOM (SFCRBOM) is designed. The impact of key dimensional parameters on the flexibility of SFCRBOM is discussed. Finally, an SFCRBOM with higher flexibility is presented.

The microstructure of trabecular bone is known to adapt its morphology in response to mechanical loads for achieving a biomechanical homeostasis. Based on this form–function relationship, previous investigators either simulated the remodeling of bone to predict the resulting density and architecture for a specific loading or retraced physiological loading conditions from local density and architecture. The latter inverse approach includes quantifying bone morphology using computed tomography and calculating the relative importance of selected load cases by minimizing the fluctuation of a tissue loading level metric. Along this concept, the present study aims at identifying an optimal, personalized, multiaxial load case at the distal section of the human radius using in vivo HR-pQCT-based isotropic, homogenized finite element (hFE) analysis. The dataset consisted of HR-pQCT reconstructions of the 20 mm most distal section of 21 human fresh-frozen radii. We simulated six different unit canonical load cases (FX palmar–dorsal force, FY ulnar–radial force, FZ distal–proximal force, MX moment about palmar–dorsal, MY moment about ulnar–radial, MZ moment about distal–proximal) using a simplified and efficient hFE method based on a single isotropic bone phase. Once we used a homogeneous mean density (shape model) and once the original heterogeneous density distribution (shape + density model). Using an analytical formulation, we minimized the deviation of the resulting strain tensors ε(x) to a hydrostatic compressive reference strain ε0, once for the 6 degrees of freedom (DOF) optimal (OPT) load case and for all individual 1 DOF load cases (FX, FY, FZ, MX, MY, MZ). All seven load cases were then extended in the nonlinear regime using the scaled displacements of the linear load cases as loading boundary conditions (MAX). We then compared the load cases and models for their objective function (OF) values, the stored energies and their ultimate strength using a specific torsor norm. Both shape and shape + density linear-optimized OPT models were dominated by a positive force in the z-direction (FZ). Transversal force DOFs were close to zero and mean moment DOFs were different depending on the model type. The inclusion of density distribution increased the influence and changed direction of MX and MY, while MZ was small in both models. The OPT load case had 12–15% lower objective function (OF) values than the FZ load case, depending on the model. Stored energies at the optimum were consistently 142–178% higher for the OPT load case than for the FZ load case. Differences in the nonlinear response maximum torsor norm ‖t‖ were heterogeneous, but consistently higher for OPT_MAX than FZ_MAX. We presented the proof of concept of an optimization procedure to estimate patient-specific loading conditions for hFE methods. In contrast to similar models, we included canonical load cases in all six DOFs and used a strain metric that favors hydrostatic compression. Based on a biomechanical analysis of the distal joint surfaces at the radius, the estimated load directions are plausible. For our dataset, the resulting OPT load case is close to the standard axial compression boundary conditions, usually used in HR-pQCT-based FE analysis today. But even using the present simplified hFE model, the optimized linear six DOF load case achieves a more homogeneous tissue loading and can absorb more than twice the energy than the standard uniaxial load case. The ultimate strength calculated with a torsor norm was consistently higher for the 6-DOF nonlinear model (OPT_MAX) than for the 1-DOF nonlinear uniaxial model (FZ_MAX). Defining patient-specific boundary conditions may decrease angulation errors during CT measurements and improve repeatability as well as reproducibility of bone stiffness and strength estimated by HR-pQCT-based hFE analysis. These results encourage the extension of the present method to anisotropic hFE models and their application to repeatability data sets to test the hypothesis of reduced angulation errors during measurement.

