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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

Next 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 2 hours for GPS and 3 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 5 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 t_k from the ephemerides reference epoch t_oe (t and t_oe are expressed in seconds in the GPS week):

t_k = t − t_oe

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

• Compute the mean anomaly for t_k,

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

M_k = E_k − esinE_k

• Compute the true anomaly v_k:

• Compute the argument of latitude u_k from the argument of perigee ω, true anomaly v_k and corrections c_uc and c_us:

• Compute the radial distance r_k, considering corrections c_rc and c_rs:

• Compute the inclination i_k of the orbital plane from the inclination i_o at reference time t_oe, and corrections c_ic and c_is:

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

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

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.