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 ],,[=
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)
300
Fig. 3 Gyro and platform frames
3.2.3 Coordinate transformation into the NED nav-
igation frame
With respect to the NED navigation frame whose
axes are defined as the first axis points the north, the
second axis points east and the third axis is aligned
with the ellipsoidal normal at a point, in the down-
ward direction. Let's consider the platform frame ax-
es point forward, to the right, and down as shown in
the above. Euler angles define the transformation,
that is, they are the roll(
R
), pitch(
P
), and yaw(
Y
)
relative to the NED axes as shown in Fig. 4. Then
the transformation matrix
n
p
C
is given by Eq. (22).
Fig. 4 Platform frame relative to NED frame
−
−=
PRPRP
RR
PRPRP
C
n
p
coscoscossinsin
sincos0
sincossinsincos
(22)
3.2.4 Determination of transformation matrices
First, for transformation matrix
p
g
C
we have to
know the rotation angles of the gyro frame,
ξ
and
η
.
They are obtained by integrating the respective
components of the spin motion rate. In doing so we
can get the transformation matrix by solving the fol-
lowing first order linear differential equation (23) as
Jeong & Park(2006) suggested.
g
p
g
gp
g
p
C
dt
dC
/
Ω−=
(23)
Here
g
gp /
Ω
is a skew-symmetric matrix and we
assume it is constant over the sampling interval. The
solution is given by Eq. (24).
2
3
2
2
2
1/
//
12
13
23
2
2
/0
00
,)(
)(
0
0
0
)cos(1)sin(
))(exp(),(
)(),(
0
0
0
aaaadia
tdt
aa
aa
aa
A
A
a
a
A
a
a
I
dttt
tCttC
t
t
g
gpi
g
gp
g
gp
t
t
t
t
g
gp
q
p
q
p
++=−=
∆Ω−=Ω−=
−
−
−
=
−
++=
Ω−=Ψ
Ψ=
∫
∫
∫
τϖ
(24)
Here
0
ttt −=∆
,
0
t
is the initial time and
i
a
is each
component of rotation angle. Once the transfor-
mation matrix
p
g
C
is obtained, the inverse matrix of
it,
g
p
C
, is immediately calculated by doing the trans-
pose of it since it is an orthogonal matrix. The rela-
tion between them is given by Eq. (25).
Tg
p
g
p
p
g
CCC )()(
1
==
−
(25)
Secondly, for transformation matrix
n
p
C
, we have
to know the rotation angles of the platform frame P
and R. They are obtained by integrating the
respective components of the motion rate of the spin
axis as shown in the above. We can also get the
transformation matrix by solving the following first
order linear differential equation (26).
p
n
p
pn
p
n
C
dt
dC
/
Ω−=
(26)
Here
p
pn /
Ω
is a skew-symmetric matrix too and
we assume it is constant over the sampling interval.
The solution is given by Eq. (27).
2
3
2
2
2
1/
//
12
13
23
2
2
/0
00
,)(
)(
0
0
0
)cos(1)sin(
))(exp(),(
)(),(
0
0
0
bbbbdib
tdt
bb
bb
bb
B
B
b
b
B
b
b
I
dttt
tCttC
t
t
p
pni
p
pn
p
pn
t
t
t
t
p
pn
p
n
p
n
++=−=
∆Ω−=Ω−=
−
−
−
=
−
++=
Ω−=Γ
Γ=
∫
∫
∫
τϖ
(27)
Here
0
ttt −=∆
,
0
t
is the initial time and
i
b
is each
component of rotation angle. Once the transfor-
mation matrix
p
n
C
is obtained, the inverse matrix of
it,
n
p
C
, is immediately calculated by doing the trans-
pose of it since it is an orthogonal matrix.
In addition, the other methods to solve the differ-
ential equations (23) and (26) are also represented
by the integration of four quaternions or three rota-
tion vectors, the integration of three Euler angle
equations, and etc. Such equations suggested in the
above are developed by referring to and using Far-
rell et al(1999), Jekeli(2001), and Rogers(2003).
