# Analysis of BDS/GPS Signals' Characteristics and Navigation Accuracy for a Geostationary Satellite
## 4 Navigation Performance
### Real-Time Orbit Determination Accuracy
#### 4.2.1 Processing Model and Strategy  
The equation of the motion of a GEO satellite can be expressed as [30]:  
\[\dot{r}=v\] \[\dot{v}=a_{m}(r,v,t)+w(t)\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Results** & **Satellite** & **X** & **Y** & **Z** & **3D** \\ \hline \multirow{3}{*}{Position (m)} & GPS & 28.0 & 4.22 & 6.06 & 28.9 \\  & GPS + BDS GEO and IGSO & 14.0 & 4.05 & 6.47 & 15.9 \\ \hline \multirow{3}{*}{Velocity (m/s)} & GPS & 0.213 & 0.0299 & 0.0347 & 0.218 \\  & GPS + BDS GEO and IGSO & 0.106 & 0.0824 & 0.125 & 0.184 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMS of position and velocity differences between single-epoch least square solutions, using only GPS observations, GPS + BDS GEO/IGSO combined, and the reference of the precision orbit determination solutions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Items**} & \multicolumn{2}{c}{**ConstellationGPS**} & \multicolumn{4}{c}{**GPS + BDS**} \\ \cline{2-7}  & **Weight** & **-** & **1:2** & **1:1** & **2:1** & **4:1** \\ \hline \multirow{4}{*}{Position (m)} & X & 31.0 & 14.8 & 14.6 & 15.1 & 15.5 \\  & Y & 4.92 & 4.38 & 3.85 & 3.73 & 3.79 \\  & Z & 6.82 & 6.88 & 6.65 & 6.61 & 6.64 \\  & 3D & 32.1 & 16.9 & 16.5 & 16.9 & 17.3 \\ \hline \multirow{4}{*}{Velocity (m/s)} & X & 0.234 & 0.157 & 0.137 & 0.133 & 0.137 \\  & Y & 0.0339 & 0.0855 & 0.0698 & 0.0556 & 0.0656 \\ \cline{1-1}  & Z & 0.0312 & 0.0827 & 0.0601 & 0.0450 & 0.0546 \\ \cline{1-1}  & 3D & 0.238 & 0.197 & 0.165 & 0.151 & 0.161 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMS of position and velocity differences over two days between single-epoch least square solutions with different combination weights and the reference of precision orbit determination solutions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Results** & **Satellite** & **X** & **Y** & **Z** & **3D** \\ \hline \multirow{3}{*}{Position (m)} & GPS & 31.1 & 4.88 & 6.83 & 32.2 \\  & GPS + BDS MEO & 26.2 & 3.87 & 6.53 & 27.3 \\ \hline \multirow{3}{*}{Velocity (m/s)} & GPS & 0.238 & 0.0343 & 0.0315 & 0.242 \\  & GPS + BDS MEO & 0.259 & 0.0570 & 0.0727 & 0.275 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RMS of position and velocity differences between single-epoch least square solutions, using only GPS observations, GPS + BDS MEO combined, and the reference of the precision orbit determination solutions. where \(r\), \(v\), and \(t\) are the position, velocity and time of the GEO satellite in a geocentric inertial coordinate frame, respectively. \(a_{m}\) is the total gravity, perturbing acceleration, including the nonspherical part of the gravitational attraction of the Earth, lunisolar gravitational perturbations, solar radiation pressure, and earth tide. \(w(t)\) is the empirical acceleration and can expressed in radial, along-track, and cross-track, which account for unmodeled forces. Various simplifications to the force models used in this filter are shown in Table 6, according to the reference in [31; 32]. Atmospheric drag, troposphere and ionosphere delay are ignored. In this mission, we adopt the 50 \(\times\) 50 order of gravity model in order to satisfy the GTO scenario of which perigee altitude is close to LEO. The state equation of EKF can be propagated by a fourth-order fixed step Runge-Kutta integrator of the adopted equation of the motion within the time update step. The filter state vector is expressed as following [30]:  
\[X_{k-1}=\left[x,\ y,\ z,\ \dot{x},\ \dot{y},\ \dot{z},\ b_{G},\ b_{B},\ \dot{b}_{u},\ C_{R},\ w_{R},\ w_{A},\ w_{C}\right]^{T} \tag{7}\]  
where \((x,\ y,\ z)\) is position vector and \((\dot{x},\ \dot{y},\ \dot{z})\) is the velocity vector, respectively. \(b_{G}\) is GPS clock offset, \(b_{B}\) is SDS clock offset, \(\dot{b}_{u}\) is clock drift of the receiver, and \(C_{R}\) is the state of solar radiation pressure parameter.