Title: Solutions of GPS Measurements by the Fixed-Point Iteration Technique
Author(s): Samsung Lim
Published in: Proceedings of the 57th Annual Meeting of The Institute of Navigation
June 11 - 13, 2001
Albuquerque, NM
Pages: 449 - 452
Cite this article: Lim, Samsung, "Solutions of GPS Measurements by the Fixed-Point Iteration Technique," Proceedings of the 57th Annual Meeting of The Institute of Navigation, Albuquerque, NM, June 2001, pp. 449-452.
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Abstract: Due to the errors of observations, the GPS position must be the mini-max solution, or the minimum of absolute deviations, or the least squares solution. In any cases, the GPS position is a solution of root-finding problems. We consider the problem of finding GPS solutions to the fixed-point problem and address the connection between the two. For this purpose, we developed three cases of fixed-point iteration algorithms from the GPS equations and proved their equivalences to the standard Newton- Raphson‰s method. One of the fixed-point iteration algorithms needs less number of floating point operations than the Newton-Raphson‰s. The other two converge slowly. The Newton-Raphson‰s method has the advantage of fast convergence if the initial value is close to the unknown solution. On the contrary, if the initial value is apart from the solution, for instance, if it is set to the center of the Earth, the fast fixed-point algorithm takes less number of iterations to converge. The overall performance of the algorithms depends upon the number of redundant observations. The more satellites observed at the epoch, the better the fixed-point algorithm gets. Position differences between the fixed-point algorithm and the Newton-Raphson‰s are less than 0.5 mm when S/A is on, but less than 0.01 mm at the present time.