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GPS and Galileo Satellite Coordinates Computation

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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]:

tk = ttoe
If t_k>302\,400 sec, subtract 604\,800 sec from tk. If t_k< -302\,400 sec, add 604\,800 sec.

M_k=M_o+\left( \frac{\sqrt{\mu }}{\sqrt{a^3}}+\Delta n\right)t_k

Mk = EkesinEk

 v_k=\arctan \left( \frac{\sqrt{1-e^2}\sin E_k}{\cos E_k-e}\right)

u_k=\omega +v_k+c_{uc}\cos 2\left( \omega +v_k\right) +c_{us}\sin
2\left( \omega +v_k\right)

r_k=a\left( 1-e\cos E_k\right) +c_{rc}\cos 2\left( \omega
+v_k\right) +c_{rs}\sin 2\left( \omega +v_k\right)

i_k=i_o+\stackrel{\bullet }{i} t_k+c_{ic}\cos 2\left(
\omega +v_k\right) +c_{is}\sin 2\left( \omega +v_k\right)

\lambda _k=\Omega _o+\left( \stackrel{\bullet }{\Omega }-\omega
_E\right) t_k-\omega _E t_{oe}

X_k \\
Y_k \\
\right] ={\mathbf R}_3\left( -\lambda _k\right)  {\mathbf R}_1\left( -i_k\right) {\mathbf R}_3\left( -u_k\right) \left [
r_k \\
0 \\

where {\mathbf R}_1 and {\mathbf R_3} are the rotation matrices defined in Transformation between Terrestrial Frames.


  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.
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