Global Positioning System Integer Ambiguity Resolution Using Factorized Least-Squares Techniques

Abstract

Factorized methods are developed for rapid solution of integer least-squares problems that arise when resolving Global Positioning System carrier cycle ambiguities. Such algorithms can enhance batch estimators and Kalman filters that use carrier-phase differential Global Positioning System data for relative spacecraft position estimation or for attitude determination. The solution of mixed real/integer linear least-squares problems is reviewed, and new algorithms are developed to speed the solution of the integer part of such problems. One new algorithm generates a candidate set of integer vectors that is bounded by an ellipsoid and that is guaranteed to contain the solution. Once generated, this set is searched by brute force to find the integer optimum. The set generator is based on the principle of backsubstitution for upper-triangular linear systems. Two new preconditioning algorithms are developed based on a principle of least-squares ambiguity decorrelation adjustment that seeks an increasing order in the magnitudes of the diagonal elements of the problem's upper-triangular square-root information matrix. These new algorithms decrease computation times in comparison to their nearest competitors by factors ranging from 2 to 4 for a random set of problems that have between 11 and 50 integer unknowns.