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

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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

Applying Calculations of Quaternionic Matrices for Formation of the Tables of Directional Cosines1 Victor Kravets1,a, Tamila Kravets1, Olexiy Burov2 1 – National Mining University, Dnipro, Ukraine 2 – Jack Baskin School of Engineering, University of California-Santa Cruz, CA, USA a – prof.w.kravets@gmail.com DOI 10.2412/mmse.78.59.591 provided by Seo4U.link

Keywords: monomial (1,0,-1)-matrices-(4x4), quaternionic matrices, parameters of Rodrigues-Hamilton, finite turn, matrices of directional cosines.

ABSTRACT. The mathematical apparatus of monomial (1,0,-1)-matrices-(4x4) 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.

Introduction. In the dynamics of the navigated moving systems [1-4], traditionally or due to the specific character of particular technical problems, it has become common to represent the solid body turn by three independent angles: plane angles and relative bearings, Euler angles, Euler-Krylov angles [5], etc. The finite turn matrices corresponding to these angles have a drawback: the turn formulas which they represent, are lacking symmetry. They are lengthy and difficult to observe [6]. The traditional methods to deduct these matrices cannot be considered simple and concise. What is more, getting the exact results is time consuming and requires much concentration [7]. Thus, it is considered appropriate to develop a common algorithm on the ground of the obtained results [8] for building the matrices of directional cosines corresponding to any independent turns. The set of systems of three independent turns in space Sequences of three independent turns in space are built on combinatory basis and make turns systems set. Turns systems set is considered as element groups: differing in turn order around three axes ( ey1 , ey 2 ,

ey 3 ) with repetition in two opposite directions. Clockwise turn towards unit vector ey1 (right propeller) is considered positive and marked as  ; in opposite direction:  .

Accordingly, for basis vectors ey 2 :   and ey 3 :  . Then systems set of three independent turns © 2017 The Authors. Published by Magnolithe GmbH. This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/

MMSE Journal. Open Access www.mmse.xyz


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