# Development and Simulation of Motion Control System for Small Satellites Formation
In the study of a new control algorithm based on the passification method, convergence was obtained at a lower initial speed than the modal control algorithm. Ref. [24] discussed maintaining a working orbit and a given configuration. For the orbit of the lead satellite, rigid (absolute) support is used. For the slave satellite, rigid and flexible (relative) support is applied. The dependence of the costs of the total characteristic velocity to maintain the master and slave vehicles on the accuracy of maintaining the working orbit was investigated, which showed the particular importance of the correct choice of the maximum accuracy of maintaining. In [25], the problem of calculating the parameters of the maneuvers of the satellite system configuration was considered. For convenience, the orbits of a group of six satellites (STARSIS) are reduced to projection circular orbits. Ref. [26] complemented the task of calculating the parameters of satellite maneuvers by examining high-elliptic orbits for four satellites. The satellite array model is brought into the main orbital plane to simplify calculations. In [27], to solve the control problem, the influence of the given initial values of the projection circular orbit for different approaches to control of a group of satellites was investigated. The design takes into account the minimization of error and fuel consumption. When performing tasks with a large number of satellites, it is more efficient to use multi-agent and decentralized control [28]. To this aim, in [4], using local relative dimensions was suggested. Another approach was used in the article [29]. It proposed a combined consensus method with a robust \(H_{\infty}\) control. Using this approach, it is possible to provide an arbitrary scalability of the system. Comparisons were made between various controllers [30]. In the course of the study, data were obtained, on the basis of which it was shown that the nonlinear controller based on the Riccati equations (_state-dependent Riccati equation_--SDRE) has better accuracy. The paper is devoted to the problem of forming and maintaining the small satellites formation in the near-earth PCO orbits. On the basis of the passification method, an algorithm is obtained that ensures the stable motion of the slave satellite relative to the orbit of the master satellite. To improve the accuracy of satellites positioning, a consensus protocol based on measurements of the relative positions of satellites is proposed and studied. In this paper, for the complete nonlinear and time-dependent dynamics of the relative motion of a pair of satellites in elliptical orbits, new combined control algorithms, including a consensus protocol, are proposed and analyzed. A comparison of the control modes using the PD controller with and without the consensus algorithm is given. The dynamic structure of the proposed control algorithm is quite simple and avoids the use of a large number of blocks. Its implementation is simple, and the computational cost is low. The new result of the paper is also a rigorous mathematical proof of the use of control laws based on the passification method for the satellite system under consideration. The rest of the paper is organized as follows. Section 2 is devoted to modeling the satellite formation. The problem statement is given in Section 3. Section 4 deals with application of the passivity concept to the control law design. The notion of projection circular orbit is described in Section 5. Section 6 presents designing the control law for PCO constellation. The simulation results are given in Section 8. Concluding remarks in Section 9 finalize the paper.