Euler Rotation Angles Research Articles

Best (orthogonal) fitting ellipsoid with quaternions

The aim of this study is the determination of the best fit ellipsoid to given points by quaternions. The problem of the fitting ellipsoid is frequently encountered in image processing, computer games, medicine, engineering and science applications, geodesy, etc. The ellipsoid fitting problem is the process of determining the ellipsoid that best fits a given set of points in 3D. In the fitting process, it is generally done over two models. The first of these is the algebraic method and the second one is orthogonal (geometric) method. In this study, we tried to solve the problem of algebraic and orthogonal ellipsoid fitting based on Euler angles for the first time over quaternions. The superiority of quaternions over Euler rotation angles is well known. In addition, the variance–covariance matrix of the parameters of the fitted ellipsoid will also be calculated. Numerical applications show that the proposed method can be used successfully.

Scattering of arbitrarily incident Laguerre-Gaussian vortex electromagnetic beams by electrically large-scaled complex targets.

We implement an algorithm, termed parallel-processing physical optics, providing an efficient high-frequency approximation method to characterize the scattering of Laguerre-Gaussian (LG) vortex electromagnetic (EM) beams by electrically large-scaled complex targets. The incident beam is described by vector expressions in terms of electric and magnetic fields, and it is combined with rotation Euler angles to achieve an arbitrary incidence of the vortex beam. The validity and capability of the proposed method are illustrated numerically, and the effects of various beam parameters as well as target geometric models such as a blunt cone and Tomahawk-A missile on monostatic and bistatic radar cross section distributions are investigated. Results show that the scattering features of the vortex beam vary significantly with the parameters of the vortex beam and the target. These results are helpful to reveal the scattering mechanism of LG vortex EM beams and provide a reference for the application of vortex beams to detect electrically large-scaled targets.

Accuracy and Precision of Agents Orientation in an Indoor Positioning System Using Multiple Infrastructure Lighting Spotlights and a PSD Sensor.

In indoor localization there are applications in which the orientation of the agent to be located is as important as knowing the position. In this paper we present the results of the orientation estimation from a local positioning system based on position-sensitive device (PSD) sensors and the visible light emitted from the illumination of the room in which it is located. The orientation estimation will require that the PSD sensor receives signal from either 2 or 4 light sources simultaneously. As will be shown in the article, the error determining the rotation angle of the agent with the on-board sensor is less than 0.2 degrees for two emitters. On the other hand, by using 4 light sources the three Euler rotation angles are determined, with mean errors in the measurements smaller than 0.35° for the x- and y-axis and 0.16° for the z-axis. The accuracy of the measurement has been evaluated experimentally in a 2.5 m-high ceiling room over an area of 2.2 m using geodetic measurement tools to establish the reference ground truth values.

Retrieval of Euler rotation angles from 3D similarity transformation based on quaternions

ABSTRACT Recently, it has been shown how quaternion-based representation of a rotation matrix has advantages over conventional Eulerian representation in 3D similarity transformations. The iterative estimation procedure in similarity transformations based on quaternions results in translations and (scaled) quaternion elements. One needs, therefore, an additional procedure for evaluating the rest of the transformation parameters (translation, scale factor and rotation angles) after this solution.This contribution shows how to evaluate the rotation angles and the full covariance matrix of the transformation parameters from the estimation results in asymmetric and symmetric 3D similarity transformations based on quaternions.

Comparison of Some Molecular Structures by Minimizing the Comparison Function

The geometric difference in the relative arrangement of atoms in molecules with the same chemical formula significantly affects the properties of the crystal. For quantitative comparison of the spatial geometric structure of two molecules, we used the method of optimal superposition of molecules through the minimization of a certain comparison function by rotating the molecules after superposing their centers of mass. The search for the minimum of the comparison function over Euler rotation angles is performed by the Rosenbrock method. The assumed criterion for comparing molecules allows us to quantify the proximity in the spatial structure of molecules. The implementation of the method is shown by comparing the structure of eight molecules in four crystalline substances.

Adaptive LEO-Phase Free-Return Orbit Design Method for Manned Lunar Mission Based on LEO Rendezvous

To design a free-return orbit for manned lunar mission based on low earth orbit (LEO) rendezvous, an adaptive LEO-phase free-return orbit design method based on high-precision dynamics model is proposed. First, the radius of perilune and the absolute value of perilune velocity are decoupled using a coordinate system rotation, which is derived from moon-centric local vertical and local horizontal instantaneous coordinate system at the time of perilune. The two Euler rotation angles and the absolute value of perilune velocity are used as independent design variables because their initial values are easy to guess. Next, a two-segment numerical integration strategy is proposed to calculate orbital elements at the moments of trans-lunar injection and free-return vacuum perigee. Subsequently, an optimization algorithm software package for solving large-scale nonlinear sequential quadratic programming problems (SQP_snopt) is employed to search the objective free-return orbit with a fixed trans-lunar injection inclination and the other two constraints on radiuses of perigee at the times of trans-lunar injection and vacuum perigee. After that, an iteration algorithm is devised to adjust trans-lunar injection window for adaptive LEO-phase. Finally, numerical results show a fast and accurate performance of the direct optimization method, which can provide valuable references to manned lunar missions based on LEO rendezvous.

