# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 2 Mathematical Model and Results for GPS, Galileo, BeiDou  
To compute the ECEF coordinates of an SV as a function of time, the GPS navigation message adopts an algorithm based on a Keplerian ellipse with secular and periodic perturbations on selected orbital parameters. The validity time of the model is nominally two hours. Outside the validity time interval, the accuracy of the coordinates tends to degrade progressively. The orbital arc of nominal duration two hours has an orientation defined by three angles: Inclination \(I_{0}\), Longitude of Ascending Node \(\Omega_{0}\) and Longitude of Perigee \(\omega\). The size of the ellipse is given by the semimajor axis \(a\) and the eccentricity \(e\). The angular position of the SV at a Reference Time Toe (Time of Ephemeris) is \(M_{0}\) and is counted from the perigee (for a pictorial view and definitions see [[https://www.gsc-europa.eu/system-service-status/orbital-and-technical-parameters](https://www.gsc-europa.eu/system-service-status/orbital-and-technical-parameters)]([https://www.gsc-europa.eu/system-service-status/orbital-and-technical-parameters](https://www.gsc-europa.eu/system-service-status/orbital-and-technical-parameters)), accessed on 10 October 2021). These six orbital parameters, which would be all constant if the total force were that of a point mass, are allowed to vary with time to account for deviations from the point mass force model. Inclination and Ascending Node are allowed to increase with a rate \(dI_{0}/dt\) and \(d\Omega_{0}/dt\) constant during the validity interval; the orbital angular velocity \(n\), which by [PERSON]'s third law is inversely proportional to the semimajor axis to the 3/2 power, is perturbed by a constant term \(\Delta n\). The radial, along track and cross track position of the satellite are assigned periodic perturbations with a period equal to one-half the orbital period. These perturbations are described by sinusoids with separate amplitudes for the sine and cosine components. There is a total of 15 parameters which refer to the Time of Ephemeris Toe. Three additional clock parameters (offset \(a0\), drift \(a1\) and drift rate \(a2\)) refer to the Time of Clock Toe. Hence, every two-hour arc is described by 18 parameters and two reference epochs, Toe and Toe, which often coincide. The Reference Precise Ephemeris are computed with rigorous constraints on the origin, orientation and scale of the Terrestrial Reference Frame (TRF), which in our case is ITRF2014 or, specifically, IGS14. In the GPS-like algorithm, only the shape and size of the orbital arc are defined. It follows that to generate spatial coordinates as close as possible to those in the SP3 file, it is appropriate to solve for a set of Helmert parameters. This can be done once per day, i.e., about two complete revolutions, because their global character requires sampling over the entire orbital arc. The mathematical model used for the GNSSs with a GPS-like message to express precise ephemeris and clocks in a broadcast form is described by Equation (1):  
\[\sum\limits_{i=1}^{96}\left[\Delta(XYZT)\right]^{2}= \tag{1}\] \[f(Tx,Ty,Tz,Rx,Ry,Rz,Sc;a0,a1,a2,Crs,\Delta n,M_{0},Cuc,e,Cus,sqrt(a), Cic,\Omega_{0},Cis,I_{0},Crc,\omega,\frac{d\Omega}{dt},\frac{d\Omega}{dt})=min\]  
We seek to minimize the sum, over the 96 epochs (384 equations) in one daily SP3 file, of the squared differences \(\Delta(XYZT)\) of \(XYZT\)(SP3) and \(XYZT\)(BRDM), that is, the difference between the SP3 and broadcast ECEF coordinates and clock correction of a given Space Vehicle. This difference is parametrized by one set of seven Helmert parameters (three translations \(T\), three rotations \(R\) and one scale \(Sc\)) valid for the entire day and 12 sets each of 18 parameters, one for each two-hour arc. The function \(f\) can be linearized by series expansion in a neighborhood of the broadcast values of the parameters (the Helmert parameters have zero a priori value).