Computer algebra solution of the GPS N-points problem

Abstract

A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss-Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the 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 GPS Toolbox

Determination of Spatial Configuration of an Underwater Swarm with Minimum Data

“…To obtain the complete configuration, it is enough to choose one member as the axes' origin and obtain the relative positions in space of three or (better) four or more of the other members as a constellation. At a later date, any other member will be able to localize from its distance from the constellation, using algorithms commonly used in GPS calculations [69, 70]; in this manner, the initial members are used like a constellation of satellites. Therefore, we now need to localize some members, with respect to one [69].…”