# Autonomous Navigation Based on the Earth-Shadow Observation near the Sun-Earth L2 Point
## 4 Simulation Analysis  
In the rotating coordinate system, the controlled periodic orbit is located in the X-Y plane. The amplitude of the periodic orbit along the X direction is 5000 km, and the orbit is composed of four uncontrolled orbital segments in one cycle. The initial parameters and the time of the uncontrolled orbits are shown in Table 1. The reference trajectory is shown in Figure 7, together with the measurement curve. It can be seen that the shadow ratiovaries from approximately 34.25% to 85.63%. Figure 8 shows the contour distribution of the measurement function near the nominal track. It can be seen that 1 th orbital period and 3 th orbital period are symmetrical and the 2 th orbital period and 4 th orbital period are symmetrical. Therefore, the 1 th orbital period and 2 th orbital period are considered and analyzed below. Firstly, for the 1 th orbital period, the simulation is performed by using CKF, where the initial position error is 100 km, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Position/AU**} & \multicolumn{2}{c}{**Velocity/VU**} \\ \hline  & **X** & **Y** & **X** & **Y** & **Time/h** \\ \hline
1 & 1.0100007094259 & \(-3.34229357677352\times 10^{-5}\) & \(8.48460734682508\times 10^{-6}\) & \(0.000122175035008374\) & 30 \\
2 & 1.0100007094259 & \(3.34229357677352\times 10^{-5}\) & \(0.000137596252572650\) & \(9.87543991711969\times 10^{-5}\) & 30 \\
3 & 1.01006754681413 & \(3.34229357677352\times 10^{-5}\) & \(-8.14563809519688\times 10^{-6}\) & \(-0.000122288334314350\) & 30 \\
4 & 1.01006754681413 & \(-3.34229357677352\times 10^{-5}\) & \(-0.000137785577674654\) & \(-9.86025641499878\times 10^{-5}\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal trajectory parameters. Figure 8: Contour lines of measurements near the nominal track. (**a**) Outline of the nominal track. (**b**) Contour map of the nominal track. Figure 7: Nominal trajectory and its measurement parameters. (**a**) Nominal trajectory. (**b**) Measurement curve. the initial speed error is 1 m/s, and the measurement error is 0.001. The simulation results are shown in Figure 9, which, respectively, shows the navigation error and covariance convergence curve. As shown in Figure 9b, the error converges about 15 days later, and the convergence speed of the Y-axis state is faster than that of the X-axis. Figure 10a shows the 500 times shooting results of the Monte Carlo simulation, in which the accuracy along the X direction is poor and the convergence speed is slow. However, it can be seen that the accuracy in the Y direction is high and the convergence speed is fast. As shown in Table 2, the standard deviation of position error along the X direction is about 4.8581 km and the standard deviation of velocity error is 0.002 m/s. The standard deviation of position error along the Y direction is about 0.1518 km, and the standard deviation of velocity error is 0.0004 m/s. For the 2 th orbital period, the initial position error is 100 km, the initial speed error is 1 m/s, and the measurement error is 0.001. The simulation results are shown in Figure 11, which, respectively, show the navigation error and covariance convergence curve. As shown in Figure 11b, the error converges about 25 days later, and the convergence speed along the Y direction is still faster than along the X direction. Figure 12a shows the 500 times shooting results of the Monte Carlo simulation, in which the accuracy along the X direction is poor and the convergence speed is slow. However, it is apparent that the accuracy along the Y direction is high and the convergence speed is fast. As shown in Table 3, the standard deviation of position error along the X direction is about 8.7276 km and the standard deviation of velocity error is 0.003 m/s. The standard deviation of position error along the Y direction is about 0.1877 km and the standard deviation of velocity error is 0.001 m/s.