On an approach to determining the orientation of the spacecraft orbit

Ilia Pankratov


The paper is dedicated to a mathematical simulation of the motion of a spacecraft in an elliptical orbit. The acceleration vector is limited in modulo and orthogonal to the plane of spacecraft orbit during its motion. The spacecraft motion is described using a linear quaternion differential equation of the spacecraft orbit orientation. This equation has no singular points. Its approximate solution was found as a linear combination of basis functions. The cases are considered when the basis functions are polynomials or trigonometric functions. Unknown quaternion coefficients of this decomposition are found using the method of mean weighted residuals. The weight functions are Dirac delta functions. The analytical calculations are compared with the numerical solution of the Cauchy problem by the Runge-Kutta method of the 4th order of accuracy. Tables of the error in determining the orientation of the spacecraft orbit are given for the case when the initial position of the spacecraft orbit corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. It is shown that the decomposition of the desired solution by polynomials gives a smaller error than by trigonometric functions. We can determine the scalar part of desired quaternion with the smaller error than its vector part.

Full Text:

PDF (Russian)


V. K. Abalakin, E. P. Aksenov, E. A. Grebennikov, V. G. Demin, Iu. A. Riabov, Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike (Reference guide on celestial mechanics and astrodynamics). Moscow: Nauka, 1976, 864 p (in Russian).

G. N. Duboshin, Nebesnaia mekhanika. Osnovnye zadachi i metody (Celestial mechanics. Main tasks and methods). Moscow: Nauka, 1968, 799 p. (in Russian).

Yu. N. Chelnokov, “Application of quaternions in the theory of orbital motion of an artificial satellite. I”, Cosmic Research, vol. 30, no. 6, pp. 612-621, 1992.

Yu. N. Chelnokov, “The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a newtonian gravitational field: I”, Cosmic Research, vol. 39, no. 5, pp. 470-484, 2001.

Yu. N. Chelnokov, Kvaternionnye i bikvaternionnye modeli i metody mekhaniki tverdogo tela i ikh prilozheniia (Quaternion and biquaternion models and methods of mechanics of solids and their applications). Moscow: Fizmatlit, 2006. 512 p (in Russian).

I. A. Pankratov, Ya. G. Sapunkov, Yu. N. Chelnokov, “About a problem of spacecraft's orbit optimal reorientation”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., vol. 12, no. 3, pp. 87-95, 2012 (in Russian).

V. N. Branets, I. P. Shmyglevskii, 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).

V. I. Zubov, Analiticheskaya dinamika giroskopicheskih sistem (Analytical dynamics of gyroscopic systems). Leningrad: Sudostroenie, 1970. 317 p (in Russian).

A. V. Molodenkov, “On the solution of the Darboux problem”, Mechanics of Solids, vol. 42, no. 2, pp. 167-176, 2007.

I. A. Pankratov, Yu. N. Chelnokov, “Analytical solution of differential equations of circular spacecraft's orbit orientation”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., vol. 11, no. 1, pp. 84-89, 2011 (in Russian).

L. S. Pontriagin, V. G. Boltianskii, R. V. Gamkrelidze, E. F. Mishchenko, Matematicheskaia teoriia optimal'nykh protsessov (The mathematical theory of optimal processes). Moscow: Nauka, 1983, 393 p. (in Russian).

Yu. N. Chelnokov, “Optimal reorientation of a spacecraft's orbit using a jet thrust orthogonal to the orbital plane”, J. Appl. Math. Mech., vol. 76, no. 6, pp. 646-657, 2012.

O. Zienkiewicz, K. Morgan, Finite elements and approximation, New York, Chichester, Brisbane, Toronto: John Wiley and Sons, 1983, 328 p.

J. J. Connor, C. A. Brebbia, Finite element techniques for fluid flow, London-Boston: Newnes-Butterworths, 1977, 310 p.

P. A. M. Dirac, The principles of quantum mechanics. Oxford: Clarendon Press, 1967, 324 p.

N. N. Moiseev, Chislennye metody v teorii optimal'nykh sistem (Numerical methods in the theory of optimal systems). Moscow: Nauka, 1971, 424 p (in Russian).

I. A. Pankratov, “Analytical solution of equations of near-circular spacecraft's orbit orientation”, Izv. Sarat. Univ. N.S. Ser. Math. Mech. Inform., vol. 15, no. 1, pp. 97-105, 2015 (in Russian).

Yu. N. Chelnokov, I. A. Pankratov, “Pereorientatsiia orbity kosmicheskogo apparata, optimal'naia v smysle minimuma integral'nogo kvadratichnogo funktsionala kachestva (The reorientation of spacecraft’s orbit, that is optimal in the sense of minimizing the integral quadratic performance functional)”, Mekhatronika, Avtomatizatsiya, Upravlenie, no 8. p. 74-78, 2010 (in Russian).


  • There are currently no refbacks.

Abava  Absolutech Convergent 2022

ISSN: 2307-8162