Although the direct GPS solution proposed by Bancroft requires the solution of nonlinear equations, these equations may be manipulated into a smaller number of linear equations that are sufficient to determine the user position when five or more satellites are visible. Because no approximations were required to arrive at this linear form, the resulting solution cannot converge to an incorrect position the way that repeated linearization methods sometimes do. When the measured pseudoranges are noisy, the linear structure allows one to estimate the solution and know the error covariance of the estimate. The conversion to the linear form excludes information present in a single scalar nonlinear measurement equation. We derive several procedures for refining the linear estimate with this remaining information. We also demonstrate extensions of our solution techniques to differential GPS problems.
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