A Modified Weighted Total Least Squares with Application in RAIM Algorithm

In this paper, first the method of solving the linear weighted total least squares, and then its generalization to nonlinear state is discussed; as the problem-solving model for determining the coordinates with pseudo-range GPS observations is fully consistent with this model. Available techniques for solving the TLS are based on the SVD and have a high computational burden. Furthermore, the other presented methods that do not use SVD, need large matrices, and there is need for placing zero in the covariance matrix of the design matrix, corresponding to the errorless columns, which increases the matrix size and, as a result, raises the volume of the calculations. But in the proposed method, problem-solving is done without need for SVD, without introducing Lagrange multipliers, and avoiding the error-free introducing of some columns of the design matrix by entering zero in the covariance matrix of the design matrix. It is performed only using easy equations and on the basis of summation principles, which results in less computing effort and high speed. In the following, an optimal method for weighting the design matrix is presented, which can yield a much better answer to the unknowns in the presence of many failures (here, up to three failures are assumed and tested). Besides, it can estimate the residuals vector so that the failure observations would have larger magnitudes than the others, and could help with detecting them in a safer and more feasible way with respect to any other method.