A New Approach to Ill-Conditioned Problems in Rapid Positioning Using Single Frequency GPS Receivers

Z. Wang

Abstract:	It is crucial to determine integer ambiguities quickly and reliably using carrier phase observations in rapid positioning of the short baselines. In traditional methods, the integer ambiguities are determined by combining the float ambiguity solutions calculated using LSE (Least-Squares Estimate) with various resolution methods of integer ambiguities. In rapid positioning only using single frequency GPS phase data, the ill-condition of the normal equation is severe because of the short observation span and few observations. So, the efficiency of rapid positioning is influenced because the differences are large between the float ambiguity solutions and the correct integer ambiguities. A new approach is proposed for this kind of ill-conditioned problems in this paper. First, an appropriate regularizer R is chosen to weaken the ill-condition of the normal equation based on TIKHONOV regularization method, according to the characteristics of the normal equation formed by DD (double difference) phase observations of the single frequency GPS receivers. The float ambiguity solutions calculated by the new approach are much closer to the correct integer ambiguities than those calculated by LSE; in the meantime, corresponding MSEM (Mean Square Error Matrix) is obtained. Then, combined with LAMBDA method, the new approach can fix the integer ambiguities correctly and quickly using MSEM instead of the cofactor matrix to determine the ambiguity search space. At last, the baseline vector is resolved. The effect of the new approach is illustrated by two numerical examples. The comparisons are made between the new approach and LSE, RE (Ridge Estimate) and TSVM (Truncated Singular Value Method) combining with LAMBDA method to fix the integer ambiguities, respectively. The results show that:(1) the success rate of fixing the integer ambiguities is high and the fixed ambiguities are reliable using the new approach;(2) the new approach shortens the observation span and thus boosts the efficiency in rapid positioning.
Published in:	Proceedings of the 16th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS/GNSS 2003) September 9 - 12, 2003 Oregon Convention Center Portland, OR
Pages:	1229 - 1239
Cite this article:	Wang, Z., "A New Approach to Ill-Conditioned Problems in Rapid Positioning Using Single Frequency GPS Receivers," Proceedings of the 16th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS/GNSS 2003), Portland, OR, September 2003, pp. 1229-1239.
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