International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number 2
June 2007
187
Analytical Model of Position Uncertainty of
Ship’s Plan Geometry in Integrated Navigation
System
A. Tomczak
Maritime University of Szczecin, Poland
ABSTRACT: For safety reason it would be essential to apply uncertainty of ship’s contour position in safety
evaluation of maneuvers carried out on basis of INS (Integrated Navigation Systems) indications, instead of
real dimensions of the ship’s contour. The paper presents analytical method of ship’s plan geometry (contour)
uncertainty area determination. The model was used to determine uncertainty area of ship maneuvering in
Świnoujście harbor for typical configuration of navigational equipment applied in existing pilot systems. The
results of experiment were discussed. The model equations were derived from measurement error propagation
theory. Potential application of uncertainty area as safety zone around ship contour was appointed.
1 INTRODUCTION
Ship’s contour, as a geometric object, presented in
Integrated Navigation Systems -INS (ECDIS, pilot
systems) and determined on basis of measured
navigational parameters is affected by some
uncertainty. Depending on type of integrated
navigation system, the number of factors influencing
ship’s location changes. The basic parameters the
ship’s contour position is determined by are her
geographical position and true course. The fact that
those parameters have a random character the ship’s
position cannot be identified in the deterministic
process and can be expressed by ship’s position
uncertainty area, which is the area horizontally
occupied by ship and its dimensions and can be
determined by probabilistic method at assigned
confidence level (Tomczak 2006). Distance of
uncertainty area outline to navigational obstruction
can be considered as criterion of maneuvering safety
assessment carried out on the basis of INS
indications. The main goal the analytical model
should attain is possibility of quick ship’s position
uncertainty determination for input standard
uncertainties of subsystems indications used in
integrated system (GNSS position system, heading
source). Additionally other input uncertainties are
inserted into the model. It results from equipment
configuration of the system, version of INS (portable
or stationary) and also the place of GNSS antenna
location on the ship’s deck.
2 MATHEMATICAL MODEL OF SHIP’S
LOCATION UNCERTAINTY AREA
The uncertainty area of ship’s position is defined by
points’ coordinates. It is the sum of consecutive
points coordinates of real model of the ship’s
contour and corresponding uncertainties (eq. 1):
riri
i
n
xcxx
σ
±=
(1)
riri
i
n
ycyy
σ
±=
where:
−
nini
yx ,
the consecutive coordinates of
points of ship’s position uncertainty area for
188
WGS-84 UTM XY projection,
−
riri
yx ,
calculated
coordinates of consecutive points of ship’s contour,
−
riri
yx
σσ
,
uncertainties of ship’s contour points
coordinates measured along x and y of Cartesian
axes.
Specifying location of the ship presented in INS
can be treated as combined measurement consisting
of parameters measured directly (coordinates,
heading) and parameters connected with practical
solutions the system works in (contour shape
approximation, the assessment of GNSS antenna
location in the ship coordinate system). In order to
examine the accuracy of the position of the ship’s
outline a mathematical model for the determination
of the area of uncertainty of ship’s position at any
level of probability/confidence has been designed
where model of measurement procedure and
uncertainty propagation rule have been involved.
One of the most important elements of combined
measurement uncertainty assessment procedure is to
define the formula for measurement result. Visual
model of measurement is presented on figure 1
where
→
riA
PP
is a vector between GNSS antenna and
consecutive point of ship’s contour.
Fig. 1. The visual measurement model of consecutive points of
ship’s contour
Ship’s location in NIS can be determined based
on following quotations:
( )
iriiAriri
dxx
αψ
++= sin
( )
iriiAriri
dyy
αψ
++= cos
where:
−
i
r
i
r
yx ,
calculated coordinates of consecutive
points of ship’s contour,
−
AriAri
yx ,
recorded
positions of GNSS antenna – assuming north up
orientation,
−
ri
ψ
heading,
−
i
d
distance between
GNSS antenna and point of ship’s contour,
−
i
α
angle between GNSS antenna and point of ship’s
contour.
Consecutive points coordinates of ship’s contour
outline (x
ri
,y
ri
) are measured values, which are two-
dimensional random variables described by two-
dimensional function vector of many partial random
variables. The estimators of measured values (x
ri
,y
ri
),
are calculated from equitation 3 for input estimators
x
Ari
, y
Ari
, d
i
, ψ
i
, λ
i
for N input values (Sanecki 2004).
