691
1 INTRODUCTION
The use of only one method and one navigation
systemtypicallyresultsinloweraccuracy,credibility
and reliability of navigational information. This
lowersthelevelofnavigationalsafetyandeconomic
efficiency of a voyage. In order to overcome this,
various independent methods and systems for
determiningnavigationalparametersareused.These
primarily include, in accordance with the SOLAS
convention, dead reckoning navigation systems
(compasses, logs, accelerometers), positioning
systems (GNSS or terrestrial systems) and electronic
chart display and informat
ion systems (ECDIS) or
papercharts.Variousmeasurementsandnavigational
parameter data are combined through joint
algorithms and information‐measurement systems.
Uptodate examples of such systems are int
egrated
navigationsystemsandintegratednavigationbridge.
Figure1illustratesdifferencesinshipʹstrajectory,
obtained from dead reckoning (DR) navigation
system and independently used positioning system
(GPS).There arevisible differences between thetwo
trajectories obtained on the basis of measurements
from these systems. Due to the influence of drift
(
systematicerror)indeadreckoningnavigation,that
was not measured, the DR trajectory is moved to
starboardinrelationtoGPSfixes.
The algorithms of integrated navigation systems
carryoutthefusionofdifferentmethods,inparticular
the parametric method is combined with the dead
reckoningmethod.
Thisprocess requires the combined processing of
measurement data, which allows us to optimize the
use of navigational informat
ion. Multi‐sensor fusion
of navigational data is widely discussed in the
literature, e.g. [10], while GPS data integration with
othernavigational measurements is described in [3],
[5].
These authors present selected va
riants of the
integration of navigational data obtained from
different navigation systems. The method of least
squaresandtheclassicKalmanfilterwereusedasthe
A Comparison of the Least Squares with Kalman Filter
Methods Used in Algorithms of Fusion with Dead
Reckoning Navigation Data
A.Banachowicz
WestPomeranianUniversityofTechnology,Szczecin,Poland
A.Wolski
PolishNavalAcademy,Gdynia,Poland
ABSTRACT:Differentcalculationmethodsandconfigurationsofnavigationsystemscanbeusedinalgorithms
ofnavigationalparameterfusionandestimation.Thearticlepresentsacomparisonoftwomethodsoffusionof
dead reckoning position with that from a positioning system. These are the least squares method and the
Kalman filt
er. In both methods the minimization of the sum of squared measurement deviations is the
optimization criterion. Both methods of navigation position parameter measurements fusion are illustrated
using the data recorded during actual sea trials. With the same probabilistic model of dead reckoning
navigation,thefusionofDRresultswithpositioningdatagivessimilaroutcome.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 4
December 2017
DOI:10.12716/1001.11.04.16
692
mathematical model of measurements integration.
The analyzed methods are illustrated with an
exampleofactualmeasurementsrecordedduringsea
trials. Both cases of measurement data integration
were based on the same measurements and similar
assumptionsconcerningtheirstatisticaldistributions.
Thismadetheircomparativeanalysisobjective.
Figure1. Comparison of shipʹs trajectory according to DR
andGPS.
Brieflylistedbelowaretheemployedmethodsof
position coordinates estimation method, the method
ofleastsquaresandtheKalmanfilter.Deadreckoning
navigation,usedinthesealgorithms,wasdescribedin
[2],[3]andotherworks.
2 THEMETHODOFLEASTSQUARES
Letusassumethatwehavemeasurementsofvarying
accuracyandwewillusethemethodofleastsquares
with weights for their fusion [11]. In this case, the
vectorof state (position coordinates) is described by
thisformula:
1
T1 T1
xGRGGRz, (1)
anditscovariancematrixiswrittenasthisrelation:
1
T1
PGRG, (2)
T
,
x
, (3)
, , ,
GPS GPS DR DR
T
z , (4)
where
x ‐vectorofstate,
z ‐vectorofmeasurements,
G ‐a matrix binding the vector of state with the
vectorofmeasurements,
P ‐covariancematrixofthestatevector,
R ‐covariancematrixofthemeasurementsvector.
