# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
Numerical simulations are performed to validate the analysis and demonstrate the results. [PERSON] and [PERSON] [84] addressed the problem of building a swarm of nanosatellites immediately after they are separated from the launch vehicle. The decentralized control is proposed using the aerodynamic drag force to eliminate the relative drift between satellites in a swarm. The effect of dividing a swarm into several independent groups is studied, which is considered a violation of the integrity of the swarm and is undesirable. An aerodynamic force application is also studied in [85], where for the formation of two satellites the control algorithms based on the modal approach (pole placement technique), on passification and variable structure control, linear and transient-speed optimal for partial stabilization algorithms are developed and examined. The stability of control systems to the plant model parameters is studied and the possibility of the occurrence of stable oscillations as a consequence of the control action (aerodynamic resistance) limitation is shown. The problem of deploying a swarm of nanosatellites immediately after their separation from the launcher was studied in [86]. Since some error in ejection velocity during the launch is inevitable, nanosatellites have different orbital periods, and as a result, gradually disperse in orbit. The decentralized differential control using the force of resistance is applied to the swarm formation problem. It is assumed that each satellite has information about the position of others in a particular communication area. The proposed control algorithm eliminates the relative drift between adjacent satellites. Ref. [PERSON] et al. [86] studied the separation effect that occurs when a swarm is divided into several independent groups. This effect depends on the size of the communication area, the number of communication satellites, and the initial conditions. The boundary values of these parameters are studied for the example of twenty 3U cubesatellites. The effect of \(J_{2}\) harmonics and uncertainty on the atmosphere's density is also examined in [86]. Knowledge of position and velocity is a practical requirement for all spacecraft in low Earth orbit. Some satellite systems require autonomous orbit determination on board. The methods for estimating the absolute positions of two satellites using relative distance and azimuth are developed by [PERSON] and [PERSON] [87]. The spherical coordinates measurements are converted to Cartesian ones with the converted offset and covariance calculated. The transformed measurements use the Unbiased Transformed Kalman Measurement (UTKM) [88, 89] algorithm to estimate the position and velocity of the satellites. A square root filter is introduced to avoid losing the positive definiteness of the state covariance matrix. The simulation results show that the algorithm proposed in [87] converges faster and has a higher estimation accuracy. [PERSON] et al. [90] introduced a new control action for the formation of satellites moving in LEO, depending on the difference between the force of aerodynamic drag and the force generated by the propeller. A parameterized algorithm for regulating the output signal for the formation of flight missions has been developed, based on the equations of relative dynamics by [PERSON] [91, 92]. It is implemented in order to accurately track different trajectories of the reference relative motion and to exclude the influence of disturbances from the second harmonic of the Earth's gravitational potential. Invention [93] relates to satellite group motion control, in which the average angular velocity of all artificial earth satellites (AES) in the group is maintained equal to the average angular velocity of the passive satellite per turn. The latter is located in the central orbit of the group. Active satellites maintain their orbital position relative to passive satellites through periodic reactive correction.