A Direct Solution to GPS-Type Navigation Equations

Abstract
One solution to the navigation equations involves iteration on the 4 by 4 augmented range-direction-cosine matrix beginning with an assumed position and so assumed direction cosines, of which there are 12 for 4 satellites. An algebraic, direct solution to this same basic equation set has recently been published. Both of these methods are reviewed. We offer a direct solution using modified functions of the range magnitude data from four satellites to yield user's clock bias correction, user's position, and true range vectors if desired. The highest order of matrix inversion used is 2 by 2. The highest order, nonlinear equation is a numeric square root. The principle of the formulation is use of differences among the range magnitudes and range magnitudes squared. An additional auxiliary difference equation is formed. A computation basis uses the ephimeride differences and an orthogonal vector. The method offers convenience, speed, simplicity, low dimensionality, and precision, with no operational constraints.

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