Nonlinear Control for Attitude Stabilization of a Rigid Body ‎Forced by Nonstationary Disturbances with Zero Mean Values

Document Type : Research Paper


1 Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, 7-9 Universitetskaya nab., Saint Petersburg, 199034, Russia‎

2 Department of Theoretical and Applied Mechanics, Saint Petersburg State University, 7-9 Universitetskaya nab., Saint Petersburg, 199034, Russia‎

3 Department of Mechanics, Saint Petersburg Mining University, 2, 21st Line, St. Petersburg, 199106, Russia‎


A rigid body forced by a nonstationary perturbing torque with zero mean value is under consideration. The control strategy for attitude stabilization of the rigid body is based on the usage of dissipative and restoring torques. It is assumed that the dissipative torque is linear, while restoring and perturbing torques are purely nonlinear. A theorem on sufficient conditions for asymptotic stability of the body angular position is proved on the basis of the decomposition method, the Lyapunov direct method and the averaging technique. Computer simulation results illustrating the theorem are presented.


Main Subjects

[1] Beletsky, V.V., Motion of an Artificial Satellite about its Center of Mass, Israel Program for Scientific Translation, Jerusalem, 1966.
[2] Broch, J.T., Mechanical Vibration and Shock Measurements, Bruel & Kjaer, 1984.
[3] Tikhonov, A.A., Resonance phenomena in oscillations of a gravity-oriented rigid body. Part 4: Multifrequency resonances, Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya, 1, 2000, 131-137.
[4] Kosjakov, E.A., Tikhonov, A.A., Differential equations for librational motion of gravity-oriented rigid body, International Journal of Non-Linear Mechanics, 73, 2015, 51-57.
[5] Beletsky, V.V., Yanshin, A.M., The Influence of Aerodynamic Forces on Spacecraft Rotation, Naukova Dumka, Kiev, 1984 (in Russian).
[6] Krasil’nikov, P.S., Applied methods for the study of nonlinear oscillations, Izhevsk Institute of Computer Science, Izhevsk, 2015 (in Russian).
[7] Mashtakov, Y., Ovchinnikov, M., Tkachev, S., Study of the disturbances effect on small satellite route tracking accuracy, Acta Astronautica, 129, 2016, 22-31.
[8] Ivanov, D., Koptev, M., Mashtakov, Y., Ovchinnikov, M., Proshunin, N., Tkachev, S., Fedoseev, A., Shachkov, M., Determination of disturbances acting on small satellite mock-up on air bearing table, Acta Astronautica, 142, 2018, 265-276.
[9] Torres, P.J., Madhusudhanan, P., Esposito, L.W., Mathematical analysis of a model for moon-triggered clumping in Saturn’s rings, Physica D: Nonlinear Phenomena, 259(15), 2013, 55-62.
[10] Emel’yanov, N., Perturbed motion at small eccentricities, Solar System Research, 49, 2015, 346-359.
[11] Krasil’nikov, P., Fast non-resonance rotations of spacecraft in restricted three body problem with magnetic torques, International Journal of Non-Linear Mechanics, 73(4), 2015, 43-50.
[12] Ivanov, D., Ovchinnikov, M., Penkov, V., Roldugin, D., Doronin, D., Ovchinnikov, A., Advanced numerical study of the three-axis magnetic attitude control and determination with uncertainties, Acta Astronautica, 132, 2017, 103-110.
[13] Joshi, R.P., Qiu, H., Tripathi, R.K., Configuration studies for active electrostatic space radiation shielding, Acta Astronautica, 88, 2013, 138-145.
[14] Zubov, V.I., Lectures on Control Theory, Nauka, Moscow, 1975 (in Russian).
[15] Rivin, E.I., Passive vibration isolation, ASME Press, New York, 2003.
[16] Gendelman, O.V., Lamarque, C.H., Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium, Chaos, Solitons and Fractals, 24, 2005, 501-509.
[17] Gourdon, E., Lamarque, C.H., Pernot, S., Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment, Nonlinear Dynamics, 50(4), 2007, 793-808.
[18] Polyakov, A., Generalized Homogeneity in Systems and Control, Springer, Cham, 2020.
[19] Aleksandrov, A.Y., Aleksandrova, E.B., Zhabko, A.P., Stability analysis for a class of nonlinear nonstationary systems via averaging, Nonlinear Dynamics and Systems Theory, 13(4), 2013, 332-343.
