Rotation Group SO Research Articles

Differentiable and accelerated spherical harmonic and Wigner transforms

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S2 and rotation group SO(3), i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d-functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity.

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z3-skew-symmetric covariant SO(3)-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO(3)↪SU(5). We find two SO(3)-invariants of Z3-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K(z,z). We study the invariant properties of K(z,z) and find its stabilizer. Making use of these invariant properties, we define an SO(3)-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO(3)-irreducible geometric structure and find its curvature and torsion.

Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially for gravity-related theories. We begin by adapting the Cho-Duan-Ge decomposition to the gauge theory of the four-dimensional rotational symmetry group SO(4), where it identifies the maximal Abelian subgroup SO(2)⊗SO(2) in a gauge covariant manner. We then find the one-loop effective theory to have a stable condensate of monopoles corresponding to the reduction of SO(4) symmetry to SO(2)⊗SO(2). The construction of the condensate ensures that the four-dimensional spatial direction of its field strength must coincide with that of this embedding, and that a magnetic potential must be worked against to divert a trajectory away from this direction. Indeed, movement along this direction represents minimal potential energy. We take it to be the time direction. The gauge-dependent nature of the condensate is such that different gauge choices may lead to different time axes and we show on very general grounds that these different coordinate systems must be relatable by transformations of Lorentz form.

Tensorial Fourier expansion oforientation distribution function defined on the orthogonal group O(3).

Recently, Man and Du, in the context of classical texture analysis where the orientation distribution function (ODF) is defined on the rotation group SO(3), presented a systematic procedure by which the classical expansion of an ODF truncated at the order [Formula: see text] can be directly rewritten as a tensorial Fourier expansion truncated at the same order. In classical texture analysis, the groups of crystallite and texture symmetries are assumed to be subgroups of SO(3), which is unreasonable, say, for aggregates of crystallites in a crystal class defined by an improper point group. In this paper, we consider ODFs defined on the orthogonal group O(3) and extend the aforementioned procedure to write down tensorial Fourier expansions for polycrystals with crystallite symmetry defined by any of the 21 improper point groups. Examples that illustrate the general procedure are given. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Averaging material tensors of any rank in textured polycrystalline materials: Extending the scope beyond crystallographic proper point groups

In many modern micromechanical applications, there is usually a need to perform the orientational averaging of certain material tensors weighted by an orientation distribution function (ODF). The computation of these averages is seen to be very simplified by means of the so-called generalized spherical harmonic method (GSHM), which is based on the classical assumption that the ODFs are defined in the rotation group SO(3). A priori, this makes the averaging strictly applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. Despite the interest in the study of materials belonging to any of the other 21 point groups, few studies have properly considered such cases. This is crucial for physical properties represented by odd-order tensors such as the third-order linear piezoelectric tensor. The goal of this work is to extend the applicability of the GSHM to crystallites with symmetry that belongs to the orthogonal group O(3). Thus, a simple formula is provided for the averaging of material tensors of any rank in textured polycrystals containing crystallites with symmetry defined by any of the 32 crystallographic point groups. Our work presents a simple yet rigorous method for indirect averaging on the orthogonal group, using averaging on SO(3). We demonstrate the full equivalence between our proposed method and averaging properly on O(3) through a detailed proof provided in this paper. This proof represents a significant contribution to the field, providing a practical and reliable approach for researchers working with crystalline materials. The results reported here confirm the validity of closed-form expressions previously derived by the authors for piezoelectric materials.

Learning View-Based Graph Convolutional Network for Multi-View 3D Shape Analysis.

View-based approach that recognizes 3D shape through its projected 2D images has achieved state-of-the-art results for 3D shape recognition. The major challenges are how to aggregate multi-view features and deal with 3D shapes in arbitrary poses. We propose two versions of a novel view-based Graph Convolutional Network, dubbed view-GCN and view-GCN++, to recognize 3D shape based on graph representation of multiple views. We first construct view-graph with multiple views as graph nodes, then design two graph convolutional networks over the view-graph to hierarchically learn discriminative shape descriptor considering relations of multiple views. Specifically, view-GCN is a hierarchical network based on two pivotal operations, i.e., feature transform based on local positional and non-local graph convolution, and graph coarsening based on a selective view-sampling operation. To deal with rotation sensitivity, we further propose view-GCN++ with local attentional graph convolution operation and rotation robust view-sampling operation for graph coarsening. By these designs, view-GCN++ achieves invariance to transformations under the finite subgroup of rotation group SO(3). Extensive experiments on benchmark datasets (i.e., ModelNet40, ScanObjectNN, RGBD and ShapeNet Core55) show that view-GCN and view-GCN++ achieve state-of-the-art results for 3D shape classification and retrieval tasks under aligned and rotated settings.

Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles

We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. The constructed loss functions ensure that the applied transformations are symmetries and the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group . Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.

