A Method of Multiple-Step Weighted Least-Square Estimate for Satellite Positioning

The method of Weighted Least-Square estimate (WLS) is often used for satellite positioning to reevaluate the weights for various linkable satellites (of more than 4) in the fitting objective function for deriving optimal position solutions. However, because of a number of complex factors construction and determination of the important weight matrix W are difficult for all algorithms involving weighting for satellites. Starting from linear equations for optimal position solutions, we have studied the patterns of propagations of errors of equivalent pseudo ranges into coordinate errors of position solutions. The propagation patterns involve accumulation of errors in the iteration processes leading to solutions. We subsequently propose a new WLS method together with the approaches of construction and practical calculation of the weight matrix. What is mainly modified by the new method from the position-solving process of the original WLS method is as follows. In the new method arguments whose values are not known beforehand are solved in separate steps, with different steps having independent weight matrices. The new method effectively avoids the weight-matrix calculation that is used to minimize the global residual (residual-squared sum) of all the arguments. Instead, it minimizes the residual of each of the arguments separately. The best estimates of all coordinates constitute the optimal position solution for the location of a user. We have carried out a GPS measurement experiment to test feasibility and accurateness of the new method. The test results show that the new WLS method can appreciably improve the accuracy and stability of a position solution in satellite positioning.