Euclidean Group Research Articles

Hilbert space of dyons

We revisit the construction of the Hilbert space of nonrelativistic particles moving in three spatial dimensions. This is given by the space of sections of a line bundle that can in general be topologically nontrivial. Such bundles are classified by a set of integers—one for each pair of particles—and arise physically when we describe the interactions of dyons, particles which carry both electric and magnetic charges. The choice of bundle fixes the representation of the Euclidean group carried by the Hilbert space. These representations are shown to recover the “pairwise helicity” formalism recently discussed in the literature. Published by the American Physical Society 2024

From the Cosserats mechanics backgrounds to modern field theory

In the paper, yet weekly known, Cosserats’ original four concepts as follow: the four-time unification of rigid body dynamics, statics of flexible rods, statics of elastic surfaces and 3D deformable body dynamics; the intrinsic formulation based on the local, von Helmholtz symmetry group of monodromy; the invariance under the Euclidean group. The concept of a set of low-dimensional branes immersed into Euclidean space are revalorized and explained in terms of the modern gauge field theory and the extended strings theory. Additionally, some useful mathematical tools that connect the continuum mechanics and the classical field theory (for instance, the convective coordinates, von Mises’ “Motorrechnung”, the Grassmann extensions, Euclidean invariance, etc.) are involved in the historical explanation that how the ideas were developing themself.

Polynomial Regression on Lie Groups and Application to SE(3).

In this paper, we address the problem of estimating the position of a mobile such as a drone from noisy position measurements using the framework of Lie groups. To model the motion of a rigid body, the relevant Lie group happens to be the Special Euclidean group SE(n), with n=2 or 3. Our work was carried out using a previously used parametric framework which derived equations for geodesic regression and polynomial regression on Riemannian manifolds. Based on this approach, our goal was to implement this technique in the Lie group SE(3) context. Given a set of noisy points in SE(3) representing measurements on the trajectory of a mobile, one wants to find the geodesic that best fits those points in a Riemannian least squares sense. Finally, applications to simulated data are proposed to illustrate this work. The limitations of such a method and future perspectives are discussed.

Topological photon, spin, and Euclidean group E(2)

The proposition that photon is a topological object [S. C. Tiwari, J. Math. Phys. 49, 032303 (2008)] is given rigorous foundation based on pure vector field theory independent of the electromagnetic fields. Holomorphy of 4-dimensional spacetime and the nature of topological obstructions are investigated to articulate singular vortex model for photon. The utility of the Euclidean symmetry group is discussed in detail for the proposed photon model.

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

Контракции калибровочных групп и спонтанное нарушение симметрии

Contractions of gauge models with orthogonal Cayley-Klein groups SO(2; ϵ), SO(3; ϵ), unitary groups SU(2; ϵ) as gauge groups are studied. In the limit of zero contraction parameters, orthogonal groups are isomorphic to the nonsemisimple Euclidean and Newton groups of the corresponding dimension, and the spaces of matter fields become fibered spaces with a degenerate metric. Particular attention is paid to the coordination of spontaneous symmetry breaking with the group contraction procedure. It is shown that contracted gauge theories describe the same set of fields with the same masses as theories with the original simple groups, if the chosen vacuum in the corresponding limit belonged to the base of the fibered space of matter fields. Lagrangians of the models depending on the contraction parameters are obtained, which makes it possible to trace the order of zeroing of terms in the Lagrangians as the contraction parameters tend to zero.

Gauge theory on ρ-Minkowski space-time

We construct a gauge theory model on the 4-dimensional ρ-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differential calculi based on twisted derivations together with a twisted notion of noncommutative connection. The twisted derivations pertain to the Hopf algebra of ρ-deformed translations, a Hopf subalgebra of the ρ-deformed Poincaré algebra which can be viewed as defining the quantum symmetries of the ρ-Minkowski space-time. The gauge theory model is left invariant under the action of the ρ-deformed Poincaré algebra. The kinetic part of the action is found to coincide with the one of the usual (commutative) electrodynamics.