Real-space computation of E/B-mode maps. Part I. Formalism, compact kernels, and polarized filaments

We derive full-sky, real-space operators that convert between polarization Stokes Q/U parameters and the coordinate-independent scalar E/B modes that are widely used in Cosmic Microwave Background (CMB) and cosmic shear analysis. We also derive real space operators that decompose the measured Stokes parameters into those corresponding to E-modes and B-modes respectively, without ever evaluating the scalar fields themselves. We cast the standard CMB polarization analysis operators in a matrix-vector notation which elucidates these derivations. For all these real space operators we show that the kernels split naturally into angular and radial parts and we show explicitly how the radial extent of these kernels depends on the targeted band-limit. We show that the kernels can be interpreted either as a complex convolving beam or as a Green's function when they are expressed in terms of the forward or inverse rotation Euler angles. We show that an arbitrary radial function can produce E/B-like maps, provided it vanishes at the origin and the antipodal point. These maps are simply filtered versions of the standard E/B maps. We can recover the standard power spectrum of the polarized CMB sky by correcting the power spectra of these maps with a simple window function, which we show how to derive for any radial dependence. For these reasons we can compute E/B maps in real space with a compactly-supported kernel, an approach that can guarantee the avoidance of known foreground regions and could be employed in a massively-parallel scheme at high-resolution. We show that the spin raising and lowering operators ð2/ð2 are special cases of these generalized radial functions, and present their band limited versions. The spatial structure of the real space operators provides great intuition for the E/B structure of polarized, filamentary galactic foregrounds. We predict a non-zero B-mode signature that is expected from polarized filaments in the sky. This paper is the first part in a series of papers that explore real-space computation of polarization modes and their applications.

Reduction of technological risks in subway and rail transport

The aim of research is ensuring the guaranteed safety of passengers traveling on subway trains. The object of the researches carried out by the authors is process of interaction of the subway arriving at high speed to the subway station with the passenger, motionlessly waiting, or, which moves to the edge of the platform as a result of the braking of a close mass of passengers. It is shown that a significant friction of the heel surface of the passenger's shoes and the friction of the upper body in the side surface of the car limits the possibility of human movement in space. Thus, in the simplest case, the upper part and extremities of the legs will form a fixed axis of rotation. But, a powerful blow deforms this axis and directs in such way that it will move over the surface, which is a collection of instantly rotational movements. And, thus, the instantaneous axis of rotation describes a conical surface with a vertex that coincides with the heels of the shoe. It is revealed that by rigidly tying the coordinate system xyz to a person it is possible to determine three Euler rotation angles relative to the instantaneous axis and to construct a table of directing cosines that allow determining the kinematic characteristics of the human forced body relative to the axes associated with it – ω x ω y ω z . The results of the model studies confirm the opinion about the expediency of replacing the translational motion of the side surface of the car with a complex movement, additionally equipped on the side surface of the car with a warning stripe. This stripe moves in the opposite direction to the car side, but with the same speed. Synthesis of these two movements gives the effect of the appearance of a fixed part of the warning stripe with respect to the passenger standing on the platform and, therefore, maximizes the confidence and reliability of the passengers' transportation by electric trains of the subway. The laboratory model completely confirms the predicted effect of the danger absence of injury to passengers and the degree of technological risk presence in operating conditions.

Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches

The solution of Kepler’s equation is accomplished via families of hybrid and table lookup techniques. A new arithmetic operation timing approach with general application to scientific programming is used to accurately estimate performance of each Kepler’s equation solution technique. The hybrid approaches couple a power series expansion starting approximation with nine different types of higher-order corrective step methods. The resulting computationally efficient non-iterative methods avoid “if” statements and directly yield in-plane Euler rotation angles necessary to map orbit elements to orbital position. The best-performing of the nine hybrid methods are up to two times faster than the original efficient Laguerre iterative method and achieve worst-case resultant true anomaly accuracies down to machine precision at 3 × 10−11° for eccentricities up to 0.999999. This matches or exceeds the performance of iterative methods and translates to less than three micro-meters at GEO altitude. Meanwhile, six digital approaches were explored, with the best table lookup approach boasting a ten-fold speed increase with a worst-case accuracy of 1 × 10−6° (154 mm) for an eccentricity of 0.999999. These combinations of accuracy and speed performance make both the hybrid and table lookup approaches well-suited for in-line incorporation into a wide range of low- to high-fidelity semi-analytic orbit propagators and multi-threaded or vector programming languages and computing hardware (GPUs, etc.). And finally, a new operator-based computational timing technique is designed and employed to estimate the performance of math-laden computer programs, with sample application to all of the aforementioned Kepler’s equation solution techniques.

