# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
Based on the constraints imposed by the decorrelation of the SAR phase in oceanographic applications for extended baselines, in [34] the upper limit of the relative baseline along the path of the spacecraft formation is \(50\text{ ms}^{-1}\text{ GHz }/fv_{\text{sc}}\), which is about \(300\text{ m}\) in the L-band and \(40\text{ m}\) in the X-band. (Recall that L-band: \(1.0\text{--}2.0\text{ GHz}\), X-band: \(8.0\text{--}12.0\text{ GHz}\).) To provide a margin of accuracy with respect to the decorrelation threshold, the appropriate separation of spacecraft in the formation along the trajectory would be \(225\pm 75\text{ m}\) in the L-band and \(30\pm 10\text{ m}\) in the X-band. There are two main options for controlling a formation: ground-in-a-loop control and autonomous formation control. Obviously, in the second case, information is required on board the spacecraft about their relative position and speed. To the relative position of the two spacecraft prediction, it is necessary to integrate the equations of their motion taking into account all relevant perturbations, including the effects of a complex form of the Earth's gravitational field, the influence of the Sun, and the Moon, atmospheric drag and solar radiation pressure [38]. For the motion of two spacecraft in the limiting case of near-circular orbits and close formations (with a distance of less than 1 km), closed forms of the Clohessy-Wiltshire equations of relative motion [23] can be obtained. A number of works are investigating the possibility of using the natural electrostatic forces of spacecraft in High Earth Orbit (HEO), or in deep space [39]. Investigation of [40], showed that the absolute charge of spacecraft can reach the level of kilovolts in the Geostationary Earth Orbit, GEO. If spacecraft fly at a distance of several tens of meters from each other, this causes disturbances ranging from micro- to millinewtons. In an orbit, these disturbances can lead to hundreds of meters of deflection. The concept of Coulomb thrust proposes to use active control of the spacecraft charge to ensure its desired values and use this disturbing force to directly control the relative motion KLA [41, 42]. The electrostatic charge of the spacecraft is partially hidden from another nearby ship due to interaction with ions and electrons of cosmic plasma. The strength of this shielding is measured by the Debye shielding length [43]. The cold and dense plasma of the environment in low-orbit orbits (LEO) has a Debye length of about a centimeter, which makes it impossible for Coulomb interaction at a distance of tens of meters. In a geostationary orbit, the Debye length is in the range of 100-1000 m, which makes it possible to use the Coulomb thrust between satellites. In deep space, at a distance of 1 AU from the Sun, the Debye length is 30-50 m [40]. Coulomb thrust usually requires only a watt level of electricity, while consuming essentially no fuel. The specific impulse \(I_{\mathrm{SP}}\) of the Coulomb thrust is in the range \(10^{9}\)-\(10^{12}\) width [40, 44]. Spacecraft charge control has been demonstrated on the SCATHA [45], ATS [46], Cluster [47, 48] missions. Coulomb structures have electrostatic forces that fully compensate for differential gravitational acceleration. As a result, a cluster of spacecraft is formed, the positions of individual vehicles which seem to be frozen relative to the rotating main locally-vertically-locally-horizontal system [39, 40, 49, 50, 51]. However, all static equilibrium solutions of relative position in orbit or deep space are unstable, and active recharging feedback is required to stabilize them. Due to the distributed nature of spacecraft formations, methods of decentralized filtration [52] are of great interest. In [53, 54, 55], the estimation is performed using the parallel operation of full-order observers with local dimensions.