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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 967248, 20 pages
Research Article

Closed Relative Trajectories for Formation Flying with Single-Input Control

1Centre for Aerospace Science and Technologies, Department of Electromechanical Engineering, University of Beira Interior, Calçada Fonte do Lameiro, 6201-001 Covilhã, Portugal
2Orientation and Motion Control Department, Keldysh Institute of Applied Mathematics, Miusskaya pl. 4, Moscow 125047, Russia
3Centre of Physics, Department of Mathematics and Applications, School of Sciences, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal
4Department of Control and Applied Mathematics, Moscow Institute of Physics and Technology, Institutskij per. 9, Dolgoprudny, Moscow 141700, Russia

Received 13 July 2011; Revised 11 November 2011; Accepted 15 November 2011

Academic Editor: Josep Masdemont

Copyright © 2012 Anna Guerman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the problem of formation shape control under the constraints on the thrust direction. Formations composed of small satellites are usually subject to serious limitations for power consumption, mass, and volume of the attitude and orbit control system (AOCS). If the purpose of the formation flying mission does not require precise tracking of a given relative trajectory, AOCS of satellites may be substantially simplified; however, the capacity of AOCS to ensure a bounded or even periodic relative motion has to be studied first. We consider a formation of two satellites; the deputy one is equipped with a passive attitude control system that provides one-axis stabilization and a propulsion system that consists of one or two thrusters oriented along the stabilized axis. The relative motion of the satellites is modeled by the Schweighart-Sedwick linear equations taking into account the effect of perturbations. We prove that both in the case of passive magnetic attitude stabilization and spin stabilization for all initial relative positions and velocities of satellites there exists a control guaranteeing their periodic relative motion.