Добірка наукової літератури з теми "Frenet-Serret Framework"

The Frenet–Serret (FS) framework stands as a pivotal tool in shape sensing for various infrastructures. However, this tool suffers from accumulative errors, particularly at inflection points where the normal vector undergoes sign changes. To minimize the error, the traditional FS framework is modified by incorporating the homogeneous matrix transformation (HMT) method for segments containing inflection points. Additionally, inclination information is also used to calculate the unit tangent vector and the unit norm vector at the start point of each segment. This novel approach, termed the FS-HMT method, aims to enhance accuracy. To validate the effectiveness of the proposed method, a simulation of a cantilever column was conducted using finite element software ANSYS 19.2. The numerical results demonstrate the capability of the proposed method to accurately predict curves with inflection points, yielding a maximum error of 1.1%. Subsequently, experimental verification was performed using a 1 m long spring steel sheet, showcasing an error of 4.9%, which is notably lower than that of the traditional FS framework. Our proposed modified FS framework exhibits improved accuracy, especially in scenarios involving inflection points. These findings underscore its potential as a valuable tool for enhanced shape sensing in practical applications.

Clasificar el movimiento humano se ha convertido en una necesidad tecnológica, en donde para definir la posición de un sujeto requiere identificar el recorrido de las extremidades y el tronco del cuerpo, y tener la capacidad de diferenciar esta posición respecto a otros sujetos o movimientos, generándose la necesidad tener datos y algoritmos que faciliten su clasificación. Es así, como en este trabajo, se evalúa la capacidad discriminante de datos de captura de movimiento en rehabilitación física, donde la posición de los sujetos es adquirida con el Kinect de Microsoft y marcadores ópticos, y atributos del movimiento generados con el marco de Frenet Serret, evaluando su capacidad discriminante con los algoritmos máquinas de soporte vectorial, redes neuronales y k vecinos más cercanos. Los resultados presentan porcentajes de acierto del 93.5% en la clasificación con datos obtenidos del Kinect, y un éxito del 100% para los movimientos con marcadores ópticos. Classify human movement has become a technological necessity, where defining the position of a subject requires identifying the trajectory of the limbs and trunk of the body, having the ability to differentiate this position from other subjects or movements, which generates the need to have data and algorithms that help their classification. Therefore, the discriminant capacity of motion capture data in physical rehabilitation is evaluated, where the position of the subjects is acquired with the Microsoft Kinect and optical markers. Attributes of the movement generated with the Frenet Serret framework. Evaluating their discriminant capacity by means of support vector machines, neural networks, and k nearest neighbors algorithms. The obtained results present an accuracy of 93.5% in the classification with data obtained from the Kinect, and success of 100% for movements where the position is defined with optical markers.

We develop in a consistent manner the Ostrogradski–Hamilton framework for gonihedric string theory. The local action describing this model, being invariant under reparametrizations, depends on the modulus of the mean extrinsic curvature of the worldsheet swept out by the string, and thus we are confronted with a genuine second-order in derivatives field theory. In our geometric approach, we consider the embedding functions as the field variables and, even though the highly nonlinear dependence of the action on these variables, we are able to complete the classical analysis of the emerging constraints for which, after implementing a Dirac bracket, we are able to identify both the gauge transformations and the proper physical degrees of freedom of the model. The Ostrogradski–Hamilton framework is thus considerable robust as one may recover in a straightforward and consistent manner some existing results reported in the literature. Further, in consequence of our geometrical treatment, we are able to unambiguously recover as a by-product the Hamiltonian approach for a particular relativistic point-particle limit associated with the gonihedric string action, that is, a model linearly depending on the first Frenet–Serret curvature.

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet–Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

We present a theoretical framework to study equilibrium configurations of filaments within a spinor representation of curves. The curve representing the filament is described by a unit two-component spinor field and its charge conjugate satisfying two-dimensional equations coupled by the curvature and torsion. The spinor field replaces the Frenet-Serret frame, whereas its structure equations replace the Frenet-Serret equations. Employing this spinorial description of curves, we derive the Euler-Lagrange equations of curves whose energies depend on their curvature and torsion. We analyze the conservation laws of the spinors representing the balance of the forces and torques along the filament. We illustrate this framework by applying these results to the Euler Elastica, whose bending energy is quadratic in the curvature.

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Abstract This paper presents a novel derivation for the governing equations of geometrically curved and twisted three-dimensional Timoshenko beams. The kinematic model of the beam was derived rigorously by adopting a parametric description of the axis of the beam, using the local Frenet-Serret reference system, and introducing the constraint of the beam cross-section planarity into the classical, first-order strain versus displacement relations for Cauchy's continua. The resulting beam kinematic model includes a multiplicative term consisting of the inverse of the Jacobian of the beam axis curve. This term is not included in classical beam formulations available in the literature; its contribution vanishes exactly for straight beams and is negligible only for curved and twisted beams with slender geometry. Furthermore, to simplify the description of complex beam geometries, the governing equations were derived with reference to a generic position of the beam axis within the beam cross-section. Finally, this study pursued the numerical implementation of the curved beam formulation within the conceptual framework of isogeometric analysis, which allows the exact description of the beam geometry. This avoids stress locking issues and the corresponding convergence problems encountered when classical straight beam finite elements are used to discretize the geometry of curved and twisted beams. Finally, the paper presents the solution of several numerical examples to demonstrate the accuracy and effectiveness of the proposed theoretical formulation and numerical implementation.

