Representation Of Rotations Research Articles

Modal analysis of cracked beam with a piezoelectric layer

Piezoelectric material was employed first as sensor/actuator for structural control and then it has got an effective use for structural health monitoring and repairing damaged structures. In this report, modal analysis of cracked beam with piezoelectric layer is carried out to investigate effect of crack and piezoelectric layer thickness on natural frequencies of the structure and output charge generated in the piezoelectric layer by vibration modes. Governing equations of the coupled structure are established using the double beam model and two-spring (translational and rotational) representation of crack and solved to obtain the modal parameters including the output charge associated with natural modes acknowledged as modal piezoelectric charge (MPC). Numerical examples have been examined for validation and illustration of the developed theory.

Space\u2013time VMS isogeometric analysis of the Taylor\u2013Couette flow

The Taylor–Couette flow is a classical fluid mechanics problem that exhibits, depending on the Reynolds number, a range of flow patterns, with the interesting ones having small-scale structures, and sometimes even wavy nature. Accurate representation of these flow patterns in computational flow analysis requires methods that can, with a reasonable computational cost, represent the circular geometry accurately and provide a high-fidelity flow solution. We use the Space–Time Variational Multiscale (ST-VMS) method with ST isogeometric discretization to address these computational challenges and to evaluate how the method and discretization perform under different scenarios of computing the Taylor–Couette flow. We conduct the computational analysis with different combinations of the Reynolds numbers based on the inner and outer cylinder rotation speeds, with different choices of the reference frame, one of which leads to rotating the mesh, with the full-domain and rotational-periodicity representations of the flow field, with both the convective and conservative forms of the ST-VMS, with both the strong and weak enforcement of the prescribed velocities on the cylinder surfaces, and with different mesh refinements. The ST framework provides higher-order accuracy in general, and the VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the flow. The ST isogeometric discretization enables exact representation of the circular geometry and increased accuracy in the flow solution. In computations where the mesh is rotating, the ST/NURBS Mesh Update Method, with NURBS basis functions in time, enables exact representation of the mesh rotation, in terms of both the paths of the mesh points and the velocity of the points along their paths. In computations with rotational-periodicity representation of the flow field, the periodicity is enforced with the ST Slip Interface method. With the combinations of the Reynolds numbers used in the computations, we cover the cases leading to the Taylor vortex flow and the wavy vortex flow, where the waves are in motion. Our work shows that all these ST methods, integrated together, offer a high-fidelity computational analysis platform for the Taylor–Couette flow and for other classes of flow problems with similar features.

Interaction algorithm of the mechanisms of relative manipulation in a 6-DOF robot control system

The paper considers the features of management of a 6-DOF robot, developed on the basis of two interacting mechanisms of a parallel structure. Each mechanism provides movement in three independent degrees of freedom: a flat 3-RPR mechanism for installing a part and a 3-RPS mechanism (tripod) for installing a tool. It is shown that when these mechanisms are jointly controlled, it becomes necessary to correct the movement of one of them in connection with the parasitic movements of the other. The authors propose the method for the description of the position of the movable platform of the tripod, which allows using only five output coordinates. According to the authors, the platform rotation representation used in this case is more convenient to describe the interaction of mechanisms than the traditional use of Tait – Bryan angles. The algorithm was developed to correct the task by the position of the planar mechanism in order to compensate parasitic movements of the movable platform of the tripod in the horizontal plane during the control of a 6-DOF robot. On the basis of a specially constructed 6-DOF robot model with mechanisms of relative manipulation, the simulation in the MATLAB package was carried out, which confirmed the effectiveness of the proposed algorithm.

