# Analytical Attitude Guidance Planner for Multiple Ground Targets Acquisitions
## 5 Simulation and Results
### Simulation Results  
In this section, the results are presented. In each figure, the APPG and the ATPG phases are indicated, respectively, with a continuous black and red line. Since the ATPG phase lasts much less than the other, in each figure a zoom of the results has been reported between point 3 and point 4 on the right side of the plot, so that the results are more intuitive and to underline the three APPG maneuvers mentioned in the previous section. Figure 10 shows the attitude of the satellite with respect to \(EC1\). The solution crosses all five desired points without any discontinuity, and the APPG phase is perfectly joined to the ATPG phase. This point can be better appreciated in the zoomed figure of the right part, where points 3 and 4 are shown more closely. Figure 11 shows the path that is followed by the payload view axis (\(Z_{PRF}\) in green) in the \(LVLH\) reference frame, in order to acquire the target points 3 (blue) and 4 (red). The trajectory is descending, and one can notice that the \(Z_{PRF}\) axis starts pointing at the blue target (3) and then the red one (4) for the necessary time (in this case \(t_{ac}=10\) s, as reported in Table 4). Furthermore, referring to Table 3, it can be seen that the points are close to each other and the observations take place in a short time, against a run time of the algorithm of just 20 milliseconds. So it is possible to affirm that with this technique a new trajectory could be evaluated each time that a new target is required, thanks to the low computational time required. In the presented simulation, the physical constraints are completely satisfied. Figure 12 shows the components of the angular velocity, which never exceeds the admissible limit \(\omega_{max}\) (indicated with a dot blue line), as well as it happens for the angular acceleration reported in Figure 13. Indeed, the angular acceleration reaches the maximum admissible value but never exceeds it. This solution agrees with the proposed analytical solution, which requires the maneuvers to be executed in near-minimum time by solving Equation (42). Moreover, the imposed kinematic constraints also guarantee that the ideal required control torque for the maneuvers implementation is confined within the maximum limits admissible. This result can be seen in Figure 14, where the ideal control policy (given as summation of \(\mathbf{u}_{c}\) and \(\mathbf{u}_{p}\) and indicated as \(\mathbf{U}=[U_{x},\,U_{y},\,U_{z}]\)) from \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline \(\overrightarrow{O_{2}O_{3}}\) & \(\begin{array}{c}PRF\text{ displacement with}\\ \text{ respect to }BF\\ \text{ }PRF\text{ rotation with respect to}\\ X_{BF}\end{array}\) & \(\begin{array}{c}[1\,0.5\,1]\,(\text{m})\\ -30\,(\text{deg})\end{array}\) \\ \(\theta\) & \(\begin{array}{c}PRF\text{ rotation with respect to}\\ Y_{BF}\end{array}\) & \(\begin{array}{c}-30\,(\text{deg})\end{array}\) \\ \(\psi\) & \(\begin{array}{c}PRF\text{ rotation with respect to}\\ Z_{BF}\end{array}\) & \(\begin{array}{c}0\,(\text{deg})\end{array}\) \\ \(t_{ac}\) & Acquisition time & \(10\,(\text{s})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Payload parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{t}_{start}\) (min) & **LAT (deg)** & **LON (deg)** & **HEIGHT (m)** \\ \hline
1 & 3.253 & \(-79.783\) & 129.459 & 91.452 \\
2 & 9.557 & \(-67.536\) & 58.597 & 167.932 \\
3 & 12.237 & \(-58.779\) & 52.500 & 942.248 \\
4 & 15.320 & \(-48.961\) & 48.114 & 849.432 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ground target points. Equation (2) for the APPG and ATPG phases are respectively represented by a continuous black and red line, while the maximum boundaries are indicated with two dot blue lines. As happened for the angular velocity and acceleration, the control also almost reaches the maximum admissible value.