# Modeling and Control of Satellite Formations: A Survey
## 3 Circle Formation Architecture
In the radial configuration of the relative equilibrium, only the Coulomb force is required to control the motion in the plane and to control the satellites from their initial to final radial positions. In this reconfiguration maneuver, the torque of the gravity gradient is used to stabilize motion in the plane. For equilibrium positions along the path and normal orbits, the reconfiguration maneuver requires hybrid control. Here, the Coulomb force is changed to control the separation distance, and the inertial micromotors are activated to control the lateral directions. Second, a reconfiguration involving hybrid steering is used to maneuver ships from any initial equilibrium position to a final one. The goal is to determine optimal maneuvers that maximize the use of the Coulomb propulsion unit while minimizing the use of the electric propulsion unit. The formulation of the two-point boundary value optimization problem is numerically solved using pseudospectral methods. The Pontryagin minimum principle checks the optimality of solutions without feedback. In [107], a quaternion-based approach is considered for solving the time-limited problem of synchronizing and stabilizing satellites while moving in a formation. Sufficient conditions are given for the finite temporal convergence and stability of the distributed consensus problem. The nonlinear control law based on the finite time control method is designed in such a way that the exact position of the spacecraft will be consistent and will converge to the leader's position, while the angular velocity will converge to zero in a finite time. A term in the form of an integral of a power function has been added to the Lyapunov function. In order to reduce the load on the network, a modified control law is proposed, where an estimate of the sliding mode with time constraints is introduced. This aims to ensure that only one satellite communicates with the leader. The modeling results are presented, showing the efficiency of the proposed method and its potential advantages. [PERSON] et al. [108] dealt with the development of a formation control strategy for a circular satellite constellation formation. In [108], the artificial potential field method is used to plan the trajectory, and the sliding mode control method is used to create a robust controller. The fuzzy inference engine is used to reduce the - chattering phenomena inherent in conventional sliding mode systems. An algorithm for adaptive tuning of tuning a fuzzy parameter is derived on the basis of the Lyapunov stability theory. The proposed method based on a fuzzy system with a sliding mode is intended to compensate for the model uncertainties in practical applications. The simulation results for a group of five satellites forming a circular formation confirm the stability and robustness of the proposed scheme. The purpose of [109] is to investigate the control of an electromagnetic tethered satellite system using Model Predictive Control (MPC). An electromagnetic tethered satellite system is powered by electromagnetic coils to generate control forces. The dynamic system model is described with high and low levels of accuracy, which are used to design the control system. Several horizons of the predictive model are used to bring the formation to the desired state. The presented control law not only satisfies the input and output constraints but also has the corresponding characteristics of optimality. The main advantage of using multiple horizons is multiple control based on predictive models, having a lower computational load compared to classical MPC. The results of numerical simulation and their comparison with sliding mode control are presented to demonstrate the effectiveness of the proposed control method and its advantages over both classical MPC control methods and sliding mode. The results obtained show a dramatic reduction in computational time and energy consumption compared to the classical MPC control methods and the sliding model, respectively.