Astronomical Applications Department, U.S. Naval Observatory Precession Memo Page 2
(3)
x
y
z
=
`(w, y, h)
X
Y
Z
where
(4)
`
z
(
h)
=
cos
h sin h 0
-
sin
h cos h 0
0
0
1
(5)
`
x
(
h)
=
1
0
0
0 cos
h sin h
0
-
sin
h cos h
is
(6)
`(w, y, h)
=
cos
h cos w
-
sin
h cos y sin w
cos
h sin w
+
sin
h cos y cos w sin h sin y
-
sin
h cos w
-
cos
h cos y sin w
-
sin
h sin w
+
cos
h cos y cos w cos h sin y
sin
y sin w
-
sin
y cos w
cos
y
We call the node angle, the inclination angle, and the azimuthal angle; or node, inclination, and
w
y
h
azimuth for short.
2.2 The Angular Velocity Vector Components in the Body Frame
The angular velocity vector may be decomposed into components along each of the rotation axes used
to construct the transformation matrix. If we transform those components to the body frame, then we
can express the angular velocity vector in the body frame in terms of the Euler angles
. The
(
w, y, h)
angular velocity vectors around the three rotation axes, as viewed in the body frame, are
(7)
W
w
=
d
w
dt `
(0,
y, h)
0
0
1
=
d
w
dt
sin
h sin y
cos
h sin y
cos
y
W
y
=
d
y
dt `
(0, 0,
h)
1
0
0
=
d
y
dt
cos
h
-
sin
h
0
W
h
=
d
h
dt `
(0, 0, 0)
0
0
1
=
d
h
dt
0
0
1
Combining the x, y, and z components, we have
(8)
W
w
+
W
y
+
W
h
=
W
x
W
y
W
z
=
d
w
dt
sin
h sin y
+
d
y
dt
cos
h
d
w
dt
cos
h sin y
-
d
y
dt
sin
h
d
w
dt
cos
y
+
d
h
dt
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