Orthogonal Group SO Research Articles

On certain generalized isometries of the special orthogonal group

In this paper we explore the structure of certain generalized isometries of the special orthogonal group SO(n) which are transformations that leave any member of a large class of generalized distance measures invariant.

Packet structure and paramodular forms

We explore the consequences of the structure of the discrete automorphic spectrum of the split orthogonal groupSO⁡(5)\operatorname {SO}(5)for holomorphic Siegel modular forms of degree22. In particular, the combination of the local and global packet structure with the local paramodular newform theory forGSp⁡(4)\operatorname {GSp}(4)leads to a strong multiplicity one theorem for paramodular cusp forms.

The Spin‐Charge‐Family Theory Offers Understanding of the Triangle Anomalies Cancellation in the Standard Model

The standard model has for massless quarks and leptons “miraculously” no triangle anomalies due to the fact that the sum of all possible traces T r[τAi τBj τCk] — where , τAi τBj and τCk are the generators of one, of two or of three of the groups SU(3), SU(2) and U(1) — over the representations of one family of the left handed fermions and anti‐fermions (and separately of the right handed fermions and anti‐fermions), contributing to the triangle currents, is equal to zero. It is demonstrated in this paper that this cancellation of the standard model triangle anomaly follows straightforwardly if the SO(3, 1), SU(2), U(1) and SU(3) are the subgroups of the orthogonal group SO(13,1), as it is in the spin‐charge‐family theory. The authors comment on the SO(10) anomaly cancellation, which works if handedness and charges are related “by hand”.

Hybrid global exponential stabilization on [formula omitted

We propose a central synergistic hybrid approach for global exponential stabilization on the Special Orthogonal group SO(3). We introduce a new switching concept relying on a central family of (possibly) non-differentiable potential functions that enjoy the following properties: (1) being quadratic (as well as their gradients) with respect to the Euclidean attitude distance, and (2) being synergistic with respect to the gradient’s singular and/or critical points. The proposed approach is used to solve the attitude tracking problem, leading to global exponential stability results.

Attitude guidance and tracking for spacecraft with two reaction wheels

ABSTRACTThis paper addresses the guidance and tracking problem for a rigid-spacecraft using two reaction wheels (RWs). The guidance problem is formulated as an optimal control problem on the special orthogonal group SO(3). The optimal motion is solved analytically as a function of time and is used to reduce the original guidance problem to one of computing the minimum of a nonlinear function. A tracking control using two RWs is developed that extends previous singular quaternion stabilisation controls to tracking controls on the rotation group. The controller is proved to locally asymptotically track the generated reference motions using Lyapunov's direct method. Simulations of a 3U CubeSat demonstrate that this tracking control is robust to initial rotation errors and angular velocity errors in the controlled axis. For initial angular velocity errors in the uncontrolled axis and under significant disturbances the control fails to track. However, the singular tracking control is combined with a nano-magnetic torquer which simply damps the angular velocity in the uncontrolled axis and is shown to provide a practical control method for tracking in the presence of disturbances and initial condition errors.

Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Systems on Lie groups naturally appear as models for physical systems with full symmetry. We consider the state estimation problem for such systems where both input and output measurements are corrupted by unknown disturbances. We provide an explicit formula for the second-order-optimal nonlinear filter on a general Lie group where optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). The resulting filter depends on the choice of affine connection which encodes the nonlinear geometry of the state space. As an example, we look at attitude estimation, where we are given a second order mechanical system on the tangent bundle of the special orthogonal group SO(3), namely the rigid body kinematics together with the Euler equation. When we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a perturbed matrix Riccati differential equation that updates the filter gain. This example demonstrates how to construct a matrix representation of the abstract general filter formula.

Robust adaptive relative motion control for spacecraft proximity operations

This paper studies the controller design for a spacecraft approaching a tumbling space target in the presence of model uncertainties and dynamic couplings between relative rotation and relative translation. An alternative relative attitude function is selected to measure the magnitude of the relative attitude motion carefully, and robust adaptive controllers are designed for relative attitude and relative position to guarantee a desirable stabilization performance uniformly for the relative motion. The strict Lyapunov analysis is presented to prove the uniformly ultimately boundedness of relative motion errors. The relative attitude controller is directly developed on the special orthogonal group SO(3) to avoid complexities and ambiguities associated with coordinate-dependent attitude representations. Performance of the controlled overall system is demonstrated via a representative numerical example.