Mass property estimation on TSE(3) via unscented Kalman filter using RCS thrusters

Estimation of spacecraft mass properties in the presence of noise is a nontrivial problem that, when properly addressed, can substantially reduce the crew time devoted to cargo management, control effort expended to manage fuel mass, or both. The estimation of such properties is treated here using a dual unscented Kalman filter (UKF) where the dynamics of the rigid-body spacecraft are formulated on the special Euclidean group SE(3) and its tangent bundle TSE(3) to avoid singularity and nonuniqueness issues found in attitude parameterization sets. The dual UKF on TSE(3) is developed, and a specific methodology is proposed for the estimation of the constant mass properties. An excitation approach is employed in the mass property estimation scheme. Arbitrary sensor suite placements are considered and implemented in the measurement model to capture real-world spacecraft behavior. Furthermore, a reaction control system with duty cycle constraints and noisy thrust values is implemented to account for uncertainty in the excitation inputs. With these features, the methods contained herein are found to be effective at estimating mass properties.

Nonlinear static and dynamic analysis of corotational shell formulated on the special Euclidean group SE(3)

Nonlinear static and dynamic analysis of corotational shell formulated on the special Euclidean group SE(3)

Nonlinear Attitude and Altitude Trajectory Tracking Control of a Bicopter UAV in Geometric Framework

In this paper, we deal with a Bicopter drone that has two thrusters and two tilting servos. Both the position and attitude dynamics of Bicopter are globally expressed on the Special Euclidean group SE3. A simple control allocation method is proposed to map between the control wrench and actuator inputs for the Bicopter. A geometric nonlinear attitude and altitude tracking controller is developed for the Bicopter and the asymptotic stability analysis is performed using the Lyapunov method for the closed-loop nonlinear system. The performance of the proposed altitude and attitude stabilization controller is validated through experimental hardware developed in-house. The attitude controller performance is validated through simulations and shown to be comparable against an linear matrix inequality-based control law.

Breast Cancer Detection in Mammography using Faster Region Convolutional Neural Networks and Group Convolution

Breast cancer is a prevalent and potentially life-threatening medical condition characterized by uncontrolled cell proliferation within breast tissue. This global health concern predominantly among women, although it can affect men as well. The timely and accurate detection of breast cancer is critical, as it significantly influenced treatment outcomes and survival rates. Despite advancements in human-based diagnostic methods, inherent limitations such as subjectivity, human error, and fatigue persist. To address these constraints, computer-vision-based techniques have been explored for breast cancer detection, aiming to enhance early detection, reduce diagnostic errors, and improve patient outcomes. This study advances previous approaches by integrating group convolution and the Special Euclidean motion group as a features extractor into the Faster Region Convolutional Neural Networks framework of Detectron2. This integration offers the advantage of enhancing the Convolutional Neural Network by incorporating a method to preserve invariant and equivariant features, effectively leveraging symmetries in mammography images. The INbreast dataset, comprising 410 images from 115 cases, including 90 instances with bilateral breast involvement in women, was utilized. To ensure data compatibility, DICOM to PNG and XML to JSON format conversions were performed. A data enhancement pipeline, encompassing techniques such as image cropping, truncation normalization, contrast-limited adaptive histogram equalization, and image synthesis, was employed for data preprocessing and cleaning. The proposed technique demonstrated competitive results in breast cancer detection, achieving a recall rate of 97.22. This underscores the efficacy of the integrated approach in improving diagnostic accuracy and holds promise for advancing computer-vision-based breast cancer detection methodologies.

An Improved Initial Alignment Method Based on SE2(3)/EKF for SINS/GNSS Integrated Navigation System with Large Misalignment Angles.

This paper proposes an improved initial alignment method for a strap-down inertial navigation system/global navigation satellite system (SINS/GNSS) integrated navigation system with large misalignment angles. Its methodology is based on the three-dimensional special Euclidean group and extended Kalman filter (SE2(3)/EKF) and aims to overcome the challenges of achieving fast alignment under large misalignment angles using traditional methods. To accurately characterize the state errors of attitude, velocity, and position, these elements are constructed as elements of a Lie group. The nonlinear error on the Lie group can then be well quantified. Additionally, a group vector mixed error model is developed, taking into account the zero bias errors of gyroscopes and accelerometers. Using this new error definition, a GNSS-assisted SINS dynamic initial alignment algorithm is derived, which is based on the invariance of velocity and position measurements. Simulation experiments demonstrate that the alignment method based on SE2(3)/EKF can achieve a higher accuracy in various scenarios with large misalignment angles, while the attitude error can be rapidly reduced to a lower level.

