Quaternion Method Research Articles

A Conformal Geometric Algebra Code Generator Comparison for Virtual Character Simulation in Mixed Reality

Over the last few years, recent advances in user interface and mobile computing, introduce the ability to create new experiences that enhance the way we acquire, interact and display information within the world that surrounds us with virtual characters. Virtual reality (VR) is a 3D computer simulated environment that gives to user the experience of being physically present in real or computer-generated worlds; on the other hand, augmented reality (AR) is a live direct or indirect view of a physical environment whose elements are augmented (or supplemented) by computer-generated sensory inputs. Both technologies use interactive devices to achieve the optimum adaptation of the user in the immersive world achieving enhanced presence, harnessing latest advances in computer vision, glasses or head-mounted-displays featuring embedded mobile devices. A common issue in all of them is interpolation errors while using different linear and quaternion algebraic methods when (a) tracking the user’s position and orientation (translation and rotation) using computer vision; (b) tracking using mobile sensors; (c) tracking using gesture input methods to allow the user to interactively edit the augmented scene (translation, rotation and scale) and (d) having animation blending of the virtual characters that augmented the mixed reality scenes (translation and rotation). In this work, we propose an efficient method for robust authoring (rotation) of Augmented reality scene using Euclidean geometric algebra (EGA) rotors and we propose two fast animation blending methods using GA and CGA. We also compare the efficiency of different GA code generators: (a) Gaigen library, (b) libvsr and (c) Gaalop using our animation blending methods and compare them with other alternative animation blending techniques: (a) quaternions and (b) dual-quaternions, so that a future user of GA libraries can choose the most appropriate one that will give the most optimal and faster results.

The Foldability of Cylindrical Foldable Structures Based on Rigid Origami

Foldable structures, a new kind of space structures developed in recent decades, can be deployed gradually to a working configuration and also can be folded for transportation, thus have potentially broad application prospects in the fields of human life, military, aerospace, building structures, and so on. Combined with the technology of origami folding, foldable structures derive more diversified models, and the foldable structures in cylindrical shape are mainly studied in this paper. Some researchers use the theory of quaternion representing spatial fixed-point rotation and construct the rotating vector model to obtain the quaternion rotation sequence method (QRS method) analyzing origami, but the method is very limited and not suitable for the cylindrical foldable structures. In order to solve the problem, a new method is developed, which combines the QRS method and the dual quaternion method. After analyzing the folding angle via the QRS method for multivertex crease system and calculating the coordinates of all vertices via the dual quaternion, the rigid foldability can be checked. Finally, two examples are carried out to confirm validity and versatility of the method.

Modeling and control of an agile tail-sitter aircraft

This paper presents a model of an agile tail-sitter aircraft, which can operate as a helicopter as well as capable of transition to fixed-wing flight. Aerodynamics of the co-axial counter-rotating propellers with quad rotors are analyzed under the condition that the co-axial is operated at equal rotor torque (power). A finite-time convergent observer based on Lyapunov function is presented to estimate the unknown nonlinear terms in co-axial counter-rotating propellers, the uncertainties and external disturbances during mode transition. Furthermore, a simple controller based on the finite-time convergent observer and quaternion method is designed to implement mode transition.

Quaternion Approach to the Measurement of the Local Birefringence Distribution in Optical Fibers

A method to structure the Jones quaternion utilizing Pauli matrices is proposed, improving the quaternion theory of polarization optics. It is proved that the Stokes quaternion is the double product of the Jones quaternion and its Hermitian transpose and that the rotation axis of the output three-adjacent-point Stokes quaternions is a mapping from the rotation axis of the three-adjacent-reflection-point Stokes quaternions in the optical fiber and that the rotation angle among the output three adjacent points is the doubleness of the rotation angle among the three adjacent reflection points in the optical fiber. The three-point quaternion method to measure birefringence distribution is proposed and implemented, the experimental result shows that the average birefringence is 0.263/m in the fiber under test, and the corresponding average beat length is 26 m. We also demonstrated the paraxial approximate quaternion Baker–Campbell–Hausdorff formula of exponential quaternions multiplying for cascaded optical components; then, a quaternion interpolation method to estimate the error of birefringence measurement is proposed, and the relative error calculated was less than 5%. The previously cited works indicate that the quaternion algorithm is very effective in the research of the polarization in optical fibers.

