Global Navigation Satellite System Ambiguity Resolution with Constraints from Normal Equations

Abstract

Carrier phase ambiguity resolution is the key to precise positioning with a global navigation satellite system (GNSS); therefore, quite a few ambiguity resolution methods have been developed in the past two decades. In this paper, a new ambiguity searching algorithm by treating part of normal equations as constraints is developed. The process starts with the truncation of the terms with respect to the small eigenvalues from the normal equations of a least-squares estimation problem. The remaining normal equations are employed as the constraint equations for the efficient searching of integer ambiguities. In the case of short single baseline rapid GNSS positioning with double differenced phase measurements, there are only three real parameters of the position to be estimated. Therefore three terms of the normal equations should be truncated off due to the fact that there is a large difference between the last three eigenvalues of the normal matrix of the float solution and the others, and then the remaining ambiguities can be trivially solved with three independent ambiguities by means of the remaining normal equations. As a result, only three independent ambiguities are necessarily searched and the searching efficiency is dramatically enhanced. Moreover, a new indicator of minimizing the conditional number of the subsquare matrix of the remaining normal equations is introduced to select three independent ambiguities. Once the correct integer values of the selected three independent ambiguities are applied to solve the remaining ambiguities, the estimated real-valued solutions are very close to their integers, which can be applied as additional strong constraints to further improve the searching efficiency. Finally, two case studies, from real dual-frequency global positioning system (GPS) data of about 10-km baseline and random simulations, respectively, are carried out to demonstrate the efficiency of the new algorithm. The results show that the new algorithm is efficient, especially for the scenarios of high-dimensional ambiguity parameters.