Дискретные ориентации космического аппарата

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Дискретные ориентации космического аппарата
Инженерный журнал: наука и инновации # 7·2017 9
Spacecraft discrete orientations
© S.A. Berestova, N.P. Kopytov, E.A. Mityushov
Ural Federal University, Ekaterinburg, 620002, Russia
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 generaliza-
tion 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 (3) SO matrixes group by the three-
dimensional 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 polyhe-
drons 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 orienta-
tions uniformly filling the orientational space on the basis of the 24-cell in the four-
dimensional space. It is shown that in the general case when creating a spacecraft dis-
crete 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 of practical applications for the results obtained.
Keywords: Rodriguez—Hamilton parameters, quaternions, orientational space, three-
dimensional hypersphere, regular polyhedrons, equable filling
REFERENCES
 Kopytov N.P., Mityushov E.A. Fundamentalnyye issledovaniya — Fundamen-
tal research, 2013, no. 4, part 3, pp. 618–622.
 Kopytov N.P., Mityushov E.A. Vestnik Permskogo gosudarstvennogo
tekhnicheskogo universiteta. Mekhanika — PNRPU Mechanics Bulletin , 2010,
no. 4, pp. 55–66.
 Kopytov N.P., Mityushov E.A. Vestnik Nizhegorodskogo universiteta im. N.I. Lo-
bachevskogo — Vestnik of Lobachevsky University of Nizhni Novgorod , 2011,
no. 4 (5), pp. 2263–2264.
 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).
 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: (ac-
cessed May 12, 2017).