Gauss-Helmert Model And Its Robust Solution In 3D Coordinate Transformation

3D coordinate transformations are widely applied in Geodesy, GNSS, Engineering Surveying, Photogrammetry, Remote sensing,3D laser scanning, and other fields. With the development of modern measuring technology, there are more and more requirements to measuring precision. At present, most of the coordinate transformations solutions are still dependent on traditional methods and models, and the precision of the models is often relatively lower. What’s more, the limitation of the theory and application is increasing obviously. Hence, researching higher precision to 3D coordinate transformation models is very important.Solving parameters is the key to 3D coordinate transformation, and the parameters’ precision depends on the precision of transformation model. The traditional 3D coordinate transformation model mainly based on the classical least squares principle, then through establishing Gauss-Markov model to solve transformation parameters. When rotation angle is small, linear Bursa-Wolf model may be used to solve transformation parameters, but when rotation angle is bigger, parameters could be caused serious distortion, even lead to the failure of transformation, under this circumstance we need to adopt nonlinear Bursa-Wolf model to solve parameters.3D coordinate transformation model which is based on least-square principle only considers the random error of observation vector, and neglect the observed values’ random error of the coefficient matrix. Therefore, the model is not very reasonable.By introducing EIV model can effectively solve the problem of coefficient matrix random error. EIV model is mainly based on TLS which can simultaneously consider random errors of observation vector and coefficient matrix. If considering the unequal precision of observation, TLS estimates can further expand as WTLS. This article firstly researching the basic principle of TLS estimation based on linear fitting simple example, implements the several calculating methods of TLS. In the case of equal precision of observation, we can use TLS direct solution based on singular value decomposition and the TLS iterative algorithm based on nonlinear Lagrange function, in the case of unequal precision of observation, we can use WTLS iterative algorithm which based on nonlinear Lagrange function and Newton-Gauss method to solve parameters. Finally, TLS and WTLS iterative algorithm is introduced into the 3D coordinate transformation, consider the Gauss-Helmert model is a generalized expression for standard EIV model, this paper deduces the Gauss-Helmert model of 3D coordinate transformation, then use the Newton-Gauss iteration algorithm, and solve the parameters. Compared with the existing researches, the new algorithm has a certain advantage in the complexity of the derivation and the number of estimated parameters. And the parameters’ precision was consistent with existing high-precision transformation model.In 3D coordinate transformations, the observed values are often polluted by gross errors. In this paper, we research the general RWTLS estimation principle and the relevant robust estimation model. RWTLS method can effectively solve the gross errors in 3D coordinate transformations, but traditional RWTLS estimation of 3D coordinate transformations directly using residual error to establish the weight factor function which couldn’t consider structural space. This paper proposes a RWTLS method based on the nonlinear Gauss-Helmert model for 3D coordinate transformations. By using the nonlinear Gauss-Helmert model to establish Bursa-Wolf model of 3D coordinate transformations, and to deduce the coordinated factor matrix of residuals, then obtain the standardized residuals, finally by using the Newton-Gauss method to deduce the iterative algorithm. Experiment results show that new algorithm under the standardized residual to establish the weight factor function could effectively solve gross errors of observation space and structural space. And with the increase of the number of gross error, the algorithm still have good robustness. Compared with the existing researches, the new algorithm have greater advantages to deal with the gross errors in 3D coordinate transformations.