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Mathematical Problems in Engineering
Volume 2012, Article ID 532708, 14 pages
Research Article

Low-Thrust Out-of-Plane Orbital Station-Keeping Maneuvers for Satellites

DMC, National Institute for Space Research (INPE), Avenida dos Astronautas 1758, 12227-010 São José dos Campos, SP, Brazil

Received 30 October 2011; Revised 30 November 2011; Accepted 8 January 2012

Academic Editor: Maria Zanardi

Copyright © 2012 Vivian M. Gomes and Antonio F. B. A. Prado. 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.


This paper considers the problem of out of plane orbital maneuvers for station keeping of satellites. The main idea is to consider that a satellite is in an orbit around the Earth and that it has its orbit is disturbed by one or more forces. Then, it is necessary to perform a small amplitude orbital correction to return the satellite to its original orbit, to keep it performing its mission. A low thrust propulsion is used to complete this task. It is important to search for solutions that minimize the fuel consumption to increase the lifetime of the satellite. To solve this problem a hybrid optimal control approach is used. The accuracy of the satisfaction of the constraints is considered, in order to try to decrease the fuel expenditure by taking advantage of this freedom. This type of problem presents numerical difficulties and it is necessary to adjust parameters, as well as details of the algorithm, to get convergence. In this versions of the algorithm that works well for planar maneuvers are usually not adequate for the out of plane orbital corrections. In order to illustrate the method, some numerical results are presented.