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# GPS and Galileo Satellite Coordinates Computation

Fundamentals
Title GPS and Galileo Satellite Coordinates Computation
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

Table 1 provides the GPS or Galileo broadcast ephemeris parameters to compute their satellite coordinates at any observation epoch. These parameters are periodically renewed (typically every  hours for GPS and  hours for Galileo) and must not be used out of the prescribed time (about four hours), because the extrapolation error grows exponentially beyond its validity period.

The algorithm provided is from the [GPS/SPS-SS, table 2-15] [footnotes 1].The Galileo satellites follow a similar scheme

Table 1: GPS and Galileo broadcast ephemeris and clock message parameters.

In order to compute satellite coordinates from navigation message, the algorithm provided as follows must be used. An accuracy of about  meters (RMS) is achieved for GPS satellites with S/A=0ff and several tens of meters with S/A=on [footnotes 2]:

• Compute the time  from the ephemerides reference epoch  ( and  are expressed in seconds in the GPS week):

If  sec, subtract  sec from . If  sec, add  sec.

• Compute the mean anomaly for ,


• Solve (iteratively) the Kepler equation for the eccentricity anomaly :


• Compute the true anomaly :


• Compute the argument of latitude  from the argument of perigee , true anomaly  and corrections  and :


• Compute the radial distance , considering corrections  and :


• Compute the inclination  of the orbital plane from the inclination  at reference time , and corrections  and :


• Compute the longitude of the ascending node  (with respect to Greenwich). This calculation uses the right ascension at the beginning of the current week (), the correction from the apparent sidereal time variation in Greenwich between the beginning of the week and reference time , and the change in longitude of the ascending node from the reference time :


• Compute the coordinates in TRS frame, applying three rotations (around ,  and ):


where  and  are the rotation matrices defined in Transformation between Terrestrial Frames.

## Notes

1. ^ [GPS/SPS-SS], DoD, USA, Global Positioning System Standard Positioning Service Performance Standard. http://www.navcen.uscg.gov/pubs/gps/sigspec/gpssps1.pdf, 1995.
2. ^ Actually, the S/A was mainly applied to the satellite clocks and, apparently, not so often to the ephemeris.