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/
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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
is defined by coil diagrams, given on Fig. 1.
Fig. 1. Coil diagrams of three independent turns with repetition. MMSE Journal. Open Access www.mmse.xyz
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
Here the following turns correspond to turn S1 system: - first positive turn about ey1 axis to angle, that is (ey1 , ); - second positive turn about ey 2 axis to angle, that is (ey 2 , ); - third positive turn about ey 3 axis to angle, that is (ey 3 , ) or in an abridged form
S1 ( , , ) , which constitutes Euler-Krylov angles system. We shall also note, that at body axes corresponding orientation, the turn system S33 ( , , ) constitutes aircraft axes – turn axes about aircraft principal axes. An aircraft in flight is free to rotate in three dimensions: pitch, nose up or down about an axis running from wing to wing; yaw, nose left or right about an axis running up and down; and roll, turn about an axis running from nose to tail, where yaw angle, pitch angle, roll (the bank angle). Turn system S67 ( , , ) constitutes Euler angles, where precession angle, nutation angle, intrinsic turn angle. Turn system S41 ( , , ) constitutes nautical angles, where hull angle (angle of trim), angle of heel (angle of roll), yaw angle of ship and so on for other angle systems, constituting turns set: S1 , S2 , S3 ,...,
S32 , S33 , S34 ,..., S64 , S65 , S66 ,..., S96 . Algorithm for building the matrices of directional cosines The offered algorithm implies the following single sequence of simple operations: 1. Addition of Rodrigues-Hamilton parameters’ set according to the accepted turn sequence of the bound reference system regarding the stable one: - first turn ai ; - second turn bi ; - third turn сi ( j 0,1, 2,3). 2. Forming four unified quaternionic matrices for each preset turn: - first turn A, t A, At , t At ; - second turn B, t B, Bt , t Bt ; - third turn C, tC, C t , tC t . 3. Defining four quaternionic matrices of the resulting turn, as a product of the respective formed quaternionic matrices: - R A B C; t R t A t B t C; - Rt C t At Bt ; t Rt tC t t Bt t At . - forming sought direct and inverse matrices for the directional cosines as a kernel of the product of, respectively, two unified matrices equivalent to the quaternion − for the direct matrix and two unified matrices equivalent to the conjugate quaternion − for the conjugate matrix: R t R or t R R MMSE Journal. Open Access www.mmse.xyz
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
Rt t Rt or t Rt Rt .
The offered algorithm, unlike other known ones, has no geometrical constructions and in general reduces the error probability during the process of calculation, owing to the symmetry of calculation formulas and quaternionic matrices’ properties [8]. Matrix of directional cosines for the angles of aircrafts’ flight direction In the dynamics of the flight, the orientation of the plane or rocket (body axes) in the inertial space (initial launch reference system) is defined by three angles , , , which respectively are called yaw attitude (angle of yaw), pitch attitude (angle of pitch) and angle of bank (or roll) (Fig. 2).
ey 2 ey 3
О e y1
Fig. 2. The sequence of turns for the set of angles of the aircrafts’ direction.
ey 2 , ; 2. ey 3 , ; 3. ey1 , .
The sequence of turn which accomplishes the transition from the initial launch reference system to the fixed one corresponds to the following Rodrigues-Hamilton parameters set:
a0 cos b0 cos
2
2
,
,
a1 0,
a2 sin
b1 0 ,
b2 0 ,
2
,
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a3 0 ;
(1)
b3 sin ; 2
(2)
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
c0 cos
2
,
c1 sin
2
c3 0 .
c2 0 ,
,
(3)
For each of three turns, there are respective quaternionic matrices:
cos A
2
0 sin 0
0 cos
2
sin
sin
cos
sin
2
cos
cos C
2 ,
0
2
cos
0
sin
2
sin
2
sin
0
B
0
2
0
cos
0
2
0
2
0
cos
2
0
2
2
0
0
0
sin
2
0
0
0
sin
2
2
2
2
0
0
cos
0
sin cos
sin
sin cos 0
2
2
2
0
, 0 cos
2
.
2
2
Quaternionic matrix of the resulting turn is found as: R A B C ,
t
R tA tB tC .
In a similar way, for the resulting turn we find the matrices equivalent to the conjugate quaternion: t
R t tC t tBt tAt ,
R t C t B t At .
Specifically, Rodrigues-Hamilton parameters of the resulting turn are defined by formula:
r tC t tB t a
or, in an extended form:
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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
cos r0 r1 r2
sin 0
2
2
sin cos
2
0
0
0
2
cos
r3 0
0
0
sin
0
sin
2
cos
sin
2
sin
2
cos
2
0
2
sin
0
sin
0
2
0
2
0
cos
2
cos
2
0
2
0
cos
2
0
sin
0
2
cos
, 2
0
2
which after transformations takes the form [1]:
cos r0 r1 r2 r3
sin cos
2
2
2
sin
cos cos cos
2
2
cos cos
2
cos
2
2
sin
2
2
sin
2
2
sin cos sin
cos
2
2
2
2
sin sin sin sin
2
2
2
2
sin
2
sin cos cos
2 .