A High Invariance Motion Representation for Skeleton-Based Action Recognition

Human action recognition is very important and significant research work in numerous fields of science, for example, human–computer interaction, computer vision and crime analysis. In recent years, relative geometry features have been widely applied to the description of relative relation of body motion. It brings many benefits to action recognition such as clear description, abundant features etc. But the obvious disadvantage is that the extracted features severely rely on the local coordinate system. It is difficult to find a bijection between relative geometry and skeleton motion. To overcome this problem, many previous methods use relative rotation and translation between all skeleton pairs to increase robustness. In this paper we present a new motion representation method. It establishes a motion model based on the relative geometry with the aid of special orthogonal group SO(3). At the same time, we proved that this motion representation method can establish a bijection between relative geometry and motion of skeleton pairs. After the motion representation method in this paper is used, the computation cost of action recognition reduces from the two-way relative motion (motion from A to B and B to A) to one-way relative motion (motion from A to B or B to A) between any skeleton pair, namely, permutation problem [Formula: see text] is simplified into combinatorics problem [Formula: see text]. Finally, the experimental results of the three motion datasets are all superior to present skeleton-based action recognition methods.

Global swarming while preserving connectivity via Lagrange–Poincarè equations

In this paper, we exploit symmetry properties of multi-agent robot systems to design control laws that preserve connectivity while swarming. We start by designing a connectivity control law for agents with configuration spaces R3 and SO(3) that is invariant under the action of the special Euclidean group SE(3) and the special orthogonal group SO(3), respectively. Therefore, the dynamics of such multi-agent systems is amenable to be reduced by these group actions. We then utilize the Lagrange–Poincaré equations that split the Euler–Lagrange equations for the multi-agent system into horizontal and vertical parts. The invariance of the connectivity controller implies that its control effort has zero vertical component. We then use the resulting vertical equations of motion to design a control law that asymptotically stabilizes the centroid and the orientation of the swarm at a desired pose.

Infinite families of harmonic self-maps of spheres

For each of the spheres Sn, n≥5, we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer degree ±1 or ±3. These self-maps are obtained by solving a singular boundary value problem. As an application we show that for each of the special orthogonal groups SO(4), SO(5), SO(6) and SO(7) there exist two infinite families of harmonic self-maps.

Half turns in characteristic 2

Let V be a nonsingular quadratic space over a field K of characteristic 2. We show that if n>4, then every element π in the special orthogonal group SO(V) is a product of an even number of half turns. If dim⁡B(π)=2k, then the length of π with respect to the half turns is k, k+1 or k+2.

The C*–algebras of Compact Transformation Groups

AbstractWe investigate the representation theory of the crossed–product C*–algebra associated with a compact group G acting on a locally compact space X when the stability subgroups vary discontinuously. Our main result applies when G has a principal stability subgroup or X is locally of finite G–orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation V of a stability subgroup is obtained by restricting V to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of V. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup; the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the C*–algebra of the motion group ℝn ⋊ SO(n) is a Fell algebra. This uses the classical branching theorem for the special orthogonal group SO(n) with respect to SO(n − 1). Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

On finite subgroups of the classical groups

Abstract In 1878, Jordan showed that a finite subgroup of GL ( n , ℂ ) ${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound ( n + 1 ) ! ${(n+1)!}$ was obtained when n ≥ 71 ${n\ge 71}$ , achieved by the symmetric group S n+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of Sp ( 2 n , ℂ ) ${\operatorname{Sp}(2n,\mathbb {C})}$ the optimal bound is ( 60 ) n · n ! ${(60)^{n}\cdot n!}$ , achieved by the wreath product SL 2 ( 5 ) wr S n ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups O ( n , ℂ ) ${\mathrm {O}(n,\mathbb {C})}$ , the generic linear group bound of ( n + 1 ) ! ${(n+1)!}$ is achieved as soon as n ≥ 25 ${n\ge 25}$ .

Some Optimization Aspects on the Lie Group SO(3)

In this paper, we investigate the LQR-like optimal control problem on the special orthogonal group SO(3). Using the dynamic programming approach, we derive a Hamilton- Jacobi-Bellman equation in the general case where a generic distance on SO(3) is used in the cost functional. We show that the geodesic distance on SO(3) yields results analogous to the well known results for linear systems. Static and dynamic Riccati-like equations for both infinite and finite time-horizon optimal control problems are obtained.

