Orthogonal Group Research Articles

Complete Systems of Galileo Invariants of a Motion of Parametric Figure in the Three Dimensional Euclidean Space

Let $E^{3}$ be the 3-dimensional Euclidean space and $S$ be a set with at least two elements. The notions of an $S$-parametric figure and the motion of an $S$-parametric figure in $E^{3}$ are defined. Complete systems of invariants of an $S$-parametric figure in $E^{3}$ for the orthogonal group $O(3,R)$ , the special orthogonal group $SO(3,R)$, Euclidean group $MO(3,R)$, the special Euclidean group $MSO(3,R)$ and Galileo groups $Gal_{1}(3,R)$ , $Gal^{+}_{1}(3,R)$ are obtained.

Biomechanical comparison of three internal fixation configurations for low transcondylar fractures of the distal humerus

BackgroundWe aimed to evaluate the biomechanical stiffness and strength of different internal fixation configurations and find suitable treatment strategies for low transcondylar fractures of the distal humerus. Methods and materialsThirty 4th generation composite humeri were used to create low transcondylar fracture models that were fixed by orthogonal and parallel double plates as well as posterolateral plate and medial screw (PPMS) configurations (n=10 in each group) using an anatomical locking compression plate-screw system and fully threaded medial cortical screws. Posterior bending (maximum 50 N), axial loading (maximum 200 N) and internal rotation (maximum 10 N·m) were tested, in that order, for each specimen. Stiffness under different biomechanical settings among different configurations were compared. Another 18 sets of fracture models were created using these three configurations (n=6 in each group) and the load to failure under axial loading among different configurations was compared. ResultsUnder posterior bending, the stiffness of parallel group was higher than orthogonal group (P<0.001), and orthogonal group was higher than PPMS group (P<0.001). Under axial loading, the stiffness of parallel group was higher than orthogonal group (P=0.001) and PPMS group (P<0.001); however, the difference between orthogonal and PPMS group was not statistically significant (P>0.05). Under internal rotation, the stiffness of parallel group was higher than orthogonal group (P=0.044), and orthogonal group was higher than PPMS group (P=0.029). In failure test under axial loading, the load to failure in the orthogonal group was lower than parallel group (P=0.009) and PPMS group (P=0.021), but the difference between parallel group and PPMS group was not statistically significant (P>0.05). All specimens in orthogonal group demonstrated “distal medial failure”; most specimens had “distal medial and trochlear failure” in the parallel group; most specimens exhibited “contact failure” in the PPMS group. ConclusionFor treating low transcondylar fractures, the overall stiffness and strength of the parallel configuration were superior to those of the orthogonal and PPMS configurations. Nevertheless, the PPMS configuration can provide adequate stability and stiffness comparable to double-plate configurations under axial loading. Therefore, the PPMS construct may have certain clinical value.

The orthogonal unit group of the trivial source ring

Let G be a finite group, p a prime, and (K,O,F) a p-modular system. We prove that the trivial source ring of OG is isomorphic to the ring of coherent G-stable tuples (χP), where χP is a virtual character of K[NG(P)/P], P runs through all p-subgroups of G, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of G with the group of coherent G-stable tuples (φP) of homomorphisms NG(P)/P→F×. The orthogonal unit group of the trivial source ring of OG is of interest, since it embeds into the group of p-permutation autoequivalences of OG.

Uncertainty-propagated Cartesian coordinated human–robot collaboration on Riemannian manifold with hidden state-space model

This paper proposes a novel Cartesian coordinated human–robot collaboration framework derived from the hidden state-space models, which are established on the behaviour cloning of both the human and robot in a nonparametric form by using dual quaternion on Riemannian manifold. Based on the Taylor linearisation, this framework could provide an analytical approximation of the posterior distribution and hence infer six DoF Cartesian hidden state variables of the collaborative manipulator given human observation and its uncertainties. As the full Cartesian pose (special Euclidean group, SE(3)) includes translation and rotation (special orthogonal group, SO(3)), which is not Euclidean, directly encoding the Cartesian motions with nonparametric regression in Euclidean space could cause significant errors. This paper addresses this issue by defining both human and robot behaviours using modified quaternion and cloning them in the tangent space of the Riemannian manifold. Not akin to other coordinated human–robot collaboration methods, the collaboration framework not only preserves the adaptation functionalities but also propagates full Cartesian state variables and their uncertainties during real-time coordinated collaboration implementation. Leveraging on the Gaussian mixture model on the Riemannian manifold, the multiple task recognition is further addressed to enhance the generalisation capability of the framework presented. The application feasibility is demonstrated in both theoretical comparison simulation and experiments.

Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions

We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves.

Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

AbstractA bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

2-Adic point counting on K3 surfaces

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p-adic method, we count the number of points on some K3 surfaces over the field mathbb {F}_{!p}, for all primes p < 10^8.

Orthogonal stability

A character (ordinary or modular) is called orthogonally stable if all non-degenerate quadratic forms fixed by representations with those constituents have the same determinant mod squares. We show that this is the case provided there are no odd-degree orthogonal constituents. We further show that if the reduction mod p of an ordinary character is orthogonally stable, this determinant is the reduction mod p of the ordinary one. In particular, if the characteristic does not divide the group order, we immediately see in which orthogonal group it lies. We sketch methods for computing this determinant, and give some examples.

