International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 3
September 2011
297
1 INTRODUCTION
A free gyro positioning system (FPS), which deter-
mines the position of a vehicle by using two free gy-
ros, was first suggested by Park & Jeong(2004). It is
originally an active positioning system like an iner-
tial navigation system (INS) in view of obtaining a
position without external source. However, a FPS is
to determine its own position by using the angle be-
tween the vertical axis of local geodetic frame and
the axis of free gyro (hereinafter called ‘nadir an-
gle’), while an INS is to do so by measuring its ac-
celeration.
The errors in the FPS were investigated broadly
by Jeong(2005). And the algorithmic designs of a
free gyroscopic compass and FPS were suggested by
measuring the earth’s rotation rate on the basis of a
free gyroscope (Jeong & Park, 2006;Jeong & Park,
2007).
This paper is to explain how to measure the nadir
angle by using the earth’s rotation rate. Firstly, the
determination of the position on or near the earth is
briefed. The motion rate of the spin axis caused by
the earth’s rotation rate is to be transformed into the
platform frame and then into the local geodetic
frame, i.e. the NED(north-east-down) navigation
frame. Finally the nadir angle is to be obtained by
using the rotation rate of the horizontal component
on the NED navigation frame. And also, a free gyro-
scopic compass is explained by measuring the
earth’s rotation rate on the basis of a free gyroscope.
2 DETERMINATION OF VEHICLE’S POSITION
BY NED NAVIGATION FRAME
First consider the transformation matrix
n
i
C
(Rogers
R M, 2000) from the inertial frame to the navigation
frame which is simply given by Eq. (1).
−+−
+
+−
+−
+−
+−
=
−
−−
−
−
−
−
=
=
φ
φ
ϖλφ
ϖλ
ϖλφ
ϖλφ
ϖλ
ϖλφ
ϖ
ϖ
ϖ
ϖ
φ
φ
λφ
λ
λφ
λφ
λ
λφ
sin
0
cos
)sin(cos
)cos(
)sin(sin
)cos(cos
)sin(
)cos(sin
1
0
0
0
cos
sin
0
sin
cos
sin
0
cos
sincos
cos
sinsin
coscos
sin
cossin
t
t
t
t
t
t
t
t
t
t
CCC
e
e
e
e
e
e
e
e
e
e
e
i
n
e
n
i
(1)
Here,
ω
e
is the (presumably uniform) rate of
Earth rotation,
λ
is the geodetic longitude,
φ
is the
geodetic latitude and t denotes time. This transfor-
mation matrix
n
i
C
denotes the transformation from
the unit vectors of axes in the inertial frame to those
in the navigation frame. Consid-
er an
arbitrary gyro vector
which is unit vector in the inertial frame. We obtain
easily the gyro vector transformed in
the navigation frame, as Eq. (2).
An Algorithmic Study on Positioning and
Directional System by Free Gyros
T.-G. Jeong & S.-C. Park
Korea Maritime University, Busan
ABSTRACT: The authors aim to establish the theory necessary for developing free gyro positioning system
and focus on measuring the nadir angle by using the motion rate of a free gyro. The azimuth of a gyro vector
from the North can be given by using the property of the free gyro. The motion rate of the spin axis in the gy-
ro frame is transformed into the platform frame and again into the NED (north-east-down) navigation frame.
The nadir angle of a gyro vector is obtained by using the North components of the motion rate of the spin axis
in the NED frame. The component has to be transformed into the horizontal component of the gyro by using
the azimuth of the gyro vector and then has to be integrated over the sampling interval. Meanwhile the au-
thors suggest north-finding principle by the angular velocity of the earth’s rotation. That is, ship's heading is
obtained by using the fore-and-aft and athwartship components of the motion rate of the spin axis in the NED
frame.
T
zyx
i
v
uuug ],,[=
T
uuu
n
v
DENg ],,[=
298
=
−+−+−
+++−
++−+−
=
−+−
+
+−
+−
+−
+−
=
=
u
u
u
zeyex
eyex
zeyex
z
y
x
e
e
e
e
e
e
D
E
N
ututu
tutu
ututu
u
u
u
t
t
t
t
t
t
i
v
g
n
i
C
n
v
g
φϖλφϖλφ
ϖλϖλ
φϖλφϖλφ
φ
φ
ϖλφ
ϖλ
ϖλφ
ϖλφ
ϖλ
ϖλφ
sin)sin(cos)cos(cos
)cos()sin(
cos)sin(sin)cos(sin
sin
0
cos
)sin(cos
)cos(
)sin(sin
)cos(cos
)sin(
)cos(sin
(2)
Fig.1 Measurement quantities in the navigation frame
As shown in Fig. 1, the azimuth angle of a gyro
vector, α, and the nadir angle, θ, can be obtained re-
spectively as Eq. (3) and Eq. (4), noting that
|
g
v
|
= 1.
φϖλφϖλφθ
sin)sin(cos)cos(coscos
zeyex
v
u
ututu
g
U
−+−+−==
(3)
φϖλφϖλφ
ϖλϖλ
α
cos)sin(sin)cos(sin
)cos()sin(
tan
zeyex
eyex
u
u
ututu
tutu
N
E
++−+−
+++−
==
(4)
If we use two free gyros whose gyro vectors in
Eq. (3) are
T
azayax
i
va
uuug ],,[=
and
T
bzbybx
i
vb
uuug ],,[=
respectively, we can determine the position (φ, λ) of
a vehicle at the given nadir angles θa, θb. Once de-
termining the position, we can also obtain the azi-
muth of a gyro vector by using Eq. (4). Park and
Jeong(2004) already suggested the algorithm of how
to determine a position.