Modifications in fMRI Representation of Mental Rotation Following a 6 Week Graded Motor Imagery Training in Chronic CRPS Patients

Complex regional pain syndrome (CRPS) is a neuropathic pain condition that is difficult to treat. For behavioral interventions, graded motor imagery (GMI) showed relevant effects, but underlying neural substrates in patient groups have not been investigated yet. A previous study investigating differences in the representation of a left/right hand judgment task demonstrated less recruitment of subcortical structures, such as the putamen, in CRPS patients than in healthy controls. In healthy volunteers, the putamen activity increased after a hand judgment task training. In order to test for longitudinal effects of GMI training, we investigated 20 CRPS patients in a wait-list crossover design with 3 evaluation time points. Patients underwent a 6 week GMI treatment and a 6 week waiting period in a randomized group assignment and treatment groups were evaluated by a blinded rater. When compared to healthy matched controls at baseline, CRPS patients showed less functional activation in areas processing visual input, left sensorimotor cortex, and right putamen. Only GMI treatment, but not the waiting period showed an effect on movement pain and hand judgment task performance. Regression analyses revealed positive associations of movement pain with left anterior insula activation at baseline. Right intraparietal sulcus activation change during GMI was associated with a gain in performance of the hand judgment task. The design used here is reliable for investigating the functional representation of the hand judgment task in an intervention study. PerspectiveTwenty chronic CRPS patients underwent a 6 week GMI intervention in a randomized wait-list crossover design. functional MRI was tested pre and post for the hand lateralization task which improved over GMI but not over WAITING. Performance gain was positively related to right parietal functional MRI activation.

Arithmetic Loophole in Bell's Theorem: Overlooked Threat to Entangled-State Quantum Cryptography

Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations. However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states. Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers, but counterfactual aspects of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and calculus. The model is classical in the sense of Einstein, Podolsky, Rosen, and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variable probaility densities. Probability densities have a Clauser-Horne product form typical of local realistic theories. However, neither the product nor the integral nor the representation of rotations are the usual ones. The integral has all the standard properties but only with respect to the arithmetic that defines the product. Certain formal transformations of integral expressions one finds in the usual proofs \`a la Bell do not work, so standard Bell-type inequalities cannot be proved. The system we consider is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, so is free of all the canonical loopholes discussed in the literature.

Space\u2013Time Variational Multiscale Isogeometric Analysis of a tsunami-shelter vertical-axis wind turbine

We present computational flow analysis of a vertical-axis wind turbine (VAWT) that has been proposed to also serve as a tsunami shelter. In addition to the three-blade rotor, the turbine has four support columns at the periphery. The columns support the turbine rotor and the shelter. Computational challenges encountered in flow analysis of wind turbines in general include accurate representation of the turbine geometry, multiscale unsteady flow, and moving-boundary flow associated with the rotor motion. The tsunami-shelter VAWT, because of its rather high geometric complexity, poses the additional challenge of reaching high accuracy in turbine-geometry representation and flow solution when the geometry is so complex. We address the challenges with a space–time (ST) computational method that integrates three special ST methods around the core, ST Variational Multiscale (ST-VMS) method, and mesh generation and improvement methods. The three special methods are the ST Slip Interface (ST-SI) method, ST Isogeometric Analysis (ST-IGA), and the ST/NURBS Mesh Update Method (STNMUM). The ST-discretization feature of the integrated method provides higher-order accuracy compared to standard discretization methods. The VMS feature addresses the computational challenges associated with the multiscale nature of the unsteady flow. The moving-mesh feature of the ST framework enables high-resolution computation near the blades. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the blade and other turbine geometries and increased accuracy in the flow solution. The STNMUM enables exact representation of the mesh rotation. A general-purpose NURBS mesh generation method makes it easier to deal with the complex turbine geometry. The quality of the mesh generated with this method is improved with a mesh relaxation method based on fiber-reinforced hyperelasticity and optimized zero-stress state. We present computations for the 2D and 3D cases. The computations show the effectiveness of our ST and mesh generation and relaxation methods in flow analysis of the tsunami-shelter VAWT.

