Averaging technique in the problem of satellite attitude stabilization in indirect position in the orbital reference frame with the use of Lorentz torque

Authors

  • Aleksandr Yu. Aleksandrov St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
  • Natalya R. Andriyanova St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
  • Aleksey A. Tikhonov St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2021.111

Abstract

A dynamically symmetric satellite in a circular orbit of small inclination is considered. The problem of the Lorentzian stabilization of the satellite in the orbital coordinate system in the indirect equilibrium position is solved under the conditions of the perturbing effect of the gravitational torque. To solve this problem, which is characterized by incomplete control, a method of averaging differential equations is developed. Using the original construction of the unsteady Lyapunov function, we obtain sufficient conditions for the asymptotic stability of the programmed satellite motion mode in the form of constructive inequalities with respect to the control parameters.

Keywords:

satellite, stabilization, attitude motion, asymptotic stability, Lyapunov function method

Downloads

Download data is not yet available.
 

References

Литература

1. Белецкий В.В. Движение искусственного спутника относительно центра масс. Москва, Наука (1965).

2. Сарычев В.А. Вопросы ориентации искусственных спутников. В: Исследование космиче- ского пространства. Т. 11. Итоги науки и техники. Москва, ВИНИТИ АН СССР (1978).

3. Сарычев В.А., Овчинников М.Ю. Магнитные системы ориентации искусственных спут- ников Земли. В: Исследование космического пространства. Т. 23. Итоги науки и техники. Москва, ВИНИТИ АН СССР (1985).

4. Ovchinnikov M.Yu., Roldugin D. S., Penkov V. I. Asymptotic study of a complete magnetic attitude control cycle providing a single-axis orientation. Acta Astronautica 77, 48–60 (2012).

5. Aleksandrov A.Yu., Tikhonov A.A. Asymptotic stability of a satellite with electrodynamic attitude control in the orbital frame. Acta Astronautica 139, 122–129 (2017). https://doi.org/10.1016/j.actaastro.2017.06.033

6. Петров К. Г., Тихонов A.A. Момент сил Лоренца, действующих на заряженный спутник в магнитном поле Земли. Ч. 1: Напряженность магнитного поля Земли в орбитальной системе координат. Вестник Cанкт-Петербургского университета. Серия 1. Математика. Механика. Астрономия, вып. 1, 92–100 (1999).

7. Петров К. Г., Тихонов A.A. Момент сил Лоренца, действующих на заряженный спутник в магнитном поле Земли. Ч. 2: Вычисление момента и оценки его составляющих. Вестник Cанкт-Пе- тербургского университета. Серия 1. Математика. Механика. Астрономия, вып. 3, 81–91 (1999).

8. Тихонов A.A. Метод полупассивной стабилизации космического аппарата в геомагнитном поле. Космические исследования 41 (1), 69–79 (2003).

9. Антипов К.А., Тихонов А.А. Параметрическое управление в задаче о стабилизации косми- ческого аппарата в магнитном поле Земли. Автомат. и телемех., (8), 44–56 (2007).

10. Aleksandrov A.Yu., Antipov K.A., Platonov A.V., Tikhonov A.A. Electrodynamic attitude stabilization of a satellite in the Konig frame. Nonlinear Dynamics 82, 1493–1505 (2015). https://doi.org/10.1007/s11071-015-2256-1

11. Sussingham J.C., Watkins S.A., Cocks F.H. Forty years of development of active systems for radiation protection of spacecraft. J. Astronaut. Sci. 47, 165–175 (1999).

12. Joshi R. P., Qiu H., Tripathi R.K. Configuration studies for active electrostatic space radiation shielding. Acta Astronautica 88, 138–145 (2013). https://doi.org/10.1016/j.actaastro.2013.03.011

13. Giri D.K., Sinha M. Three-axis attitude control of Earth-pointing isoinertial magnetoCoulombic satellites. Int. J. Dynam. Control 5, 644–652 (2017).

14. Митропольский Ю.А. Метод усреднения в нелинейной механике. Киев, Наукова думка (1971).

15. Гребеников Е.А. Метод усреднения в прикладных задачах. Москва, Наука (1986).

16. Найфэ А. Введение в методы возмущений, пер. с англ. Москва, Мир (1984).

17. Красильников П.С. Об усреднении дифференциальных уравнений с двумя независимыми малыми параметрами. Доклады Академии наук 436 (3), 332–335 (2011).

18. Красильников П.С. Прикладные методы исследования нелинейных колебаний. Ижевск, Институт компьютерных исследований (2015).