Optimal attitude maneuvers in the presence of prohibited directions

AbstractWe consider the problem of realigning a space telescope from one observation target to the next in the presence of prohibited directions. More specifically, we want to steer the space telescope – modelled as a gyrostat – from an initial attitude at rest to a final attitude at rest within a fixed time interval. During the motion, the line of sight of the telescope must be kept away from forbidden directions towards bright objects like the sun, moon or earth due to power or thermal requirements. The kinematics of the spacecraft motion are governed by a differential equation on the rotation group SO(3). Treating the angular velocities as control variables, this equation takes the form of a controlled dynamical system. To ensure reorientation maneuvers satisfying these pointing constraints, we introduce a cost functional penalizing proximity of the line of sight of the telescope to any of the forbidden directions. Furthermore, we include penalty terms which provide a smooth motion of the satellite and ensure the execution of a rest‐to‐rest maneuver. The chosen cost functional is minimized over all possible trajectories of the controlled dynamical system between the prescribed initial and target attitudes, which leads to an optimal control problem on SO(3), which is solved by applying a version of Pontryagin's Maximum Principle tailor‐made for optimal control problems on Lie groups. Parametrizing SO(3) in terms of Cardan angles, the solution is formulated as a boundary‐value problem on Euclidean space and hence can be solved numerically by conventional methods. The existence of two first integrals is established and exploited to reduce the computational effort. The applicability of this approach is shown in concrete examples.

On Perpendicular Curve-Based Task Space Trajectory Tracking Control With Incomplete Orientation Constraint

The Incomplete Orientation Constraint (IOC), which does not require a controlled motion constrained by all three spatial directions, exists widely in a lot of robotic tasks. However, the IOC remains a challenge for existing methods, due to the nonlinear structure of the rotation group SO(3). Moreover, the IOCs are time varying in the trajectory tracking problem, which makes it more challenging than the set-point control. To address the IOC problems, we define, identify and prove the closed-form solution of the perpendicular curve in SO(3). Based on the proposed perpendicular curve, we develop a new trajectory tracking controller considering the IOC. Compared with existing methods, the proposed method can achieve faster and more accurate tracking results. Moreover, the proposed method can be applied not only to manipulators with redundancy (including both functional and intrinsic redundancy), but also to manipulators that are non-redundant. Furthermore, it is easier to incorporate a secondary optimization objective into consideration for the intrinsic redundant case, which is difficult for existing methods. The proposed method has been implemented on both simulations and experiments. The numerous simulation and experimental results validate the effectiveness and advantages of the proposed method. Note to Practitioners—The trajectory tracking control with IOC is important and can be applied in a wide spectrum of robotic-assisted manufacturing, e.g. arc-welding, engraving, etc.. In this paper, a perpendicular curve-based trajectory tracking method is proposed to consider the IOC automatically. It is no longer necessary to carefully plan a reachable orientation trajectory. Users only need to focus the planning of the tool direction, which is determined by the tasks. In addition, the proposed method can achieve faster and more accurate tracking result. Moreover, it is universal and can be applied to both redundant and non-redundant cases.

A Haptic Shared Autonomy With Partial Orientation Regulation for DoF Deficiency in Remote Side.

Orientation regulation permits an autonomous controller to regulate the operators' orientation commands automatically. Although kinds of orientation regulation strategies have been proposed for various purposes, few works have focused on the partial orientation regulation (POR), which requires an autonomous controller to prevent the unreachable rotational motion for safety, while preserving the remaining motions for intuitiveness. However, the POR is deeply demanded for systems with Degree-of-Freedom (DoF) deficiency in remote side. The POR is a more challenging task owing to: First, it is difficult to decompose an orientation into reachable and unreachable components due to the non-linear structure of the rotation group SO(3). Second, it is non-trivial to design a haptic rendering algorithm which can indicate the missing DoF information to human operators. To address the rotational DoF deficiency, we propose a haptic shared autonomy with POR ability, based on the perpendicular curve theory in SO(3). The proposed method can partially regulate the operator's orientation command to discard the unreachable motions and preserve the remaining motions for follower robots. Here the conventional "master" and "slave" are all replaced by "leader" and "follower" to avoid the concern of association to racism and human subjugation. In addition, a haptic rendering algorithm is designed to display correct haptic cues about the missing DoF to operators. The simulation and experimental results validate the effectiveness of the proposed method.

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.

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

A particle that is constrained to freely move on a hyperspherical surface in an N≥2 dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago (Ohnuki and Kitakado, 1993). We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential and its components obey the commutation relations pi,pj=−iħJij/r2, in which ħ is the Planck’s constant, and pi (i,j=1,2,3,…N) denotes the i-th component of the geometric momentum, and Jij specifies the ij-th component of the generalized angular momentum containing both the orbital part and the coupling of the generators of continuous rotational symmetry group SO(N−1) and curvature, and r denotes the radius of the N−1 dimensional hypersphere.

