# Modeling and Control of Satellite Formations: A Survey
## 3 Circle Formation Architecture  
An approach to communicating satellites in a ring is also discussed by [PERSON] [94]. To maintain the shape of the formation and the integrity of the bonds, the system rotates along an axis perpendicular to the plane of the ring. In [94], the influence of the Earth on this system is considered, therefore the canonical Likins-Pringle Relation (LPR) [95, 96] is used to construct the orbit for maintaining the rotation of the group around the axis in the strict direction to the Earth. The equations of motion are constructed for an annular formation with an arbitrary number of satellites. Simulation is performed for the formation of three satellites and the assessment of the system stability is provided. The first virtual structures with feedback and charge stabilization were proposed within the framework of the Coulomb cable concept with two devices [97, 98, 99]. While the physical tether must always be taut, the Coulomb tether can exert forces of attraction and repulsion between two ships. However, the concept of the Coulomb cable is viable only for relatively short distances up to 100 m, while the size of a typical space with a leash length of kilometers is considered within the framework of other concepts. These Coulomb mission concepts consider static scenarios in which ships are in nominally fixed locations. The first passively stable virtual Coulomb structure is a rotating system of two devices [51]. Without the screening effect of plasma charges, the force of attraction between two oppositely charged bodies is mathematically equivalent to the gravitational force of two bodies. The resulting trajectories of the two vehicles are also conical solutions, which are orbitally stable. Taking into account the finite Debye lengths and charge confinement, the relative trajectories are no longer closed conical solutions. In [51], it is shown that if ordinary circular relative motion has a diameter less than the Debye length, then the resulting non-linear motion remains stable. In [100], the generality of the problems of three bodies rotation under the action of gravitational and electrostatic forces is investigated for finding invariant forms. A schematic illustration of the structure discussed in [100] is shown in Figure 1. [PERSON] and [PERSON] [39] examined the general relative equilibria of such a system, and also discusses the stability of an open-loop system. It is assumed that the charged spacecraft moves in circular heliocentric orbits, far from the gravitational potential fields of planets or other celestial, bodies, therefore [PERSON] and [PERSON] [39] did not take into account the relative gravitational forces and focuses on the motion of freely moving bodies under the influence of electrostatic forces. In [39], general conditions are obtained, the solutions of which are relative equilibria for a rotating group of three spacecraft with Coulomb cables. Using the methods of linear control theory, stabilizing feedback is obtained in the form of PD and PID controllers for a nonlinear system, which guarantees its convergence to the neighborhood of the desired relative equilibrium. It is noted that for asymptotic stabilization of the desired relative equilibrium, independent of the value of the angular momentum \(h\), it will be necessary to change the nominal equilibrium value of the charge in order to reflect the actual value of the angular momentum of the system. It is also indicated that the use of general methods of nonlinear stabilization can provide larger regions of stability than the methods based on linearization considered in [39]. The decentralized state estimation algorithms and their application to the spacecraft group are discussed in [101], where three main architectures are considered:  
Centralized--only one point is involved, which performs primary calculations for a group of devices based on data collected at remote sites. Decentralized--each apparatus in the group is involved, making an equal contribution to the assessment process.