The inverse transformation matrix in this case may be obtained by use of Eq. (E-4), or by replacing qt,q2, and q3by -qx, — q* and - q^ respectively, in Eq. (E-8) and leaving q4 unaltered.

The Euler symmetric parameters may be regarded as components of a quaternion, q, defined by

where i, j, and k are as defined in Appendix D. The multiplication rule for successive rotations represented by Euler symmetric parameters is given in Appendix D. The Euler symmetric parameters in terms of the 3-1-3 Euler angle rotation <>, 0, if» (defined below) are as follows:

Gibbs Vector. The Gibbs vector (components g„ g^ and g3) representation (see Section 12.1) for finite rotations is defined by gi = <7i/94 = eitan("I>/2)

The transformation matrix A in terms of the Gibbs vector representation is as follows:

The inverse of A can be obtained in this case by the method of Eq. (E-4), or by replacing g, by -g{ in Eq. (E-12).

Euler Angle Rotation. The Euler angle rotation (<J>,0, <//) is defined by successive rotations by angles <j>, 9, and <p, respectively, about coordinate axes /, j, k (Section 12.1). The i-j-k Euler angle rotation means that the first rotation by angle <J> is about the i axis, the second rotation by angle 0 is about the j axis, and the third rotation by angle is about the k axis. There are 12 distinct representations for the Euler angle rotation which divide equally into two types:

TYPE 1. In this case, the rotations take place successively about each of the three coordinate axes. This type has a singularity at 9= ±90 deg, because for these values of 9, the <J> and \j/ rotations have a similar effect.

TYPE 2. In this case, the first and third rotations take place about the same axis and the second rotation takes place about one of the other two axes. This type has a singularity at 9=0 deg and 180 deg, because for these values of 9, the <f> and t¡i rotations have a similar effect.

Table E-l gives the transformation matrix, A, for all of the 12 Euler angle representations. The 3-1-3 Euler angle representation is the one most commonly used in the literature. The Euler angles <¡>, 9, and if/ can be easily obtained from the elements of matrix A. A typical example from each type is given below.

TYPE 1: 3-1-2 Euler Angle Rotation

<>=arctan(-i42,/i422) 0<«¡><360° (E-13a)

if>=arc tan( - A ,3/.«433) 0<^<360° (E-13c)

The correct quadrants for <j> and if/ are obtained from the relative signs of the elements of A in Eqs. (E-l3a) and (E-l3c), respectively.

TYPE 2: 3-1-3 Euler Angle Rotation

<¡>=arcUn(v43l/-^32) 0<<¡><360° (E-14a)

Table E-l. The Attitude Matrix, A, for the 12 Possible Euler Angle Representations (S=sine, C5cosine, 1=Jt axis, 2sy axis, 3s z axis)

type -1 euler angle representation |
matrix a |
type-2 euler angle REPRESENTATION |
matrix a | ||||||||||

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