Symmetric similarity 3D coordinate transformation based on dual quaternion algorithm

Nowadays a unit quaternion is widely employed to represent the three-dimensional (3D) rotation matrix and then applied to the 3D similarity coordinate transformation. A unit dual quaternion can describe not only the 3D rotation matrix but also the translation vector meanwhile. Thus it is of great potentiality to the 3D coordinate transformation. The paper constructs the 3D similarity coordinate transformation model based on the unit dual quaternion in the sense of errors-in-variables (EIV). By means of linearization by Taylor's formula, Lagrangian extremum principle with constraints, and iterative numerical technique, the Dual Quaternion Algorithm (DQA) of 3D coordinate transformation in weighted total least squares (WTLS) is proposed. The algorithm is capable to not only compute the transformation parameters but also estimate the full precision information of computed parameters. Two numerical experiments involving an actual geodetic datum transformation case and a simulated case from surface fitting are demonstrated. The results indicate that DQA is not sensitive to the initial values of parameters, and obtains the consistent values of transformation parameters with the quaternion algorithm (QA), regardless of the size of the rotation angles and no matter whether the relative errors of coordinates (pseudo-observations) are small or large. Moreover, the DQA is advantageous to the QA. The key advantage is the improvement of estimated precisions of transformation parameters, i.e. the average decrease percent of standard deviations is 18.28%, and biggest decrease percent is 99.36% for the scaled quaternion and translations in the geodetic datum transformation case. Another advantage is the DQA implements the computation and precision estimation of traditional seven transformation parameters (which still are frequent used yet) from dual quaternion, and even could perform the computation and precision estimation of the scaled quaternion.Graphical

Coordinate transformation is a fundamental issue in the related studies of measurement. However, existing methodologies often need to pay more attention to the available spatial information, leading to suboptimal results. This paper addresses this issue by incorporating geometric constraints into the symmetric coordinate transformation. We propose the so-called geo-constrained transformation method based on the joint adjustment of the coordinate transformation in conjunction with the geometric constraints over a set of points. By formulating the geometric constraints as the conditional model, we analyze the effects of geometric constraints on the estimated transformation parameters and point locations. By removing such effects during the symmetric transformation algorithm, the results show better statistical performance and satisfy geometric constraints. Two numerical examples are given to demonstrate the expected improvement in the statistical accuracy. It is shown that the improvement of the point determination accuracy can go beyond 50%.

The aim of a coordinate transformation is to determine the coordinates of a point in a second coordinate system, given their known values in the first system. The subject of coordinate transformations has applications in a wide range of areas. There are different types of coordinate transformation methods: orthogonal, similarity, affine, projective, and polynomial transformations. In this study, we considered a symmetric 12-parameter 3D affine transformation based on the dual quaternion. The performances of different transformation models were compared. We also sought to determine which transformation model to use. A transformation implementation including gimbal lock has been added as an Appendix to clearly demonstrate the advantages of quaternions over Euler angles.

Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible.Graphical

The traditional phase-locked loop (PLL) technique widely used in three-phase four-wire system can only obtain one phase angle, and cannot obtain other parameters of unbalanced voltages, such as the amplitude and phase angle of each phase voltage. In order to improve the performance of PLL in unbalanced three-phase four-wire system with disturbance and reduce the order of PLL to improve the stability of the system, a PLL technique based on three-dimensional (3D) coordinate transformation was proposed in this paper. Firstly, the phase detector error of three-phase unbalanced voltage was deduced based on the 3D coordinate transformation. The error can be regarded as the expression form in Hilbert space after linearizing of it. Based on the correspondence between the orthonormal basis in Hilbert space and the unit vectors in Euclidean space, the error can be mapped to Euclidean space, and the detection control law for each voltage parameter can be designed respectively based on the principle of perpendicular decoupling. Then, the voltage parameter detection control law in Euclidean space is mapped back to Hilbert space, so the PLL technique scheme can be obtained which is beneficial to the implementation in the actual system. The control law of each unbalance indices, such as the amplitude and initial phase of each phase voltage, can be decoupled into a first-order or second-order system, so the system structure is simple, and the control parameter design is convenient. The detection of unbalance indices of the three-phase unbalanced voltages with zero-sequence component can be realized with fast and stable characteristics. The simulation and experiments of the proposed PLL technique are carried out finally under several transient conditions of grid voltages, stable within a quarter cycle at the fastest and always stable in 50 ms, so as to validate the rapidness and stability of the proposed PLL technique.

Total least L1- and L2-norm estimations of a symmetrical coordinate transformation model with a structured parameter matrix are proposed, with the aim to account for the relationships between the transformation parameters. In the model, the errors in the coordinates of the measured points in both the source and target coordinate systems in the transformation model are taken into account. The solution of the proposed symmetrical coordinate transformation model is derived by the use of the total least L1- and L2-norm estimations. In addition, the variance-covariance matrices of the estimated parameters and the adjusted coordinates of the points are further derived in the two proposed methods. A numerical experiment in coordinate transformation is conducted to test the proposed methods. The results show that the proposed total least L2-norm estimation method is suitable for resolving the transformation model when the coordinates of the points in both the source and target systems are contaminated only by random errors. However, in the case of gross errors in the coordinates of the points, the proposed total least L1-norm estimation method performs better than the total least L2-norm estimation, resulting in higher precision of the estimated parameters.