Direction Finding and Information Detection Algorithm with an L-Shaped CCD Array

ABSTRACTThis paper proposes a quaternion-ESPRIT (estimate signal parameters via rotation invariance) algorithm for estimating frequency, direction of arrival (DOA) and polarization, using a collocated crossed dipole (CCD) L-shaped array configuration. First, two sets of synchronous sampling data are used to construct the data covariance matrix, the array steering vector of the whole array is estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Second, the direction cosine of the x-axis and y-axis are acquired by applying block operation to the spatial steering vectors, hence the estimates of two-dimensional DOA are obtained. Finally, the polarization parameters are estimated using the relationship between the dipole sub-array steering vectors of the x-axis and y-axis. The closed-form solutions for each parameter are obtained and parameters can automatically match. The simulation results verify that the performance of quaternion method is obviously better than that of the long-vector method.

A new grid deformation technology with high quality and robustness based on quaternion

Quality and robustness of grid deformation is of the most importance in the field of aircraft design, and grid in high quality is essential for improving the precision of numerical simulation. In order to maintain the orthogonality of deformed grid, the displacement of grid points is divided into rotational and translational parts in this paper, and inverse distance weighted interpolation is used to transfer the changing location from boundary grid to the spatial grid. Moreover, the deformation of rotational part is implemented in combination with the exponential space mapping that improves the certainty and stability of quaternion interpolation. Furthermore, the new grid deformation technique named “layering blend deformation” is built based on the basic quaternion technique, which combines the layering arithmetic with transfinite interpolation (TFI) technique. Then the proposed technique is applied in the movement of airfoil, parametric modeling, and the deformation of complex configuration, in which the robustness of grid quality is tested. The results show that the new method has the capacity to deal with the problems with large deformation, and the “layering blend deformation” improves the efficiency and quality of the basic quaternion deformation method significantly.

The aggregation and diffusion of asphaltenes studied by GPU-accelerated dissipative particle dynamics

The heavy crude oil consists of thousands of compounds and much of them have large molecular weights and complex structures. Studying the aggregation and diffusion behavior of asphaltenes can facilitate the understanding of the heavy crude oil. In previous studies, the fused aromatic rings were treated as rigid bodies so that dissipative particle dynamics (DPD) integrated with the quaternion method can be used to study asphaltene systems. In this work, DPD integrated with the quaternion method is implemented on graphics processing units (GPUs). Compared with the serial program, tens of times speedup can be achieved when simulations performed on a single GPU. Using multiple GPUs can provide faster computation speed and more storage space for simulations of significant large systems. By using large systems, simulations of the asphaltene–toluene system at extremely dilute concentrations can be performed. The determined diffusion coefficients of asphaltenes are similar to that in experimental studies. At last, the aggregation behavior of asphaltenes in heptane was investigated, and the simulation results agreed with the modified Yen model. Monomers, nanoaggregates and clusters were observed from the simulations at different concentrations.

Concurrent validity of accelerations measured using a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces.

Although accelerometers are extensively used for assessing gait, limited research has evaluated the concurrent validity of these devices on less predictable walking surfaces or the comparability of different methods used for gravitational acceleration compensation. This study evaluated the concurrent validity of trunk accelerations derived from a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces and contrasted two methods used to remove gravitational accelerations; i) subtraction of the best linear fit from the data (detrending); and ii) use of orientation information (quaternions) from the inertial measurement unit. Twelve older and twelve younger adults walked at their preferred speed along firm, compliant and uneven walkways. Accelerations were evaluated for the thoracic spine (T12) using a tri-axial inertial measurement unit and an eleven-camera Vicon system. The findings demonstrated excellent agreement between accelerations derived from the inertial measurement unit and motion analysis system, including while walking on uneven surfaces that better approximate a real-world setting (all differences <0.16 m.s−2). Detrending produced slightly better agreement between the inertial measurement unit and Vicon system on firm surfaces (delta range: −0.05 to 0.06 vs. 0.00 to 0.14 m.s−2), whereas the quaternion method performed better when walking on compliant and uneven walkways (delta range: −0.16 to −0.02 vs. −0.07 to 0.07 m.s−2). The technique used to compensate for gravitational accelerations requires consideration in future research, particularly when walking on compliant and uneven surfaces. These findings demonstrate trunk accelerations can be accurately measured using a wireless inertial measurement unit and are appropriate for research that evaluates healthy populations in complex environments.