3 SHIP’S HEADING, AZIMUTH AND NADIR
ANGLE OF GYRO VECTOR
3.1 Relation between ship’s heading and azimuth of
gyro vector
As Jeong & Park(2006) mentioned, let’s consider
the earth's rate
e
ϖ
. Then its north component
is
φϖ
cos
e
, where
φ
depicts the geodetic latitude of an
arbitary position. Fig. 2 shows that the angular ve-
locities of the fore-aft and the athwartship compo-
nents are given by Eq.(5) (Titterson, et al., 2004),
where
ψ
is ship's heading. And it also shows that
ς
is the azimuth of a gyro vector from ship’s head.
ψφϖω
ψφϖω
sincos
coscos
ey
ex
−=
=
(5)
Fig. 2 Relation between ship’s heading and azimuth of a gyro
vector
By taking the ratio of the two independent gyro-
scopic measurement, the heading,
ψ
, is computed by
Eq. (6).
x
y
ω
ω
ψ
arctan=
(6)
Meanwhile assuming that a gyro vector is
ς
away
from ship’s head, its azimuth from North is
represented by Eq. (7). Therefore the angular
velocity of the horizontal axis of a gyro (hereinafter
called ‘
H
ϖ
’) is given by Eq. (8) on the navigation
frame or local geodetic frame.
ςψα
+=
(7)
αφϖω
sincos
eH
−=
(8)
Eq.(8) shows that if the North component of the
earth’s rotation rate can be known on the navigation
frame, the nadir angle of a gyro vector,
θ
,is obtained
by Eq.(9), by integrating Eq. (8) incrementally over
a time interval.
∫
=
2
1
t
t
H
dt
ϖθ
(9)
3.2 Representation of the motion rate of the spin
axis in the frames
3.2.1 The motion rate of the spin axis
Let the motion rate of the spin axis in the gyro
frame,
T
gzgy
g
gi
]0[
/
ωωω
=
, where we denote:
g
gi /
ϖ
=
the motion rate of the gyro frame(g) relative to the
inertial frame(i), with coordinates in the gyro
frame(g), and hereafter the same notation of the an-
gular velocity is applied. In fact this angular velocity
is all you can get from a free gyro and has to be
transformed into the local geodetic frame through
the platform frame or the body frame (Jeong & Park,
2006).
299
First, the motion rate of the spin axis in the gyro
frame,
g
gi /
ϖ
, is transformed into that in the platform
frame,
p
gi /
ϖ
, as follows.
g
gi
p
g
p
gi
C
//
ϖω
=
(10)
Next, the motion rate of the spin axis in the plat-
form frame,
p
gi /
ϖ
, is also transformed into that in the
navigation frame,
n
gi /
ϖ
, as Eq. (11).
p
gi
n
p
n
gi
C
//
ϖω
=
(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,
n
gi /
ϖ
(hereinafter
called ‘
L
ϖ
’), is equal to the angular velocity on the
navigation frame,
n
ni /
ϖ
, which is composed of “earth”
and “transport” rates and rewritten in Eq.
(13)(Rogers,2003). Considering that the earth’s rota-
tion rate,
n
ei /
ϖ
, 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,
n
ne /
ϖ
, is shown in Eq. (15). In
Eq. (15)
λ
denotes the time rate of change of the
longitude while
φ
is the time rate of change of the
latitude. And
E
v
is the east velocity,
N
v
is the north
velocity,
R
is the radius of the earth and
h
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,
ψ
, 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.
Nx
Ny
ϖ
ϖ
ψ
arctan=
(16)
The azimuth of the gyro vector from the ship’s
head,
ς
, 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).
∫
=
2
1
t
t
Lz
dt
ϖς
(17)
The northward angular velocity of the local geo-
detic frame,
LH
ϖ
, which is determined by the sum of
the rates the earth’s rotation and the ship’s transport,
is represented by Eq. (18).
22
LyLxLH
ϖϖϖ
+=
(18)
And the horizontal component of the motion rate
of the free gyro,
H
ϖ
, can be obtained by Eq. (19). It
is evident that the azimuth of the gyro vector,
α
, is
given by the sum of the ship’s heading
ψ
and the
azimuth of the gyro vector from the ship’s head
ς
.
ςψα
αϖϖ
+=
−= sin
LHH
(19)
Therefore the nadir angle of the gyro vector,
θ
,
can be obtained by integrating Eq. (19).
∫
=
2
1
t
t
H
dt
ϖθ
(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(
g
x
), horizontal(
g
y
), and downward(
g
z
) direc-
tions. The platform frame refers to the vehicle to be
navigated, whose axes are defined along the for-
ward(
p
x
), right(
p
y
), and through-the floor(
p
z
) direc-
tions.
The angle
ξ
is a rotation angle about the down-
ward axis
p
z
and is positive in the counterclockwise
sense as viewed along the axis toward the origin ,
while the angle
η
is a rotation angle about the hori-
zontal axis(
p
y
) and is positive in the same manner as
above. Here the transformation matrix
p
g
C
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)