2
2
The sought matrix of directional cosines is found by the following formula expansion:
Rt R A B C tA tB tC .
According to the commutative property of the examined matrices, this formula takes the form:
R tR A tA B tB C tC .
The inverse matrix of directional cosines is found in a similar way:
Rt tRt C t Bt At tC t tBt tAt
or
Rt t Rt C t tC t Bt t Bt At t At . Obtaining the product of matrices R t tR t and accomplishing simple trigonometric transformations, we obtain in the kernel of the resulting matrix the matrix of directional cosines:
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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
cos cos
sin
sin sin cos cos sin cos sin sin cos sin
cos cos cos sin
sin cos sin cos cos sin sin , cos cos sin sin sin
which corresponds to the known matrix form for the directional cosines of the angles of aircrafts’ flight direction [6] and, thus, the correctness of the offered algorithm is confirmed. Matrix of directional cosines for Euler-Krylov angles When Euler-Krylov angles are used, three successive turns of the moving system regarding a fixed one are accomplished via independent angles which are denoted respectively , , [6] (Fig. 3).
ey 3
e y1 О e y 2
Fig. 3. Sequence for turns of Euler-Krylov angles system.
ey1 , ; 2. ey 2 , ; 3. ey 3 , . To the provided turns’ sequence, the following set of Rodrigues-Hamilton corresponds:
a0 cos
2
, a1 sin
2
, a2 0, a3 0 ;
(4)
b0 cos , b1 0, b2 sin , b3 0 ; 2 2
(5)
c0 cos , c1 0, c2 0, c3 sin . 2 2
(6)
According to the set sequence of three turns, the quaternion equivalent matrices are formed. Then, the resulting turn’s parameters are found as:
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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
cos r0 r1 r2
0
2
0
cos
sin
sin
2
sin
0
r3
2
cos
0
2
sin
0
cos
2
0
2
0
cos
sin
2
cos
sin
sin
cos
2
sin
0
2
0
2
0
2
0
0
2
0
2
cos
sin
0
2
0
0
2
cos
2
2
.
0
0
2
Hence, we find the sought Rodrigues-Hamilton parameters for the resulting turn [1]:
cos
r0 r1 r2
cos
2
2
sin
r3 sin
cos
2
2
cos cos
cos
2
cos
2
2
2
sin
sin
cos
2
2
2
2
sin sin
2
sin
2
cos cos
2
2
sin
2
2
sin sin
sin cos
2
2
cos
2
sin
2
.
2
2
After simple trigonometric transformations, taking into consideration the found Rodrigues-Hamilton parameters, the kernel of the resulting matrix takes the following form: cos cos
sin sin cos cos sin
cos sin cos sin sin
cos sin sin
sin sin sin cos cos sin cos
cos sin sin sin cos cos cos
,
which corresponds to the known result [6]. Hence, for the first column and the third row of the finite turn matrix, a simple connection is found in form of:
cos cos r1 cos sin r2 sin r3
sin
sin cos
cos cos
r3
r0 r3 r2
r2
r3 r0 r1
r2 r1 r0
r1 r0
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r1 r0 , r3 r2
r1 r0 r3 r2
r2 r3 r0 r1
r3 r2 r1 r0
.
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
The found correspondence may be useful for the inverse problem solution — the search of EulerKrylov angles by the known Rodrigues-Hamilton parameters of the resulting turn [1]. Matrix of the directional cosines for Euler angles Let us assume that the movement (turn) of the moving axes from the initial position to the final position is accomplished with a help of set sequence of three turns with the predefined angles. The angles , , of these turns, Euler angles, are three independent values and are nominated, respectively, precession angle, nutation angle and intrinsic turn angle, i.e., the sequence of turns for system of Euler angles is the following: ey 3 , ; 2. ey1 ,; 3. ey 3 , . The provided turns’ sequence is characterized by the following set of Rodrigues-Hamilton parameters:
a0 cos
2
, a1 0, a2 0, a3 sin
2
;
(7)
b0 cos , b1 sin , b2 0, 2 2
b3 0; (8)
c0 cos , c1 0, c2 0, c3 sin . 2 2
(9)
The resulting turn’s parameters are found with the formula:
cos r0 r1 r2
2
0
cos
sin
2
sin
0
2
sin
2
sin
0
r3
0
2
0
2
0
cos
sin
2
sin
0
2
cos
0
cos
cos
2
0
0
2
0 cos
0
2
0
0
2
cos
0
2
sin
0
2
which is transformed to the form of:
cos
r0 r1 r2
2
cos
sin
2
2
sin
cos
cos
2
sin
2
sin
r3
cos
cos
2
2
2
cos
cos
2
2
2
2
sin sin
2
sin
2
cos
cos
cos
2
2
2
2
sin
cos
2
2
or
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sin sin
2
2
sin sin
2
2
sin cos
2
2
2
0
,
0 sin
2
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
cos
2
r0 r1
r2
sin sin
2
2
r3
cos
2
cos cos sin sin
2
2
.
