Research On Coordinate System Transformation Based On Total Least Squares Algorithm

The Gauss Markov(G-M)model based on the least square principle is widely used in the field of Geodesy and engineering survey.The premise of this model is to consider the observation vector error.However,with the development of surveying and mapping theory for many years,if only the error of observation vector is considered,there will be a big deviation between the estimated parameters and the true values.The limitation of traditional transformation model and method in application is more and more obvious,in view of this,this paper does the following work on the application of total least squares algorithm in coordinate transformation:(1)In this paper,the total least squares(TLS)algorithm considering both the errors of observation vector and coefficient matrix is studied.The total least squares method based on singular value decomposition and the total least squares iteration method based on Lagrange function are derived.The error in variables(TLS)is verified by a linear fitting example,EIV)model is more reasonable,and the mean square error is reduced by 45.32%compared with the least square algorithm,which can obtain more reliable parameter estimation.(2)Considering that the coefficient matrix contains both fixed elements and random elements,and there is no error in the coefficient matrix,the least squares-total least squares(LS-TLS)algorithm solves this problem,and can obtain the parameter estimation with better accuracy.LS,TLS and LS-TLS are used to solve the parameter estimation,and the unit weight mean square error is used to evaluate the accuracy.The results show that the accuracy of LS-TLS is 33.33%higher than that of LS and 24.10%higher than that of TLS.(3)Under unequal precision,WTLS iterative method based on Gauss Newton and partial errors in variables(PEIV)model are derived.The equivalence of the two WTLS methods is verified by a line fitting example.The accuracy of WTLS is 29.54%,41.87%and 49.72%higher than that of LS-TLS,TLS and LS respectively.The three-dimensional space coordinate transformation parameters solved by the four algorithms show that the parameters of WTLS are the best,and the root mean square error of checking points is the smallest,while the other three methods are used the root mean square error of the three algorithms is greater than the root mean square error of WTLS,which shows that WTLS can effectively solve the problem of unequal precision of observation vector and coefficient matrix.(4)This paper focuses on the case of gross errors in the observed values.Because TLS,LS-TLS and WTLS cannot achieve good results in solving the gross error problem,this paper deduces the residual covariance matrix,obtains the standardized residual,and then uses the standardized residual to construct the cofactor factor function and the median method to obtain the unit weight mean square error.The robust total least squares algorithm(robust least squares-total least squares)is proposed,In the solution of RLS-TLS and RWTLS,the weight function of IGG series is selected,and the weight of the observation value containing gross error is gradually modified in the continuous iteration,so as to gradually realize the ability to resist gross error.Through the analysis of example 6 and engineering example,it is found that when there are gross errors,the estimated parameters of LS-TLS and WTLS are quite different from the real values,while the results of RLS-TLS and RWTLS are closer to the real values,through the unit weight error,we can see that the accuracy of the parameters obtained by other algorithms is improved by more than 30%.With the increase of the number of gross errors,the two algorithms are still robust and have certain advantages in dealing with gross errors.Figure[20]Table[27]Reference[75]...