Three-dimensional (3D) coordinate transformations, generally consisting of origin shifts, axes rotations, scale changes, and skew parameters, are widely used in many geomatics applications. Although in some geodetic applications simplified transformation models are used based on the assumption of small transformation parameters, in other fields of applications such parameters are indeed large. The algorithms of two recent papers on the weighted total least-squares (WTLS) problem are used for the 3D coordinate transformation. The methodology can be applied to the case when the transformation parameters are generally large of which no approximate values of the parameters are required. Direct linearization of the rotation and scale parameters is thus not required. The WTLS formulation is employed to take into consideration errors in both the start and target systems on the estimation of the transformation parameters. Two of the well-known 3D transformation methods, namely affine (12, 9, and 8 parameters) and similarity (7 and 6 parameters) transformations, can be handled using the WTLS theory subject to hard constraints. Because the method can be formulated by the standard least-squares theory with constraints, the covariance matrix of the transformation parameters can directly be provided. The above characteristics of the 3D coordinate transformation are implemented in the presence of different variance components, which are estimated using the least squares variance component estimation. In particular, the estimability of the variance components is investigated. The efficacy of the proposed formulation is verified on two real data sets.

Rotation in space is one of the most important problems in mathematics and other sciences. Respectively, the three-dimensional coordinate transformations from one system to another and more specifically, the Helmert transformation problem, is one of the most well-known transformations in the field of engineering. After analyzing the mathematical context of point rotation in space, this chapter presents an investigation of specific data using three different transformation methods. The method of Euler angles, quaternion, and dual-quaternion algebra is used. After research, three artificial sets of data, which were structured in a specific way and forced into specific transformations, were used to find out the sensitivity of each method. In addition, three real transformation problems, concerning monitoring and deformation, were tested, to have an accurate result of which method is best. The problems  Statistical analysis of the results was performed by each method, while it was found that there were significant deviations in rotations and translations in the method of Euler angles and dual quaternions, respectively.

3D coordinate transformation is a problem frequently encountered in many different fields, for example, computer graphics, robotics, aeronautics and computer vision, in addition, in surveying, datum transformation and the transformation of lidar point cloud. This study aims to introduce a completely new expanded dual quaternion method that performs 3D coordinate transformation, which can also calculate the variance-covariance (VCV) matrix of the transformation parameters. The new dual quaternion algorithm (DQA) presented here will be given with two numerical examples as simply and clearly as possible.

This paper proposes a real time symmetrical coordinate transformation applied for the fault-ride-through when the imbalanced fault occurs in the utility power system. The power conditioner of the photovoltaic system in Japan is required to remain connected when the utility power system fault occurs under certain condition. The most popular fault is 1 line-to-ground fault. The next is 2 line-short-circuit fault. Both cause the imbalance voltages and currents in 3-phase power system. This paper presents the novel fault-ride-through control using the real time symmetrical coordinate transform from the memorized 3-phase voltage data. Simulation results show the effectiveness of the proposed method when the dc side voltage is constant and the 2 Line-Short circuit fault. Simulation results also certify that the proper change of the controller of the dc-dc converter from the MPPT control to the constant dc voltage control can perform the good FRT operation.

A generalization of the analytical least-squares solution to the 3D symmetric Helmert coordinate transformation problem with an approximate error analysis

This paper proposes a real time symmetrical coordinate transformation applied for the fault-ride-through when the imbalanced fault occurs in the utility power system. The power conditioner of the photovoltaic system in Japan is required to remain connected when the utility power system fault occurs under certain condition. The most popular fault is 1 line-to-ground fault. This fault causes the imbalance voltages and currents in the utility system. To have better fault ride through ability, it is indispensable quick and precise detection of the imbalanced voltages and phases. This paper presents the novel fault-ride-through control using the real time symmetrical coordinate transform from the memorized 3-phase voltage data. Simulation results show the effectiveness of the proposed method when the dc side voltage is constant and the 1 Line-Low Voltage fault. Simulation results certify that the proper change of the controller of the dc-dc converter from the MPPT control to the constant dc voltage control can perform the good FRT operation. Extended discussion about the role of the dc energy storage and the dc circuit breaker for the future dc system is also shown.

Dual quaternion hand-eye calibration algorithm for hunter-prey optimization based on twice opposition-learning and random differential variation

Using several kinds of coordinate transformations, we standardize the noncanonical symplectic structure of the Ablowitz–Ladik model (A–L model) of nonlinear Schrödinger equation (NLSE), then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the A–L model and also the conserved quantities of the original NLSE. In comparison with a higher order non-symplectic scheme applied directly to the A–L model, we show the overwhelming superiorities of the symplectic method. We also compare the implementation of the same symplectic scheme to different standardized Hamiltonian systems resulting from different coordinate transformations, and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one, in preserving the invariants of the A–L model and the original NLSE.

This paper proposes novel 3-D symmetric RF passive components, including inductors, transformers, and baluns. Layout areas of these components are drastically reduced by means of stacked structure while the symmetry of input and output ports is also maintained. The area saving of a 3-D inductor is up 70%. The 1:1 transformer shows less than 0.1 % inductance mismatch in a 18 GHz range, and K is up to 0.87 at 17 GHz. The 3-D balun manifests less than 0.8 dB gain mismatch from 5.25 GHz to 6 GHz and phase error is about 4/spl deg/ at 5.25 GHz according to measurement results. All the components are fabricated in a 0.18 /spl mu/m standard CMOS process.