Multispectral processing for color particle image velocimetry

Since the adoption of digital video cameras and cross-correlation methods for particle image velocimetry (PIV), the use of color images has largely been abandoned. Recently, however, with the re-emergence of color-based stereo and volumetric techniques, and the extensive use of color microscopy, color imaging for PIV has again become relevant. In this work, we explore the potential advantages of color PIV processing by developing and proposing new methods for handling multi-color images. The first method uses cross-correlation of every color channel independently to build a color vector cross-correlation plane. The vector cross-correlation can then be searched for one or more peaks corresponding to either the average displacement of several flow components using a color ensemble operation, or for the individual motion of colored particles, each with a different behavior. In the latter case, linear unmixing is used on the correlation plane to separate each known particle type as captured by the different color channels. The second method introduces the use of quaternions to encode the color data, and the cross-correlation is carried out simultaneously on all colors. The resulting correlation plane can be searched either for a single peak, corresponding to the mean flow or for multiple peaks, with velocity phase separation to determine which velocity corresponds to which particle type. Each of these methods was tested using synthetic images simulating the color recording of noisy particle fields both with and without the use of a Bayer filter and demosaicing operation. It was determined that for single-phase flow, both color methods decreased random errors by approximately a factor of two due to the noise signal being uncorrelated between color channels, while maintaining similar bias errors as compared to traditional monochrome PIV processing. In multi-component flows, the color vector correlation technique was able to successfully resolve displacements of two distinct yet coupled flow components with errors similar to traditional grayscale PIV processing of a single phase. It should be noted that traditional PIV processing is bound to fail entirely under such processing conditions. In contrast, the quaternion methods frequently failed to properly identify the correct velocity and phase and showed significant cross talk in the measurements between particle types. Finally, the color vector method was applied to experimental color images of a microchannel designed for contactless dielectrophoresis particle separation, and good results were obtained for both instantaneous and ensemble PIV processing. However, in both the synthetic color images that were generated using a Bayer filter and the experimental data, a significant peak-locking effect with a period of two pixels was observed. This effect is attributed to the inherent architecture of the Bayer filter. In order to mitigate this detrimental artifact, it is suggested that improved image interpolation or demosaicing algorithms tuned for use in PIV be developed and applied on the color images before processing, or that cameras that do not use a Bayer filter and therefore do not require a demosaicing algorithm be used for color PIV.

Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model

This paper proposes a quaternion-ESPRIT algorithm for closed-form estimation of direction of arrival (DOA) and polarization, using a cylindrical conformal array configuration. First, the array steering vector and the elevation are estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Second, the azimuth is estimated. Finally, the polarization parameters are estimated using the relationships between the dipoles and the loops. These estimates are automatically matched. Simulation results show that the performance of quaternion method is obviously better than that of the long-vector method.

