## 5 Guidance and Control Algorithm
### Differential Air Drag Control  
After rendezvous, satellites in periodic orbit drift by perturbation. Therefore, it is necessary to control the formation of satellites to maintain stability. The difference in atmospheric drag between the two satellites allows the formation to be maintained for months without thrust consumption. In this study, differential atmospheric drag control was performed by controlling the radial and along-track positions that had the greatest impact on drift. Figure 6: Block diagram of orbit and attitude GNC loop for rendezvous. Figure 7: (**a**) Controlled and reference trajectory and (**b**) control input of rendezvous described in the LVLH frame. Averaged In-Plane and Drift Terms  
Factors that affect differential air drag include the structural properties of the spacecraft, atmospheric density, solar flux, geomagnetic activity, and operational altitude [43]. The air drag is described as  
\[\mathbf{a}_{drag}=-\frac{1}{2}\frac{C_{D}\mathbf{A}}{m}\rho V_{rel}^{2}\hat{\mathbf{V}}_{rel}, \tag{13}\]  
where \(C_{D}\) is the drag coefficient, \(\mathbf{A}\) is the effective cross-sectional area, \(m\) is the mass of the spacecraft, \(\rho\) is the atmospheric density, and \(V=V_{rel}\hat{V}_{rel}\) is the velocity relative to the atmosphere. The differential air drag that occurs when these factors are different is described as  
\[\ddot{y}(t)=\frac{3}{2}V_{rel}^{2}\rho C_{D}(\frac{\mathbf{A}_{deputy}}{m_{deputy }}-\frac{\mathbf{A}_{chief}}{m_{chief}}). \tag{14}\]  
Atmospheric density is affected by solar flux, geomagnetic activity, and altitude. The intensity of the solar flux is expressed as \(F10.7\), and the geomagnetic activity is the \(K_{p}\) or \(\mathbf{A}_{p}\) index. According to [43], the along-track separation increases nonlinearly as \(F10.7\) and \(\mathbf{A}_{p}\) increase. These values change under the influence of solar activity, with a solar cycle of approximately \(11\) years, and show a similar level within a few months. Therefore, the air density was calculated using Equation (15) by obtaining observation data for a few months before the operation period from Celestrak. \[ \begin{cases}T=900+2.5(F10.7-70)+1.5\mathbf{A}_{p}\\ m=27-0.012(h-200)\\ H=T/m\\ \rho=6\times 10^{-10}\exp\left(-(h-175)/H\right),\end{cases}  \tag{15}\]  
where \(T\) (Kelvin) is a fixed exospheric temperature, \(m\) is the actual variation in molecular mass, and \(H\) (km) is a variable scale height to describe the variation of the molecular mass, with the altitude \(h\) being \(180\) to \(500\) km. Because the air drag depends on the velocity with respect to the atmosphere, the DADC strategy was established by analyzing the radial and along-track direction components of the HCW equation. The average value during one orbital cycle of the radial and along-track components was calculated using an averaging operator to remove the oscillatory motion as follows:  
\[ \begin{split}\bar{x}&=\langle x(t)\rangle\triangleq \frac{1}{T}\int_{0}^{T}x(\tau)d\tau\\ &=\frac{1}{T}\int_{0}^{T}\left[4x(0)+\frac{2\dot{y}(0)}{n}\right] +\frac{\dot{x}(0)}{n}\sin n\tau-\left[3x(0)+\frac{2\dot{y}(0)}{n}\right]\cos n \tau d\tau\\ &=4x(0)+\frac{2\dot{y}(0)}{n},\end{split}  \tag{16}\]  
\[ \begin{split}\bar{y}&=\langle y(t)\rangle\triangleq \frac{1}{T}\int_{0}^{T}y(\tau)d\tau\\ &=\frac{1}{T}\int_{0}^{T}-[6 nx(0)+3\dot{y}(0)]\tau+\left[y(0)- \frac{2\dot{x}(0)}{n}\right]+\left[6x(0)+\frac{4\dot{y}(0)}{n}\right]\sin n \tau+\frac{2x(0)}{n}\cos n\tau d\tau\\ &=-\frac{1}{2}(6 nx(0)+3\dot{y}(0))T+\left[y(0)-\frac{2\dot{x}(0)} {n}\right]\\ &=\bar{y}_{1}+\bar{y}_{2}.\end{split}  \tag{17}\]  
Periodic functions such as sine and cosine functions are removed, and as a result the remaining terms are only the radial and along-track initial distances. The transverse component has a drift term that increases linearly with time. The control drag area (\(\delta\mathbf{A}\)) was determined by multiplying the three values by the appropriate gains.