The bar and hinge framework uses truss elements and rotational springs to efficiently model the structural behaviour of origami. The framework is especially useful to investigate origami metamaterials as they have repeating geometry, which makes conventional finite element simulations very expensive due to a large number of degrees of freedom. This work proposes improvements to the parameters of bar and hinge model within the context of structural dynamics, specifically modal analysis under small deformations, which has not been carried out previously in the literature. A range of low-frequency modes involving origami folding and panel bending deformations that can be accurately captured by the bar and hinge framework are identified. Within this range, bar and hinge parameters like the lumped masses and the rotational spring stiffness values are derived using conservation laws and finite element tests. The best among the proposed schemes is found to predict natural frequencies of the considered origami structures to within 10% maximum error, improving the accuracy by more than three times from existing schemes. In most cases, the errors in natural frequencies are less than 5%. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Stabilization of stokes pulse energies in the nonlinear regime of stimulated Raman scattering

The visual servoing formalism provides a framework for achieving tasks in an active closed-loop fashion based on visual information. We consider a typical visual servoing ap proach that uses geometric information about image features for controlling the position and attitude of a camera and extend the applicability of this approach by using image motion infor mation. We present two different approaches to visual tasks that use motion information. The first uses the focus of expansion, which is related to the translational velocity of the camera. The second incorporates the parameters of the 2D affine motion model in control equations. Both of these approaches are il lustrated by means of a task of aligning the optical axis of the camera with the unknown direction of translational motion of the system on which it is mounted. Results of simulation experi ments and real experiments on a six degrees of freedom (DOF) robot with a camera on its end effector are presented.

A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics

In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ(ω) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.

Using a recently developed algorithm for generic rigidity of two-dimensional graphs, we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to $L=4096.$ We compare three different universality classes: the generic rigidity class, the connectivity class, and the generic ``braced square net''(GBSN). We analyze the spanning cluster density ${P}_{\ensuremath{\infty}},$ the backbone density ${P}_{B}$, and the density of dangling ends ${P}_{D}.$ In the generic rigidity (GR) and connectivity cases, the load-carrying component of the spanning cluster, the backbone, is fractal at ${p}_{c},$ so that the backbone density behaves as $B\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}^{\ensuremath{'}}}$ for $p>{p}_{c}.$ We estimate ${\ensuremath{\beta}}_{\mathrm{gr}}^{\ensuremath{'}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ for generic rigidity and ${\ensuremath{\beta}}_{c}^{\ensuremath{'}}=0.467\ifmmode\pm\else\textpm\fi{}0.007$ for the connectivity case. We find the correlation length exponents ${\ensuremath{\nu}}_{\mathrm{gr}}=1.16\ifmmode\pm\else\textpm\fi{}0.03$ for generic rigidity compared to the exact value for connectivity, ${\ensuremath{\nu}}_{c}=\frac{4}{3}.$ In contrast the GBSN undergoes a first-order rigidity transition, with the backbone density being extensive at ${p}_{c},$ and undergoing a jump discontinuity on reducing p across the transition. We define a model which tunes continuously between the GBSN and GR classes, and show that the GR class is typical.

The aim of this paper is to show how to predict the accurate machining technology for the particular free form NURBS or B-spline surface. Since that kind of a surface is very hard to describe in an analytical manner, the topological and geometrical information about the surface was acquired with the help of self-organized neural networks (NNs) and first- or second-order statistic parameters. It is proved that the most significant parameter in this process is the curvature, especially when rapid changes of curvature on a free form surface occurred. As the Gaussian distribution of surface curvatures and slope gradient data were presumed, the mean and variance was used for one-dimensional data presentation, and the Hebbian output data vector was used to assess probability, density function and distribution of the presented data. For collecting the maximum amount of surface information, the principal component analysis method inside the Hebbian NN was used.

ABSTRACTA high-precision direct georeferencing method based on multi-sensor information for unmanned aerial vehicle (UAV) remote sensing is proposed in this research. The method makes contributions in three areas. First, the direct georeferencing model for the most popular system structure of fixed wing UAV loading of a camera with two degrees of freedom is established by using multi-sensor parameters. Second, based on this model, it performs image geographical reconstruction under complex motion and ultimate imaging conditions, which constructs the resulting image from the image frame, geographical frame and image pixels, and provides real geometric and geographic information. Third, in order to improve the accuracy of the georeferencing of the corrected image, a method of error prediction and compensation is presented. It converts the six factors affecting the accuracy of image georeferencing, to three factors for reducing the complexity. Then, the error prediction model is constructed, the parameters of which could be determined by supervised machine learning method. A medium-altitude UAV is employed to collect large amount of images and other sensor data. The experiments show quite encouraging results.