19. Aleksandrov A.Yu., Tikhonov A.A. Rigid body stabilization under time-varying perturbations with zero mean values. Cybernetics and Physics 7 (1), 5–10 (2018). https://doi.org/10.35470/2226-4116- 2018-7-1-5-10

20. Александров А.Ю., Тихонов А.А. Одноосная стабилизация вращательного движения твердого тела при наличии возмущений с нулевыми средними значениями. Вестник Санкт-Петер- бургского университета. Математика. Механика. Астрономия 6 (64), вып. 2, 270–280 (2019). https://doi.org/10.21638/11701/spbu01.2019.209

21. Александров А.Ю. Об устойчивости равновесия нестационарных систем. Прикладная ма- тематика и механика 60, вып. 2, 205–209 (1996).

22. Aleksandrov A.Yu., Aleksandrova E.B., Zhabko A.P. Stability analysis for a class of nonlinear nonstationary systems via averaging. Nonlinear Dynamics and Systems Theory 13 (4), 332–343 (2013).

23. Бранец В.Н.,Шмыглевский И.П. Применение кватернионов в задачах ориентации твер- дого тела. Москва, Наука (1973).

24. Тихонов А.А., Петров К. Г. Мультипольные модели магнитного поля Земли. Космические исследования 40 (3), 219–229 (2002).

25. International Geomagnetic Reference Field. Доступно на: http://www.ngdc.noaa.gov/IAGA /vmod/igrf.html (дата обращения: 05.12.2020).

26. Shamolin M.V. Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium. Journal of Mathematical Sciences 110 (2), 2528–2557 (2002). https://doi.org/10.1023/a:1015026512786

27. Зубов В.И. Динамика управляемых систем. Москва, Высшая школа (1982).

28. Смирнов Е.Я. Некоторые задачи математической теории управления. Ленинград, Изд-во Ленингр. ун-та (1981).

29. Александров А.Ю., Тихонов А.А. Электродинамическая стабилизация ИСЗ на экватори- альной орбите. Космические исследования 50 (4), 335–340 (2012).

References

1. Beletsky V.V. Motion of an Artificial Satellite about its Center of Mass. Moscow, Nauka Publ. (1965). (In Russian)

2. Sarychev V.A. Problems of artificial satellites orientation. In: Space Research, vol. 11, Advances in Science and Technology, Moscow, VINITI of the USSR Academy of Sciences (1978). (In Russian)

3. Sarychev V.A., Ovchinnikov M.Yu. Magnetic systems for orientation of satellites. In: Space Research. vol. 23, Advances in Science and Technology, Moscow, VINITI of the USSR Academy of Sciences (1985). (In Russian)

4. Ovchinnikov M.Yu., Roldugin D. S., Penkov V. I. Asymptotic study of a complete magnetic attitude control cycle providing a single-axis orientation. Acta Astronautica 77, 48–60 (2012).

5. Aleksandrov A.Yu., Tikhonov A.A. Asymptotic stability of a satellite with electrodynamic attitude control in the orbital frame. Acta Astronautica 139, 122–129 (2017). https://doi.org/10.1016/j.actaastro.2017.06.033

6. Petrov K.G., Tikhonov A.A. The moment of Lorentz forces, acting upon the charged satellite in the geomagnetic field. Part 1. The strength of the Earth’s magnetic field in the orbital coordinate system. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, iss. 1, 92–100 (1999). (In Russian)

7. Petrov K.G., Tikhonov A.A. The moment of Lorentz forces, acting upon the charged satellite in the geomagnetic field. Part 2. The determination of the moment and estimations of its components. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, iss. 3, 81–91 (1999). (In Russian)

8. Tikhonov A.A. A method of semipassive attitude stabilization of a spacecraft in the geomagnetic field. Kosmicheskie issledovaniya 41 (1), 69–79 (2003). (In Russian) [Engl. transl.: Cosmic Research 41 (1), 63–73 (2003). https://doi.org/10.1023/A:1022355730291].

9. Antipov K.A., Tikhonov A.A. Parametric control in the problem of spacecraft stabilization in the geomagnetic field. Avtomat. i Telemekh., (8), 44–56 (2007). (In Russian) [Engl. transl.: Autom. Remote Control 68 (8), 1333–1345 (2007)].

10. Aleksandrov A.Yu., Antipov K.A., Platonov A.V., Tikhonov A.A. Electrodynamic attitude stabilization of a satellite in the Konig frame. Nonlinear Dynamics 82, 1493–1505 (2015). https://doi.org/10.1007/s11071-015-2256-1

11. Sussingham J.C., Watkins S.A., Cocks F.H. Forty years of development of active systems for radiation protection of spacecraft. J. Astronaut. Sci. 47, 165–175 (1999).

