299
First, the motion rate of the spin axis in the gyro
frame,
, is transformed into that in the platform
frame,
, as follows.
(10)
Next, the motion rate of the spin axis in the plat-
form frame,
, is also transformed into that in the
navigation frame,
, as Eq. (11).
(11)
By the way assuming that there is no error in the
free gyro and rate sensors, the motion rate of the
spin axis in the local geodetic frame,
(hereinafter
called ‘
’), is equal to the angular velocity on the
navigation frame,
, which is composed of “earth”
and “transport” rates and rewritten in Eq.
(13)(Rogers,2003). Considering that the earth’s rota-
tion rate,
, is shown in Eq,(14), it is represented
by Eq. (13). Therefore the north component of the
earth’s rotation rate is computed by using the meas-
ured horizontal components, i.e. the fore-aft and
athwartship ones.
===
Lz
Ly
Lx
n
ni
n
giL
ϖ
ϖ
ϖ
ϖωϖ
//
(12)
=−==
+=
Nz
Ny
Nx
n
ne
n
gi
n
eiN
n
ne
n
ei
n
ni
ϖ
ϖ
ϖ
ϖϖϖϖ
ϖϖω
///
///
(13)
−
=
λϖ
λϖ
ω
sin
0
cos
/
e
e
n
ei
(14)
The transport rate,
, is shown in Eq. (15). In
Eq. (15)
denotes the time rate of change of the
longitude while
is the time rate of change of the
latitude. And
is the east velocity,
is the north
velocity,
is the radius of the earth and
is the
height above ground.
+
−
+
−
+
=
−
−=
φ
φλ
φ
φλ
ϖ
tan
sin
cos
/
hR
v
hR
v
hR
v
E
N
E
n
ne
(15)
Meanwhile the ship’s heading,
, can be com-
puted by using Eq. (13) and given by Eq. (16). Of
course Eq. (15) has to be transformed by multiplying
the transformation matrix,
H
n
C
, which changes from
the NE frame to the fore-aft and rightward one.
(16)
The azimuth of the gyro vector from the ship’s
head,
, can be obtained by integrating the vertical
component of the motion rate of the spin axis, i.e.
Eq. (12) and given by Eq. (17).
(17)
The northward angular velocity of the local geo-
detic frame,
, which is determined by the sum of
the rates the earth’s rotation and the ship’s transport,
is represented by Eq. (18).
(18)
And the horizontal component of the motion rate
of the free gyro,
, can be obtained by Eq. (19). It
is evident that the azimuth of the gyro vector,
, is
given by the sum of the ship’s heading
and the
azimuth of the gyro vector from the ship’s head
.
(19)
Therefore the nadir angle of the gyro vector,
,
can be obtained by integrating Eq. (19).
(20)
3.2.2 Coordinate transformation from gyro frame
to platform frame
In Fig. 3 the gyro frame refers to free gyro itself
on the platform, whose axes are defined along the
spin(
), horizontal(
), and downward(
) direc-
tions. The platform frame refers to the vehicle to be
navigated, whose axes are defined along the for-
ward(
), right(
), and through-the floor(
) direc-
tions.
The angle
is a rotation angle about the down-
ward axis
and is positive in the counterclockwise
sense as viewed along the axis toward the origin ,
while the angle
is a rotation angle about the hori-
zontal axis(
) and is positive in the same manner as
above. Here the transformation matrix
from the
gyro frame to platform frame is given by Eq.(21),
using Euler angles and direction cosines.
−
−
=
ηη
ηξξηξ
ηξξηξ
cos0sin
sinsincoscossin
sincossincoscos
p
g
C
(21)