Paleogeodynamics of the Bransfield Strait

The axes of zones split off between the northern continental slope of the Bransfield Basin and its southern slope belonging to the Antarctic Peninsula are reconstructed by determining the Euler poles and rotation angles that describe the splitting-off process. The revealed difference in the depths of joined isobaths of up to a hundred of meters reflects the different-scale slippage of peripheral parts of the continental crust from the main body of the Antarctic Peninsula along the plane of the through-lithosphere fracture. On the basis of paleogeodynamic reconstruction, it is possible to reconstruct also the initial bottom topography before the splitting off of the slipping fragments. It is shown that peripheral areas towered initially over the main surface of the shelf of the Antarctic Peninsula for many tens of meters. The locations of tectonic distortions along both borders of the strait due to transtension are reconstructed.

Calculation of the robot trajectory for the optimum directional orientation of fibre placement in the manufacture of composite profile frames

This article deals with theissue of calculating the trajectory of the end-effector of an industrial robot in the manufacture of composites. In the introduction to the article we describe the basic approaches used in the manufacture of composites. Robots are used to define the winding orientation of carbon fibre strands on an uneven polyurethane 3D core. The core is attached to the robot-end-effector and is led through a fibre-processing head according to a suitably defined robot trajectory during dry carbon fibre winding on the core. The model of a passage of the polyurethane core through a fibre-processing head is described in the article. The placement of the fibre-processing head is defined in the basic Euclidean coordinate system E3 of the robot. The core is specified in the local coordinates of the Euclidean coordinate system E3, the origin of this local system is in the robot-end-effector. The positioning of the local system in the basic system of the robot is entered using the “tool centre point” of the robot. A matrix calculus is used when calculating the trajectory robot-end-effector to determine the desired passage of the core through the fibre-processing head. Gradually, the required rotation and translation matrices of the local coordinate system of the robot-end-effector relative to the basic system are calculated and subsequently the Euler angles of rotation are determined corresponding to the transformation matrices. This is used to determine the sequence of values of the “tool centre point” for defining the desired trajectory of the robot-end-effector. The calculation for the trajectory was programmed in the Delphi development environment. The article also solves practical tasks of the polyurethane core passage through the fibre-processing head. The calculations of the trajectory of the robot-end-effector were used as input values for the graphic software simulator and at the same time winding of carbon strands on the polyurethane core was verified for the calculated trajectory of the robot-end-effector in the experimental laboratory.

Numerical Investigation on Scattering of an Arbitrarily Incident Bessel Beam by Fractal Soot Aggregates

Scattering of an arbitrarily incident Bessel beam by fractal soot aggregates is numerically investigated. The incident beam is described by the vector expressions of the zero-order Bessel beam in combination with rotation Euler angles. The scattering problems involving fractal soot aggregates are formulated with a hybrid vector finite element-boundary integral-domain decomposition method. Some numerical results are included to show the scattering behaviors of fractal soot aggregates when they are illuminated by Bessel beams.

Scattering of Bessel beam by arbitrarily shaped composite particles with core–shell structure

This study investigates the scattering of Bessel beam by composite particles with core–shell structure. Specifically, the vector expressions of zero-th order Bessel beam that satisfy well Maxwell׳s equations in combination with the rotation Euler angles are used to represent the arbitrarily incident Bessel beams. An efficient numerical method based on surface integral equations is introduced to formulate the scattering problems involving arbitrarily shaped composite particles with core–shell structure. Solutions are performed iteratively by using the multilevel fast multipole algorithm. The numerical results for differential scattering cross sections of several selected composite particles are presented and analyzed. This investigation is expected to provide useful guidance for techniques of laser detection on particle, diagnosis, and manipulation.

Multiple scattering of arbitrarily incident Bessel beams by random discrete particles

In this paper, we introduce an efficient numerical method to characterize the multiple scattering by random discrete particles illuminated by Bessel beams with arbitrary incidence. Specifically, the vector expressions of Bessel beams that perfectly satisfy Maxwell's equations in combination with rotation Euler angles are used to represent the arbitrarily incident Bessel beams. A hybrid vector finite element-boundary integral-characteristic-basis function method is utilized to formulate the scattering problems involving multiple discrete particles with a random distribution. Due to the flexibility of the finite element method, the adopted method can conveniently deal with the problems of multiple scattering by randomly distributed homogeneous particles, inhomogeneous particles, and anisotropic particles. Some numerical results are included to illustrate the validity and capability of the proposed method and to show the scattering behaviors of random discrete particles when they are illuminated by Bessel beams.

