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motion of the TRS will start to deviate from the described initial state. Hence, the
instantaneous rotation vector of the body-fixed reference systemwill exhibit changes
in its magnitude and direction, denoted as 'Perturbation' in Fig.
10
. In particular, the
rotation axis will no longer coincide with the polar axis of either the TRS, the GCRS
or the IRS, see Gross (
1992
). Provided the perturbing force is small, the induced
rotational variations are small, too, and the coordinate conversion between the IRS
and the TRS can be realized by employing an infinitesimal transformation matrix
x
=
Bx
u
(70)
⎛
⎞
1
p
3
p
1
⎝
⎠
.
B
=
−
p
3
1
p
2
(71)
−
p
1
−
p
2
1
T
, which apparently represents
the position vector of the CIP within the IRS, the corresponding vector in the TRS
becomes
x
Introducing in Eq. (
70
) the quantity
x
u
=
(
0
,
0
,
1
)
T
. Thus,
p
1
and
p
2
locate the CIP with respect to the body-
fixed rotating reference system and are equivalent to the routinely reported polar
motion parameters
x
p
and
y
p
=
(
p
1
,
p
2
,
1
)
p
1
=
x
p
,
p
2
=−
y
p
.
(72)
The negative sign in the second component considers that the y-coordinate is mea-
sured positively towards 90
◦
W longitude, whereas the y-axis of the TRS, to which
p
2
is referred to, points to 90
◦
E. The quantity
p
3
is associated with changes in the
rotation rate of the Earth, see Sect.
3.3
for a detailed derivation.
Combining Eqs. (
68
) and (
70
) supplies
x
=
BUx
i
=
Rx
i
.
Both
B
and
U
are known matrices, so that their product yields the desired repre-
sentation of the transformation matrix
R
.ViaEq.(
66
), after neglecting second-order
terms, the elements of
C
read
⎛
⎞
d
p
3
d
t
d
p
1
d
t
0
+
Ω
−
p
2
Ω
⎝
⎠
.
d
p
3
d
t
d
p
2
d
t
C
=
(73)
−
−
Ω
0
+
p
1
Ω
d
p
1
d
t
d
p
2
d
t
−
+
p
2
Ω
−
−
p
1
Ω
0
d
p
2
d
t
⇒
ω
1
=
Ω
p
1
+
(74)
d
p
1
d
t
.
ω
2
=
Ω
p
2
−
(75)
If Eqs. (
11
) and (
72
) are taken into account, the last two expressions can also be
written as