Quaternion statistics applied to the classification of motion capture data

Unit quaternions give quite new possibilities in an analysis of motion capture data. They provide a compact, holistic axis-angle representation of 3D rotations. An application of descriptive statistics – measures of location and dispersion – is common in numerous problems related to an assessment of joint movements. For those reasons, the paper proposes new approaches to the extraction of motion descriptors on the basis of descriptive statistics of 3D rotations represented by unit quaternions as well as the appropriate classification schemes operating on motion descriptors. Mean and median values and standard deviations are calculated for time series with raw rotational data as well as angular velocities and accelerations. The problem of human gait identification is addressed in the numerical validation of the introduced methods and highly precise marker-based motion capture data are utilized. The results obtained – the accuracy of gait recognition – are compared to the ones achieved by descriptive statistics calculated for time series of Euler angles. The general conclusion is that unit quaternions are effective in the calculation of descriptive statistics. They preserve robust discriminative features of joint movements and they can be applied in numerous challenges of expert and intelligent systems.

A robust tracker integrating particle filter into correlation filter framework

The location and scale filters in discriminative correlation filter methods are lack of accurate rotation representation capability and updated with fixed intervals, which leads to tracking failure and time-consuming in complex scenarios. In this manuscript, a robust tracker integrating particle filter into correlation filter is presented to cope with sharp rotation and remarkable deformation. The target position and scale factor are firstly estimated from the correlation filter, and then the rotation factor is determined by similarity between candidates and template based on the particle filter. As a result, target variation can be accurately described with position, scale and rotation factor. Moreover, a long-time and short-time update scheme is proposed to solve target template drifting problem. Extensive experimental results conducted on OTB-2013, OTB-2015 and VOT-2016 show that the proposed tracker improves the accuracy and robustness of discriminative correlation filter methods.

On geometric applications of quaternions

Quaternions have become a popular and powerful tool in various engineering fields, such as robotics, image and signal processing, and computer graphics. However, classical quaternions are mostly used as a representation of rotation of a vector in $3$-dimensions, and connection between its geometric interpretation and algebraic structures is still not well-developed and needs more improvements. In this study, we develop an approach to understand quaternions multiplication defining subspaces of quaternion $\mathbb{H}$, called as $\mbox{Plane}(N)$ and $\mbox{Line}(N)$, and then, we observe the effects of sandwiching maps on the elements of these subspaces. Finally, we give representations of some transformations in geometry using quaternion.

The adjoint trigonometric representation of displacements and a closed‐form solution to the IKP of general 3C chains

AbstractBased on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3‐DOF, general, spherical 3R‐chains into a closed‐form solution to the IKP of 6‐DOF, general, affine 3C‐chains.

Rotating 3He droplets.

Motivated by recent experiments, we study normal-phase rotating 3He droplets within density functional theory in a semi-classical approach. The sequence of rotating droplet shapes as a function of angular momentum is found to agree with that of rotating classical droplets, evolving from axisymmetric oblate to triaxial prolate to two-lobed shapes as the angular momentum of the droplet increases. Our results, which are obtained for droplets of nanoscopic size, are rescaled to the mesoscopic size characterizing ongoing experimental measurements, allowing for a direct comparison of shapes. The stability curve in the angular velocity-angular momentum plane shows small deviations from the classical rotating drop model predictions, whose magnitude increases with angular momentum. We attribute these deviations to effects not included in the simplified classical model description of a rotating fluid held together by surface tension, i.e., to surface diffuseness, curvature, and finite compressibility, and to quantum effects associated with deformation of the 3He Fermi surface. The influence of all these effects is expected to diminish as the droplet size increases, making the classical rotating droplet model a quite accurate representation of 3He rotation.

A consistent strain-based beam element with quaternion representation of rotations

We present a novel consistent singularity-free strain-based finite element formulation for the analysis of three-dimensional frame-like structures. Our model is based on a geometrically exact finite-strain beam theory, quaternion parametrization of spatial rotations, assumption that the strain measures are constant along the length of the element and a proper choice of basis for the translational strain vector representation. As it is common for strain-based elements, the present formulation does not suffer from shear locking. A comparison of our results with the results from the literature and a commercial finite element analysis software demonstrates the advantages of the proposed formulation, especially when the structure is subjected to larger shear deformations. This stems from the fact that our model ensures a mathematically consistent updating procedure for all the quantities describing the beam. This aspect is often overlooked, since most of the numerical cases from other studies on this topic engage rather small-shear strains for which the consistent update is not crucial as the number of elements is increased.

