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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, Volume 21, Issue 2, Pages 194–201
DOI: https://doi.org/10.18500/1816-9791-2021-21-2-194-201
(Mi isu886)
 

This article is cited in 1 scientific paper (total in 1 paper)

Scientific Part
Mechanics

Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method

I. A. Pankratovab

a Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia
b Institute of Precision Mechanics and Control, Russian Academy of Sciences, 24 Rabochaya St., Saratov 410028, Russia
Full-text PDF (176 kB) Citations (1)
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Abstract: In the quaternion formulation, the problem of mathematical modeling of the spacecraft movement in an elliptical orbit was considered. Control is an acceleration vector from jet thrust. Control modulus is constant. The control is directed orthogonally to the plane of the spacecraft orbit. The quaternion differential equation of an orbital coordinate system orientation was used to describe spacecraft movement. An approximate analytical solution of the quaternion differential equation of the orbital coordinate system orientation in the form of an expansion system of linearly independent basis functions was constructed. The method of pointwise collocation was used to find unknown quaternion coefficients of this decomposition. The above decomposition was simplified taking into account the well-known solution of the orientation equation of the orbital coordinate system for the case when the spacecraft orbit is circular. With respect to the desired coefficients, a system of linear algebraic equations is obtained in which the components of the stiffness matrix and the column of free terms are quaternions. A program in Python was written to conduct numerical simulations of the spacecraft movement. A comparison of calculations for analytical formulas obtained in the paper and the numerical solution of the Cauchy problem by the Runge – Kutta method of the 4th order accuracy was done. Error tables have been obtained for determining the orientation of the orbital coordinate system for cases when basic functions are polynomials and trigonometric functions. Examples of numerical solution of the problem are given for the case when the initial orientation of the orbital coordinate system corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. Graphs describing changes in the components of error quaternion in determinig the orientation of the orbital coordinate system are constructed. The analysis of the received results is given. The features and regularities of the spacecraft movement on an elliptical orbit are established.
Key words: spacecraft, orbit, optimal control, quaternion.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00205
This work was supported by the Russian Foundation for Basic Research (project No. 19-01-00205).
Received: 24.08.2020
Revised: 07.10.2020
Bibliographic databases:
Document Type: Article
UDC: 629.78;519.6
Language: Russian
Citation: I. A. Pankratov, “Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method”, Izv. Saratov Univ. Math. Mech. Inform., 21:2 (2021), 194–201
Citation in format AMSBIB
\Bibitem{Pan21}
\by I.~A.~Pankratov
\paper Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2021
\vol 21
\issue 2
\pages 194--201
\mathnet{http://mi.mathnet.ru/isu886}
\crossref{https://doi.org/10.18500/1816-9791-2021-21-2-194-201}
\elib{https://elibrary.ru/item.asp?id=45797873}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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    Full-text PDF :42
    References:23
     
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