Terms Of Rotation Matrices Research Articles

Pose consensus control of multi‐agent rigid body systems with homogenous and heterogeneous communication delays

AbstractA new strategy for full pose and velocity consensus control of multi‐agent rigid body systems in the presence of communication delays is presented. Specifically, consensus protocols are proposed for a system of N heterogeneous rigid bodies on the Banach manifold associated with the tangent bundle , under a fixed and undirected communication topology, where the attitudes of the rigid bodies are described in terms of rotation matrices. The stability argument is strengthened from that used in prior studies by using an extension of Morse–Lyapunov–Krasovskii approach, and sufficient conditions are derived to achieve almost global asymptotic stability of the consensus subspace. Finally, illustrative examples are given to demonstrate the proposed method where both homogenous and heterogeneous communication delays are considered.

Velocity-to-velocity human motion forecasting

Forecasting human motion from a sequence of human poses is an important problem in the fields of computer vision and robotics. Most previous approaches merely consider learning the temporal dynamics of body joints or joint angles, while neglect derivatives of body joints (i.e., pose velocities) which could reasonably reduce noise impact and improve stability. To exploit the benefits of pose velocities, we propose the velocity-to-velocity learning paradigm for human motion prediction which attempts to directly build the sequence-to-sequence model in the velocity space. Two variant architectures based on recurrent encoder-decoder networks are introduced under this paradigm. Considering human motion as kinematics of rigid bodies, joint angles which denote transformation are the computations of inverse kinematics. Accordingly, a novel loss function in terms of rotation matrices is designed during training for human motion prediction through a rotation matrix transformation (RMT) layer. Finally, we present an effective training algorithm which exploits sequence transformation to improve model generalization. Our approaches substantially outperform state-of-the-art approaches on two large-scale datasets, Human3.6M and CMU Motion Capture, for both short-term prediction and long-term prediction. In particular, our model can competently forecast human-like and meaningful poses up to 1000 milliseconds. The code is available on GitHub: https://github.com/hongsong-wang/RNN_based_human_motion_prediction.

A version of the connection representation of Regge action

We define for any 4-tetrahedron (4-simplex) the simplest finite closed piecewise flat manifold consisting of this 4-tetrahedron and of one other 4-tetrahedron identical up to reflection to the present one (call it a bisimplex built on the given 4-simplex, or a two-sided 4-simplex). We consider an arbitrary piecewise flat manifold. The gravity action for it can be expressed in terms of the sum of the actions for the bisimplices built on the 4-simplices constituting this manifold. We use a representation of each bisimplex action in terms of rotation matrices (connections) and area tensors. This gives some representation of any piecewise flat gravity action in terms of connections. The action is a sum of terms each depending on the connection variables referring to a single 4-tetrahedron. Application of this representation to the path integral formalism is considered. Integrations over connections in the path integral reduce to independent integrations over finite sets of connections on separate 4-simplices. One of the consequences is exponential suppression of the result at large areas or lengths (compared to the Plank scale). This is important for the consistency of the simplicial description of spacetime.

Uniform Semiclassical Approximation for the Wigner 6j-Symbol in Terms of Rotation Matrices

A new uniform asymptotic approximation for the Wigner 6j-symbol is given in terms of Wigner rotation matrices (d-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near caustics. The derivation of the new approximation is not given, but the geometrical ideas supporting it are discussed and numerical tests are presented, including comparisons with the exact 6j-symbol and with the Ponzano-Regge approximation.

Cyclorotation Models for Eyes and Cameras

The human visual system obeys Listing's law, which means that the cyclorotation of the eye (around the line of sight) can be predicted from the direction of the fixation point. It is shown here that Listing's law can conveniently be formulated in terms of rotation matrices. The function that defines the observed cyclorotation is derived in this representation. Two polynomial approximations of the function are developed, and the accuracy of each model is evaluated by numerical integration over a range of gaze directions. The error of the simplest approximation for typical eye movements is less than half a degree. It is shown that, given a set of calibrated images, the effect of Listing's law can be simulated in a way that is physically consistent with the original camera. This condition is important for robotic models of human vision, which typically do not reproduce the mechanics of the oculomotor system.