Robust point cloud registration for map-based autonomous robot navigation

Robust point cloud registration for map-based autonomous robot navigation

Equivariant neural network force fields for magnetic materials

Neural network force fields have significantly advanced ab initio atomistic simulations across diverse fields. However, their application in the realm of magnetic materials is still in its early stage due to challenges posed by the subtle magnetic energy landscape and the difficulty of obtaining training data. Here we introduce a data-efficient neural network architecture to represent density functional theory total energy, atomic forces, and magnetic forces as functions of atomic and magnetic structures. Our approach incorporates the principle of equivariance under the three-dimensional Euclidean group into the neural network model. Through systematic experiments on various systems, including monolayer magnets, curved nanotube magnets, and moiré-twisted bilayer magnets of CrI3, we showcase the method’s high efficiency and accuracy, as well as exceptional generalization ability. The work creates opportunities for exploring magnetic phenomena in large-scale materials systems.

Heisenberg Double of the Generalized Quantum Euclidean Group and Its Representations

Heisenberg Double of the Generalized Quantum Euclidean Group and Its Representations

Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)

Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)

Jordan automorphisms and derivatives of symmetric cones

Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that the derivative automorphisms of a symmetric cone are closely related to the automorphisms of its associated Euclidean Jordan algebra. In the process, we find the automorphism group of the quaternion positive-semidefinite cone and list explicitly the Jordan-automorphisms of the quaternion Hermitian matrices. We also address the path-connectedness of the simple Euclidean Jordan-automorphism groups.

Geometric stochastic ray propagation using the special Euclidean group.

This paper describes a stochastic model of ray trajectory propagation through a medium-such as the ocean-which has an uncertain sound speed profile. We frame ray propagation as a geometric fractal Brownian motion process on the special Euclidean group of dimension two, SE(2). The framing includes diffusion parameters to describe how the stochastic rays deviate from the expected rays, and these diffusion parameters are a function of the uncertainty in the sound speed profile. We demonstrate this framing for the classical Munk profile and a double-ducted profile in the Beaufort.

Mathematical Modeling of Robotic Locomotion Systems

This article deals with the presentation of an alternative approach that uses methods of geometric mechanics, which allow one to see into the geometrical structure of the equations and can be useful not only for modeling but also during the design of symmetrical locomotion systems and their control and motion planning. These methods are based on extracting the symmetries of Lie groups from the locomotion system in order to simplify the resulting equations. In the second section, the special two-dimensional Euclidean group SE2 and its splitting into right and left actions are derived. The physical interpretation of the local group and spatial velocities is investigated, and by virtue of the fact that both of these velocities represent the same velocity from a physical point of view, the dependence between them can be found by means of the adjoint action. The third section is devoted to the modeling and analysis of the planar locomotion of the symmetrical serpentine robot; the positions and local group velocities of its links are derived, the vector fields for the local connections are given, and the trajectories of the individual variables in the lateral movement of the kinematic snake are shown. At the end of the article, the overall benefits of the scientific study are summarized, as is the comparison of the results from the simulation phase, while the most significant novelty compared to alternative publications in the field can be considered the realization of this study with a description of the relevant methodology at a detailed level; that is, the locomotion results confirm the suitability of the use of geometric mechanics for these symmetrical locomotion systems with nonholonomic constraints.

Efficient learning of Scale-Adaptive Nearly Affine Invariant Networks

Recent research has demonstrated the significance of incorporating invariance into neural networks. However, existing methods require direct sampling over the entire transformation set, notably computationally taxing for large groups like the affine group. In this study, we propose a more efficient approach by addressing the invariances of the subgroups within a larger group. For tackling affine invariance, we split it into the Euclidean group E(n) and uni-axial scaling group US(n), handling invariance individually. We employ an E(n)-invariant model for E(n)-invariance and average model outputs over data augmented from a US(n) distribution for US(n)-invariance. Our method maintains a favorable computational complexity of O(N2) in 2D and O(N4) in 3D scenarios, in contrast to the O(N6) (2D) and O(N12) (3D) complexities of averaged models. Crucially, the scale range for augmentation adapts during training to avoid excessive scale invariance. This is the first time nearly exact affine invariance is incorporated into neural networks without directly sampling the entire group. Extensive experiments unequivocally confirm its superiority, achieving new state-of-the-art results in affNIST and SIM2MNIST classifications while consuming less than 15% of inference time and fewer computational resources and model parameters compared to averaged models.