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    Computer Algebra Solution of GPS N-points Problem

    Access Status
    Fulltext not available
    Authors
    Palancz, Bela
    Awange, Joseph
    Grafarend, Erik
    Date
    2007
    Type
    Journal Article
    
    Metadata
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    Citation
    Palancz, Bela and Awange, Joseph and Grafarend, Erik. 2007. Computer Algebra Solution of GPS N-points Problem. GPS Solutions.
    Source Title
    GPS Solutions
    DOI
    10.1007/s10291-007-0066-8
    ISSN
    15211886
    Faculty
    Department of Spatial Sciences
    Faculty of Science and Engineering
    WA School of Mines
    URI
    http://hdl.handle.net/20.500.11937/13584
    Collection
    • Curtin Research Publications
    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 (http://www.ngs.noaa.gov/gps-toolbox/exist.htm) website or the Wolfram Information Center (http://library.wolfram.com/infocenter/MathSource/6629/).

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