用于声矢量阵列波达方向估计的四元数最小范数法

To reduce the calculation amount of quaternion Multiple Signal Classification(MUSIC)algorithm,a Minimum Norm(MN)quaternion method based on the quaternion form of an acoustic sensor vector was proposed for estimating Direction of Arrival(DOA).First,based on the MN principle,the(M-N)×Mdimension(Mis the number of array sensors,Nis the number of signal sources)orthogonal noise subspace vectors were converted to a 1×M dimension quaternion noise vector.Then,a simplified formula for peak search was proposed.The searching computation amount of Quaternion MN(Q-MN)was analyzed theoretically.Compared with the Q-MUSIC algorithm,the Q-MN algorithm can save at least 50% of the peak search volume.Experimental results show that the orthogonality of the steering vector with the low-dimensional noise vector derived by proposed methodis superior to that with high dimensional noise subspace.When the Signal to Noise Ratio(SNR)is 0dB,the Vandermonde norm and the spatial spectrum peak value of Q-MN algorithm are 1/3 times and 3 times as much as those of Q-MUSIC.The Q-MN algorithm can successfully identify the DOA of signal sources,and its resolution is equivalent to that of Q-MUSIC algorithm.Moreover,it can be applied to not only L-shaped arrays,but also plane arrays and parallel linear arrays.

Direction Finding and Positioning Algorithm with COLD-ULA Based on Quaternion Theory

Electromagnetic vector sensor arrays have been widely applied in the communication, radio, navigation, and so on. The application of electromagnetic vector sensor antenna array will greatly improve the overall performance of the communication system. In this paper, a novel quaternion- ESPRIT (estimation of signal parameters via rotational invariance techniques) algorithm for direction finding and positioning based on COLD (cocentered orthogonal loop and dipole) uniform linear array (COLD-ULA) is proposed. First, quaternion data model of COLD-ULA is deduced and constructed. Second, the array steering vector and the angle of arrival (DOA) are estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Finally, the estimation of polarization parameters are required using the relationships between the dipoles and the loops. The proposed technique not only decouples the DOA estimation information from the polarization estimation information, but also improves the ability of signal detection. Moreover, the proposed technique has the advantage of small amount of calculation and parameter automatic matching. Simulation results show that the performance of quaternion method is obviously better than that of the long-vector method.

Temperature Dependence on Structure and Self-Diffusion of Water: A Molecular Dynamics Simulation Study using SPC/E Model

In this study, molecular dynamics simulations of SPC/E (extended simple point charge) model have been carried out in the canonical NVT ensemble over the range of temperatures 300 to 550 K with and without Ewald summation. The quaternion method was used for the rotational motion of the rigid water molecule. Radial distribution functions gOO(r), gOH(r), and gHH(r) and self-diffusion coefficients D for SPC/E water were determined at 300-550 K and compared to experimental data. The temperature dependence on the structural and diffusion properties of SPC/E water was discussed.

Doing a Good Turn: The Use of Quaternions for Rotation in Molecular Docking

Much work has been done on algorithms for structure-based drug modeling in silico, and almost all these systems have a core need for three-dimensional geometric models. The manipulation of these models, particularly their transformation from one position to another, is a substantial computational task with design questions of its own. Solid body rotation is an important part of these transformations, and we present here a careful comparison of two established techniques: Euler angles and quaternions. The relative superiority of the quaternion method when applied to molecular docking is demonstrated by practical experiment, as is the crucial importance of proper adjustment calculations in search methods.

Research on the Hardware-in-the-Loop Simulation for High Dynamic SINS/GNSS Integrated Navigation System

In the process of developing guided munitions, ground test can only verify the performance of integrated navigation system in low dynamic condition, and its costly and risky to use means of authentication such as flight test and throw experiment. This paper proposes a kind of hardware-in-the-loop simulation (HILS) scheme with tri-axial turntable for verifying the performance of navigation system in high dynamic condition. It respectively uses quaternion method and four-sample rotation vector algorithm as attitude updating algorithms for comparison. On the basis of analyzing the characteristics of some tactical missile and the HILS system, the error sources of integrated navigation system in the simulation with turntable and that without turntable are discussed in detail. The results of HILS show that integrated navigation system is of good performance under high dynamic environment; moreover, for the fiber optic gyroscope (FOG) inertial measurement unit (IMU) which outputs angular rate, quaternion method is better than four-sample rotation vector algorithm.

