Computer Algebra Solution of GPS N-points Problem
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, s...
| Main Authors: | , , |
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| Format: | Journal Article |
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John Wiley and Sons, Inc.
2007
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| Online Access: | http://hdl.handle.net/20.500.11937/13584 |
| _version_ | 1848748384264912896 |
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| author | Palancz, Bela Awange, Joseph Grafarend, Erik |
| author_facet | Palancz, Bela Awange, Joseph Grafarend, Erik |
| author_sort | Palancz, Bela |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | 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>/). |
| first_indexed | 2025-11-14T07:04:11Z |
| format | Journal Article |
| id | curtin-20.500.11937-13584 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T07:04:11Z |
| publishDate | 2007 |
| publisher | John Wiley and Sons, Inc. |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-135842017-09-13T16:04:16Z Computer Algebra Solution of GPS N-points Problem Palancz, Bela Awange, Joseph Grafarend, Erik Algebra Positioning CAS GPS 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>/). 2007 Journal Article http://hdl.handle.net/20.500.11937/13584 10.1007/s10291-007-0066-8 John Wiley and Sons, Inc. restricted |
| spellingShingle | Algebra Positioning CAS GPS Palancz, Bela Awange, Joseph Grafarend, Erik Computer Algebra Solution of GPS N-points Problem |
| title | Computer Algebra Solution of GPS N-points Problem |
| title_full | Computer Algebra Solution of GPS N-points Problem |
| title_fullStr | Computer Algebra Solution of GPS N-points Problem |
| title_full_unstemmed | Computer Algebra Solution of GPS N-points Problem |
| title_short | Computer Algebra Solution of GPS N-points Problem |
| title_sort | computer algebra solution of gps n-points problem |
| topic | Algebra Positioning CAS GPS |
| url | http://hdl.handle.net/20.500.11937/13584 |