Geodesic Motions in SO(2,1)

In this study,we have considered the rotational motions of a particle around the origin of the unit 2-sphere $S_2^2$ with constant angular velocity in semi-Euclidean 3-spacewithindextwo $E^3_2$, namely geodesic motions of $SO(2,1)$. Then we have obtained the vector and the matrix representations of the spherical rotations around the origin of a particle on $S_2^2$. Furthermore, we consider some relations between semi-Riemann spaces $SO(2,1)$ and $T_1S_2^2$ such as diffeomorphism and isometry. We have obtained the system of differential equations giving geodesics of Sasaki semi-Riemann manifold $(T_1S_2^2,g^S)$ . Moreover, we consider the stationary motion of a particle on $S_2^2$ corresponding to one parameter curve of $SO(2,1)$, which determines a geodesic of $SO(2,1)$. Finally, we obtain the system of differential equations giving geodesics of the semi-Riemann space $(SO(2,1),h)$ and we show that the system of differential equations giving geodesics of Riemann space $(SO(2,1),h)$ is equal to that of $(T_1S_2 2,g^S)$.

Scalar Coupling Constant Prediction Using Graph Embedding Local Attention Encoder

Scalar coupling constant (SCC) plays a key role in the analysis of three-dimensional structure of organic matter, however, the traditional SCC prediction using quantum mechanical calculations is very time-consuming. To calculate SCC efficiently and accurately, we proposed a graph embedding local self-attention encoder (GELAE) model, in which, a novel invariant structure representation of the coupling system in terms of bond length, bond angle and dihedral angle was presented firstly, and then a local self-attention module embedded with the adjacent matrix of a graph was designed to extract effectively the features of coupling systems, finally, with a modified classification loss function, the SCC was predicted. To validate the superiority of the proposed method, we conducted a series of comparison experiments using different structure representations, different attention modules, and different losses. The experimental results demonstrate that, compared to the traditional chemical bond structure representations, the rotation and translation invariant structure representations proposed in this work can improve the SCC prediction accuracy; with the graph embedded local self-attention, the mean absolute error (MAE) of the prediction model in the validation set decreases from 0.1603 Hz to 0.1067 Hz; using the classification based loss function instead of the scaled regression loss, the MAE of the predicted SCC can be decreased to 0.0963 HZ, which is close to the quantum chemistry standard on CHAMPS dataset.

MathDL: mathematical deep learning for D3R Grand Challenge 4.

We present the performances of our mathematical deep learning (MathDL) models for D3R Grand Challenge 4 (GC4). This challenge involves pose prediction, affinity ranking, and free energy estimation for beta secretase 1 (BACE) as well as affinity ranking and free energy estimation for Cathepsin S (CatS). We have developed advanced mathematics, namely differential geometry, algebraic graph, and/or algebraic topology, to accurately and efficiently encode high dimensional physical/chemical interactions into scalable low-dimensional rotational and translational invariant representations. These representations are integrated with deep learning models, such as generative adversarial networks (GAN) and convolutional neural networks (CNN) for pose prediction and energy evaluation, respectively. Overall, our MathDL models achieved the top place in pose prediction for BACE ligands in Stage 1a. Moreover, our submissions obtained the highest Spearman correlation coefficient on the affinity ranking of 460 CatS compounds, and the smallest centered root mean square error on the free energy set of 39 CatS molecules. It is worthy to mention that our method on docking pose predictions has significantly improved from our previous ones.