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

The collocation method is extended to the special orthogonal group SO(3) with application to optimal attitude control (OAC) of a rigid body. A left-invariant rigid-body attitude dynamical model on SO(3) is established. For the left invariance of the attitude configuration equation in body-fixed frame, a geometrically exact numerical method on SO(3), referred to as the geometric collocation method, is proposed by deriving the equivalent Lie algebra equation inso(3)of the left-invariant configuration equation. When compared with the general Gauss pseudo-spectral method, the explicit RKMK, and Lie group variational integrator having the same order and stepsize in numerical tests for evolving a free-floating rigid-body attitude dynamics, the proposed method is higher in accuracy, time performance, and structural conservativeness. In addition, the numerical method is applied to solve a constrained OAC problem on SO(3). The optimal control problem is transcribed into a nonlinear programming problem, in which the equivalent Lie algebra equation is being considered as the defect constraints instead of the configuration equation. The transcription method is coordinate-free and does not need chart switching or special handling of singularities. More importantly, with the numerical advantage of the geometric collocation method, the proposed OAC method may generate satisfying convergence rate.

Attitude Estimation in SO(3): A Comparative UAV Case Study

This paper introduces a novel algorithm to obtain attitude estimations from low cost inertial measurement units including 3-axis accelerometer, 3-axis gyroscope and 3-axis magnetometer. This nonlinear attitude estimator is derived from Lyapunov's theory and formulated in the special orthogonal group SO(3). The impact of the gyroscope bias is also assessed and an online estimator provided. The performance of the proposed estimator is validated and compared to current commonly used methods, namely the classical extended Kalman filter and two other nonlinear estimators in SO(3). Realistic simulations consider a quadcopter unmanned aerial vehicle subject to wind disturbances and whose sensors parameters have been identified from flight tests data.

Quantization of borderline Levi conjugacy classes of orthogonal groups

We construct an equivariant quantization of a special family of Levi conjugacy classes of the complex orthogonal group SO(N), whose stabilizer contains a Cartesian factor SO(2) × SO(P), 1 ⩽ P < N, P ≡ N mod 2.

The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation.

For any pair of three-dimensional real unit vectors and with and any rotation U, let denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either or . This work gives the number as a function of U. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number are also given explicitly.

A new method of constructing a grid in the space of 3D rotations and its applications to texture analysis

In computational work, data sets must often be represented on the surface of a sphere or inside a ball, requiring uniform grids. We construct a new volume-preserving projection between a cube and the set of unit quaternions. The projection consists of two steps: an equal-volume mapping from the cube to the unit ball, followed by an inverse generalized Lambert projection to either of the two unit quaternion hemispheres. The new projection provides a one-to-one mapping between a grid in the cube and elements of the special orthogonal group SO(3), i.e., 3D rotations. We provide connections to other rotation representation schemes, including the Rodrigues–Frank vector and the homochoric parameterizations, and illustrate the new mapping through example applications relevant to texture analysis.

A global approach for image orientation using Lie algebraic rotation averaging and convex L∞ minimisation

Abstract. In this paper we present a new global image orientation approach for a set of multiple overlapping images with given homologous point tuples which is based on a two-step procedure. The approach is independent on initial values, robust with respect to outliers and yields the global minimum solution under relatively mild constraints. The first step of the approach consists of the estimation of global rotation parameters by averaging relative rotation estimates for image pairs (these are determined from the homologous points via the essential matrix in a pre-processing step). For the averaging we make use of algebraic group theory in which rotations, as part of the special orthogonal group SO(3), form a Lie group with a Riemannian manifold structure. This allows for a mapping to the local Euclidean tangent space of SO(3), the Lie algebra. In this space the redundancy of relative orientations is used to compute an average of the absolute rotation for each image and furthermore to detect and eliminate outliers. In the second step translation parameters and the object coordinates of the homologous points are estimated within a convex L∞ optimisation, in which the rotation parameters are kept fixed. As an optional third step the results can be used as initial values for a final bundle adjustment that does not suffer from bad initialisation and quickly converges to a globally optimal solution. We investigate our approach for global image orientation based on synthetic data. The results are compared to a robust least squares bundle adjustment. In this way we show that our approach is independent of initial values and more robust against outliers than a conventional bundle adjustment.