Certificate of Registration Media number Эл #ФС77-53688 of 17 April 2013. ISSN 2308-6033. DOI 10.18698/2308-6033
  • Русский
  • Английский

Spacecraft discrete orientations

Published: 20.06.2017

Authors: Berestova S.A., Kopytov N.P., Mityushov E.A.

Published in issue: #7(67)/2017

DOI: 10.18698/2308-6033-2017-7-1661

Category: Aviation and Rocket-Space Engineering | Chapter: Aircraft Dynamics, Ballistics, Motion Control

We consider the problem of modeling the variety of spacecraft discrete orientations which can be used in testing the systems of controlling the spacecraft positions in space. The criterion of equable filling the orientational space forms the basis of this model. We use the proprietary universal methodology for the random distribution of the points on the smooth regular surfaces in the three-dimensional Euclidean space and its generalization for the hypersurfaces defined by the parameter mode in multidimensional spaces. We have identified the function of the orientational parameters joint distribution density in the form of Euler angles with the uniform distribution of the points on the surface in the three-dimensional space. It is established that the uniformly distributed points on the surface of the three-dimensional unit hypersphere in the four-dimensional Euclidean space define the corresponding Rodriguez-Hamilton parameters set, that confirms the fact of two-sheeted covering the special orthogonal SO(3) matrixes group by the threedimensional hypersphere. We have carried out the transition from the continuous discrete distribution to the uniform one. The article introduces an algorithm for discrete filling the orientational space based on the application of regular centrosymmetrical polyhedrons in the four-dimensional space. The vertices of these polyhedrons form the sets of needed Rodriguez-Hamilton parameters or quaternions. We provide a constructive proof of the formulated algorithm correctness and its illustrating by means of the body position visualization in the three-dimensional space exemplified by creating 12 discrete orientations uniformly filling the orientational space on the basis of the 24-cell in the fourdimensional space. It is shown that in the general case when creating a spacecraft discrete orientations system we can use the information on the vertices coordinates of five regular four-dimensional polyhedrons (hypercube, 16-cell, 24-cell, 120-cell, 600-cell). The article describes the potential area ofpractical applications for the results obtained.

[1] Kopytov N.P., Mityushov E.A. Fundamentalnyye issledovaniya - Fundamental research, 2013, no. 4, part 3, pp. 618-622.
[2] Kopytov N.P., Mityushov E.A. Vestnik Permskogo gosudarstvennogo tekhnicheskogo universiteta. Mekhanika - PNRPU Mechanics Bulletin, 2010, no. 4, pp. 55-66.
[3] Kopytov N.P., Mityushov E.A. Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo - Vestnik of Lobachevsky University of Nizhni Novgorod, 2011, no. 4 (5), pp. 2263-2264.
[4] Kopytov N.P., Mityushov E.A. Universal algorithm of uniform distribution of points on arbitrary analytic surfaces in three-dimensional space. Intellectual Archive, 2012. Available at: (accessed May 12, 2017).
[5] Kopytov N.P., Mityushov E.A. The method for uniform distribution of points on surfaces in multi-dimensional Euclidean space. Intellectual Archive, 2012. Available at: (accessed May 12, 2017).
[6] Kopytov N.P., Mityushov E.A. Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternyye nauki - The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 1, pp. 29-35.
[7] Bauer R. Uniform Sampling of SO3. Proceedings of NASA Space Flight Mechanics Symposium. NASA Goddard Space Flight Center, Greenbelt, Maryland, June 19–21, 2001, pp. 347–359.
[8] Shuster M.D. The Journal of the Astronautical Sciences, 2003, vol. 51, no. 4, pp. 451–475.
[9] Gelfand I.M., Shapiro Z.Ya. Uspekhi matematicheskikh nauk - Russian Mathematical Surveys, 1952, vol. 7, no. 1 (47), pp. 3–117.
[10] Dubrovin B.A., Novikov S.P., Fomenko A.T. Sovremennaya geometriya. Metody i prilozheniya. V 3 tomakh. T. 1: Geometriya poverkhnostey, grupp preobrazovaniy i poley [Modern Geometry. Methods and Applications. In 3 vols. Vol. 1: Geometry of surfaces, transformation groups and fields]. Moscow, URSS Publ., Librokom Publ., 2013, 336 p.
[11] Miles R.E. Biometrika, 1965, vol. 52 (3–4), pp. 636–639.
[12] Volkov S.D., Klinskikh N.A. Doklady akademii nauk SSSR - Proceedings of the USSR Academy of Sciences, 1962, vol. 146, no. 3, pp. 565–568.
[13] Borisov A.V., Mamayev I.S. Dinamika tverdogo tela [Rigid-body dynamics]. Moscow-Izhevsk, NITs Reguliarnaya i khaoticheskaya dinamika Publ., 2001, 384 p.
[14] Golubev Yu.F. Preprint IPM imeni M.V. Keldysha - Keldysh Institute Preprints, 2013, no. 39, p. 23.
[15] Roberts P.H., Winch D.E. Advances in Applied Probability, 1984, vol. 16, pp. 638–655.
[16] Borovkov M.V., Savelova T.I. Normalnyye raspredeleniya na SO(3) [Normal distributions on SO(3)]. Moscow, MEPhI Publ., 2002, 94 p.