12. Joshi R. P., Qiu H., Tripathi R.K. Configuration studies for active electrostatic space radiation shielding. Acta Astronautica 88, 138–145 (2013). https://doi.org/10.1016/j.actaastro.2013.03.011

13. Giri D.K., Sinha M. Three-axis attitude control of Earth-pointing isoinertial magnetoCoulombic satellites. Int. J. Dynam. Control 5, 644–652 (2017).

14. Mitropolsky Y.A. The Averaging Method in Nonlinear Mechanics. Kiev, Naukova Dumka Publ. (1971). (In Russian)

15. Grebennikov E.A. The Averaging Method in Applied Problems. Moscow, Nauka Publ. (1986). (In Russian)

16. Nayfeh A.H. Introduction to Perturbation Techniques. New York, Wiley Interscience (1981). [Russ. ed.: Vvedenie v metody vozmushhenij. Moscow, Mir Publ. (1984)].

17. Krasil’nikov P. S. On the average of differential equations with two independent small parameters. Doklady Akademii Nauk 436 (3), 332–335 (2011). (In Russian) [Engl. transl.: Doklady Physics 56, 58–61 (2011). https://doi.org/10.1134/S1028335811010113].

18. Krasil’nikov P. S. Applied Methods for the Study of Nonlinear Oscillations. Izhevsk, Institute of Computer Science Press (2015). (In Russian)

19. Aleksandrov A.Yu., Tikhonov A.A. Rigid body stabilization under time-varying perturbations with zero mean values. Cybernetics and Physics 7 (1), 5–10 (2018). https://doi.org/10.35470/2226-4116- 2018-7-1-5-10

20. Aleksandrov A.Yu., Tikhonov A.A. Uniaxial attitude stabilization of a rigid body under conditions of nonstationary perturbations with zero mean values. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 6 (64), iss. 2, 270–280 (2019). https://doi.org/10.21638/11701/spbu01.2019.209 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 52 (2), 187–193 (2019). https://doi.org/10.1134/S106345411902002X].

21. Aleksandrov A.Yu. On the stability of equilibrium of unsteady systems. Prikladnaya matematika i mekhanika 60 (2), 205–209 (1996). (In Russian)

22. Aleksandrov A.Yu., Aleksandrova E.B., Zhabko A.P. Stability analysis for a class of nonlinear nonstationary systems via averaging. Nonlinear Dynamics and Systems Theory 13 (4), 332–343 (2013).

23. Branets V.N., Shmyglevsky I. P. Application of Quaternions in the Problems of the Rigid Body Attitude Determination. Moscow, Nauka Publ. (1973). (In Russian)

24. Tikhonov A.A. Petrov K.G. Multipole models of the Earth’s magnetic field. Kosmicheskie issledovaniya 40 (3), 219–229 (2002). (In Russian) [Engl. transl.: Cosmic Research 40 (2), 203–212 (2002). https://doi.org/10.1023/A:1015916718570].

25. International Geomagnetic Reference Field. Available at: http://www.ngdc.noaa.gov/IAGA /vmod/igrf.html (accessed: Dec. 05, 2020).

26. Shamolin M.V. Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium. Journal of Mathematical Sciences 110 (2), 2528–2557 (2002). https://doi.org/10.1023/a:1015026512786

27. Zubov V. I. Dynamics of Controlled Systems. Moscow, Vysshaya Shkola Publ. (1982). (In Russian)

28. Smirnov E.Ya. Some Problems of the Mathematical Control Theory. Leningrad, Leningrad University Press (1981). (In Russian)

29. Aleksandrov A.Yu., Tikhonov A.A. Electrodynamic stabilization of Earth-orbiting satellites in equatorial orbits. Kosmicheskie issledovaniya 50 (4), 335–340 (2012). (In Russian) [Engl. transl.: Cosmic Research 50 (4), 313–318 (2012). https://doi.org/10.1134/S001095251203001X].

Downloads

Published

2021-05-29

How to Cite

Aleksandrov, A. Y., Andriyanova, N. R., & Tikhonov, A. A. (2021). Averaging technique in the problem of satellite attitude stabilization in indirect position in the orbital reference frame with the use of Lorentz torque. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 123–137. https://doi.org/10.21638/spbu01.2021.111

Issue

Vol. 8 No. 1 (2021)

Section

Mechanics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.