Abstract
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.
Similar content being viewed by others
References
Beletsky, V.V.: Motion of an Artificial Satellite About its Center of Mass. Israel Program for Scientific Translation, Jerusalem (1966)
Zubov, V.I.: Theorie de la Commande. Mir, Moscow (1978). (in French)
Wertz, J.R.: Spacecraft Attitude Determination and Control. D. Reidel Publishing Co., Dordrecht (1985)
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)
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). https://doi.org/10.1134/S0010952512010078
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). https://doi.org/10.1007/s11071-016-2805-2
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). https://doi.org/10.1007/s11071-015-2256-1
Hughes, P.C.: Spacecraft Attitude Dynamics. Wiley, New York (1988)
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). https://doi.org/10.1016/j.actaastro.2013.06.030
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). https://doi.org/10.1134/S000511791308002X
Doroshin, A.V.: Evolution of the precessional motion of unbalanced gyrostats of variable structure. J. Appl. Math. Mech. 72(3), 259–269 (2008). https://doi.org/10.1016/j.jappmathmech.2008.07.003
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). https://doi.org/10.1134/S1064230712060032
Antipov, K.A., Tikhonov, A.A.: Electrodynamic control for spacecraft attitude stability in the geomagnetic field. Cosm. Res. 52(6), 472–480 (2014). https://doi.org/10.1134/S001095251406001X
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). https://doi.org/10.12988/ams.2015.59584
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). https://doi.org/10.1016/S0021-8928(01)00076-4
Rouche, N., Habets, P., Laloy, M.: Stability Theory by Liapunov’s Direct Method. Springer, New York (1977)
Rumyantsev, V.V., Oziraner, A.S.: Stability and Stabilization of Motion with Respect to a Part of Variables. Nauka, Moscow (1987). (in Russian)
Hatvani, L.: On the stability of the zero solution of nonlinear second order differential equations. Acta Sci. Math. 57, 367–371 (1993)
Cantarelli, G.: The stability of the equilibrium position of scleronomous mechanical systems. J. Appl. Math. Mech. 66(6), 943–956 (2002). https://doi.org/10.1016/S0021-8928(02)00136-3
Sugie, J., Amano, Y.: Global asymptotic stability of nonautonomous systems of Lienard type. J. Math. Anal. Appl. 289(2), 673–690 (2004). https://doi.org/10.1016/j.jmaa.2003.09.023
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). https://doi.org/10.1016/j.jappmathmech.2007.07.015
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). https://doi.org/10.1134/S1064230708030027
Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Marcel Dekker, New York (1989)
Siljak, D.D.: Decentralized Control of Complex Systems. Academic Press, New York (1991)
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). https://doi.org/10.1016/0362-546X(95)00150-T
Krasil’nikov, P.S.: A generalized scheme for constructing Lyapunov functions from first integrals. J. Appl. Math. Mech. 65(2), 195–204 (2001). https://doi.org/10.1016/S0021-8928(01)00023-5
Smirnov, E.Y.: Control of rotational motion of a free solid by means of pendulums. Mech. Solids 15(3), 1–5 (1980)
Haller, G., Ponsioen, S.: Exact model reduction by a slow-fast decomposition of nonlinear mechanical systems. Nonlinear Dyn. 90(1), 617–647 (2017). https://doi.org/10.1007/s11071-017-3685-9
Aleksandrov, A.Yu., Kosov, A.A., Chen, Y.: Stability and stabilization of mechanical systems with switching. Autom. Remote Control 72(6), 1143–1154 (2011). https://doi.org/10.1134/S0005117911060026
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). https://doi.org/10.1007/s11071-015-2491-5
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). https://doi.org/10.3103/S1063454117040021
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). https://doi.org/10.1016/j.chaos.2004.09.088
Luongo, A., Zulli, D.: Nonlinear energy sink to control elastic strings: the internal resonance case. Nonlinear Dyn. 81(1–2), 425–435 (2015). https://doi.org/10.1007/s11071-015-2002-8
Kozmin, A., Mikhlin, Y., Pierre, C.: Transient in a two-DOF nonlinear system. Nonlinear Dyn. 51(1–2), 141–154 (2008). https://doi.org/10.1007/s11071-007-9198-1
Beards, C.F.: Engineering Vibration Analysis with Application to Control Systems. Edward Arnold, London (1995)
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)
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). https://doi.org/10.1007/s00034-015-0227-x
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). https://doi.org/10.1016/j.jappmathmech.2007.07.011
Acknowledgements
The research was supported by the Russian Foundation for Basic Research (Grant Nos. 16-01-00587-a and 17-01-00672-a).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aleksandrov, A.Y., Tikhonov, A.A. Attitude stabilization of a rigid body under the action of a vanishing control torque. Nonlinear Dyn 93, 285–293 (2018). https://doi.org/10.1007/s11071-018-4191-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-018-4191-4