Displacement Analysis of Spherical Mechanisms Having Three or Fewer Loops

Spherical linkages, having rotational joints whose axes coincide in a common center point, are sometimes used in multi-degree-of-freedom robot manipulators and in one-degree-of-freedom mechanisms. The forward kinematics of parallel-link robots, the inverse kinematics of serial-link robots and the input/output motion of single-degree-of-freedom mechanisms are all problems in displacement analysis. In this article, loop equations are formulated and solved for the displacement analysis of all spherical mechanisms up to three loops. We show how to solve each mechanism type using either a formulation in terms of rotation matrices or quaternions. In either formulation, the solution method is a modification of Sylvester’s elimination method, leading directly to numerical calculation via standard eigenvalue routines.

Quaternion-based algorithm for micromagnetics

We describe an algorithm for the integration of the Landau-Lifshitz equation for the precession of a magnetic moment in the presence of dissipation. The algorithm describes the rotation of the magnetization vector in terms of rotation matrices (implemented using quaternions). Its major advantage is that it separates precessional and dissipational rotations, which allows the former to be computed analytically over long time intervals. This allows the use of a much longer time increment $\ensuremath{\Delta}t$ than is possible with conventional algorithms, especially for problems with low anisotropy and weak exchange coupling. The spirit of the method is similar to that of the exact solution of the single-particle problem by Kikuchi [J. Appl. Phys. 27, 1352 (1956)], who also separated the precessional and dissipational motions.

A class of 6-j symbols for SO(2l+1) in terms of rotation matrices for SO(3)

It is shown that a 6-j symbol for SO(2l+1) in which four primitive spinor representations (1/21/2. . .1/2) appear is directly related to an SO(3) rotation matrix possessing a rank of l+1/2 and characterised by the Euler angles (0, 1/2 pi , 0). An interpretation is given in terms of a rotation operator exp(1/2i pi Ty) acting in the combined spin and quasispin space of an atomic l shell, whose states mod T, MT) are defined in a quasiparticle scheme in which four coupled spinors (1/21/2. . .1/2) are used.

Nuclear spin relaxation by translational diffusion in solids. IX. Orientation dependence of single-crystal relaxation rates

Nuclear spin rates of relaxation due to magnetic dipole coupling between spins undergoing relative translational diffusion in single crystals will depend on the orientation of the applied static magnetic field with respect to the crystal axes. The general form of the orientation dependence is calculated for the nuclear spin correlation functions J(q)( omega ) and hence for the relaxation rates and the second moment of a rigid-lattice absorption line broadened by dipolar interactions. The only assumption concerning the diffusional motion is the principle of detailed balance and the orientation dependence is obtained using the transformation properties of spherical harmonics in terms of rotation matrices. Explicit results are obtained for the correlation functions and relaxation rates for each point group and the form of the relaxation rates in the low-frequency (high-temperature) limit is deduced.

A program calculating the formulae for polarization effects in nuclear reactions

In the R-matrix theory of nuclear reactions, properly normalized observables can be expanded in terms of rotation matrices or Legendre functions. The corresponding expansion coefficients are then bilinear functions of the reaction matrix elements in the (1,s,J) coupling scheme. The numerical factor for each bilinear combination of matrix elements in a given coefficient is a function involving Clebsch-- Gordan, Racah, and X coefficients. By obtaining numerical values for the angular momentum functions and taking into account all relevant symmetries, the program FATSO computes the explicit formulas for any observable quantity for a given set of (1,s,J)-matrix elements. A set of 40 matrix elements can be accommodated, with up to 20 expansion coefficients. Approximate running time for cross sections involving vector- and tensorpolarized deuteron beams for a set of 20 matrix elements is 2 minutes. The program, in FORTRAN IV for the UNIVAC 1108 and the operating system EXEC 8, requires 44030 36- bit words of computer memory. (RWR)

Magnetic multipoles in nonresonance time dependent fields

The time dependence of the density operator for an arbitrary number of uncoupled spins in a resonant field has recently been presented [B. C. Sanctuary, J. Chem. Phys. 73, 1048 (1980). In this paper, the solutions are extended to the nonresonant case and are given in terms of rotation matrices.