2
2
It is easy to check that for Euler angles the matrix of directional cosines R t t R t acquires the known form: cos cos sin sin cos
cos sin sin cos cos
sin sin
sin cos cos sin cos sin sin
sin sin cos cos cos sin cos
cos sin cos
and constitutes the kernel of the resulting matrix Rt t Rt . In particular, the following equations are correct
sin sin r1 cos sin r2 cos r3
sin sin
sin cos
r0 r3 r2
cos r3
r2
r3 r2 , r1 r0
r2 r1 r0
r3 r0 r1
r1
r0
r1 r0 r3 r2
r2 r3 r0 r1
r3 r2 r1 r0
.
This equation is used for inverse problem solving: finding Euler angles , , the known Rodrigues-Hamilton parameters of the resulting turn. Matrix for directional cosines of relative bearing The matrix for directional cosines of relative bearings, used by Aleksey Krylov in ship oscillation theory, is found on the ground of the system of angles , , , which define respectively pitch, list and yaw of the ship. For the turn sequence offered by Aleksey Krylov, we obtain:
ey 2 , ; 2. ey1 ,; 3. ey 3 , . Then, Rodrigus-Hamilton parameters have respectively the form: MMSE Journal. Open Access www.mmse.xyz
Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
a0 cos
2
b0 cos
2
, a1 0, a2 sin , b1 sin
2
2
, a3 0 ;
(10)
, b2 o, b3 0 ;
(11)
, c1 0, c2 0, c3 sin . 2 2
c0 cos
(12)
The resulting turn which can be found via relative bearing is characterized by Rodrigues-Hamilton parameters which can be found with the following formula:
cos r0 r1 r2
0
2
0
sin
sin
2
sin
0
r3
cos
2
cos
0
2
sin
0
0
2
0
2
cos
sin
2
sin
0
2
cos
cos
2
0
2
2
0
0
2
0
0
0 cos
0 cos
0
2
sin
2
sin cos
2
2
0 sin
,
2
0
2
or, in an expanded form
cos r0 r1 r2
cos
2
2
sin
r3 sin
2
2
cos sin
2
2
sin cos
2
2
cos cos
2
cos cos
2
2
2
sin sin
2
2
cos cos
2
2
sin cos
2
2
cos sin
2
2
sin
2
sin
2
sin sin
.
2
2
The found matrix for directional cosines, which constitutes the matrix kernel Rt t Rt , which corresponds to the defined Rodrigues-Hamilton parameters of the resulting turn, can be reduced to the known form: cos cos sin sin sin
sin cos
cos sin sin cos sin
cos sin sin cos sin cos cos cos sin sin
sin sin cos cos sin cos cos
It is to mention that the inverse problem − finding the relative bearings by the known RodriguesHamilton parameters of resulting turn, − is easily solved with the following equations:
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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954
cos sin
sin
cos cos
sin cos r1 cos cos r2 sin r3
r3
r2
r1
r0 r3 r2
r3 r0 r1
r2 r1 r0
r0
r1 r0 r3 r2
r2 r3 r0 r1
r3 r2 ; r1 r0
r2 r3 . r0 r1
The provided algorithm is applied to other systems of independent turns in a similar way. Summary. The methods were developed to represent the theory of solid body finite turn by quaternionic matrices. With a help of Rodrigues-Hamilton parameters, the formulas are obtained for forward and backward transformations of the moving reference system regarding a fixed one. The concise formulas are provided for adding the sequence of finite turns of the solid body in threedimensional space. The algorithm is offered for composing the matrices of directional cosines. This algorithm was tested on the examples of plane angles, Euler-Krylov angles, relative bearings and Euler angles. In contrast to other known methods, the offered algorithm is based on the mathematical apparatus of quaternionic matrices, and the lengthy transformations and geometrical construction are not necessary. Due to the properties of quaternionic matrices, the algorithm contains a calculation formula distinguished by an ordered record, which reduces the error rate during calculation and provide the ability to build effective computational algorithms. 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: 9667985-81-4. [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-(4х4). 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.
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