381
1 INTRODUCTION
Inageneralcasedatafusionisa processofcombining
dataforthepurposeof:
supplementing data to get a complete
mathematicalmodelofanexaminedprocess,
dataverificationandconsistency,
estimationandprediction.
Datafusioninnavigationismostlyassociatedwith
the top
level of fusion. However, with modern
navigationalandcomputer technologies, data fusion
canbeappliedatalllevels,notonlyfortheestimation
ofnavigationalmeasurements.
Navigation makes use of many engineering and
computing methods to determine position
coordinates in an established reference system.
Basically, these methods can be divided
into three
types:
model, based on a model of navigating object
movement – dead reckoning (DR) and inertial
navigationsystem(INS),
parametric,inwhichapositionisdeterminedfrom
ameasurementofnavigationalparameters,thatis
spatial relations betweennavigating object
coordinatesandnavigationalmarks,
comparative navigation, in
which images of
measured Earthʹs physical fields are compared
withcartographicimages(databases).
Cartographic data are directly used for object
positiondeterminationinthelastmentionedmethod
only. However, these measurements are not
combinedwithotherpositiondeterminationmethods.
We present herein possibilities of the fusion of data
from
cartographic database with a running fix
(parametricnavigation).
Theauthorsgotinspiredtodealwiththeissueby
thefactthatthereoccursastatisticalincompatibility
of shipʹ position with cartographic data in cases of
vessels berthing, docking or proceeding along a
fairway.
The Fusion of Point and Linear Objects in Navigation
A.Banachowicz
WestPomeranianUniversityofTechnology,Szczecin,Poland
A.Wolski
M
aritimeUniversityofSzczecin,Poland
ABSTRACT: There are great many human activities where problems dealt with are based on data from a
numberofsourcesorwherewelackcertaindata tosolveaproblem correctly.Suchsituationsalso occurin
navigation,wherewe have tocombinedatafromdiverse
navigationaldeviceswith archivaldata,including
images. This article discusses a problem of the fusion of position data from shipboard devices with those
retrieved from a hydrographic data base, the data being of varying accuracy. These considerations are
illustratedwithexamplesofthefusionofshipboardmeasurementswiththepierline
(oranothercartographic
object).
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 3
September 2013
DOI:10.12716/1001.07.03.09
382
2 FORMULATIONOFTHEPROBLEM
All combined navigational data should always be
broughttoajointreferencesystem. Atpresent,WGS‐
84 fulfills this function due to a wide use of the
satellitenavigationalGPSandECDISsystem.Forthis
reason all navigational measurement and
cartographic data from a navigational
‐hydrographic
database should be brought to this reference system
unless original data have been determined in this
system. Failing to satisfy this condition results in a
systematic error substantially exceeding random
errorsofthedata.
Thefollowingassumptionshavebeenmadeinthe
measurement(position)andcartographicdatafusion
problem
tobesolved:
dataaredeterminedinthesamereferencesystem,
data are of random character with a specific
probabilitydistribution,
dataarenotburdenedwithsystematicerrors,
data will undergo fusion by means of the least
squares method with or without measurement
covariancematrixbeingconsidered.
Therelativepositionsofashipandthepier(chart
feature)areshowninFigure1.
Figure1.Ashipberthingalongapier.
Theshipislyingalongside,sothepierlinecanbe
regardedasaconventionallineofpositionparallelto
theshipʹsplaneofsymmetryshiftedbyavectorfrom
aconventionalshipʹspoint,towhichallnavigational
measurements are brought. The vector can be
determined by direct measurement
or indirectly,
calculating its elements on the basis of a known
position of conventional point on shipʹs plane and
distanceofshipʹssidetothepierline.
3 DATAFUSION
High accuracy of satellite navigational systems and
autonomous shipboard systems (dead reckoning,
inertial navigational systems) creates high standard
requirements for methods of navigational data
processing.
We will perform a fusion of navigational and
cartographicdatausingthemethodof leastsquares.
In the method, we will regard the line of a
cartographic object (chart feature) as an additional
lineofposition.AKalmanfiltercanbeusedifa
ship
isproceeding.Thereisalsoapossibilityofmeasuring
therelativepositionofcartographicobjects.
