# Applying Calculations of Quaternionic Matrices for Formation of the Tables of Directional Cosines

<- Back to II. Mechanical Engineering & Physics Vol. 11

Cite the paper

 Kravets, Victor; Kravets, Tamila; Burov, OlexiyMechanics, Materials Science & Engineering, 11 , 2017, ISSN: 2412-5954.

Authors: Victor Kravets, Tamila Kravets, Olexiy Burov

ABSTRACT. The mathematical apparatus of monomial (1,0,-1)-matrices-(4×4) is applied to the description of the turn in space in the moving (bound) and fixed (inertial) frames of reference. A general algorithm for the formation of transformation matrices of the sequence of three independent turns with repetition and in opposite directions is proposed. A finite set of systems of three independent turns is constructed, consisting of 96 variants and including known systems of angles. The algorithm is approved for the formation of tables of directing cosines of the Euler-Krylov angles systems, aircraft angles, Euler angles, nautical angles. The proposed algorithm for generating directional cosine tables meets both the aesthetic criteria, expressed in orderliness, laconism, convenience of analytical transformations, and the utilitarian needs of computer technologies, providing a mathematically elegant, compact, universal matrix algorithm that, on whole, increases the productivity of intellectual labor.

Keywords: monomial (1,0,-1)-matrices-(4×4), quaternionic matrices, parameters of Rodrigues-Hamilton, finite turn, matrices of directional cosines

References

[1] Ishlinskij, A.Yu. Orientatsiya, giroskopy i inertsial’naya navigatsiya [Orientation, gyroscopes and inertial navigation], Nauka Publ., Moscow, 1976, 672 p. (in Russian).

[2] Raushenbax, B.V., Tokar’, E.N. Upravlenie orientatsiej kosmicheskix apparatov [The orientation of the spacecraft management], Nauka Publ., Moscow, 1974, 600 p. (in Russian).

[3] Lysenko, L.N. Navedenie i navigatsiya ballisticheskix raket [Guidance and navigation of ballistic missiles], Bauman university Publ., Moscow, 2007, 672 p. (in Russian).

[4] Igdalov, I.M., Kuchma, L.D., Polyakov, N.V., Sheptun, Yu.D. Raketa kak ob”ekt upravleniya [The missile as an object of control], Art-Press Publ., Dnipro, 2004, 544 p. (in Russian). ISBN: 966-7985-81-4.

[5] Korn, G., Korn, T. Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov [Mathematical Handbook for Scientists and Engineers], Nauka Publ., Moscow, 1984, 832 p. (in Russian).

[6] Lur’e, A.I., Analytical mechanics, Springer Science & Business Media Publ., 2002, 824 p.

[7] Pavlovskij, M.A. Teoretichna mexanika [Theoretical Mechanics], Technika Publ., Kyiv, 2002, 512 p. [in Ukrainian]

[8] Kravets, V., Kravets, T., Burov, O. Monomial (1, 0, -1)-matrices-(4Ń…4). Part 1. Application to the transfer in space. Lap Lambert Academic Publishing, Omni Scriptum GmbH&Co. KG., 2016, 137 p. ISBN: 978-3-330-01784-9.