301
4 ALGORITHMIC DESIGN OF FREE GYRO
POSITIONING & DIRECTIONAL SYSTEM
Fig. 5 and Fig. 6 show the algorithmic design of free
gyros positioning system mechanization. First, let’s
look into the ship’s heading (Fig.5). In this mechani-
zation two sensors for sensing the motion rate of the
spin axis are mounted in the free gyro. Three sensors
for sensing the motion rate of the platform are
mounted in orthogonal triad. From the sensors in the
gyro frame, the spin motion rate,
g
gi /
ϖ
, is obtained
and from the ones in the platform frame,
p
pi /
ϖ
, is de-
tected. By using the sum,
g
gp /
ϖ
, of the rates from the
free gyro and the ones detected from the platform
sensors, the transformation matrix
g
p
C
is calculated
and its inverse is determined. Therefore the spin mo-
tion rate,
g
gi /
ϖ
, sensed from the free gyro is trans-
formed into
p
gi /
ϖ
by using the inverse matrix,
p
g
C
.
Meanwhile the rate of the earth's rotation
n
ei /
ϖ
and
the rate of the vehicle movement
n
ne /
ϖ
are summed
and transformed into
n
ni /
ϖ
. It is subtracted from the
sensed rate from the platform,
p
pi /
ϖ
. As a result,
p
pn /
ϖ
is generated. By using this, the transformation ma-
trix,
p
n
C
, is calculated and the inverse of it,
n
p
C
, is ob-
tained. And the rate
p
gi /
ϖ
is transformed into
n
gi /
ϖ
by
using the transformation matrix,
n
p
C
. By using Eq.
(13), the spin motion rate in the NED frame,
N
ϖ
, is
obtained from the rate,
n
gi /
ϖ
. Finally, the ship's head-
ing is calculated by using the components of the spin
motion rate according to Eq. (13) and Eq. (16).
Next let’s look into the nadir angle (Fig.6). Be-
cause the motion rate of the spin axis in the local ge-
odetic frame,
n
gi /
ϖ
is represented by
L
ϖ
, The azimuth
of the gyro vector from the ship’s head,
ς
, can be
obtained by using Eq. (17). Then the azimuth of the
gyro vector from the North,
α
, can be easily taken
by Eq. (19).
The northward angular velocity of the local geo-
detic frame,
LH
ϖ
, is represented by Eq. (18). And
the horizontal component of the motion rate of the
free gyro,
H
ϖ
, can be obtained by Eq. (19). As a re-
sult the nadir angle of the gyro vector,
θ
, can be ob-
tained by Eq. (20).
Fig. 5 Free gyro positioning system mechanization (1)
Fig. 6 Free gyro positioning system mechanization (2)
5 RESULTS AND DISCUSSIONS
This paper investigated and developed the algorithm
regarding free gyro positioning system theoretically
and analytically. As a result conclusions are the fol-
lowing.
1 Once the spin motion rate of free gyro is known,
the ship's heading is determined by using Eq.
(16).
2 The azimuth of the gyro vector from the ship’s
head,
ς
, can be obtained by Eq. (17). And the
northward angular velocity of the local geodetic
frame,
LH
ϖ
, can be given by Eq. (18).
3 The horizontal component of the motion rate of
the free gyro,
H
ϖ
, can be obtained by Eq. (19).
Finally the nadir angle of the gyro vector,
θ
, can
be obtained by Eq. (20).
4 In order to transform the spin motion rate of the
gyro frame into the one of the NED navigation
frame, the differential equations of Eq. (23) and
Eq. (26) are solved by using Eq. (24) and Eq. (27)
and the transformation matrices are obtained re-
spectively.
This paper ascertained the feasibility to set a
stepping stone to the development of the free gyro
positioning system. However, several problems re-
main unsolved in the aspect of the following. Firstly
a two-degree-of-freedom gyro is very expensive and
302
is commercially disadvantageous in practice. Sec-
ondly the inherent errors caused by many elements
complicated. Errors caused by free gyro itself, sen-
sors of the platform, sensors of the free gyro, sam-
pling time and etc. will be dealt with in the next
study.
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