# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}  \tag{63}\]  
with the rotation matrix \(\mathbf{R}\), depending on the longitude of the ascending node of the lead satellite \(\Omega\), the angle between the equatorial plane and the orbital plane of the lead satellite \(i\), latitude of the lead satellite \(\omega=\omega_{0}+n(t-t_{0})\), \(n=\sqrt{\dfrac{GM}{r_{0}^{3}}}\), and the radius \(r_{0}\) of the circular orbit of the lead satellite. This circular is based on linearized equation solutions, and therefore non-linear behavior of relative motion will violate this configuration. To avoid this, additional control force is used. The approach of [130] provides a general methodology that can be easily applied to any type of relative configuration. To solve the problem in [130], further transformations of constraints are performed (61) and (62) to the form (36) and then use (42). For this purpose, equations (59) and (60) are first reduced to the form  
\[ \begin{bmatrix}y&z\end{bmatrix}\begin{bmatrix}\dot{y}\\ \dot{z}\end{bmatrix} =- \begin{bmatrix}\dot{y}&z\end{bmatrix}\begin{bmatrix}\dot{y }\\ \dot{z}\end{bmatrix} , \tag{64}\] \[ \begin{bmatrix}2&-1\end{bmatrix}\begin{bmatrix}\ddot{x}\\ \dot{z}\end{bmatrix} =0. \tag{65}\]  
Finally, the system equations are reduced to the form [81]:  
\[\ddot{\mathbf{x}}+\mathbf{a}+\mathbf{A}^{\dagger}(\mathbf{b}- \mathbf{A}\mathbf{a}), \tag{66}\] \[\mathbf{M}\ddot{\mathbf{x}}=\mathbf{F}+\mathbf{M}\mathbf{A}^{ \dagger}(\mathbf{b}-\mathbf{A}\mathbf{a}), \tag{67}\]  
whence the expression for the control force \(\mathbf{F}^{c}\) is derived:  
\[\mathbf{F}^{c}=\mathbf{M}\mathbf{A}^{\dagger}(\mathbf{b}-\mathbf{A}\mathbf{a }). \tag{68}\]  
The presentation is illustrated by numerical examples. It is worth mentioning that in [130] there are no explicit restrictions on either the control action or the fuel consumption. Simple and accurate formation control schemes are presented by [PERSON] and [PERSON] [133], where a method of analytic dynamics by [PERSON] [76], [PERSON] and [PERSON] [77], [PERSON] [78, 79] is employed. In [133], the explicit control inputs for accurately maintaining a given formation configuration, using continuous thrust systems, are found. A much simpler and more explicit expression is obtained to accurately satisfy the formation conservation constraints for Keplerian reference orbits. [PERSON] and [PERSON] [133] also includes explicit control results when the follower is put into orbit with incorrect initial conditions, as is usually the case in practice. Extensive computational simulations demonstrate that the approach of [133] is easy to implement and numerically precise. The problems of attitude synchronization and tracking in the spacecraft formation in the presence of model uncertainties and external interference are considered by [PERSON] et al. [134]. A decentralized control law for the adaptive slip mode using the topology of undirected communication between spacecraft is proposed, which is analyzed on the basis of the theory of algebraic graphs. A sliding manifold is obtained for several spacecraft, on which each spacecraft approaches the desired time-varying orientation and angular velocity, while maintaining orientation synchronization with other spacecraft in the formation. Next, a control law is developed to ensure convergence to a sliding manifold. The stability of the resulting closed-loop system is proved by applying the Barbalat lemma. The simulation results demonstrate the effectiveness of the proposed synchronization and tracking methodology for orientation problems. The problem of designing a distributed controller for a continuous time system consisting of a number of identical dynamically coupled subsystems is discussed in [135]. The main mathematical conclusion is based on the Kronecker product and a special similarity transformation built from a matrix of interconnection patterns that breaks the system into modal subsystems.