## 4 Numerical Simulation Results
### Monte-Carlo Performance Study  
The example above shows that the proposed aerodynamic-based control can establish the required relative trajectories and form images by means of satellites with sunlight reflectors. However, the considered system depends on multiple parameters, and the performance study of the controlled motion is carried out using the Monte Carlo approach. For this study, two characteristics are considered: the change in the orbit height \(\Delta H\) of the formation during the controlled motion and the convergence time taken to achieve the required relative trajectories. The formation is assumed to be converged if the mean error of trajectory deviation is less than 50 m during one orbital period. The single-image demonstration scenario without reconfiguration as in the previous section is simulated. One of the important system parameters is the error of the separation velocity, which is considered to be a random normally distributed variable with standard deviation \(\sigma_{\delta\mathbf{V}_{e}}\). The results of 20 numerical simulations for each random initial condition set with a defined \(\sigma_{\delta\mathbf{V}_{e}}\) are analyzed. For each simulation run, the mean value of \(\Delta H\) and the convergence time are calculated. In Figure 10, the results of the experiment are presented in boxplots. Figure 8: Calculated (**a**) by LQR control and implemented (**b**) by aerodynamic forces control accelerations for the first satellite. Figure 9: Orbital height (**a**) and atmosphere density (**b**) for the considered example. Each box contains 50% of the results, the red line corresponds to the mean value, above and below the box are 25% of the results, and red crosses are outliers. As one can see from Figure 10, the greater the errors in the launch velocity, the longer the convergence time and the broader the box of the change in the orbit height \(\Delta H\). It is interesting that with random errors in launch velocity at some simulations, the resulting \(\Delta H\) is less than in the case of low error values of 0.1 m/s. It can be explained that at some conditions the errors lead to less initial relative orbit deviations--that is why even the mean value of \(\Delta H\) is 1 km less than \(\Delta H\approx\) 14 km at \(\sigma_{\
abla\mathbf{v}_{e}}=0.1\) m/s. Nevertheless, the convergence time mean value is evenly increasing with \(\sigma_{\
abla\mathbf{v}_{e}}\). The lower the orbit height, the higher the atmospheric density and the greater the magnitude of the aerodynamic control force. Other system parameters being the same, the variations in the initial orbit height significantly influence the performance of the formation flying controlled motion. With the fixed value of the launch error \(\sigma_{\
abla\mathbf{v}_{e}}=0.1\) m/s, the initial circular orbit height is varied, and the Monte Carlo simulations are carried out. Figure 11 demonstrates the results of the study in boxplots of the random values of \(\Delta H\) and convergence time. As expected, the convergence time is much shorter at lower orbit height. The change in the height \(\Delta H\approx\) 6.5 km if the initial orbit height is 300 km, although the lifetime of the formation at such low orbits could only be a couple of weeks. The convergence time at an orbit height of 400 km is dramatically longer and amounts to nearly 155 h, although the change in the orbit height is about the same as at an orbit height of 350 km. Figure 11: Change in orbit height (**a**) \(\Delta H\) and (**b**) convergence time depending on the initial orbit height. Figure 10: Change in orbit height (**a**)\(\Delta H\) and (**b**) convergence time depending on the standard deviation launch velocity error. The performance study showed that the proposed aerodynamic control for the formation with sunlight reflectors is quite effective at orbit altitudes lower than 400 km, although the change in the orbit height during the required image configuration results in a very limited lifetime of such formation of less than about a month without propulsion.