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Attitude stabilization of a rigid body under disturbances with zero mean values

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Abstract

The problem of attitude stabilization of a rigid body exposed to a nonstationary perturbing torque is investigated. The control torque consists of a restoring component and a dissipative one. Linear and nonlinear variants of restoring and perturbing torques are analyzed. Conditions of the asymptotic stability of the programmed orientation of the body are found with the use of the Lyapunov direct method and the averaging technique. The results of computer modeling, illustrating the conclusions obtained analytically, are presented.

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Abbreviations

\(A_1, A_2, A_3\) :

Satellite principal central moments of inertia with respect to body frame axes \(x_1, x_2, x_3\), kg \(\cdot \) m\(^2\)

\(a_1, a_2\) :

Positive constants

\(\mathbf{B}\) :

Constant symmetric and negative definite matrix

\(b_1, b_2\) :

Positive constants

c :

Positive constant

\(c_1,\ldots , c_9\) :

Positive constants

\(\mathbf{D}_1(t)\) :

Continuous and bounded matrix for \(t\in [0,+\infty )\)

\(\mathbf{D}_2(t)\) :

Continuous and bounded matrix for \(t\in [0,+\infty )\)

\(\mathbf{J}\) :

Satellite inertia tensor in body frame \(x_1, x_2, x_3\), kg \(\cdot \) m\(^2\)

h :

Positive parameter

\( h_0\) :

Positive number

\(\vec L\) :

Control torque vector in body frame, N\(\cdot \)m

\(\vec L_d\) :

Dissipative component of control torque, N\(\cdot \)m

\(\vec L_p\) :

Perturbing torque vector in body frame, N\(\cdot \)m

\(\vec L_r\) :

Restoring component of control torque, N\(\cdot \)m

t :

Time, s

V :

Lyapunov function

\(V_1,\ldots , V_4\) :

Lyapunov functions

\(\alpha \) :

Positive constant

\(\delta \) :

Positive parameter

\(\bar{\delta }\) :

Positive parameter

\(\varepsilon \) :

Positive parameter such that \(\varepsilon \in (0,1)\)

\({\vec {\eta }}_1, {\vec {\eta }}_2, {\vec {\eta }}_3\) :

Unit vectors of body-fixed frame

\(\theta \) :

“Aircraft” angle

\(\lambda \) :

Auxiliary positive parameter

\(\lambda _0\) :

Positive number

\({\vec {\xi }}_1, {\vec {\xi }}_2, {\vec {\xi }}_3 \) :

Unit vectors of inertial frame

\(\varphi \) :

“Aircraft” angle

\(\psi \) :

“Aircraft” angle

\(\vec \omega \) :

Angular velocity of satellite in inertial reference frame, rad/s

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Acknowledgements

Sections 3 and 5 were supported by the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-573).

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Authors and Affiliations

  1. Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, 61 Bolshoj prosp. V.O., Saint Petersburg, Russia, 199178

    A. Yu. Aleksandrov

  2. Saint Petersburg State University, 7–9 Universitetskaya nab., Saint Petersburg, Russia, 199034

    A. Yu. Aleksandrov & A. A. Tikhonov

  3. Saint Petersburg Mining University, 2, 21st Line, Saint Petersburg, Russia, 199106

    A. A. Tikhonov

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Correspondence to A. A. Tikhonov.

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Aleksandrov, A.Y., Tikhonov, A.A. Attitude stabilization of a rigid body under disturbances with zero mean values. Acta Mech 233, 1231–1242 (2022). https://doi.org/10.1007/s00707-022-03163-0

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