#### For citation:

Pankratov I. A. Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 2, pp. 194-201. DOI: 10.18500/1816-9791-2021-21-2-194-201, EDN: LUXVFF

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

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.

- Chelnokov Yu. N. Application of quaternions in the theory of orbital motion of an artificial satellite. I.
*Cosmic Research,*1992, vol. 30, no. 6, pp. 612–621. - Pankratov I. A., Sapunkov Ya. G., Chelnokov Yu. N. Solution of a problem of spacecraft’s orbit optimal reorientation using quaternion equations of orbital system of coordinates orientation.
*Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics,*2013, vol. 13, iss. 1, pt. 1, pp. 84–92 (in Russian). https://doi.org/10.18500/1816- 9791-2013-13-1-1-84-92 - Branets V. N., Shmyglevskii I. P.
*Primenenie kvaternionov v zadachakh orientatsii tverdogo tela*[Use of Quaternions in the Problems of Orientation of Solid Bodies]. Moscow, Nauka, 1973. 320 p. (in Russian). - Zubov V. I.
*Analiticheskaya dinamika giroskopicheskih sistem*[Analytical Dynamics of Gyroscopic Systems]. Leningrad, Sudostroenie, 1970. 317 p. (in Russian). - Molodenkov A. V. On the solution of the Darboux problem.
*Mechanics of Solids,*2007, vol. 42, no. 2, pp. 167–176. https://doi.org/10.3103/S002565440702001X - Pontriagin L. S., Boltianskii V. G., Gamkrelidze R. V., Mishchenko E. F.
*Matematicheskaia teoriia optimal’nykh protsessov*[The Mathematical Theory of Optimal Processes]. Moscow, Nauka, 1983. 393 p. (in Russian). - Chelnokov Yu. N. Optimal reorientation of a spacecraft’s orbit using a jet thrust orthogonal to the orbital plane.
*Journal of Applied Mathematics and Mechanics,*2012, vol. 76, iss. 6, pp. 646–657. https://doi.org/10.1016/j.jappmathmech.2013.02.002 - Chelnokov Yu. N. On determining vehicle orientation in the Rodrigues – Hamilton parameters from its angular velocity.
*Mechanics of Solids,*1977, vol. 37, no. 3, pp. 8–16. - Zienkiewicz O., Morgan K.
*Finite Elements and Approximation.*New York, Chichester, Brisbane, Toronto, John Wiley and Sons, 1983. 328 p. (Russ. ed.: Moscow, Mir, 1986. 318 p.). - Connor J. J., Brebbia C. A.
*Finite Element Techniques for Fluid Flow.*London, Boston, Newnes-Butterworths, 1977. 310 p. (Russ. ed.: Leningrad, Sudostroenie, 1979. 264 p.). - Dirac P. A. M.
*The Principles of Quantum Mechanics.*Oxford, Clarendon Press, 1967. 324 p. (Russ. ed.: Moscow, Nauka, 1979. 408 p.). - Moiseev N. N.
*Chislennye metody v teorii optimal’nykh sistem*[Numerical Methods in the Theory of Optimal Systems]. Moscow, Nauka, 1971, 424 p. (in Russian). - Pankratov I. A. Analytical solution of equations of near-circular spacecraft’s orbit orientation.
*Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics,*2015, vol. 15, iss. 1, pp. 97–105 (in Russian). https://doi.org/10.18500/1816- 9791-2015-15-1-97-105

- 1515 reads