The dipole moment expansion for a tetrahedral molecule in the ground vibronic state

The expansion of the electric dipole moment operator in terms of angular momentum operators is examined in detail for a tetrahedral molecule in the ground vibronic state. The components of this centrifugal distortion moment in the molecule fixed frame are formally expanded to arbitrary order, with the expansion coefficients being given in terms of rotation matrices. For terms of rank j less than 15, general expressions are given for the matrix elements in the symmetric top basis of the space-fixed components of the dipole moment. The dependence of these matrix elements on the K-quantum number is shown to factor in such a way that previous first order calculations can be extended to second order by replacing the first order dipole coupling constant μ 2 (2) with a function of J which involves μ 2 (2) and two second order constants, μ 2 (4) and μ 4 (4) . Different functions are required for Q- and R-branch matrix elements and explicit expressions are given for both. The third order terms are examined in detail in the Appendix. A recurrence relation is derived for j < 15 between the tetrahedral harmonics of types A 2 and F 2. The implications of this work for distortion moment spectroscopy in tetrahedral molecules are discussed.

Point-Group Selection Rules and Polarization for an Alternative Singlet—Triplet Transition Mechanism

A radiative transition mechanism which involves the direct interaction between the electron spin and the radiation field is studied in detail in the limit of small spin—orbit coupling. In this mechanism, the odd-parity, spin-dependent multipoles of the lowest order, the spin—electric dipole and the spin—magnetic quadrupole, are considered as alternative operators for the usual spin—orbit assisted electric-dipole operator in singlet—triplet transitions. It is shown that, although these two spin-dependent spherical operators transform very differently in the three-dimensional rotation group, they contribute comparably to the radiative transition and they must be considered together to preserve the transverse nature of light. For applications to (plane) polarized light spectroscopy in oriented molecular crystals, expressions of these spin-dependent spherical operators are found in terms of the quadratic products of the Cartesian coordinates x, y, and z with the spin components Sx, Sy, and Sz. The explicit forms of the transition operators for propagation of light and polarization along x, y, or z directions are tabulated. The selection rules of the coordinate part of these operators are illustrated for the D6 point group. The selection rules of the spin part are given for the ``uncoupled'' spin quantized with respect to the principal molecular axis or with respect to an external magnetic field. General formulas for deriving the angular distribution of intensity are given in terms of rotation matrices. Specific formulas for the angular dependence and polarization of the ``direct'' phosphorescence from each and all of the three triplet components via this mechanism are derived. These are expressed in terms of the cosine of Euler angles [open phi] and ψ and functions of reduced rotation matrices. The explicit values of these functions of reduced rotation matrices in terms of the Euler angle θ are tabulated. The tabulation is extended for possible use in transitions involving symmetry-adapted spin wavefunctions and transitions in systems with any (lower than axial) point-group symmetry.

Unitary Symmetry of Oscillators and the Talmi Transformation

The Hamiltonian of an isotropic harmonic oscillator is invariant under unitary transformations in three dimensions. This well-known invariance is exploited in a treatment of the Talmi transformation, viz., the transformation of two-particle oscillator functions to center-of-mass and relative coordinates. A simple and transparent form of this transformation in terms of rotation matrices and Wigner coefficients of SU3 is given. The calculation of these Wigner coefficients is described and the problem of degeneracies discussed.