# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
To coordinate the subsystems, the third step uses the projection mapping. A reference projection is defined as a transformation \(s=\gamma(x)\) such that \(\gamma\big{(}x_{d}(s)\big{)}=s\), that is, if the state \(x_{d}(s)\) is on the desired trajectory, then \(\gamma\) must give the corresponding value of \(s\) also on the desired trajectory \(x_{d}(s)\). For example, for any state \(x_{0}\), vector \(x_{d}(s_{0})\) can be an orthogonal projection of \(x_{0}\) onto \(x_{d}(s_{0})\). Orthographic design is not the only way to define \(\gamma\), and changing it can fundamentally change the way of subsystems mutual coordination. The last stage of the controller design is the synthesis of a non-time-based feedback law that is used to control a multi-vehicle system. The control law is obtained by simple substitution and has the form  
\[u_{i}(x)=u_{i}\big{(}x,T(x)\big{)},\quad T(x)=v^{-1}\big{(}\gamma(x)\big{)}. \tag{44}\]  
A number of examples for selecting reference projections to ensure coordination of the formation are presented in [111]. In particular, the satellite clusters, forming a high-resolution synthetic aperture imaging formation are described. A flat formation consists of a group of satellites occupying the same orbital plane \(OXY\) in the inertial terrestrial coordinate system \(OXYZ\) and separated by the mean anomaly. The desired trajectory of movement of each satellite is considered to be a circular orbit of radius \(r_{0}\) as  
\[x_{d}=r_{0}\cos(\omega t),\quad y_{d}=r_{0}\sin(\omega t). \tag{45}\]  
As is known, \(\omega=\left(\dfrac{\mu}{r_{0}^{3}}\right)^{1/2}\), \(\mu=3.986\times 10^{14}\) m\({}^{3}\) / s\({}^{2}\). Following [112], the synthesis of the control law in [111] is performed by the feedback linearization method. For a near-circular orbit, the reference action \(s\) is defined as \(s=\theta\), where \(\theta\) is the angle between the satellite radius vector and \(X\)-axis on the \(OXY\) plane. For the desired trajectory, \(\theta=\omega t\). The relative position of satellites in the formation is determined through \(\gamma_{i}\). For example, one can accept  
\[\gamma_{1}=\theta_{1},\quad\gamma_{2}=\theta_{1}-\theta_{0},\quad\gamma_{3}= \theta_{3},\quad\gamma_{4}=\theta_{3}-\theta_{0}, \tag{46}\]  
where \(\theta_{0}\) is the desired angle between two satellites in the formation. According to (46), satellites #1, #3 are the leaders of the formation. [PERSON] et al. [111] presented the theorems on the stability of the formation and the simulation results of the formation reconfiguration during movement. [PERSON] et al. [113] discussed the positioning and reorientation control of a formation of fully steerable low-thrust microsatellites. A general control methodology based on optimization is proposed for solving the problems of constructing a limited trajectory for positioning and reorientation. Taking advantage of the all-wheel drive microsatellites, onboard calculations can be performed on them. The software package Nonlinear Trajectory Generation (NTG) was used for typical space missions. A systematic approach for studying the formations of multi-agent systems is presented in [114]. In particular, undirected formations for centralized formations and directed formations for decentralized formations, are considered. Paper [114] focuses on the feasibility problem: given the kinematics of multiple agents together with inter-agent constraints, one should find out if there are agent trajectories satisfying constraints. If the answer is yes, the task is to find a \"smaller\" control system that supports the movement of the formation along its trajectory, which, from the side of higher control levels, allows one to consider the formation as a whole. In [114], \(n\) heterogeneous agents with states \(x_{i}\in M_{i}\), \(i=1,\ldots,n\) are considered.