Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Zavoti and Kalmar (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6
 ).

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A dual quaternion algorithm of the Helmert transformation problem

Nonlinear geodetic datum transformation is particular suitable for the case of large rotation angles, and has been an active area of research in recent years. In this paper, we propose a new analytical solution. The method expresses the scale parameter and translation parameters as the function of rotation matrix, through the derivation process. Then, according to the orthogonality of rotation matrix, the method adopts Rodrigues matrix to represent it. Using the relative formula of Rodrigues matrix with anti-symmetric matrix which is formed by a rotation vector, we derive the solution formula of the rotation vector. Finally, we obtain the analytical solution to the seven parameters. The case studies indicate that the proposed method is valid, regardless of whether the coordinates are polluted by noise or not, and whether rotation angles are big or small.

A new analytical solution of nonlinear geodetic datum transformation

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 estimat...

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

The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

Iterative solution of Helmert transformation based on a unit dual quaternion

As the gravity formation flying spacecraft jointly developed by National Aeronautics and Space Administration and German Research Centre for Geosciences, the Gravity Recovery and Climate Experiment (GRACE) and GRACE follow-on (GRACE-FO) satellites have adopted the same K-band ranging system and orbit design in order to detect the Earth’s gravity field information. Different from the BlackJack receiver onboard GRACE, a TriG GNSS receiver is loaded on the GRACE-FO satellites. The orbit determination accuracy of GRACE is better than 2.5 cm, but for GRACE-FO there is no comprehensive assessment of orbit accuracy. We discuss the quality of Global Positioning System data, as well as effects of phase center variation (PCV) model and attitude for GRACE and GRACE-FO. The results show that there is no significant difference in terms of rate of change in ionospheric delay (IOD) and multipath effect, which suggests that the performance of TriG receiver is as excellent as that of BlackJack receiver. After using PCV corrections, the root mean square (RMS) errors of kinematic and reduced-dynamic (RD) orbit residuals decrease by 0.4–0.5 and 0.6–0.9 mm, respectively. Satellite laser ranging RMS errors for RD orbit solutions are lower than 2.59 cm whether PCV corrections are used or not. The effect of attitude data on kinematic and RD orbits indicates that nominal attitude data can reliably replace measured attitude data in GRACE-FO orbit determination.

On GPS data quality of GRACE-FO and GRACE satellites: effects of phase center variation and satellite attitude on precise orbit determination

The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a non-isotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a non-isotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

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Iterative approach of 3D datum transformation with a non-isotropic weight

To solve the problem that traditional solution of coordinate transformation parameters based on rotation angles depends strongly on initial value of parameter, the paper adopts quaternion to represent 3D rotation matrix, then puts forward a quaternion-based solution of non-linear 3D coordinate transformation parameters. The case study shows that the solution has a weak dependence on the initial value of parameter and satisfactory result with fast speed, and is suitable for coordinate transformation of any rotation angle.

Yue Wu

Papers

Iterative solution of Helmert transformation based on a unit dual quaternion

Iterative approach of 3D datum transformation with a non-isotropic weight

A Quaternion-Based Solution of Non-Linear 3D Coordinate Transformation Parameters