# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
The results of the study show that in the developed method it is advisable to use control in a sliding mode. [PERSON] et al. [120] discussed achieving a given formation using thrust only in the direction along the trajectory. A system consisting of the leading satellite and the following satellites is considered, and linear control law is developed to obtain a formation of a given type. The orbital motion of the leader satellite is determined by the radial distance \(r_{c}\) from the center of the Earth and the true anomaly \(\theta\). The movement of the slave satellite is described with respect to the movement of the leader satellite using the relative r.m.s. S-xyz, anchored in the center of the leader satellite. The \(x\)-axis is directed along the local vertical, the z-axis is taken along the normal to the orbital plane, and the y-axis represents the third axis of this selected right RMS. The geometry of the orbital motion of the lead and slave satellites according to [120] is shown in the Figure 2. The orbital equations of motion for the lead satellite and the equations of motion of the slave satellite relative to the lead satellite are of the form [21, 23]  
\[\ddot{r}_{c}-r_{c}\dot{\theta}^{2}+\frac{\mu}{r_{c}^{2}}=0, \tag{48}\] \[r_{c}\ddot{-}\theta+2\dot{\theta}r_{c}=0,\] (49) \[\ddot{x}-x\dot{\theta}^{2}-2y\dot{\theta}-y\ddot{\theta}=-\frac{ \mu}{r^{3}}(r_{c}+x)+\frac{\mu}{r_{c}^{2}}+f_{dx},\] (50) \[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]0 (51) \[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]1  
where \(r=\left((r_{c}+x)^{2}+y^{2}+z^{2}\right)^{1/2}\); \(f_{y}\) is the acceleration of the control force in the longitudinal direction; \(f_{dy}\), \(j\in\{x,y,z\}\) are the accelerations of the disturbing force along the directions \(x\), \(y\) and \(z\), respectively. To synthesize the control law, the following linearized model of the (50)-(52) system is used, obtained under the assumption that the leading satellite is moving in a circular orbit  
\[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]2 \[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]3 (54) \[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]4  
where \(\dot{\theta}=\text{const}\) is taken. In [120] the following linear proportional-differential (PD) control law is proposed  
\[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]5  
where \(x_{d}\), \(y_{d}\) are the desired (specified) values of the coordinates \(x\), \(y\), \(k_{p1}\), \(k_{d1}\), \(k_{p2}\), \(k_{d2}\) are the regulator coefficients. The performance of the proposed regulator was investigated by the Routh-Hurwitz criterion and tested using the simulation of nonlinear equations of motion of the system taking into account several factors, including changes in the initial conditions, the size of the formation, the presence of disturbing forces and eccentricity of the orbit of the lead satellite. It was found that a regulator using only thrust along the trajectory can provide a bounded relative position error, without damping, with a maximum control acceleration (in m/s\({}^{2}\)) of \(0.0718\,\dot{\theta}^{2}\) times the error along the trajectory and, therefore, has a limited ability to  
\[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]6 Geometry of the orbital motion of the lead and trailing satellites according to [120]. neutralize disturbances. However, by modeling, it was found that the proposed controller successfully provided limited relative position errors below \(\pm\)10 m.  
[PERSON] et al. [121] presented expressions describing the average relative motion of two satellites in adjacent orbits around a planet flattened at the poles.