A Uniformly Accurate Scheme for the Numerical Integration of Penalized Langevin Dynamics

In molecular dynamics, penalized overdamped Langevin dynamics are used to model the motion of a set of particles that follow constraints up to a parameter $\varepsilon$. The most used schemes for simulating these dynamics are the Euler integrator in $\mathbb{R}^d$ and the constrained Euler integrator. Both have weak order one of accuracy, but work properly only in specific regimes depending on the size of the parameter $\varepsilon$. We propose in this paper a new consistent method with an accuracy independent of $\varepsilon$ for solving penalized dynamics on a manifold of any dimension. Moreover, this method converges to the constrained Euler scheme when $\varepsilon$ goes to zero. The numerical experiments confirm the theoretical findings, in the context of weak convergence and for the invariant measure, on a torus and on the orthogonal group in high dimension and high codimension.

Solving orthogonal group synchronization via convex and low-rank optimization: tightness and landscape analysis

Group synchronization aims to recover the group elements from their noisy pairwise measurements. It has found many applications in community detection, clock synchronization, and joint alignment problem. This paper focuses on the orthogonal group synchronization which is often used in cryo-EM and computer vision. However, it is generally NP-hard to retrieve the group elements by finding the least squares estimator. In this work, we first study the semidefinite programming (SDP) relaxation of the orthogonal group synchronization and its tightness, i.e., the SDP estimator is exactly equal to the least squares estimator. Moreover, we investigate the performance of the Burer-Monteiro factorization in solving the SDP relaxation by analyzing its corresponding optimization landscape. We provide deterministic sufficient conditions which guarantee: (i) the tightness of SDP relaxation; (ii) optimization landscape arising from the Burer-Monteiro approach is benign, i.e., the global optimum is exactly the least squares estimator and no other spurious local optima exist. Our result provides a solid theoretical justification of why the Burer-Monteiro approach is remarkably efficient and effective in solving the large-scale SDPs arising from orthogonal group synchronization. We perform numerical experiments to complement our theoretical analysis, which gives insights into future research directions.

On the eigenstructure of rotations and poses: commonalities and peculiarities

Locating vehicles, targets and objects in three-dimensional space is key to many fields of science and engineering such as robotics, aerospace, computer vision and graphics. Rotations and poses (position plus orientation) of bodies can be expressed in a variety of ways. Rotation matrices constitute one of the classic matrix Lie groups, the special orthogonal group— SO ( 3 ) . Poses can likewise be represented by matrices. One such representation is embodied in a 4 × 4 matrix establishing another famous matrix Lie group, the special Euclidean group— SE ( 3 ) . An alternative representation of pose uses 6 × 6 matrices and is referred to as the group of pose adjoints— Ad ( SE ( 3 ) ) . The eigenstructures of these representations reveal much about them from Euler’s theorem for rotations to the Mozzi–Chasles theorem for the general displacement of a rigid body. While the eigenstructure of SO ( 3 ) has been extensively studied, those of SE ( 3 ) and Ad ( SE ( 3 ) ) have hardly received the same scrutiny yet their structure is much richer. Motivated by their importance in kinematics and dynamics, we provide here a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations. An eigendecomposition of pose matrices reveals that they can be cast into a form similar to that of rotations although the structure of the former can vary depending on the nature of the pose involved. In particular, the pose matrices of SE ( 3 ) and Ad ( SE ( 3 ) ) cannot generally be diagonalized as can rotation matrices but they of course do yield to a Jordan normal form, from which we can identify a principal-axis pose in much the same manner that we can a principal-axis rotation. We also address the minimal polynomials for poses and derive a novel expression for the Jacobian in Ad ( SE ( 3 ) ) . Finally, we argue that the true counterpart to SO ( 3 ) for poses is not SE ( 3 ) but Ad ( SE ( 3 ) ) .

H ∞ inverse optimal attitude tracking on the special orthogonal group SO(3)

The problem of attitude tracking in the presence of disturbances is addressed by combining inverse optimality and disturbance attenuation. Conditions are provided which ensure that an inverse optimal nonlinear attitude control problem is solved and a meaningful cost function is minimised. The approach results in a PD-type attitude control law on the Special Orthogonal Group which guarantees that, for almost all initial conditions, the energy gain from an exogenous disturbance to a specified error signal respects a given upper bound. For numerical simulations, a satellite attitude tracking problem from literature is considered. The controller gains are tuned using a decoupled linearised single-axis model and the structured control synthesis method. Results indicate that the proposed controller has good tracking and disturbance attenuation capability. In particular, for the problem under consideration, the inverse optimal controller is seen to have better transient performance than its continuous-time quaternion counterpart. Moreover, for initial errors up to radians, it has comparable performance to the shortest-path quaternion PD controller.