An Angle–Axis Space-Based Orientability Index Characterizing Complete Orientations

The development of new materials/actuators in robotics leads to the rapid growth of new types of robots, including soft/continuum robots, reconfigurable robots, exoskeleton robots, haptic master devices, and so on. When designing a new robot, orientability is an important performance index to evaluate its dexterity. However, a theoretical investigation of the orientability is still lacking. The existing orientability index can only characterize the reachability of a line, rather than a complete orientation. To evaluate the reachability of complete orientations, we propose a novel orientability index based on the Angle-Axis Space, which is bijective to the rotation group SO(3). Furthermore, a relative orientability index is proposed to characterize the matching degree of the workspace of a given robot to its target workspace. The proposed orientability and relative orientability indices are then used to evaluate and compare the dexterity of three 6-DoF continuum robots and a haptic master devices. The results validate the effectiveness of the proposed orientability in evaluating the dexterity.

The Spin group in superspace

There are two well-known ways of describing elements of the rotation group SO(m). First, according to the Cartan–Dieudonné theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix.In this paper, we study similar descriptions of a group of rotations SO0 in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp(m|2n) on a Grassmann algebra. While still being connected, the group SO0 is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices.At the same time, SO0 strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO0. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO0. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group.

Detecting symmetries with neural networks

Identifying symmetries in data sets is generally difficult, but knowledge about them is crucial for efficient data handling. Here we present a method how neural networks can be used to identify symmetries. We make extensive use of the structure in the embedding layer of the neural network which allows us to identify whether a symmetry is present and to identify orbits of the symmetry in the input. To determine which continuous or discrete symmetry group is present we analyse the invariant orbits in the input. We present examples based on rotation groups SO(n) and the unitary group SU(2). Further we find that this method is useful for the classification of complete intersection Calabi-Yau manifolds where it is crucial to identify discrete symmetries on the input space. For this example we present a novel data representation in terms of graphs.

Incomplete Orientation Mapping for Teleoperation With One DoF Master-Slave Asymmetry

Teleoperation systems require a human-centered approach in which the kinematic mapping is intuitive and straightforward for the operators. However, a mismatch in the degrees-of-freedom (DoFs) between master and slave could result in an asymmetrical teleoperation system. That is an obstacle for intuitive kinematic mapping. In particular, it is even more challenging when the missing DoF is a pure rotational DoF, since the rotation group SO(3) is a nonlinear Riemannian manifold. This letter is concerned with an asymmetric teleoperation system, where the master subsystem can provide 6-DoF pose sensing while the slave subsystem only has 5 DoFs. The rotation along the missing DoF, which is configuration-dependent, is mapped to a geodesic curve in SO(3). We define and prove the closed-form solution of the perpendicular curve to the geodesic curve. Based on the perpendicular curve, we develop a novel Incomplete Orientation Mapping (IOM) approach to avoid the motion in the missing DoF. By comparing with two baseline methods, the experimental results demonstrate that the proposed method can discard the rotational motion along the missing DoF for all configurations, while preserving the remaining rotations.

Approximation to uniform distribution in $$\mathrm {SO}(3)$$

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3). The variance of rotation matrices sampled by the determinantal point process is estimated, and formulas for the L2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest. Also a simple but effective algorithm to sample points in SO(3) is given.

Metric-Affine Geometries with Spherical Symmetry

We provide a comprehensive overview of metric-affine geometries with spherical symmetry, which may be used in order to solve the field equations for generic gravity theories which employ these geometries as their field variables. We discuss the most general class of such geometries, which we display both in the metric-Palatini formulation and in the tetrad/spin connection formulation, and show its characteristic properties: torsion, curvature and nonmetricity. We then use these properties to derive a classification of all possible subclasses of spherically symmetric metric-affine geometries, depending on which of the aforementioned quantities are vanishing or non-vanishing. We discuss both the cases of the pure rotation group SO ( 3 ) , which has been previously studied in the literature, and extend these previous results to the full orthogonal group O ( 3 ) , which also includes reflections. As an example for a potential physical application of the results we present here, we study circular orbits arising from autoparallel motion. Finally, we mention how these results can be extended to cosmological symmetry.

Coherent Polarization States Multiplexer and Its Feasibility in Quantum Communication

In this paper, a multiplexing technique of coherent polarization states based on three-dimensional rotation group is proposed. The technique applies two Stokes parameters in one polarized light as signal and local oscillator (LO), and combines them in the single spatial mode. Three-dimensional rotation group SO(3) is used to describe the transformation of Stokes parameters. SO(3) is decomposed into three subgroups, with each subgroup represents a rotation along different axes. A multiplexer is designed with three electro-optical crystals to realize the function of three-dimensional rotation group. The multiplexer achieves an ergodicity of polarization states. Signal and LO propagates in the same spatial mode, multiplexing of polarization states is realized. Finally, the multiplexer is tested in a free space quantum communication system and its feasibility in quantum communication is confirmed.

Temporally Coupled Dynamical Movement Primitives in Cartesian Space

Control of robot orientation in Cartesian space implicates some difficulties, because the rotation group SO(3) is not contractible, and only globally contractible state spaces support continuous and globally asymptotically stable feedback control systems. In this paper, unit quaternions are used to represent orientations, and it is first shown that the unit quaternion set minus one single point is contractible. This is used to design a control system for temporally coupled dynamical movement primitives (DMPs) in Cartesian space. The functionality of the control system is verified experimentally on an industrial robot.