The Segal-Bargmann transform on classical matrix Lie groups

In this tutorial, exponentiation and factorization (decomposition) formulas are derived and discussed for common matrix operators that arise in studies of classical dynamics, linear and nonlinear optics, and special relativity. To understand the physical properties of systems of common interest, one first needs to understand the mathematical properties of the symplectic group Sp(2), the special unitary groups SU(2) and SU(1,1), and the special orthogonal groups SO(3) and SO(1,2). For these groups, every matrix (of practical interest) can be written as the exponential of a generating matrix, which is a linear combination of three fundamental matrices (generators). For Sp(2), SU(1,1) and SO(1,2), every matrix also has a Schmidt decomposition, in which it is written as the product of three simpler matrices. The relations between the entries of the matrix, the generator coefficients and, where appropriate, the Schmidt-decomposition parameters are described in detail. It is shown that Sp(2) is homomorphic to (has the same structure as) SU(1,1) and SO(1,2), and SU(2) is homomorphic to SO(3). Several examples of these homomorphisms (relations between Schmidt decompositions and product rules) are described, which illustrate their usefulness (complicated results can be anticipated or derived easily). This tutorial is written at a level that is suitable for senior undergraduate students and junior graduate students.

The relationship between stress rates and tangent stiffness values with special orthogonal group SO (3) for finite-element formulation was shown in Part1, and stiffness elements of finite beam element using SO (3) were shown in Part2. Those stress rates are the Truesdell stress rate, the Jaumann stress rate, the Neo-Green stress rate and the Ishihara stress rate, a nonsymmetric stress rate that will be defined by this author in this report. In this study, the constitutive equation of the Ishihara stress rate will be considered. It will be shown that although the stress rate and the strain rate are nonsymmetric, the material stiffness matrix is symmetric. Next, it is shown that although the geometric stiffness of a rigid rotation is nonsymmetric, even if only the symmetric part of the stiffness is used, the second-order convergent rate determined by the Newton-Raphson method is conserved. Finally, it will be shown that the tangent stiffness formulated with SO (3) is different from that of the conventional formulation with linear rotation.

In this paper, we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n) due to its practical importance. Our technique is based on the exponential representation of the elements of the group, which allows as to transfer the problem to the corresponding Lie algebra (n) and benefit from the linearity of this space. Due to the exponential representation, we can obtain a high degree of smoothness of an interpolating curve. The underlying problem is challenging because the standard SO(n) → (n) map is multivalued.

Although tightly coupled Precise Point Positioning/Inertial Navigation Systems (PPP/INS) are capable of decimeter-level accuracy, conventional filtering frameworks suffer from a theoretical disconnect: attitude errors are mapped onto the special orthogonal group SO(3), while position and velocity errors are treated in Euclidean space. This mathematical heterogeneity often induces error accumulation during state propagation. To resolve this, this paper presents a multi-frequency PPP-AR/INS framework based on the Left-Invariant Lie Group. By strictly defining all state errors on the Lie group manifold, estimation consistency is significantly enhanced. Field experiments confirm the superiority of the proposed approach over traditional methods. Specifically, under open-sky conditions, the left-invariant formulation outperforms the right-invariant and conventional method by reducing 3D positioning errors by 3.3% and 9.3%, respectively. In challenging environments with partial signal blockage, the method yields improvements of 4.8% for 2D and 13.1% for 3D. Furthermore, during complete GNSS outages, the enhanced accuracy of the IMU state estimation mitigates drift, lowering 2D and 3D errors by 11.2% and 6.3%, respectively. Notably, these gains are achieved with only a marginal 2.4% increase in computational load, validating the efficiency of the method for real-time applications.

In this paper, we construct three ternary linear codes associated with the orthogonal group O − (2, q) and the special orthogonal groups SO − (2, q) and SO − (4, q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of “Gauss sums” for the orthogonal and special orthogonal groups O − (2n, q) and SO − (2n, q).

The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Finite element analysis of 3-dimensional large rotation problem has been possible with nonlinear beam element by J.C.Simo,L.Vu-Quoc[1,2]. Their beam model is described by a configulation manifold which involves special orthogonal group SO(3). But the special orthogonal group is noncomutative, so the tangent stiffness matrix becomes non-symmetric away from equilibrium.