Classical group matrix models and universal criticality

We study generalizations of the Gross-Witten-Wadia unitary matrix model for the special orthogonal and symplectic groups. We show using a standard Coulomb gas treatment — employing a path integral formalism for the ungapped phase and resolvent techniques for the gapped phase with one coupling constant — that in the large N limit, the free energy normalized modulo the square of the gauge group rank is twice the value for the unitary case. Using generalized Cauchy identities for character polynomials, we then demonstrate the universality of this phase transition for an arbitrary number of coupling constants by linking this model to the random partition based on the Schur measure.

Definite orthogonal modular forms: computations, excursions, and discoveries

We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.

Presentations of projective quantum groups

Given an orthogonal compact matrix quantum group defined by intertwiner relations, we characterize by relations its projective version. As a sample application, we prove that $PU_n^+=PO_n^+$.

Near-optimal performance bounds for orthogonal and permutation group synchronization via spectral methods

Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great popularity due to their efficiency and convenience. We will study the performance guarantees of the spectral methods in solving these two synchronization problems by investigating how well the computed eigenvectors approximate each group element individually. We establish our theory by applying the recent popular leave-one-out technique and derive a block-wise performance bound for the recovery of each group element via eigenvectors. In particular, for orthogonal group synchronization, we obtain a near-optimal performance bound for the group recovery in presence of additive Gaussian noise. For permutation group synchronization under random corruption, we show that the widely-used two-step procedure (spectral method plus rounding) can recover all the group elements exactly if the SNR (signal-to-noise ratio) is close to the information theoretical limit. Our numerical experiments confirm our theory and indicate a sharp phase transition for the exact group recovery.

On the Observability of Attitude With Single Direction Measurements

The attitude of a rigid body evolves on the three-dimensional (3-D) special orthogonal group, and it is often estimated by measuring reference directions, such as gravity or magnetic field, using onboard sensors. As a single direction measurement provides a 2-D constraint, it has been widely accepted that at least two nonparallel reference directions should be measured, or the reference direction should change over time, to determine the attitude completely. This article uncovers an intriguing fact that the attitude can actually be estimated by using multiple measurements of a single, fixed reference direction, provided that the angular velocity and the direction measurements are resolved in appropriate frames, respectively. More specifically, after recognizing that the attitude uncertainties propagated over the left-trivialized stochastic kinematics are distinct from those over the right-trivialized one, stochastic attitude observability with single direction measurements is formulated by an information theoretic analysis. These are illustrated by experiments.

A Novel DVL Calibration Method Based on Robust Invariant Extended Kalman Filter

The calibration accuracy of the Doppler Velocity Log (DVL) error directly affects the performance of the SINS/DVL integrated navigation system. Previous studies of DVL calibration have not dealt with DVL measurement outliers. Therefore, in this paper, a novel DVL calibration method is proposed to reduce the influence of outliers on calibration accuracy. Introducing the Lie group theory to the DVL calibration study for the first time, and taking advantage of the Special Orthogonal Group of order 3 [SO(3)] representation, a new DVL calibration model based SO(3) group with the ability to represent large misalignment angles is established. And an invariant extended Kalman filter (IEKF) design for the DVL calibration model is introduced. Then, a robust IEKF is proposed by combining the linear error propagation equation of the SO(3)-based DVL calibration model with the statistical similarity measure (SSM) theory. The proposed robust IEKF algorithm enriches the IEKF theory and has the ability to accurately estimate the state of the Lie group model in outlier environments. The simulation and lake trial are used to illustrate the effectiveness and superiority of the proposed DVL calibration method.

Uncertainty Propagation for General Stochastic Hybrid Systems on Compact Lie Groups

This paper deals with uncertainty propagation of general stochastic hybrid systems (GSHS) where the continuous state space is a compact Lie group. A computational framework is proposed to solve the Fokker-Planck (FP) equation that describes the time evolution of the probability density function for the state of GSHS. The FP equation is split into two parts: the partial differential operator corresponding to the continuous dynamics, and the integral operator arising from the discrete dynamics. These two parts are solved alternatively using the operator splitting technique. Specifically, the partial differential equation is solved by the spectral method where the density function is decomposed into a linear combination of a complete orthonormal function basis brought forth by the Peter-Weyl theorem, thereby resulting an ordinary differential equation. Next, the integral equation is solved by approximating the integral by a finite summation using a quadrature rule. The proposed method is then applied to a three-dimensional rigid body pendulum colliding with a wall, evolving on the product of the three-dimensional special orthogonal group and the Euclidean space. It is illustrated that the proposed method exhibits numerical results consistent with a Monte Carlo simulation, while explicitly generating the density function that carries the complete stochastic information of the hybrid state.

On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds

The systematic study of harmonic self-maps on cohomogeneity one manifolds has recently been initiated by Püttmann and the second named author in [19]. In this article we investigate the corresponding Jacobi equation describing the equivariant stability of such harmonic self-maps. Besides several general statements concerning their equivariant stability we explicitly solve the Jacobi equation for some harmonic self-maps in the cases of spheres, special orthogonal groups and SU(3). In particular, we show by an explicit calculation that for specific cohomogeneity one actions on the sphere the identity map is equivariantly stable.