Matching pursuit based on the rotated time-frequency atomic dictionary and its applications to seismic non-stationary data

AbstractThe time-frequency spectrum of an attenuated seismic wave rotates due to the influence of intrinsic quality factor. Traditional time-frequency analysis methods are either limited by time-frequency resolution or incapable of rotated time-frequency representation. To solve it, a matching pursuit (MP) method based on the rotated time-frequency atomic dictionary (MPFR) was proposed. By introducing a frequency rotation factor to the four-parameter dictionary in the conventional MP method, a five-parameter dictionary with frequency rotation characteristics was constructed, and the calculation process of each parameter was given. We tested the method on the synthetic traces and field data sets. The results showed that the proposed method can effectively realize rotated time-frequency characterization of attenuated seismic waves. Compared with the conventional atoms, the rotated time-frequency atoms are more compliant to the local features of non-stationary seismic data. Moreover, the effective description of rotated time-frequency characteristics can be used as an auxiliary technique to predict the reservoirs related to the attenuation. A care has to be taken when thin layers are encountered, since wavelet interference may cause rotated time-frequency characteristics independent of dispersion and attenuation in MPFR method.

Fluid dynamics of heavy ion collisions with mode expansion

The fluid dynamics of a relativistic fireball with longitudinal and transverse expansion is described using a background-fluctuation splitting. Symmetry representations of azimuthal rotations and longitudinal boosts are used for a classification of initial state configurations and their fluid dynamic propagation in terms of a mode expansion. We develop an accurate and efficient numerical scheme based on the pseudo-spectral method to solve the resulting hyperbolic partial differential equations. Comparison to the analytically known Gubser solution underlines the high accuracy of this technique. We also present first applications of FluiduM to central heavy ion collisions at the LHC energies featuring a realistic thermodynamic equations of state as well as shear and bulk viscous dissipation.

3-D Rotation Representation of Multiple Reflections and Parametric Model for Bistatic Scattering From Arbitrary Multiplate Structure

Based on geometrical optics (GO), this paper develops a representation theory of multiple specular reflections from perfect electric conducting target with an arbitrary patchwise smooth surface. In the course of rays' multiple reflections, the polarization of each ray is found to change in a manner consistent with the variations of its propagating direction, which can be described with an equivalent 3-D rotation transformation. This transformation provides a new representation of the polarization rotation effect induced by multiple reflections. More importantly, the parameters of this transformation characterize the target's GO field response, and their values can classify six classes of geometrical structures. Furthermore, we derive explicit formulas relating these parameters to target's geometrical attributes for double and triple reflections, by using a quaternion algebra method. In order to validate this representation theory, we construct a parametric model of bistatic scattering from an arbitrary multiplate structure, by combining the GO-physical optics (PO) method with this representation. Through two special cases, the representation theory is validated by comparing the RCSs calculated by the parametric model with those by a shooting and bouncing ray (SBR) algorithm. The representation and derived concise parametric model are useful for extracting geometrically relevant features from bistatic radar measurement.

Quaternion Electromagnetism and the Relation with Two-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation.

A Novel Geometric Hierarchical Approach for Dynamic Visual Servoing of Quadrotors

This paper proposes a novel nonlinear geometric hierarchical dynamic visual servoing approach to drive a quadrotor to the desired pose defined by a previously captured image of a planar target. Different from existing works, the key novelty is to extend the positionbased nonlinear hierarchical control to image-based nonlinear hierarchical control. More specifically, by seamlessly integrating the nonlinear hierarchical control with the geometric control, and taking full advantage of the cascade property of the system, the proposed visual servoing strategy does not require the thrust force or its derivative to be measurable when compared with the existing backstepping methods, which brings much convenience for practical applications. For the attitude loop, the axis-angle rotation representation is adopted to design a tracking control law on the vector space s (3). In the outer loop, perspective image moments in the virtual image plane are employed as image feedback to construct the outer-loop image-based visual servoing controller with geometric control and backstepping techniques. Based on Lyapunov techniques and the theory of cascade systems, it is rigorously proven that the proposed image-based controller achieves asymptotic stability. Comparative experiments are conducted to show that the proposed approach has advantages of better transient performance, better steady-state performance, and stronger robustness.