A Direct Solution to GPS-Type Navigation Equations

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.