Astronomical Applications Department, U.S. Naval Observatory Precession Maple Page 2
Eulerian Angle Transformation
Rotation Matrix
Construct a rotation matrix that transforms coordinates from the fixed frame (X,Y,Z) to the
body frame (x,y,z). First, rotate the coordinates ccw around the Z axis.
:=
r1
( )
cos
( )
sin
0
-
( )
sin
( )
cos
0
0
0
1
Next, rotate ccw around the X' axis.
:=
r2
1
0
0
0
(
)
cos
(
)
sin
0
-
(
)
sin
(
)
cos
Next, rotate ccw around the Z'' axis.
:=
r3
( )
cos
( )
sin
0
-
( )
sin
( )
cos
0
0
0
1
Now combine the rotations into a single rotation matrix.
R :=
(
)
, ,
p q r
(
)
evalm (
)
(
)
subs
,
=
r
(
)
eval r3
`&*`
(
)
subs
,
=
q
(
)
eval r2
`&*`
(
)
subs
,
=
p
(
)
eval r1
Hence, we have the coordinate transformation
=
(
)
mat , ,
x y z
(
)
R
,
,
&*
(
)
mat
, ,
X Y Z
x
y
z
=
[
,
,
]
-
( )
cos
( )
cos
( )
sin
(
)
cos
( )
sin
+
( )
cos
( )
sin
( )
sin
(
)
cos
( )
cos
( )
sin
(
)
sin
-
-
( )
sin
( )
cos
( )
cos
(
)
cos
( )
sin
-
+
( )
sin
( )
sin
( )
cos
(
)
cos
( )
cos
[
,
,
( )
cos
(
)
sin
]
[
,
,
]
(
)
sin
( )
sin
-
(
)
sin
( )
cos
(
)
cos
&*
X
Y
Z
(
)
latex
,
% "d:/dynamics/precession/FixedToBody.tex"
The diagram below illustrates the three rotations.
Page 2
