Nonlinear Dynamics

, Volume 93, Issue 2, pp 285–293 | Cite as

Attitude stabilization of a rigid body under the action of a vanishing control torque

  • A. Yu. Aleksandrov
  • A. A. Tikhonov
Original Paper


The problem of attitude stabilization of a rigid body with the use of restoring and dissipative torques is studied. The possibility of implementing a control system in which the restoring torque tends to zero as time increases, and the only remaining control torque is a linear time-invariant dissipative one, is investigated. Both cases of linear and essentially nonlinear restoring torques are considered. With the aid of the Lyapunov direct method and the comparison method, conditions are derived under which we can guarantee stability or asymptotic stability of an equilibrium position of the body despite the vanishing of the restoring torque. A numerical simulation is provided to demonstrate the effectiveness of analytical results.


Rigid body Attitude stabilization Time-varying control Asymptotic stability Lyapunov function Decomposition 



The research was supported by the Russian Foundation for Basic Research (Grant Nos. 16-01-00587-a and 17-01-00672-a).


  1. 1.
    Beletsky, V.V.: Motion of an Artificial Satellite About its Center of Mass. Israel Program for Scientific Translation, Jerusalem (1966)Google Scholar
  2. 2.
    Zubov, V.I.: Theorie de la Commande. Mir, Moscow (1978). (in French)zbMATHGoogle Scholar
  3. 3.
    Wertz, J.R.: Spacecraft Attitude Determination and Control. D. Reidel Publishing Co., Dordrecht (1985)Google Scholar
  4. 4.
    Sazonov, V.V., Sarychev, V.A.: Effect of dissipative magnetic moment on rotation of a satellite relative to the center of mass. Mech. Solids 18(2), 1–9 (1983)Google Scholar
  5. 5.
    Ovchinnikov, M.Y., Pen’kov, V.I., Roldugin, D.S., Karpenko, S.O.: Investigation of the effectiveness of an algorithm of active magnetic damping. Cosm. Res. 50(2), 170–176 (2012). CrossRefGoogle Scholar
  6. 6.
    Tikhonov, A.A., Tkhai, V.N.: Symmetrical oscillations of charged gyrostat in weakly elliptical orbit with small inclination. Nonlinear Dyn. 85(3), 1919–1927 (2016). CrossRefGoogle Scholar
  7. 7.
    Aleksandrov, A.Yu., Antipov, K.A., Platonov, A.V., Tikhonov, A.A.: Electrodynamic attitude stabilization of a satellite in the Konig frame. Nonlinear Dyn. 82(3), 1493–1505 (2015).
  8. 8.
    Hughes, P.C.: Spacecraft Attitude Dynamics. Wiley, New York (1988)Google Scholar
  9. 9.
    Ovchinnikov, M.Y., Ivanov, D.S., Ivlev, N.A., Karpenko, S.O., Roldugin, D.S., Tkachev, S.S.: Development, integrated investigation, laboratory and in-flight testing of Chibis-M microsatellite ADCS. Acta Astronaut. 93(1), 23–33 (2014). CrossRefGoogle Scholar
  10. 10.
    Aleksandrov, A.Yu., Tikhonov, A.A.: Monoaxial electrodynamic stabilization of earth satellite in the orbital coordinate system. Autom. Remote Control 74(8), 1249–1256 (2013).
  11. 11.
    Doroshin, A.V.: Evolution of the precessional motion of unbalanced gyrostats of variable structure. J. Appl. Math. Mech. 72(3), 259–269 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ivanov, D.S., Ovchinnikov, M.Y., Pen’kov, V.I.: Laboratory study of magnetic properties of hysteresis rods for attitude control systems of minisatellites. J. Comput. Syst. Sci. Int. 52(1), 145–164 (2013). CrossRefzbMATHGoogle Scholar
  13. 13.
    Antipov, K.A., Tikhonov, A.A.: Electrodynamic control for spacecraft attitude stability in the geomagnetic field. Cosm. Res. 52(6), 472–480 (2014). CrossRefGoogle Scholar
  14. 14.
    Melnikov, G.I., Dudarenko, N.A., Melnikov, V.G., Alyshev, A.S.: Parametric identification of inertial parameters. Appl. Math. Sci. 9(136), 6757–6765 (2015). Google Scholar
  15. 15.
    Hatvani, L.: The effect of damping on the stability properties of equilibria of non-autonomous systems. J. Appl. Math. Mech. 65(4), 707–713 (2001). MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rouche, N., Habets, P., Laloy, M.: Stability Theory by Liapunov’s Direct Method. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  17. 17.