Cette thèse, réalisée dans le cadre d'une collaboration avec MocapLab, une entreprise spécialisée en motion capture, vise à déterminer le cadre mathématique le plus adapté et des descripteurs pertinents pour l'analyse des trajectoires de mouvement en langue des signes. En nous appuyant sur les principes du contrôle moteur, nous avons identifié le cadre défini par les formules de Frenet-Serret, incluant les paramètres de courbure, torsion et vitesse, comme particulièrement pertinent pour cette tâche. Ainsi, en introduisant de nouvelles approches d'analyse de courbes basées sur le cadre de Frenet, cette thèse contribue au développement de nouvelles méthodes dans les domaines de l'analyse de données fonctionnelles et de l'analyse de forme. La première partie de ce travail aborde le défi de l'estimation lisse des paramètres de courbures de Frenet, en traitant le problème comme une estimation de paramètres d'une équation différentielle dans SO(d), (d ≥ 1). Nous introduisons un algorithme Expectation-Maximization fonctionnel qui définit une méthode d'estimation unifiée des variables dans le groupe SE(3), fournissant des estimateurs lisses, plus fiables et robustes que les méthodes existantes. Dans la deuxième partie, deux nouvelles représentations des courbes sont introduites : les courbures de Frenet non paramétrisées et la Square Root Curvatures (SRC) transform, établissant de nouveaux cadres géométriques riemanniens pour les courbes lisses dans ℝᵈ, (d ≥ 1). En utilisant les informations géométriques d'ordre supérieur et dépendant de la paramétrisation, la Square Root Curvatures transform surpasse la représentation state-of-the-art Square-Root Velocity Function (SRVF) sur des résultats synthétiques. Étant donné une collection de courbes, ce type de géométrie nous permet de définir des critères statistiques efficaces pour estimer les formes moyennes de Karcher sur les espaces de formes riemanniens associés, qui se révèlent particulièrement performants sur des données bruitées. Enfin, ce cadre développé ouvre la voie à des applications plus pratiques dans le traitement de la langue des signes, comprenant l'étude des lois puissances sur nos données et le développement d'un modèle génératif pour le mouvement d'un point en langue des signes
This thesis, conducted in collaboration with MocapLab, a company specializing in motion capture, aims to determine the optimal mathematical framework and relevant descriptors for analyzing sign language motion trajectories. Drawing on principles of motor control, we identified the framework defined by the Frenet-Serret formulas, including curvature, torsion, and velocity parameters, as particularly suitable for this task. By introducing new curve analysis approaches based on the Frenet framework, this thesis contributes to developing novel methods in functional data analysis and shape analysis. The first part of this thesis addresses the challenge of smoothly estimating Frenet curvature parameters, treating the problem as parameter estimation of differential equation in SO(d), (d ≥ 1). We introduce a functional Expectation-Maximization algorithm that defines a unified variable estimation method in the SE(3) group, providing smoother estimators that are more reliable and robust than existing methods. In the second part, two new curve representations are introduced: unparametrized Frenet curvatures and the Square Root Curvatures (SRC) transform, establishing new Riemannian geometric frameworks for smooth curves in ℝᵈ, (d ≥ 1). Leveraging higher-order geometric information and parametrization dependence, the Square Root Curvatures transform outperforms the state-of-the-art Square-Root Velocity Function (SRVF) representation on synthetic results. Given a collection of curves, this type of geometry allows us to define efficient statistical criteria for estimating Karcher mean shapes on the associated Riemannian shape spaces, proving particularly effective on noisy data. Finally, this developed framework opens the door to more practical applications in sign language processing, including the study of power laws on our data and the development of a generative model for a point motion in sign language

The present dissertation develops an invariant framework for 3D gesture comparison studies. 3D gesture comparison without Lagrangian models is challenging not only because of the lack of prediction provided by physics, but also because of a dual geometry representation, spatial dimensionality and non-linearity associated to 3D-kinematics. In 3D spaces, it is difficult to compare curves without an alignment operator since it is likely that discrete curves are not synchronized and do not share a common point in space. One has to assume that each and every single trajectory in the space is unique. The common answer is to assert the similitude between two or more trajectories as estimating an average distance error from the aligned curves, provided that the alignment operator is found. In order to avoid the alignment problem, the method uses differential geometry for position and orientation curves. Differential geometry not only reduces the spatial dimensionality but also achieves view invariance. However, the nonlinear signatures may be unbounded or singular. Yet, it is shown that pattern recognition between intrinsic signatures using correlations is robust for position and orientation alike. A new mapping for orientation sequences is introduced in order to treat quaternion and Euclidean intrinsic signatures alike. The new mapping projects a 4D-hyper-sphere for orientations onto a 3D-Euclidean volume. The projection uses the quaternion invariant distance to map rotation sequences into 3D-Euclidean curves. However, quaternion spaces are sectional discrete spaces. The significance is that continuous rotation functions can be only approximated for small angles. Rotation sequences with large angle variations can only be interpolated in discrete sections. The current dissertation introduces two multi-scale approaches that improve numerical stability and bound the signal energy content of the intrinsic signatures. The first is a multilevel least squares curve fitting method similar to Haar wavelet. The second is a geodesic distance anisotropic kernel filter. The methodology testing is carried out on 3D-gestures for obstetrics training. The study quantitatively assess the process of skill acquisition and transfer of manipulating obstetric forceps gestures. The results show that the multi-scale correlations with intrinsic signatures track and evaluate gesture differences between experts and trainees.