View source for GPS and Galileo Satellite Coordinates Computation

{{Article Infobox2
|Category=Fundamentals
|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
|Level=Intermediate
|YearOfPublication=2011
|Title={{PAGENAME}}
}}
Table 1 provides the [[GPS General Introduction|GPS]] or [[GALILEO General Introduction|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 General Introduction|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 [IS-GPS-200, table 20-IV] <ref group="footnotes"> [IS-GPS-200], NAVSTAR GPS Space Segment/Navigation User Interfaces https://www.gps.gov/technical/icwg/IS-GPS-200M.pdf </ref>.The Galileo satellites follow a similar scheme

::[[File: GPS_Galileo_Coord_Comp_Table_1.png|none|thumb|640px|'''''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 4 meters (RMS) is achieved for GPS satellites with S/A=0ff and several tens of meters with S/A=on <ref group="footnotes"> Actually, the S/A was mainly applied to the satellite clocks and, apparently, not so often to the ephemeris.</ref>:


*  Compute the time <math>t_k</math> from the ephemerides reference epoch <math>t_{oe}</math> (<math>t</math> and <math>t_{oe}</math> are expressed in seconds in the GPS week):

::<math>t_k=t-t_{oe}</math>

:If <math>t_k>302\,400</math> sec, subtract <math>604\,800</math> sec from <math>t_k</math>. If <math>t_k< -302\,400</math> sec, add <math>604\,800</math> sec.


*  Compute the mean anomaly for <math>t_k</math>,
::<math>M_k=M_o+\left( \frac{\sqrt{\mu }}{\sqrt{a^3}}+\Delta n\right)t_k
</math>


* Solve (iteratively) the Kepler equation for the eccentricity anomaly <math>E_k</math>: 
::<math>M_k=E_k-e\sin E_k</math>


*  Compute the true anomaly <math>v_k</math>:
::<math> v_k=\arctan \left( \frac{\sqrt{1-e^2}\sin E_k}{\cos E_k-e}\right)</math>


*  Compute the argument of latitude <math>u_k</math> from the argument of perigee <math>\omega </math>, true anomaly <math>v_k</math> and corrections <math>c_{uc}</math> and <math>c_{us}</math>:
::<math>u_k=\omega +v_k+c_{uc}\cos 2\left( \omega +v_k\right) +c_{us}\sin
2\left( \omega +v_k\right)</math>


*  Compute the radial distance <math>r_k</math>, considering corrections <math>c_{rc}</math> and <math>c_{rs}</math>:
::<math>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)
</math>


*  Compute the inclination <math>i_k</math> of the orbital plane from the inclination <math>i_o</math> at reference time <math>t_{oe}</math>, and corrections <math>c_{ic}</math> and <math>c_{is}</math>:
::<math>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)
</math>


*  Compute the longitude of the ascending node <math>\lambda_k</math> (with respect to Greenwich). This calculation uses the right ascension at the beginning of the current week (<math>\Omega _o</math>), the correction from the apparent sidereal time variation in Greenwich between the beginning of the week and reference time <math>t_k=t-toe</math>, and the change in longitude of the ascending node from the reference time <math>t_{oe}</math>:
::<math>\lambda _k=\Omega _o+\left( \stackrel{\bullet }{\Omega }-\omega
_E\right) t_k-\omega _E t_{oe}
</math>


*  Compute the coordinates in TRS frame, applying three rotations (around <math>u_k</math>, <math>i_k</math> and <math>\lambda _k</math>):
::<math>\left[
\begin{array}{c}
X_k \\
Y_k \\
Z_k
\end{array}
\right] ={\mathbf R}_3\left( -\lambda _k\right)  {\mathbf R}_1\left( -i_k\right) {\mathbf R}_3\left( -u_k\right) \left [
\begin{array}{c}
r_k \\
0 \\
0
\end{array}
\right]
</math>


:where <math>{\mathbf R}_1</math> and <math>{\mathbf R_3}</math> are the rotation matrices defined in [[Transformation between Terrestrial Frames]].



==Notes==
<references group="footnotes"/>

[[Category:Fundamentals]]
[[Category:GNSS Time Reference, Coordinate Frames and Orbits]]
[[Category:GPS]]
[[Category:GALILEO]]