Scattering of arbitrarily incident Gaussian beams by fractal soot aggregates

A hybridization of the finite element and boundary integral methods is applied to the study of light scattering by fractal soot aggregates illuminated by Gaussian beams with arbitrary incidence. In particular, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is employed to represent the arbitrarily incident Gaussian beams. The finite element method is used to obtain the solution of the vector wave equation inside each primary particle and the boundary integral equations are applied on the surfaces of all the particles as a global boundary condition. The resultant matrix equation is solved by an iterative method, where the multilevel fast multipole algorithm can be employed to speed up the matrix–vector multiplication. Some numerical results are included to illustrate the validity of the present method and to show the scattering behaviors of fractal soot aggregates when they are illuminated by Gaussian beams.

Scattering of Gaussian beam by arbitrarily shaped inhomogeneous particles

A hybrid finite element-boundary integral method is applied to characterize the scattering of an arbitrarily incident-focused Gaussian beam by arbitrarily shaped inhomogeneous particles. Specifically, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is used to represent the arbitrarily incident Gaussian beams. The finite element method is employed to formulate the fields in the interior region of the inhomogeneous particle, while the boundary integral equation is applied to represent the fields in the exterior region. The interior and exterior fields are coupled by means of the field continuity conditions. To reduce the computational burden, the frontal method and the multilevel fast multipole algorithm are adopted to solve the resultant matrix equation. Numerical results for differential scattering cross sections of several selected inhomogeneous particles are presented and can be served as further study on this subject.

Scattering of Gaussian beam by arbitrarily shaped particles with multiple internal inclusions

In this paper, we introduce an efficient numerical method based on surface integral equations to characterize the scattering of an arbitrarily incident Gaussian beam by arbitrarily shaped particles with multiple internal inclusions. The incident Gaussian beam is described by the Davis-Barton fifth-order approximation in combination with rotation Euler angles. For numerical purposes, the surfaces of the host particle and the inclusions are modeled using small triangular patches and the established surface integral equations are discretized with the method of moments. The resultant matrix equation is solved by using a parallel implementation of conjugate gradient method on distributed-memory architectures. Some numerical results are included to illustrate the validity and capability of the developed method. These results are also expected to provide useful insights into the scattering of Gaussian beam by composite particles.

Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams

In this paper, we present an efficient numerical method for the simulation of multiple scattering by random discrete particles illuminated by focused Gaussian beams with arbitrary incidence. Specifically, the Davis first-order approximation in combination with rotation Euler angles is used to represent the arbitrarily incident Gaussian beams. The surface integral equations are applied to formulate the scattering problems involving multiple discrete particles with a random distribution and are numerically discretized by the method of moments. The resultant matrix equation is solved by employing the characteristic basis function method based on the use of macrobasis functions constructed according to the Foldy-Lax multiple scattering equations. Since this method only requires the solution of small-size matrix equations associated with isolated particles and it is also readily parallelized, the computational burden can be significantly relieved. Some numerical results are included to illustrate the validity of the present method and to show the scattering behaviors of random discrete particles when they are illuminated by focused Gaussian beams.

Factors determining the spin axis of a pitched fastball in baseball

In this study, we wished to investigate the factors that determine the direction of the spin axis of a pitched baseball. Nineteen male baseball pitchers were recruited to pitch fastballs. The pitching motion was recorded with a three-dimensional motion analysis system (1000 Hz), and the orientations of the hand segment in a global coordinate system were calculated using Euler rotation angles. Reflective markers were attached to the ball, and the direction of the spin axis was calculated on the basis of their positional changes. The spin axis directions were significantly correlated with the orientations of the hand just before ball release. The ball is released from the fingertip and rotates on a plane that is formed by the palm and fingers; the spin axis of the ball is parallel to this plane. The lift force of the pitched baseball is largest when the angular and translational velocity vectors are mutually perpendicular. Furthermore, to increase the lift forces for the fastballs, the palm must face home plate.

Pattern analysis of conformal array based on geometric algebra

In this study, a general and systematic method for three-dimensional pattern analysis of arbitrary conformal arrays is presented; this analysis is based on the mathematical framework of geometric algebra and considers both directional and polarised element patterns. A compact representation of the conformal array pattern is presented. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors that can easily be formulated for any arbitrary conformal array geometry. Analysing conformal arrays using geometric algebra is simpler than the traditional Euler rotation angles and matrix representations. As well as presenting the new derivation and comparing it with conventional methods, the authors also present simulation results on cylindrical and conical arrays to illustrate the practicality and conciseness of the proposed method.