Special Orthogonal Group Research Articles

Symmetry-Projected Nuclear-Electronic Hartree-Fock: Eliminating Rotational Energy Contamination.

We present a symmetry projection technique for enforcing rotational and parity symmetries in nuclear-electronic Hartree-Fock wave functions, which treat electrons and nuclei on equal footing. The molecular Hamiltonian obeys rotational and parity inversion symmetries, which are, however, broken by expanding in Gaussian basis sets that are fixed in space. We generate a trial wave function with the correct symmetry properties by projecting the wave function onto representations of the three-dimensional rotation group, i.e., the special orthogonal group in three dimensions SO(3). As a consequence, the wave function becomes an eigenfunction of the angular momentum operator which (i) eliminates the contamination of the ground-state wave function by highly excited rotational states arising from the broken rotational symmetry and (ii) enables the targeting of specific rotational states of the molecule. We demonstrate the efficiency of the symmetry projection technique by calculating the energies of the low-lying rotational states of the H2 and H3+ molecules.

On the Whittaker range of the generalized metaplectic theta lift

The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r+1 groups for the lift from the r-fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower.

Continuous appointed‐time prescribed performance attitude tracking control for rigid spacecraft with actuator faults on SO(3)

AbstractIn this article, the attitude tracking problem for rigid spacecraft subject to external disturbance, inertia uncertainties, and actuator faults is investigated. A novel continuous control strategy is proposed to guarantee that the attitude tracking errors converge with desired performance before an appointed time. In addition, the proposed control law can enable the closed‐loop system to be unwinding‐free. For the purpose, a modified appointed‐time performance function (APF) is used to construct a disturbance observer‐based adaptive attitude control law based on the special orthogonal group (SO(3)). With the aid of the modified APFs, desired performance metrics, including transient and steady‐state performance, on the attitude tracking error can be specified in advance without the requirement for initial values of the tracking system to avoid singularity. By resorting to error transformation and backstepping technique, the control design process is simplified and a few control gains are involved in the proposed control law. Rigorous stability analysis based on the Lyapunov theory is provided to prove the appointed‐time stability of the attitude tracking error as well as the boundedness of all signals in the closed‐loop system despite external disturbance, inertia uncertainties, and actuator faults. Finally, numerical simulations are conducted to support the analysis.

An intrinsic Bayesian bound for estimators on the Lie groups [formula omitted] and [formula omitted

In the past decade, estimation on manifolds has been paid an increased attention for making it possible to intrinsically account for constraints on the unknown parameters. Applications are widespread from computer vision to target tracking. To evaluate the performance of the algorithms, it is of interest to derive bounds on the estimation error. This issue has been mostly investigated for parametric estimation on Riemannian manifolds. This paper addresses the problem of computing a Bayesian lower bound on the norm of the estimation error when both the unknown variables and the observations belong to specific manifolds called Lie groups (LGs). For that purpose, a metric is considered that preserves the geometric properties of the latter. We firstly derive an inequality on the correlation matrix of this intrinsic error under the mild assumption of unimodular matrix LGs. Then, we obtain a closed-form expression of the bound for the special orthogonal group SO(3) and the special Euclidean group SE(3). Finally, we detail its derivation in the case when both the prior distribution and the likelihood are concentrated Gaussian distributions. The proposed Bayesian bound is implemented and tested on two estimation problems involving both unknown parameters and observations on SE(3): the inference of the pose of a camera and that of the centroid of a cluster of space debris.

A new tableau model for representations of the special orthogonal group

A new tableau model for representations of the special orthogonal group

Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates.

Currently, managing a group of satellites or robot manipulators requires coordinating their motion and work in a cooperative way to complete complex tasks. The attitude motion coordination and synchronization problems are challenging since attitude motion evolves in non-Euclidean spaces. Moreover, the equation of motions of the rigid body are highly nonlinear. This paper studies the attitude synchronization problem of a group of fully actuated rigid bodies over a directed communication topology. To design the synchronization control law, we exploit the cascade structure of the rigid body's kinematic and dynamic models. First, we propose a kinematic control law that induces attitude synchronization. As a second step, an angular velocity-tracking control law is designed for the dynamic subsystem. We use the exponential coordinates of rotation to describe the body's attitude. Such coordinates are a natural and minimal parametrization of rotation matrices which almost describe every rotation on the Special Orthogonal group SO(3). We provide simulation results to show the performance of the proposed synchronization controller.