Ifwedo nottakedataaccuracyinto account, the
method of least squares (LS) can be written in this
form[6],[8],[9],[10]:
1
TT
,
xGGGz (1)
where
x – m ‐dimensional state vector (of shipʹs
coordinates,searched‐forposition),
z – n ‐dimensionalvector,
u –n‐dimensionalvectorofmeasurednavigational
parameters,
'( )
Gfx – Jacobian matrix of the function f in
respectto
x .
1
11
12
22 2
12
12
x
xx
xx x
x
xx
m
m
n
nn
m
f
ff
f
ff
f
ff
G , (2)
f –n‐dimensionalvectorfunction,
u –vectorofdirectmeasurements,
()
zufx
– generalized vector of
measurements.
The position x coordinates vector covariance
matrixisexpressedbythisformula[6],[9],[10],[11]:
1
T1
x
PGRG (3)
Whenwetakedataaccuracyintoaccount,wedeal
withthemethodofweightedleastsquares(WLS)
1
T1 T1
,
xGRGGRz (4)
where
22
22
2
0
0
00
xxy
xy y
pier
R
‐navigationaldata
covariancematrix.
Wemakeafusionofpositionsorlinesofposition
withthepierlinefollowingthisprocedure:
determine the position coordinates (or lines of
position) together with their accuracy assessment
(variancesandcovar iances),
383
determinethedirectionandaccuracyofberthline
(usingarelevantchart,ordatabase,andpossibly,
usingtherelativeerrortoestablishtheaccuracyof
thatline,
shift the berth line parallel towards the shipʹs
positionbythevectorrepresentingthedistanceof
that line from the
assumed reference point
connectedwithshipʹsposition–centreofmasses,
geometric centre, GPS antenna position or
another),
calculatetheshipʹspositioncoordinates,regarding
theberthlineasanadditionalpositionline.
Thecovariancematrixoftherunningfixincaseof
aGPSiscalculatedfrom
aseriesofpositionsorfrom
Kalman filter. If there are terrestrial navigational
systems,wecanusethefollowingrelations[2].
An average error of geographic latitude
determinationforthecommonmiddlestation
2222
23
12
12 23
0,5 cos ec cos cos ec cos cos ec
22
D
AA
(5)
where
A
ij–averageazimuthbetweenthei‐th and the j‐th
station,
ij –baseanglebetweenthei‐thandthej‐thstation,
D–measurementerrorofdistancedifference.
An average error of geographic longitude
determinationforthecommonmiddlestation
2222
23
12
12 23
0,5 cos ec sin cos ec sin cos ec
22
D
AA
(6)
The covariance between geographic coordinates
forthecommonmiddlestation
22 2
23
12
2
12
23
1
cos ec sin( )(cos ec
82
sin( ) cos ec )
2
D
AA
AA
(7)
Also,wecanchangeaGPS‐obtainedpositioninto
a system of two position lines by calculating their
elementsbyusinga vectorofmeancoordinates and
elementsofitscovariancematrix.Inthiscaseposition
lines are regression lines running in the same
direction(parallel)(tangentnearthe
actual position)
[8]:
a)
2
,
xy
x
y
yxx
(8)
b)
2
()
xy
y
x
xyy
,
2
()
y
xy
y
yxx
, (9)
c)
(, )
x
y
–centreofgravityofthepopulation(mean
position).
In the geographical coordinate system these lines
areexpressedasfollows:
2
l
śr śr
l
ll
1
tg ,
śr śr
ll NR
(10)
2
ΔΔ
l
śr śr
ll
2
Δ tg ,
śr śr
lNR
(11)
2
1
2
tg
.
tg
l
NR
NR
(12)
Let us illustrate the above considerations of the
fusionofshipʹspositionandacartographiclinebythe
followingexamples.
EXAMPLE1.
The first example refers to the fusion of a ship’s
positionfrom GPS (point)withalinear cartographic
object (pier line or depth contour). Such situations
oftenoccurwhenashipismoored,dockedorisclose
tohydrotechnicalobjects.
