Computer Algebra Solution of GPS N-points Problem

Abstract

A computer algebra solution is applied here todevelop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four ormore pseudoranges at one epoch (the GPS N-points problem).Using Mathematica 5.2 software, the GPS N-pointsproblem is solved numerically, symbolically, semi-symbolically,and with Gauss-Jacobi, on a work station. Forthe case of N > 4, two minimization approaches based onresiduals and distance norms are evaluated for the direct numerical solution and their computational duration iscompared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative directnumerical method. For N = 6, the direct numerical solutionis twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared tothe minimization of the distance norm. Gauss-Jacobi requires eight times more computation time than the direct numerical solution.It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file)for these algorithms (i.e., Sturmfel's resultant, Dixon's resultants, Groebner basis, reduced Groebner basis and Gauss-Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPSToolbox (<a href="http://www.ngs.noaa.gov/gps-toolbox/exist.htm">http://www.ngs.noaa.gov/gps-toolbox/exist.htm</a>) website or the Wolfram Information Center (<a href="http://library.wolfram.com/infocenter/MathSource/6629">http://library.wolfram.com/infocenter/MathSource/6629</a>/).