## 4 Numerical Simulation Results  
The proposed control of satellite formation flying with sunlight reflectors is based on many assumptions, and there hardly be any analytical proof that the control algorithm would unfailingly result in the achievement of the required relative configuration. The algorithm performance study is conducted using the numerical simulations of the controlled motion of the satellites. In numerical simulation, the initial conditions for orbital motion are determined by the cluster launch. It is assumed that the separation velocity \(\mathbf{V}_{e}\) from the launch vehicle is aligned with orbital velocity, and the satellites are ejected from the launch containers with a fixed time interval. The random launch velocity error \(\delta\mathbf{V}_{e}\) is modeled as a normally multivariate distributed vector with a standard deviation of \(\sigma_{\delta\mathbf{V}_{e}}\) without correlation in component values. In the equations of motion (7), the \(J_{2}\) disturbance is taken into account and the atmospheric density is given by the GOST model [32]. The integration of Equation (7) is carried out by the 4 th-order Runge-Kutta method with a constant integration step. The simulation control loop is presented in Figure 5. The inputs are the initial conditions for the satellite motion, the image, and the control parameters. The image to be demonstrated is discretized and the reference trajectory is assigned to each satellite according to Equation (28). Then, for each pair of satellites, the deviation \(\mathbf{e}_{ij}\) and the corresponding acceleration vectors \(\mathbf{w}_{ij}\) by (16) and \(\mathbf{u}_{ij}\) by (21) are calculated. Then, the modification (23) is applied to calculate \(\mathbf{u}_{i}\). Using the implementation strategy (30), the aerodynamic acceleration \(\mathbf{f}_{i}^{ero}\) is calculated, which is used for each satellite orbital motion integration. The following mission scenario is considered. After launching in a near-circular orbit according to the initial conditions, the reference trajectory assignment problem is to be solved. The example image to be constructed is chosen as that consisting of three English alphabet letters. Each letter consists of a certain number of satellites with sunlight reflectors, and the distance between the neighbor satellites is set by 700 m, which is in accordance with the visibility conditions calculated in [5]: The magnitude of the reflectors is about \(-2.8\)  
Figure 5: Simulation scheme. and the minimum distance is 380 m at orbit altitude of 350 km. After the convergence to the reference trajectories at a given point with proper lighting conditions, the attitudes of all the satellites are synchronized, and the sunlight is reflected by the satellites to be visible to the Earth observer. After the demonstration of the first image (three letters), the satellite's relative positions are to be reconfigured to compose another image (three different letters). The main parameters of the simulated motion scenario are provided in Table 1. The LQR weight matrices are selected as follows:  
\[\mathbf{Q}= \begin{pmatrix}10a_{0}^{2}&0&0&0&0&0\\ 0&10a_{0}^{2}&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&10a_{0}^{2}&0&0\\ 0&0&0&0&10a_{0}^{2}&0\\ 0&0&0&0&0&1\end{pmatrix} ,\ \mathbf{R}= \begin{pmatrix}10^{14}a_{0}^{2}&0&0\\ 0&10^{15}a_{0}^{2}&0\\ 0&0&10^{15}\end{pmatrix} .\]  
Values of elements in these matrices should ensure the absence of saturation in limited aerodynamic control. The drag force is about 10 times greater than the lift force; therefore, the elements of \(\mathbf{R}\) for the control component \(\bar{\varphi}^{\bar{\varphi}^{\bar{\varphi}}}\) should be less than those for \(\bar{\theta}^{\bar{\varphi}}\) and \(\bar{\rho}^{\bar{\varphi}}\). The values of the elements of the matrix \(\mathbf{R}\) are selected in order to avoid the saturation of the aerodynamic value control. In matrix \(\mathbf{Q}\), values corresponding to the deviation of \(\varphi\) and \(\theta\) are greater than those of \(\rho\) and its derivatives.