∂∂
∂∂
∂∂
∂∂
⋅
⋅
∂∂∂∂
∂∂∂∂
=
i
ri
i
ri
i
ri
i
ri
Ari
ri
Ari
ri
Ari
ri
Ari
ri
i
i
AriAriAri
AriAriAri
i
ri
i
ri
Ari
ri
Ari
ri
i
ri
i
ri
Ari
ri
Ari
ri
yx
d
y
d
x
y
y
y
x
x
y
x
x
d
yyx
yxx
y
d
y
x
y
y
y
x
d
x
y
x
x
x
M
ψψ
δψ
σ
σδσ
σσ
ψ
ψ
000
000
00
00
2
2
2
cov
(3)
Based on general formula of uncertainty
propagation theory (eq. 3) the standard uncertainties
of input values were determined.
Covariance matrix of two-dimensional probability
density function M
cov
presents equitation 4:
=
2
2
cov
ririri
ririri
yyx
yxx
M
σσ
σσ
(4)
where:
riri
yx
σ
-covariance of random variables
(x
ri
,y
ri
).
Multiplying matrixes of equitation 3, combined
standard uncertainties of relevant consecutive points
coordinates, forming ship’s contour shape and its
covariance were obtained( eq. 5):
( ) ( )
iiiiiiiAriri
dxx
αψσασψαψσσσ
+++++=
2
22
222
cossin
( ) ( )
iiiiiiiAiri
dyy
αψσασψαψσσσ
+++++=
2
2
2222
sincos
(5)
iiiiiiAriAririri
ddyxyx
ψσψψψσσσ
2222
coscoscos −+=
Determined uncertainties are directional errors of
points coordinates. Graphical presentation of these
uncertainties (Fig. 6) enables general errors evaluation
and quick indication of sectors with significant
errors magnitude. The standard uncertainties of
distance (σd
i
) and direction (σλ
i
) input values,
describing vectors
→
riA
PP
= [d
i
,λ
i
] are sum of
uncertainties coming from inaccuracy of the
assessment of GNSS antenna location in the ship’s
coordinate frame (σd
hfi
, σα
hfi
) and also uncertainties
propagated from ship’s contour model approximation
process respectively (σd
apri
, σα
apri
) (eqt.: 6, 7):
iaprhfii
ddd
.
σσσ
+=
(6)
iaprhfii .
σασασα
+=
(7)
(2)
189
The direction (α
i
)
of
→
riA
PP
vector (fig.1) is
calculated according to formula (8):
i
i
i
x
y
arctan=
α
(8)
where:
−
ii
yx ,
coordinates of consecutive points of
real model of ship’s contour taken from ship’s plan.
According to general rule of errors propagation
after partial derivatives of indirectly measured values
had been calculated, as a result obtained combined
standard uncertainty α
I
of
→
riA
PP
vector:
2
2
22
2
2
2
1
hf
ii
i
hf
i
ii
hfi
y
xy
x
x
y
xy
σσσα
+
−
+
+
=
(9)
The distance d
i
of
→
AriA
PP
vector is expressed by
square root of relevant coordinates (x
i
,y
i
) sum, raised
to a power of two:
22
iii
yxd +=
(10)
As a result of a differential calculus of equitation
10 with respect to (x
i
, y
i
) combined distance standard
uncertainty (d
i
) of
→
AriA
PP
vector was obtained:
22
2
2
2
2
ii
ii
hf
yx
yyxx
d
+
+
=
σσ
σ
(11)
After combined standard uncertainties of each
point the ship’s contour is built from had been
provided to equitation 12, the formulas to calculate
point’s coordinates of ship’s position uncertainty
area in INS at a given confidence level was obtained:
( ) ( )
iiiiiiiAriri
i
n
dxcxx
αψσασψαψσσ
++++++=
2
22
222
cossin
(12)
( ) ( )
iiiiiiiAriri
i
n
dycyy
αψσασψαψσσ
++++++=
2
2
2222
sincos
(13)
The error ellipse is the most precise measure of
ship’s position and can be used to asses the accuracy
of points the ship’s contour is built from. It comes
from her specific characteristics which are as follows
(Gucma 2006): it is the only figure with constant
probability density on her circumference, it enables
to conclude from which direction the errors have
greater values, parameters of ellipse allows to
calculate directional errors, it gives the most
probable location of ship’s shape points among other
figures with the same area.
