The Study And Application Of Nonlinear Homotopy Least Squares Theory

Traditional method of surveying data processing is that generally the nonlinear least squares model are expanded into linear ones in the place of the approximate values of unknown parameters by Taylor, neglecting the higher-order terms, then the linear models obtained are used to replace the nonlinear model and data are processed in linear theory and method. To obtain enough accuracy parameters to be estimated, this linear model must meet two preconditions:First, the nonlinear strength of nonlinear models is enough weak; Second, the approximation of unknown parameters is enough accurate. If the two preconditions are difficult to satisfy, the model error caused by the linear approximation will become difficult to control. Most surveying positioning models have strong nonlinear strength, and various surveying control networks to meet special requirements that vary widely make extremely difficult to calculate coordinates of approximation. With the development of surveying and mapping technology, meanwhile with high accuracy demand on surveying data processing increasing in the related field of surveying and mapping services, for these reasons, this linear model is difficult to meet these requirements, and actively carrying out nonlinear model data processing research becomes increasingly important.At the beginning of this thesis, the purpose and significance of nonlinear data processing is stated; a more systematic and comprehensive analysis of the current study on nonlinear surveying data processing is done. Then a discussion in the nonlinear data processing features and application range of the linear approximate method, several Newton-like method (including Newton method, Gauss-Newton method and modified Gauss-Newton method, etc.), direct method of high order partial derivative, sequential quadratic programming (SQPM) and random search methods (including Monte-Carlo method and genetic algorithm) are made respectively.Aimed at the above methods shortcomings-such as limit range of convergence, strong initial values dependence, long computing time and so on, according to the advantages of homotopy method, this thesis proposes a nonlinear homotopy least squares algorithm based on curvature ratio adaptive step. The basic theory of homotopy method is introduced in detail, focusing on the Li-Yorke continuous homotopy algorithm. Then in the algorithm Newton iteration termination criterion is improved, meanwhile, this thesis puts forward a step adaptive control strategy based on curvature ratio, for the two steps improved, the homotopy curve tracking speed and stability is increased, and tracking homotopy curve fast and reliably is achieved.Finally, the proposed nonlinear homotopy least squares algorithm based on curvature ratio adaptive step have been programmed. Through plane network case calculation and analysis, compared with traditional homotopy method, linear approximation method and Newton-like method and so on, this method has a lower dependence on the approximation, a wider range of convergence and a higher stability of parameter estimation. And this method operation time is shorter than classic homotopy method. According to computing and analyzing the three-dimensional coordinate transformation case, this method can satisfy to the arbitrary rotation angle of the coordinate transformation. When Bursa-Wolf model is applied to the coordinate transformation, the rotation angle must be less than 660 sec. If not, coordinate transformation accuracy will decline sharply. However the application of the improved homotopy method is not limited by the rotation angle, and accuracy and stability of parameter estimation is higher than Bursa-Wolf model.