# Analytical Attitude Guidance Planner for Multiple Ground Targets Acquisitions
## 5 Simulation and Results
### Simulation Environment  
The most important components of the SPICE system are the SPICE Toolkit software and the SPICE data files, often called \"kernels\". In this work the following kernels are used: naif0012.tls, earth-200101-990628-predict.bpc, de438.bsp, and gm-de431.tpc. In particular, these files contain the ephemeris of several satellites and the real displacement between \(ECEF\) and \(ECI\) reference frames as a function of seconds elapsed since midnight on November 17, 1858 (it is the date from which the counting of the modified Julian date begins). In particular, the simulation time was set from 26 Nov 2020 19:26:20 to 26 Nov 2020 19:56:20, and the SPOT-7 satellite has been chosen from SPICE files as a test case. SPOT-7, together with SPOT-6, forms a constellation of Earth-imaging satellites designed to provide continuity of high-resolution (the satellite physical parameters are reported in Table 2). Furthermore, the shape of the Earth is considered ellipsoidal with an equatorial radius equal to 6378.137 km and a polar radius equal to 6356.752 km by using the WGS84 model from SPICE. In this way, a high-precision geometry of the problem has been considered (it is reported in Figure 1).  
Figure 8: Maneuver strategy.  
The _ground targets_ positions are identified by means of coordinates with respect to \(ECEF\) reference frame, expressed in terms of latitude (\(\mu\)), longitude (\(\lambda\)), and altitude/elevation (\(h\)) above mean sea level. The _ground targets_ are generated in a limited spatial area around the sub-satellite path. In particular, Figure 9 shows the location of the ground target points (on the south pole) and the thickness \(d\), which identifies the allowed area within the ground points. The thickness \(d\) is set equal to 400 km, which corresponds to a rotation of 30 degrees with respect to the nadir. The number of target points \(N_{t}\) has been set equal to 4 (as shown in Figure 9). The latitude and longitude of the targets are then reported in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & _Attitude Parameters_ & \\ \hline \(I_{x}\) & Inertia along \(X_{BF}\) & 603.896 (kg/m\({}^{2}\)) \\ \(I_{y}\) & Inertia along \(Y_{BF}\) & 565.396 (kg/m\({}^{2}\)) \\ \(I_{z}\) & Inertia along \(Z_{BF}\) & 318.792 (kg/m\({}^{2}\)) \\ \(u_{max}\) & Maximum torque & 0.5 (Nm) \\ \(\dot{\omega}_{max}\) & Maximum angular acceleration & 0.0474 (deg/s\({}^{2}\)) \\ \(\omega_{max}\) & Maximum angular velocity & 1 (deg/s) \\ \hline \multicolumn{3}{c}{_Orbit Parameters_} \\ \hline \(a\) & Semi-major axis & 7075.945 (km) \\ \(e\) & Eccentricity & \(1.251\times 10^{-4}\) \\ e-04 \(i\) & Inclination & 98.165 (deg) \\ \(RAAN\) & Right ascension of the ascending node & 38.184 (deg) \\ \(AoP\) & Argument of periapsis & 102.289 (deg) \\ \(v\) & True anomaly & 155.692 (deg) \\ \hline \hline \end{tabular}
\end{table}
Table 2: SPOT-7 parameters.  
Figure 9: Sub-satellite trajectory (continuous green line) and ground targets (red cross) are reported on the top. The spatial band around the sub-satellite path (dashed red line) is shown in the enlarged graph below along with the thickness \(d\).  
Finally, the _Payload_ is an optical sensor in which the \(PRF\) is translated with respect to the \(BF\) reference frame by the vector \(\overrightarrow{O_{2}O_{3}}\), while is rotated with a 3-2-1 set of Euler angles with respect to the \(BF\) reference frame by the mounted rotational matrix. The payload's parameters are summarized in Table 4, where \(t_{ac}\) is the exposure time required to acquire the ground target (i.e., it is the duration of the \(ATPG\) phase).