Determining the geometrical centre, direction of
axis and both semiaxis are essential in ellipse
building process. The point the model ship’s contour
is built from and determined uncertainties of its
coordinates were used to characterize the semiaxis
and geometrical centres of error ellipses. The bigger
semiaxis – a corresponds to direction error along X
axis of cartesian reference frame. The smaller
semiaxis – b corresponds to direction error along Y
axis. Figure 2 presents hypothetical ellipse formed
by 16 points described by parametric quotation:
x
j
= acos
φ
j
, y
j
= bsin
φ
j
(
φ
j
– angle between X-axis and
radius of j-th point of ellipse, a, b – length of bigger
and smaller semiaxis of ellipse).
-60
-40
-20
0
20
40
60
-110 -90 -70 -50 -30 -10 10 30 50 70 90
[m]
[m]
ellips 95%
ellips 95%
model ship's outline
Fig. 2. The errors ellipses of chosen points the ship’s shape
outline is built from with semiaxes a = σxri i b = σyri formed
in result of continuous line discretization into 16 points
Providing directional uncertainties to mentioned
quotations obtained:
rijriei
rijriei
yycy
xxcx
+=
+=
φσ
φσ
sin
cos
(14)
)
πφ
2;0∈
j
where:
−
eiei
yx ,
consecutive points the ellipse is built
from,
−
riri
yx ,
calculated coordinates of consecutive
points of ship’s contour.
Having determined the ellipse errors for every
points describing ship’s contour the two-dimensional
matrix of points P
i
(x
i
, y
i
) is formed. The outline of
the area covered by points of ellipses is found by
searching through every sector with angle width
∆
α
around the ship’s shape. The extreme point in each
sector is found on the basis of distances calculated
between these points and reference point
(geometrical centre of ship’s shape). The extreme
points create the limit of uncertainty area around the
model ship at assumed confidence level (Fig. 3).
190
-60
-40
-20
0
20
40
60
-110 -90 -70 -50 -30 -10 10 30 50 70 90
[m]
[m]
95%
ellips 95%
ellips 95%
obw iednia w zorcow a
Fig. 3. The uncertainty area of ship’s location around the model
ship outline formed after extreme points of error ellipse had
been found for 95% confidence level and antenna GNSS placed
in fore part of the ship
Ship’s location uncertainty area determining
process is based on input data that do not change
while calculations are being done. The dimensions
of uncertainty area depend on heading the ship
proceeds while maneuvering on research restricted
area. That is why in practical approach the recorded
ship’s path coming from real experiment or
simulated data are used. In order to achieve accurate
results it is recommended to have this information
inserted into model with the GNSS positioning
frequency (1s). Directional errors of points the ship’s
contour is built from are determined for courses the
ship is expected to proceed. In next step after
statistical analysis the mean directional errors and
errors at assigned confidence level are determined.
This approach enables to take into consideration
changeability the dimensions of uncertainty area
depending on courses the ship is going to keep
in real conditions. Picture 4 presents the ship
Jan Śniadecki uncertainty area determined for input
data assumed for pilot navigation system that
uses EGNOS as a source of position and two
synchronically working DGPS IALA receivers with
reference station situated in Dziwnów. The reference
DGPS antenna was placed in fore part of the ferry
next to navigation bridge on starboard side.