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B.

In-motion initial alignment method using Cayley–Kalman filter on special orthogonal group

Aiming at the in-motion initial alignment problem of strapdown inertial navigation system (SINS), this paper proposed a precise direct in-motion alignment method based on attitude matrix representation and Cayley–Kalman filter on the special orthogonal group. There are two innovations in this paper. First, as an element of the special orthogonal group [Formula: see text], the error attitude matrix is used to construct an in-motion alignment model, and it is transformed into the vector space to represent the attitude error state. The error matrix was used as a state to decrease the influence of the state-dependent problem. Second, this method uses the Cayley transform to replace the Taylor expansion to derive the update equation, which is a no approximation mapping relation to assure accuracy when deriving the filtering algorithm. Simulation and experimental results demonstrate the proposed alignment method has advantages over the existing methods in alignment accuracy and alignment time and performs well in the in-motion alignment process of SINS.

Security of medical images based on special orthogonal group and Galois field

Security of medical images based on special orthogonal group and Galois field

Aritmethic lattices of SO(1,n) and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group SO(1,n) are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of mathbb {R}^d. For restrictions to the Euclidean ball in odd dimensions, to the rotation group textrm{SO}(3), and to the Grassmannian manifold mathcal {G}_{2,4}, we compute the kernels’ Fourier coefficients and determine their asymptotics. The L_2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For textrm{SO}(3), the nonequispaced fast Fourier transform is publicly available, and, for mathcal {G}_{2,4}, the transform is derived here. We also provide numerical experiments for textrm{SO}(3) and mathcal {G}_{2,4}.

Controllability and Accessibility on Graphs for Bilinear Systems Over Lie Groups

This article presents graph theoretic conditions for the controllability and accessibility of bilinear systems over the special orthogonal group, the special linear group and the general linear group, respectively, in the presence of drift terms. The controlled terms are assumed to take place between pairwise states. Such bilinear systems naturally induce two interaction graphs: one graph from the drift, and another from the controlled dynamics. As a result, the system controllability or accessibility becomes a property of the two graphs in view of the classical Lie algebra rank condition. We establish a systemic way of transforming the Lie bracket operations in the underlying Lie algebra, into specific operations of removing or creating links over the drift and controlled interaction graphs. As a result, we establish a series of graphical conditions for the controllability and accessibility of such bilinear systems, which rely only on the connectivity of the union of the drift and controlled interaction graphs. We present examples to illustrate the validity of the established results, and show that the proposed conditions are in fact considerably tight.

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

Knowledge graph embedding with the special orthogonal group in quaternion space for link prediction

Graph embedding is an important technique for improving the quality of link prediction models on knowledge graphs. Although embedding based on neural networks can capture latent features with high expressive power, geometric embedding has other advantages, such as intuitiveness, interpretability, and few parameters. Several recent works have used rotation in quaternion space to enhance performance, yet these rotations have been limited to 3D representations. This paper proposes Rotate4D, a novel model that performs 4D rotations in quaternion space using a special orthogonal group. Specifically, Rotate4D begins by embedding entities in quaternion space, then rotates the head embedding to the tail embedding in two different ways (isoclinic and double rotation). The group structure helps decompose the relation into unit vectors to perform 4D rotations in quaternion space. In addition, quaternion scaling is applied to solve hierarchical relations. With this approach, the proposed model improved link prediction performance compared with state-of-the-art models on standard datasets, especially for the Hits@1 metric (an improvement of 2.72% to 3.49%). In addition to these experiments, we also theoretically prove that Rotate4D can model all relational patterns, including symmetry, anti-symmetry, inversion, and composition.

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies

The Alesker–Bernig–Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker–Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge–Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge–Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.

The explicit Zelevinsky–Aubert duality

In this paper, we give an explicit computable algorithm for the Zelevinsky–Aubert duals of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.

