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Direct polynomial approach to nonlinear distance (ranging) problems

Earth, Planets and Space volume 55, pages 231–241 (2003)Cite this article

Abstract

In GPS atmospheric sounding, geodetic positioning, robotics and photogrammetric (perspective center and intersection) problems, distances (ranges) as observables play a key role in determining the unknown parameters. The measured distances (ranges) are however normally related to the desired parameters via nonlinear equations or nonlinear system of equations that require explicit or exact solutions. Procedures for solving such equations are either normally iterative, and thus require linearization or the existing analytical procedures require laborious forward and backward substitutions. We present in the present contribution direct procedures for solving distance nonlinear system of equations without linearization, iteration, forward and backward substitution. In particular, we exploit the advantage of faster computers with large storage capacities and the computer algebraic softwares of Mathematica, Maple and Matlab to test polynomial based approaches. These polynomial (algebraic based) approaches turn out to be the key to solving distance nonlinear system of equations. The algebraic techniques discussed here does not however solve all general types of nonlinear equations but only those nonlinear system of equations that can be converted into algebraic (polynomial) form.

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Authors and Affiliations

  1. Department of Geophysics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto City, Kyoto, 606-8502, Japan

    Joseph L. Awange, Yoichi Fukuda & Shuzo Takemoto

  2. Department of Geodesy and GeoInformatics, Geschwister-Scholl str. 24D, 70174, Stuttgart, Germany

    Erik W. Grafarend

Authors
  1. Joseph L. Awange
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  2. Erik W. Grafarend
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  3. Yoichi Fukuda
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  4. Shuzo Takemoto
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Correspondence to Joseph L. Awange.

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Awange, J.L., Grafarend, E.W., Fukuda, Y. et al. Direct polynomial approach to nonlinear distance (ranging) problems. Earth Planet Sp 55, 231–241 (2003). https://doi.org/10.1186/BF03351754

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  • DOI: https://doi.org/10.1186/BF03351754

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