-40
-20
0
20
40
-110 -90 -70 -50 -30 -10 10 30 50 70 90 110
outline 68%
outline 95%
outline 99%
model ship's outline
Fig. 4. The uncertainty area for Jan Śniadecki ferry, at given
confidence levels
3 RESULTS OF EXPERIMENT
The analysis of research results was based on the
evaluation of the size of uncertainty area occupied
by the Jan Śniadecki ferry’s contour, estimated with
the use of uncertainty propagation theory (Tylor
1999). The magnitude of errors influencing ship’s
uncertainty area as the directly measured values was
verified. The calculated error of waterline contour
position where GNSS antenna was situated in
geometric centre of ship’s contour plane did not
exceed 6m at the confidence level 0.95% (dashed
line in fig. 5). In case when antenna was situated in
the fore part of deck the error did not exceed 11m
assuming directly measured errors as in tab. 1:
Table 1. Magnitude of directly measured errors
GNSS position error (DGPS
IALA)
mymx
ArAr
84,0,96,0 ±=±=
σσ
Accuracy of the assessment
of antenna location
mymx
HfHf
1,1 ±=±=
σσ
Heading error (2 sets of
synchronized DGPS
receivers)
4,2±=
σψ
Ship’s model approximation
error in i-th sector
mymx
ii
4,0,5,0 ±=±=
σσ
,
10190,9079 <<<< ii
mymx
ii
35,0,3,0 ±=±=
σσ
,
290270,270255 <<<< ii
-25
-20
-15
-10
-5
0
5
10
15
20
25
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
[m]
[m]
wz
ant. in fore part of the deck
ant. in geometrical centre
Fig. 5. Comparision of ship’s location uncertainty area
determined for INS using DGPS IALA position source and two
synchronically working DGPS IALA receivers as a heading
source with GNSS antenna situated in geometrical centre and
out of it
0
2
4
6
8
10
12
14
0 60 120 180 240 300 360
point of model ship outline
uncertainty [m]
ant. in fore part of deck
ant.in geom. centre
Fig. 6. Errors comparision of consecutive ship’s outline points
determined for INS using DGPS IALA position source and two
synchronically working DGPS IALA receivers as a heading
source with GNSS antenna situated in geometrical centre and
out of it
191
The significant errors appear in bow and aft
sectors of ship’s contour due to GPS antenna is
placed in the geometric centre of the contour. The
calculated error of waterline contour position differs
when GPS antenna is not situated in geometric
centre of ship’s contour plane (Tomczak 2006).
Antenna reference – ship’s (0,0) point was
established 28m from the bow and 5m right from the
centre line of the ship. The significant influence of
heading error is clearly seen. The determined area is
much wider in aft part of ship’s shape and errors
reach 11m at the confidence level 0.95 (dashed line
in fig. 5) assuming directly measured errors as
above.
4 CONCLUSIONS
The research has provided results that can be
summarized as follows:
− Uncertainty error propagation theory may be
applied to ship’s location uncertainty area
determination at assigned probability level in
pilot navigation system,
− Worked out mathematical model of ship’s
location uncertainty area, allows to identify the
position of ship’s waterline with an error up to
6 metres at the confidence level 0.95 for the
directly measured errors when GNSS antenna is
placed in geometrical centre of ship’s contour
plane and in case when the GNSS antenna is
shifted out of geometrical centre with error up to
11 metres at the confidence level 0.95,
− The determined uncertainty area strongly depends
on GNSS antenna placing in relating to ship’s
coordinate frame when the directly measured
errors remain unchanged,
− Worked out mathematical model of ship’s
location uncertainty area and the results obtained,
can be used in the process of designing pilot
navigation systems in respect of the ship
visualization in a given area.
REFERENCES
Gucma S.: Nawigacja pilotażowa, Fundacja Promocji
Przemysłu Okrętowego i Gospodarki Morskiej, Gdańsk,
2004.
Sanecki J.: Elementy rachunku wyrównawczego, Fundacja
Rozwoju Wyższej Szkoły Morskiej w Szczecinie, Szczecin
2004.
Tomczak A., Zalewski P.: Evaluation of navigation safety
criteria in Świnoujście harbour by means of GNSS
technology, The European Navigation Conference GNSS
2004 Proc., Rotterdam, 2004.
Tomczak A., Zalewski P.: Method of probabilistic evaluation
of ship’s contour inclusive area for pilot navigation system,
Internation Congress “Sea and Oceans”. Szczecin-Świno-
ujście 20-24.09.05.
Tomczak A.: Mathematical model of ship’s uncertainty
locateon area in pilot navigation system, Komputerowe
systemy wspomagania nauki, przemysłu i transportu.
Transkomp 2006, Zakopane, 2006.
Tylor John R.: Wstęp do analizy błędu pomiarowego,
Wydawnictwo Naukowe PWN, Warszawa 1999.