Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 326

Fig. 7.2 Euler angles

Euler angles are usually described as shown in Fig. 7.2 . Assign ʵ X , ʵ Y , and ʵ Z as
Euler angles, and the rotations of the coordinate system are as follows:
First, rotate OX 0

to OX and correspondingly OY 0 to OY around the OZ 0 axis by

ʵ Z . Then rotate OZ 0 to OZ around the OY axis, and corre-
spondingly OX to OX, and the rotation angle is

the rotation angle of

ʵ Y . Finally, rotate OZ to axis OZ
around the OX axis; OY is correspondingly rotated to OY, and the angle of rotation
is ʵ X .

Hence, we can write:

2 3 2 3

X 0
Y 0
Z 0

X
Y
Z

4 5 ﾼ 4 5 ,

R X ʵðÞ R Y ʵðÞ R Z ʵðÞ ð 7 : 4 Þ

where R X ( ʵ X ), R Y ( ʵ Y ), and R Z ( ʵ Z ) are the rotation matrices, and the expressions are

given by:

2
4

3
5

9
=

1 0 0

R X ʵðÞﾼ 0 cos ʵ X sin ʵ X

0 sin ʵ X cos ʵ X

2
4

3
5

cos ʵ Y 0 sin ʵ Y

R Y ʵðÞﾼ : ð 7 : 5 Þ

0 1 0

;

sin ʵ Y 0 cos ʵ Y

2
4

3
5

cos ʵ Z sin ʵ Z 0

R Z ʵðÞﾼ sin ʵ Z cos ʵ Z 0

0 0 1

Substituting ( 7.5 ) into ( 7.4 ) yields: