Orthogonal Group SO Research Articles

An accurate and locking-free geometric exact beam formulation on the special orthogonal group SO(3)

An accurate and locking-free geometric exact beam formulation (GEBF) on the special orthogonal group SO(3) is developed for slender beams with large deformations and large rotations. Due to the nonlinear nature of the spatial rotations, the classical GEBFs usually have a non-constant mass matrix and complex expressions of inertia forces; meanwhile, the singularity and locking issues are also key matters of concern. Aiming at resolving these drawbacks, two main contributions are achieved in the present work. First, the angular velocity is independently interpolated, resulting in a constant mass matrix and explicitly representable inertial forces, which can offer significant advantages for efficient dynamic simulations. Second, inspired by the assumed natural strain method in shell theory, the stretch-shear strain is interpolated to obtain simplified and locking-free elastic forces, which is a new attempt to alleviate the locking problems for the GEBFs. In addition, the special orthogonal group SO(3) is utilized for updating incremental rotation vectors to eliminate singularities, and objective strain measurement is achieved by employing relative rotation vector interpolation. The effectiveness and superiority of the developed low-order and high-order elements are demonstrated through numerical simulations of standard benchmark examples. The present work contributes to the advancement of accurate and reliable formulation for slender beams; the constant mass matrix, the locking-free characteristics, and the elegant form of the formulation make it particularly suitable for multibody dynamic analysis.

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.

On the Structure of SO(3): Trace and Canonical Decompositions

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is described by trace mapping. Second, we show that every special orthogonal matrix can be expressed as the product of three elementary special orthogonal matrices. Explicit formulas for the decomposition as needed for applications in differential geometry and physics as symmetry transformations are given.

Hecke theory for SO+(2,n + 2)

We describe the foundations of a Hecke theory for the orthogonal group SO+(2,n+2). In particular we consider the Hermitian modular group of degree 2 as a special example of SO+(2,4). As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients.

Multimode Control Strategy for Robotic Rehabilitation on Special Orthogonal Group SO(3)

Robot-assisted rehabilitation for three-degree-of-freedom joints, such as hip and ankle, is significant for patients with motor function injuries. The control of such robots involves attitude control. To adapt to different disease stages, multi-mode hybrid control is considered to be one of the best choices. Passive mode is based on trajectory tracking control, while active mode is based on field-based assist-as-needed (AAN) control. The key to AAN control is the solution of the closest attitude point. However, the attitude point belongs to a special orthogonal group SO(3), and its topology is completely different from Euclidean space, which causes difficulties to the solution. Both passive and active control methods are affected by the inaccuracy of model parameters and external disturbances. Therefore, this paper proposes a multi-mode hybrid control method on SO(3). First, the expressions of trajectory tracking and contour tracking errors are proposed. To solve the contour tracking error, a feedback linearization algorithm based on sliding surface was used. A radial basis function (RBF) neural network was used for adaptive compensation. Subsequently, a controller for different modes was designed and its stability was analysed. Experiments were conducted using a hip exoskeleton, and the results verified the effectiveness of the proposed control method.

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

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.

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.

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.

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.

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.

Second-order sliding-mode on SO(3) and fault-tolerant spacecraft attitude control

As one of attitude representation methods for a rigid body, the rotation matrices defined on the special orthogonal matrix group SO(3) can give the unique, global and nonsingular solution. For the attitude control systems evolving on Lie group SO(3)×R3, a second-order sliding surface is constructed and proved to be a Lie subgroup, along which the reduced-order dynamics is almost globally asymptotically stable at the equilibrium state. Based on the new sliding-mode surface, a fault-tolerant controller is designed to deal with the actuator faults and external disturbances in attitude control systems. The new method is numerically simulated and further implemented in a semi-physical simulation system of an air-bearing platform to verify its effectiveness in spacecraft practical engineering.

On the attitude estimation of nonholonomic wheeled mobile robots

This paper addresses the attitude estimation problem of wheeled mobile robots (WMRs) with nonholonomic constraints moving in a plane. We consider the case that only the robot’s Cartesian position and linear velocity are available from sensor measurements. Based on the kinematic model, three attitude observers are proposed. The first observer is designed directly on the Special Orthogonal group SO(2) and guarantees that for almost all initial conditions, the estimated attitude converges to the actual one with an exponential convergence rate. The second observer estimates the heading direction of the mobile robot, i.e., its dynamics evolves on the unit circle S1. The first and second observers rely on linear velocity measurements. On the other hand, the third observer presents a globally exponentially stable error dynamics and requires only Cartesian position measurements but does not evolve on S1. Additionally, a control-observer algorithm is proposed to solve the trajectory-tracking problem of WMRs. Finally, the performances of the proposed attitude observers and control law were evaluated by a set of experiments on a commercial WMR.

Schwarzschild Spacetimes: Topology

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3∖{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.

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.

Distributed attitude tracking control of multiple rigid bodies with global exponential stability on SO(3)

AbstractIn this article, the leader‐following attitude tracking problem is considered for a group of rigid bodies, and a distributed control algorithm is proposed with global exponential stability on the rotational orthogonal group SO(3). In particular, we consider that only a subset of followers has access to the leader's trajectory, and a semi‐global exponential distributed observer is first proposed for each follower to estimate the leader's trajectory. Then, we introduce the shifted reference attitude of the leader to circumvent the topological obstruction on SO(3), and a combined distributed observer algorithm is developed by synthesizing different reference attitude trajectories. By constructing two Lyapunov functions, it is demonstrated that the leader's trajectory can be accurately estimated by the proposed distributed observer with exponential convergence. Next, an observer‐based attitude tracking control algorithm is synthesized, which is continuous in time but is discontinuous with respect to initial conditions. It is shown that the trajectory of the leader can be tracked by all followers with global exponential stability on SO(3). Finally, numerical simulation examples are provided to illustrate the effectiveness of the proposed control algorithms.

Finding geodesics joining given points

Finding a geodesic joining two given points in a complete path-connected Riemannian manifold requires much more effort than determining a geodesic from initial data. This is because it is much harder to solve boundary value problems than initial value problems. Shooting methods attempt to solve boundary value problems by solving a sequence of initial value problems, and usually need a good initial guess to succeed. The present paper finds a geodesic gamma :[0,1]rightarrow M on the Riemannian manifold M with γ(0) = x0 and γ(1) = x1 by dividing the interval [0,1] into several sub-intervals, preferably just enough to enable a good initial guess for the boundary value problem on each subinterval. Then a geodesic joining consecutive endpoints (local junctions) is found by single shooting. Our algorithm then adjusts the junctions, either (1) by minimizing the total squared norm of the differences between associated geodesic velocities using Riemannian gradient descent, or (2) by solving a nonlinear system of equations using Newton’s method. Our algorithm is compared with the known leapfrog algorithm by numerical experiments on a 2-dimensional ellipsoid Ell(2) and on a left-invariant 3-dimensional special orthogonal group SO(3). We find Newton’s method (2) converges much faster than leapfrog when more junctions are needed, and that a good initial guess can be found for (2) by starting with Riemannian gradient descent method (1).

The Hilbert modular group and orthogonal groups

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.