# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture  
[PERSON] and [PERSON] [18] developed a feedback controller to maintain a movement of a swarm of satellites using the calculated optimal trajectory. Chemical thrusters are used for the slave satellite as the actuators. The situation is considered when two satellites fly at a distance of 700 m from the geophysical orbit. The results of [18] show that movement in the swarm can be maintained at a distance of 21 m of satellites from each other. [PERSON] [19] studied keeping the satellite formation in near-circular orbits. The tangential maneuvers of a group of satellites are determined by continuous relative motion equations. During the entire movement, the satellites move relative to each other, and these movements are presented in the form of a graph. In [19], (\(2N-1\)) linearized equations, where \(N\) is the total number of satellites are used. A linear-quadratic controller has been developed to control satellite formation. An analytical solution of linear-quadratic equations for the formation of two satellites is found. The simulation results for low Earth orbits (LEO) are presented where atmospheric drag is taken into account. [PERSON] et al. [20] focused on the planning of missions that provide the necessary conditions for the initialization of a periodic formation of a satellite constellation that provides a good distributed image of the Earth, for example, using synthetic aperture radar. These passive apertures were previously designed using the closed form solution provided by the Hill equations (see [21; 22]), also known as the Clohessy-Wiltshire equations [23], which are linearized with respect to the near-circular reference orbit. A study was carried out to develop apertures that are insensitive to differential noise \(J_{2}\) (the second zonal harmonic \(J_{2}\) characterizes the polar compression). Based on nonlinear dynamic models (see [16]), [PERSON] et al. [20] used the results by [24; 25; 26] on the derivation and solution of homogeneous equations of relative motion for several spacecraft. They present an initialization procedure for a large constellation of satellites with an eccentric reference orbit. The main result is derived from homogeneous solutions of the linearized relative equations of satellite motion. These solutions are used to find necessary conditions on the initial states for generations of \(T\)-periodic solutions so that the satellites return to their initial relative states at the end of each orbit. This periodicity condition and the resulting initialization procedure, originally defined (in compact form) at the perigee of the reference orbit, are generalized to include initialization at any point relative to the reference orbit. In particular, in [20], an algorithm is presented that minimizes the fuel consumption associated with the initialization of the satellite state to values that are consistent with periodic relative motion. Periodicity conditions and homogeneous solutions can also be used to estimate relative motion errors and approximate costs of fuel associated with neglect of the reference orbit eccentricity. The simulation results are presented, showing that ignoring the eccentricity leads to an error close to the error from disturbances, caused by differential accelerations of gravity. The movement of a satellite formation relative to an eccentric reference orbit is discussed in [27], where optimal fuel/time ratio algorithms for low-level maintenance of station movement in the presence of interference are given. The algorithm by [PERSON] and [PERSON] [27] is optimized by applying the linear programming method with respect to a time-varying dynamics model linearized with respect to an eccentric reference orbit. Numerous results of nonlinear system simulation are given, demonstrating the efficiency of this general approach to control.