electronics Article Development and Simulation of Motion Control System for Small Satellites Formation Alexander M. Popov 1 , Ilya Kostin 1 , Julia Fadeeva 1 and Boris Andrievsky 1,2,3, *   Citation: Popov, A.M.; Kostin, I.; Fadeeva, J.; Andrievsky, B. Development and Simulation of Motion Control System for Small Satellites Formation. Electronics 2021, 10, 3111. https://doi.org/10.3390/ electronics10243111 Academic Editor: Katarzyna Antosz Received: 11 November 2021 Accepted: 9 December 2021 Published: 14 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Control Systems and Computer Technologies Department, Faculty of Information and Control Systems, Baltic State Technical University “VOENMEH” Named after D.F. Ustinov, 190005 Saint Petersburg, Russia; popov_am@voenmeh.ru (A.M.P.); kostin_id@voenmeh.ru (I.K.); fadeeva_iuv@voenmeh.ru (J.F.) 2 Control of Complex Systems Lab., Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia 3 Applied Cybernetics Department, Faculty of Mathematics and Mechanics, Saint Petersburg University, 198504 Saint Petersburg, Russia * Correspondence: b.andrievsky@spbu.ru; Tel.: +7-812-321-4776 Abstract: In the paper, the problem of forming and maintaining the small satellites formation in the near-earth projected circular orbits is considered. The satellite formation reconfiguration and formation-keeping control laws are proposed by employing the passivity-based output feedback control. 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 passivity-based output feed- back control and the proportional-differential controller with and without the consensus algorithm is given. On the basis of the passification method, the algorithm is obtained ensuring the stable motion of the slave satellite relative to the orbit of the master satellite. To improve the accuracy of the satellites’ positioning, a consensus protocol based on measurements of the relative positions of the satellites is proposed and studied. Computer simulations of the proposed algorithms for options to construct formations are provided for two projected circular orbits of 8 satellites, demonstrating the efficiency of the proposed control schemes. It is shown that the resulting passivity-based output feedback control provides better accuracy than the PD controller. It is also shown that the use of the consensus protocol further increases the positioning accuracy of the satellite constellation. Keywords: satellite formation; keeping; control; passification; consensus; projected circular orbit 1. Introduction Space research has led to the emergence of the idea of spacecrafts’ motion in a group. Group motion makes it possible to perform a wider range of tasks. This approach has reduced the costs of production and operation, and increased the reliability of the entire sys- tem. There are several satellite constellation formation methods for creating time-invariant and time-varying configurations. In the literature, various options to create satellite constel- lations are proposed, cf. [1–3]. The constant tuning method [4] implies periodic adjustment of active satellites relative to the spatial position. The variable formation method [5] forms a swarm, which periodically changes its configuration as satellites move in non-coplanar orbits. It is possible to form the configuration of satellites by means of their tether con- nection; such formations are possible in low orbits [6]. A grouping can be formed using several methods; this method of formation is called combined [2]. Among the groupings with a constant structure, a class stands out that uses projected circular orbits (PCO) [7] to form the structure. When using this construction method, a group is considered, which consists of the head and surrounding slave satellites [8]. The orbit of the master satellite is considered to be the reference orbit, and the remaining orbits are called projection circular orbits, which are circular when projected onto the local horizontal plane [9,10]. The first Electronics 2021, 10, 3111. https://doi.org/10.3390/electronics10243111 https://www.mdpi.com/journal/electronics