    Rumyantsev, V.V., Oziraner, A.S.: Stability and Stabilization of Motion with Respect to a Part of Variables. Nauka, Moscow (1987). (in Russian)zbMATHGoogle Scholar
  18. 18.
    Hatvani, L.: On the stability of the zero solution of nonlinear second order differential equations. Acta Sci. Math. 57, 367–371 (1993)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cantarelli, G.: The stability of the equilibrium position of scleronomous mechanical systems. J. Appl. Math. Mech. 66(6), 943–956 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sugie, J., Amano, Y.: Global asymptotic stability of nonautonomous systems of Lienard type. J. Math. Anal. Appl. 289(2), 673–690 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Aleksandrov, A.Yu.: The stability of the equilibrium positions of non-linear non-autonomous mechanical systems. J. Appl. Math. Mech. 71(3), 324–338 (2007).
  22. 22.
    Aleksandrov, A.Yu., Kosov, A.A.: Asymptotic stability of equilibrium positions of mechanical systems with a nonstationary leading parameter. J. Comput. Syst. Sci. Int. 47(3), 332–345 (2008).
  23. 23.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Marcel Dekker, New York (1989)zbMATHGoogle Scholar
  24. 24.
    Siljak, D.D.: Decentralized Control of Complex Systems. Academic Press, New York (1991)zbMATHGoogle Scholar
  25. 25.
    Krasil’nikov, P.S.: Functional extensions of a solution germ space of first order differential equation and their applications. Nonlinear Anal. Theory Methods Appl. 28(2), 359–375 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Krasil’nikov, P.S.: A generalized scheme for constructing Lyapunov functions from first integrals. J. Appl. Math. Mech. 65(2), 195–204 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Smirnov, E.Y.: Control of rotational motion of a free solid by means of pendulums. Mech. Solids 15(3), 1–5 (1980)Google Scholar
  28. 28.
    Haller, G., Ponsioen, S.: Exact model reduction by a slow-fast decomposition of nonlinear mechanical systems. Nonlinear Dyn. 90(1), 617–647 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Aleksandrov, A.Yu., Kosov, A.A., Chen, Y.: Stability and stabilization of mechanical systems with switching. Autom. Remote Control 72(6), 1143–1154 (2011).
  30. 30.
    Aleksandrov, A.Y., Aleksandrova, E.B.: Asymptotic stability conditions for a class of hybrid mechanical systems with switched nonlinear positional forces. Nonlinear Dyn. 83(4), 2427–2434 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Aleksandrov, A.Yu., Tikhonov, A.A.: Attitude stabilization of a rigid body in conditions of decreasing dissipation. Vestn. St. Petersburg Univ. Math. 50(4), 384–391 (2017).
  32. 32.
    Gendelman, O.V., Lamarque, C.H.: Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium. Chaos Solitons Fractals 24, 501–509 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Luongo, A., Zulli, D.: Nonlinear energy sink to control elastic strings: the internal resonance case. Nonlinear Dyn. 81(1–2), 425–435 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kozmin, A., Mikhlin, Y., Pierre, C.: Transient in a two-DOF nonlinear system. Nonlinear Dyn. 51(1–2), 141–154 (2008). zbMATHGoogle Scholar
  35. 35.
    Beards, C.F.: Engineering Vibration Analysis with Application to Control Systems. Edward Arnold, London (1995)Google Scholar
  36. 36.
    Samba, Y.C.M., Pascal, M.: Nonlinear effects in dynamic analysis of flexible multibody systems. In: Proceedings of the ASME Design Engineering Technical Conference, 18th Biennial Conference on Mechanical Vibration and Noise, Pittsburgh, PA, USA, vol. 6 A, pp. 453–459 (2001)Google Scholar
  37. 37.
    Aleksandrov, A.Yu., Aleksandrova, E.B., Zhabko, A.P.: Asymptotic stability conditions and estimates of solutions for nonlinear multiconnected time-delay systems. Circuits Syst. Signal Process. 35, 3531–3554 (2016).
  38. 38.
    Kosov, A.A.: The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces. J. Appl. Math. Mech. 71(3), 371–384 (2007). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySt. PetersburgRussia
  2. 2.Saint Petersburg Mining UniversitySt. PetersburgRussia

Personalised recommendations