A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates |
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Authors: | James D Turner |
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Institution: | (1) Aerospace Engineering Department, Consortium for Autonomous Space Systems, Texas A&M University, 745 H.R. Bright Building, College Station, TX 77843, USA |
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Abstract: | The Cartesian-to-Geodetic coordinate transformation is re-cast as a minimization algorithm for the height of the Satellite
above the reference Earth surface. Optimal necessary conditions are obtained that fix the satellite ground track vector components
in the Earth surface. The introduction of an artificial perturbation variable yields a rapidly converging second order power
series solution. The initial starting guess for the solution provides 3–4 digits of precision. Convergence of the perturbation
series expansion is accelerated by replacing the series solution with a Padé approximation. For satellites with heights < 30,000 km
the second-order expansions yields ~mm satellite geodetic height errors and ~10−12 rad errors for the geodetic latitude. No quartic or cubic solutions are required: the algorithm is both non-iterative and
non-singular. Only two square root and two arc-tan calculations are required for the entire transformation. The proposed algorithm
has been measured to be ~41% faster than the celebrated Bowring method. Several numerical examples are provided to demonstrate
the effectiveness of the new algorithm. |
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Keywords: | Geodetic coordinates Coordinate transformation Constrained minimization Artificial perturbation parameter Padé approximant Non-singular |
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