In this post I’ll show implementation of algorithm for calculating position of GPS satellite on its orbit (see download link at the bottom). I followed equations described in ICD-GPS-200C (Interface Control Document of U.S. Department of Homeland Security).
In order to calculate the position we’ll need satellite almanac that defines current orbital parameters of the satellite. What information is contained in almanacs:
GPS Week Number: The almanac reference week (VINA). 0000..1024 – every 7 days since 22 Aug 1999;
GPS Time of Applicability: The number of seconds in the orbit when the almanac was generated;
Eccentricity: This shows the amount of the orbit deviation from circular (orbit). It is the distance between the foci divided by the length of the semi-major axis (our orbits are very circular);
Inclination Offset: The satellite almanac orbital “inclination angle offset” (dik). This does not include orbital inclination – the angle to which the SV orbit meets the equator (GPS is at approx. 55 degrees);
Rate of Right Ascension: The satellite almanac orbital “rate of right ascension” (OMEGADOT);
Square Root of Semi-Major Axis: This is defined as the measurement from the center of the orbit to either the point of apogee or the point of perigee;
Longitude of Orbital Plane: The satellite almanac orbital “geographic longitude of the orbital plane at the weekly epoch” (OMEEGAO);
Argument of Perigee: An angular measurement along the orbital path measured from the ascending node to the point of perigee, measured in the direction of the SV’s motion;
Mean Anomaly: Angle (arc) traveled past the longitude of ascending node (value= 0..180 degrees or 0..-180 degrees).
Definition of orbital parameters
Definition of orbital parameters using terminology typical of Keplerian orbit parameters. Click images to view full-size:
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Equations
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Result
I calculated position of GPS satellite for 24 hours and built its trajectory (in orbital plane, ECEF coordinates, ECI coordinates):
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Sources of satellite almanacs:
Related file formats:
* TLE
Click HERE to download Mathematica *.nb file with implementation.