Theoriginofalocalcoordinatesystem
0
x
y isat
anestablishedpointontheship(forsimplification).In
thiscasetheshipismooringalongapierdescribedby
the equation
2yx
(after a displacement by a
vector representing the distance from pier line to
assumedcoordinateorigin)andaccuracy
1 .
pier
m
Therunningfix, determinedbyGPSontheship,had
thiscovariancematrix:
20
02
R .
Thus we get
0, 2
xy x y
. The GPS
positioncanbeconsideredasapointofintersectionof
a meridian (vertical line) and parallel (horizontal
line).
Thematrices
G and R areasfollows:
10
01
11
G
,
200
020
001
R
,
whiletheresultantcoordinatevector
LS–
T
0, 667; 0, 667 ,x
WLS–
T
0,8; 0,8 .x
ThissituationisdisplayedinFigure2.Wecansee
that taking into account the accuracy of individual
positionlinesleadstoadisplacementofLSpositionto
384
the point WLS (due to higher accuracy of the
cartographic line than that of the GPS‐obtained
position).
Figure2.AfusionofaGPSpositionandpierline.
EXAMPLE2.
Intheotherexampleweperformafusionoflinear
objects, e.g. pier line with lines of position of the
radionavigationalsystem.
Similarly to Example 1, we adopt
0
x
y at an
establishedpointontheship(forsimplification).Now
the ship is mooring at a pier described by the
equation
10yx
(after a relevant displacement)
and accuracy
10 .
pier
m
A running fix was
determined on the ship by using a terrestrial
hyperbolic system with the following covariance
matrix:
10,5
0,5 1
R
.
We havethen
0,5; 1
xy x y
. The
matrices
G and R areasfollows:
61
51
11
G
,
10,50
0,5 1 0
0010
R
,
whiletheresultantcoordinatevector
LS–
T
0,161; 3,333 ,x
WLS–
T
0,115; 7,022 .x
Thesituationisillustrated by Figure3.Thistime
thedifferencesbetween both estimated positions are
larger due to another geometric configurationof
positionlinesandothervaluesoftheirerrors.
Figure3. A fusion of positions from a terrestrial
radionavigationalsystemwithapierline.
4 CONCLUSIONS
In the above considerations we have shown how
additional data, in this case data on cartographic
features position and accuracy, can be utilized for
estimating shipʹs position and its covariance matrix.
Inthiswayweincreasetheaccuracyandreliabilityof
shipʹspositionbeingdetermined.Thisis
ofpa rticular
importanceforshipslocatedinimmediateproximity
of port and marine facilities and other navigational
dangers. Similar situations also occur in aircraft
navigation on the airfields. In rail navigation
additional requirement is to take into account the
spatialpositionoftrainortramtracks, whileintruck
navigation
–positionofroads.
Anotherapproachconsistsinimposingconstraints
resulting from the Rao‐Cramer inequality [7], [8] on
thecovariancematrix.However,furtherresearchinto
these problems should take into account both
deterministic and probabilistic constraints on the
coordinate vector, that is the random character of
determining coordinates of cartographic
objects. The
point is to avoid superimposition of two disjoint
objects such as a ship and a pier. In the simplest
solutionthenavigatorcaninterferewiththeresultsof
calculations. However, such solution is far from
satisfactoryasitdoesnottakeintoconsiderationall
possible situations and cannot
be automated. It
shouldbeborneinmindthatcartographicobjectsare
alsodeterminedwithadefiniteaccuracy.
The presented problem of the fusion of ship
position data and data on the location of
hydrotechnical objects does not cover all issues of
navigationaldatafusion(spatialdata,inmoregeneral
terms).
Problemsofintegrationofmeasurementdata
from navigational systems have a long history,
starting from a well‐known article by R. E. Kalman
[5], then his successors, with a period of rapid
development of integrated navigational systems in
the1980sand‘90s,alsoinPoland[1].Atpresent,the
focus
ofdatafusioninnavigationisshiftingtowards
preliminaryprocessingandfusionofvarioustypesof
data – measurement, image and text [4]. Also,
385
methods and applications of data and information
fusionhavebeenwidelyextended.
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