A research on novel direct vector method for ray tracing

Ray tracing is the foundation of optical system design and analysis. Matrix method is one of the classical methods for ray tracing, while quaternion method gradually becomes an important means to trace ray. However the expression forms of ray are so complex that matrix method and quaternion method are hard to be applied in ray tracing in complicated optical system. In this work, a ray tracing method which uses space vector to describe beams in three-dimensional space directly is demonstrated. First, rotation transformation formula for space vector is derived, then, the direct vector formulae for optical laws are set up. At last, in a ray-tracing example of universal 3D optical system, the direct vector method is much clearer and simpler, comparing with the quaternion method and matrix method. What's more, it is proved that through a complex derivation the quaternion method can be translated into the direct vector method finally.

Relative position and attitude estimation of spacecrafts based on dual quaternion for rendezvous and docking

The capacity to acquire the relative position and attitude information between the chaser and the target satellites in real time is one of the necessary prerequisites for the successful implementation of autonomous rendezvous and docking. This paper addresses a vision based relative position and attitude estimation algorithm for the final phase of spacecraft rendezvous and docking. By assuming that the images of feature points on the target satellite lie within the convex regions, the estimation of the relative position and attitude is converted into solving a convex optimization problem in which the dual quaternion method is employed to represent the rotational and translational transformation between the chaser body frame and the target body frame. Due to the point-to-region correspondence instead of the point-to-point correspondence is used, the proposed estimation algorithm shows good performance in robustness which is verified through computer simulations.

Maneuvering modeling and simulation of AUV dynamic systems with Euler-Rodriguez quaternion method

The purpose of this study is to develop maneuvering models and systems of a simulator to improve the motion performance of autonomous underwater vehicles (AUVs) at the preliminary design stages in advance. The AUVs simulation systems based on the standard submarine equations of motion in six-degree-of-freedom (6-DOF) integrated with the Euler-Rodriguez quaternion method for representing singularity-free AUV attitude and time-saving calculation, and with a nonlinear control model for maneuvering and depth control simulations, time-marching in the fourth-order Runge-Kutta scheme. For validation of the simulation codes, results of the ISiMI AUV open-loop tests including turning test and zigzag test as well as an AUV simulator on the basis of Euler-angle method were used to compare with the quaternion-based AUV simulator. The computational results from the proposed simulator agree well with those from both the ISiMI AUV experiments and the Euler-angle based simulations. Additionally, a new maneuvering procedure, namely “put-out” was implemented to test directional stability for a large-scale AUV in the proposed AUV simulator that can be considered for vehicles in space as well as in constrained planes.

Modeling and Simulation of an AUV Simulator With Guidance System

An intelligent autonomous underwater vehicle (AUV) simulator with path-planning capability was developed. A guidance system of the simulator automatically generates continuous-curvature paths of a cubic B-spline class constrained by the minimum turning radius of the marine vehicle and waypoints. The simulator of the guidance system includes a line-of-sight (LOS) algorithm and a horizontal proportional–derivative (PD) controller, adapting the Euler–Rodriguez quaternion method on the base of 3-D Euler–Lagrange formulation. A web-based interactive simulation system can animate the attitudes and position of the AUV in real time. A 3000-T AUV was used to test the guidance system. Comparisons of linear and cubic path-planning strategies were discussed, including a straight line and a conventional cubic spline method, three parametric methods for planning cubic B-spline paths, and an iterative method for improving and expanding the function of the path generator. Simulation results of the tracking performance tests show that the AUV can precisely approach targets and waypoints using the proposed method. The improvement in the cross-tracking error was approximately 80%, whereas reduction in traveling time was 5%.

Electromagnetic fields in dispersive chiral media generated by modulated nonuniformly moving sources

A representation for the fields generated by moving sources in chiral media in the form of double time-frequency oscillating integrals is obtained by using quaternionic analysis methods. Some additional assumptions concerning the source allow us to introduce a large dimensionless parameter λ > 0 which characterizes simultaneously the slowness of variations of the amplitude and of the velocity of the source. Application of the two-dimensional stationary phase method to the integral representation of the field leads to asymptotic formulas for the electromagnetic field for large λ > 0, and efficient formulas for the frequency and the time Doppler effects in dispersive chiral media. As an application of the proposed method, we consider the Vavilov-Cherenkov radiation in chiral dispersive media.