Neighboring extremal nonlinear model predictive control of a rigid body on SO(3)

AbstractThe issue of implementing nonlinear model predictive control (NMPC) on mechanical systems evolving on special orthogonal group (SO(3)) is taken into consideration in the first place. Necessary conditions of optimality are extracted based on Lie group variational integrators, leading to a two-point boundary value problem (TPBVP) which is solved using sensitivity derivatives and indirect shooting methods. Fast Newton-like methods referred to as fast solvers which are commonly used to solve the TPBVP are established based on the repetition of a nonlinear process. The numerical schemes employed to alleviate the computation burden consist of eliminating some constraint-related but non-essential terms in the trend of sensitivity derivatives calculation and for solving the TPBVP equations. As another claim, assuming that a first attempt to resolve the NMPC problem is accessible, the problem subjected to some changes in its initial conditions (due to some re-planning schemes) can be resolved cost-effectively based on it. Instead of solving the whole optimization process from the scratch, the optimal control inputs and states of the system are updated based on the neighboring extremal (NE) method. For this purpose, two approaches are considered: applying NE method on the first solution that leads to a neighboring optimal solution, or assisting this latter by updating the NMPC-related optimization using exact TPBVP equations at some predefined intermediate steps. It is shown through an example that the first method is not accurate enough due to error accumulations. In contrast, the second method preserves the accuracy while reducing the computation time significantly.

Improvement of RSS-Based Measurement Based on Adaptive Kalman Filter Considering the Anisotropy on Antenna in Dynamic Environment

The classic propagation model of received signal strength (RSS) measurement is significantly affected by the environment. This paper divides the error sources of RSS ranging into three parts and deals with them in different methods. First, considering the influence of UAV pose on RSS, the compensation model is established through Gaussian Process Regression (GPR) for the first time by mapping the quaternion of the UAV and the attenuation caused by the influence of anisotropy on the antenna in the corresponding the special orthogonal group <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</i> (3). On this basis, the distribution of random compensation variables is derived by unscented transformation (UT) and covariance propagation. Besides, a polynomial function is employed for fitting the distance-dependent environment noise. Then, to deal with the RSS measurement instability caused by the environment noise and the uncertainty of compensation, the adaptive Kalman filter (AKF) is introduced. Finally, an UWB-based environmental adaptive algorithm (UEAA) is proposed to improve ranging precision by using the UWB device in a dynamic environment. By setting the threshold parameter, the UEAA enables the classic RSS ranging model to have a certain adaptive ability for the dynamic environment of UAV, which significantly reduces the error caused by model parameter mismatch. Several experiments are carried out on the UAV platform, and the results show that the improvement of the RSS propagation model is adequate.

Constrained Attitude Control of Uncertain Spacecraft With Appointed-Time Control Performance

This article studies the appointed-time attitude tracking control of the spacecraft on the special orthogonal group, with the attitude forbidden zone, the parameter uncertainties, and the external disturbances. A novel projection function is proposed, such that the normalized boresight vector of the sensitive instrument is mapped to a reduced dimensional vector in the Euclidean space. If the reduced dimensional vector is uniformly bounded, the constraint on the attitude forbidden zone will be satisfied at all the time. By virtue of the designed reduced dimensional vector and the associated auxiliary vectors, a set of vector-based error functions, the appointed-time performance constraints, and the according switching law is carefully constructed. The proposed vector-based adaptive control scheme ensures that the spacecraft attitude can satisfy the attitude constraint and appointed-time control performance simultaneously, in the presence of parameter uncertainties and external disturbances. Simulation results show the effectiveness of the designed control scheme.

Anti-disturbance Trajectory Tracking Control for a Quadrotor UAV with Input Constraints⋆

This paper presents the trajectory tracking control for a quadrotor unmanned aerial vehicle(UAV) with disturbances and input constraints. The UAV's system is decomposed into an outer position loop and an inner attitude loop. The position tracking control is achieved by a class of saturated proportional-derivative(PD) control laws. Assuming that the unknown disturbances mainly effect on the translational dynamics, an adaptive control term is investigated to compensate the effects of the disturbances. The position tracking control force implies a coupled-attitude system that connects the inner attitude loop with the outer position loop. An intrinsic PD control law, which is directly designed on the special orthogonal group SO(3), is proposed to track the desired attitude. In order to reduce the calculational difficulty of the attitude tracking control law, an observer-based control law is further developed while the control law and observer can be designed separately. Simulation results complete this work.