In the previous papers, the relationship between the stress rate and the tangent stiffness was derived using the special orthogonal group SO (3) in the finite element formulation. The stress rates used are the Truesdell stress rate, the Jaumann stress rate, the Neo-Green stress rate and the Ishihara stress rate. In that work, the stiffness elements of the tangent stiffnesses of beam elements with SO (3) were given. Those elements include material stiffness ΔδΠm, geometric stiffness of rigid rotation ΔδΠ, geometric stiffness of stretch to stress direction ΔδΠ, geometric stiffness of stretch perpendicular to area ΔδΠ and geometric stiffness of stretch to deformation rate ΔδΠ. In this paper, post-buckling analyses are made using the beam element of Ishihara stress rate. It is shown that the results are almost the same as the results of both 2-dimensional and 3-dimensional analyses of elastica, because the beam element has only geometric stiffness of rigid rotation as geometric stiffness and it can exactly show the nonlinearity of rigid rotation. It is also shown that the beam element without SO (3) can't be used for 3-dimensional large rotation analysis.

Since the exactly solvable higher-dimensional quantum systems with certain central potentials are usually related to the real orthogonal group O(N) defined by orthogonal n×n matrices, we shall give a brief review of some basic properties of group O(N) based on the monographs and textbooks. We first outline the development in order to make the reader recognize its importance in physics and then review the tensor and spinor representations of the group SO(N) and Γ matrix group.

GES Attitude Observers – Part I: Multiple General Vector Observations

The part of a one-electron coefficient of fractional parentage (cfp) depending, for the atomic l shell, on the spin S, seniority υ, and electron number n of the daughter state, is factored into two parts, the first with dependence on n and υ, and the second with dependence on υ and S. Various techniques are used to extend these results to the two-electron cfp. These entail deriving analogues of the 6-j symbols for the symplectic group Sp(4l + 2) and the special orthogonal group SO(2l + 1). Explicit algebraic expressions are provided, and the choice of multiplicity indices and their circumvention are discussed.

Fusion or not: Learning visual relocalization with matrix Fisher distribution

In this paper, a method to compute the best geodesic approximation of a set of points that belong to a Riemannian manifold is proposed. This method is based on a gradient descent technique on the tangent bundle of the manifold. An expression for the gradient is derived using the theory of Jacobi fields and an efficient numerical technique is proposed to compute these Jacobi fields. The presented approach is valid on any locally symmetric space, and the sphere S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , the set of symmetric positive definite matrices P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , the special orthogonal group SO(3) and the Grassmann manifold Grass(n; p) are considered.

Attitude and gyro bias estimation for a VTOL UAV

The disturbance rejection problem of aerial vehicles is a widely concerned issue. For attitude dynamics, we propose a method for directly designing a geometric observer (GOB) based on the special orthogonal group SO(3), thus reducing the impact of model mismatch and being unrestricted by the aircraft attitude. Considering the type of measurement available, we have developed two GOBs: GOB-I, which uses both attitude and angular velocity measurements, and GOB-II, which uses only attitude measurement. The stability of GOB is theoretically proved, with the GOB-I exhibiting almost global asymptotic stability and the GOB-II demonstrating local asymptotic stability. We have compared the estimation performance of our proposed GOB with four other disturbance observers (DOBs) for five types of disturbances. The simulation results show that, in the small attitude scenarios, our proposed GOB-I observer performs comparably with the compensation function observer (CFO) and the nonlinear disturbance observer (NDOB) that use both attitude and angular velocity measurements, while the GOB-II performs comparably with the extended state observer (ESO) that uses only attitude measurements. In the agile attitude scenarios, our proposed GOB-I observer significantly outperforms CFO and NDOB, and GOB-II also performs much better than ESO. Compared with the reduced-order observer, which uses only angular velocity measurement, the GOB-I can handle more complex disturbances and provides compensation for sensor errors. Flight experiments conducted with a lifting-wing quadcopter demonstrate the efficiency of our method in resisting unknown payload in hover flight and model uncertainty in agile maneuver.