[20] Aleksandrov, A.Y., On the asymptotical stability of solutions of nonstationary differential equation systems with homogeneous right hand sides, Dokl. Akad. Nauk Rossii, 349(3), 1996, 295-296 (in Russian).
[21] Aleksandrov, A.Y., Kosov, A.A., Chen, Y., Stability and stabilization of mechanical systems with switching, Automation and Remote Control, 72(6), 2011, 1143-1154.
[22] Peuteman, J., Aeyels, D., Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying, SIAM Journal on Control and Optimization, 37(4), 1999, 997-1010.
[23] Peuteman, J., Aeyels, D., Averaging techniques without requiring a fast time-varying differential equation, Automatica, 47, 2011, 192-200.
[24] Bogoliubov, N.N., Mitropolsky, Y.A., Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach, New York, 1961.
[25] Khalil, H.K., Nonlinear Systems, Prentice-Hall, Upper Saddle River NJ, 2002.
[26] Aleksandrov, A.Yu., Tikhonov, A.A., Rigid body stabilization under time-varying perturbations with zero mean values, Cybernetics and Physics, 7(1), 2018, 5-10.
[27] Siljak, D.D., Decentralized Control of Complex Systems, Academic Press, New York, 1991.
[28] Zubov, V.I., Methods of A.M. Lyapunov and Their Applications, P. Noordhoff Ltd., Groningen, 1964.
[29] Aleksandrov, A.Y., Aleksandrova, E.B., Tikhonov, A.A., Stabilization of a programmed rotation mode for a satellite with electrodynamic attitude control system, Advances in Space Research, 62(1), 2018, 142-151.
[30] Aleksandrov, A.Yu., To the question of stability with respect to nonlinear approximation, Siberian Mathematical Journal, 38(6), 1997, 1039-1046.
[31] Gavartin, E., Verlot, P., Kippenberg, T.J., Stabilization of a linear nanomechanical oscillator to its thermodynamic limit, Nature Communications, 4, 2013, 2860.
[32] Hanan, N.P., Anchang, J.Y., Satellites could soon map every tree on Earth, Nature, 587(7832), 2020, 42-43.
[33] Aleksandrov, A.Yu., Antipov, K.A., Platonov, A.V., Tikhonov, A.A., Electrodynamic Stabilization of Artificial Earth Satellites in the Konig Coordinate System, Journal of Computer and Systems Sciences International, 55(2), 2016, 296-309.
[34] Aleksandrov, A.Y., Tikhonov, A.A., Asymptotic stability of a satellite with electrodynamic attitude control in the orbital frame, Acta Astronautica, 139, 2017, 122-129.
[35] Tikhonov, A.A., Petrov, K.G., Multipole models of the Earth’s magnetic field, Cosmic Research, 40(3), 2002, 203-212.
[36] Ovchinnikov, M.Y., Penkov, V.I., Roldugin, D.S., Pichuzhkina, A.V., Geomagnetic field models for satellite angular motion studies, Acta Astronautica, 144, 2018, 171-180.
[37] 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 EarthТs magnetic field in the orbital coordinate system, Vestnik of St. Petersburg State University, Ser. 1, 1, 1999, 92-100.
[38] 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 St. Petersburg State University, Ser. 1, 3, 1999, 81-91.
[39] Abdel-Aziz, Y.A., Attitude stabilization of a rigid spacecraft in the geomagnetic field, Advances in Space Research, 40(1), 2007, 18–24.
[40] Mashtakov, Y., Ovchinnikov, M., Woske, F., Rievers, B., List, M., Attitude determination & control system design for gravity recovery missions like grace, Acta Astronautica, 173, 2020, 172-182.
[41] Melnikov, G.I., Dudarenko, N.A., Malykh, K.S., Ivanova, L.N., Melnikov, V.G. Mathematical models of nonlinear oscillations of mechanical systems with several degrees of freedom, Nonlinear Dynamics and Systems Theory, 17(4), 2017, 369-375.
[42] Dosaev, M., Klimina, L., Lokshin, B., Shalimova, E., Lin, C.-H., Double-Frequency Averaging in the Dynamical Model of a Double-Rotor Wind Turbine, Proceedings of 2020 15th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference), STAB 2020, Moscow, Russia, 9140515, 2020.
[43] Chowdhury, A., Clerc, M.G., Barbay, S., Robert-Philip, I., Braive, R. Weak signal enhancement by nonlinear resonance control in a forced nano-electromechanical resonator, Nature Communications, 11(1), 2020, 2400.