One of the simple measurement situations is a
combination of GPS position coordinates with dead
reckoning(DR)position.Inthiscase
G
,thematrixwill
beablockmatrixinthefollowingform:
T
22 22
GI I
. (5)
The matrix of measurements covariance
R will
alsobeablockmatrix
0
0
GPS
D
R
P
R
P
, (6)
where
GPS
P the matrices
D
R
P are, respectively,
matrices of GPS and DR covariances. With these
assumptions,, the matrix inverse to the covariance
matrixofmeasurementswillhavethisform
1
1
1
0
.
0
GPS
DR
P
R
P
(7)
Ultimately, with the above assumptions, we get
thevectorofstate(positioncoordinates)
1
11 11
GPS DR GPS DR
xP P P Pz
, (8)
anditscovariancematrix
1
11
.
GPS DR
PP P
(9)
Inthiscase,theDRposition istreatedasaseparate
measurement(alongwithitsevaluationofaccuracy).
ItshouldbenotedthatDRisperformedatshort,one
second intervals, which makes the DR error
comparabletothefixerror.
3 KALMANFILTER
Kalmanfilteringiscommonlyused
today[5],[6],[7],
[10], [12]. It is implemented at various levels of
navigational information processing, from physical
measurements by sensors (preliminary processing),
through the combination of measurements from
different sensors (intermediate processing), to the
estimation of position coordinates and other
navigationalparameters(finalprocessing).Ateachof
theselevelswe
usethesamemathematicaltoolsand
thesamecomputingalgorithm.
The discrete Kalman filter, in a particula r case,
describesthesystemoftwoequations[1],[8],[9]:
thestateequation(structuralmodel)
11,
xAxw
iiiii
, (10)
measurementequation(measurementmodel)
693
11
zCxv
iiii
(11)
where
x ‐n‐dimensionalvectorofstate,
w
‐r‐dimensionalvectorofstatedisturbances,
z ‐m‐dimensionalvectorofmeasurements,
v ‐ p‐dimensional vector of measurement
disturbances(identifiedwithmeasurementnoise),
A ‐nn‐dimensionaltransitionmatrix,
C
‐mn‐dimensionalmeasurementmatrix,
rn,pm.
Weassumethatthevectorsofdisturbanceswand
v are Gaussian noise with normal distribution, with
zeromeanvectorandaremutuallynon‐correlated.In
the case of colour noise (with a trend) the extended
Kalmanfilteris
applied,wherethedisturbancetrend
is included as additional components of the state
vector.
Theequationofstatedescribestheevolutionofthe
dynamicsystemdescribedinthestatespace,andthe
model of measurements functionally combines
measurementswiththesystemstate.Thesolutionto
the equations (10), (11), taking
into account the
constraintsimposedonthevectorsofdisturbances,is
theKalmanfilter.Calculationofthestatevectorinthe
Kalmanfilterisdescribedbythefollowingalgorithm:
projectionofthestatevector:
i1,1 i1,i i
ˆ
xAx
, (12)
where
x
is the projected value of the state
vector,
x isestimatedvalueofthestatevector,
covariancematrixoftheprojectedstatevector
T
1, 1, 1,1
ii iii i i
PAPAQ, (13)
where Q is the covariance matrix of disturbances of
thestate(ofvectorw),
innovationprocess
11 11,ii iii
εzCx
, (14)
covariancematrixoftheinnovationprocess
T
1111iiiii
SRCPC, (15)
where R is the covariance matrix of measurement
disturbances(ofvectorv),
filtergainmatrix(Kalmanmatrix)
T1
11,111
iiii
KPCS
, (16)
estimated value of the state vector from filtering
aftermeasurement
1
i
z
11, 11
ˆ
iiiii
xx Kε
, (17)
covariancematrixoftheestimatedstatevector
1111,
.
iiiii
PIKCP (18)
4 THESTRUCTUREOFTHEINTEGRATING
FILTER
Theadopted mathematicalmodel of ship movement
and the configuration of navigational devices affect
the structure of the Kalman filter algorithm. Let us
assume,asinthepositionestimationalgorithmbythe
methodofleastsquares,thatposition coordinatesare
determined using
GPS (parametric navigation), and
measurements in dead reckoning navigation are
obtainedfromagyroscopiccompassandDopplerlog.
Letusdefinethestatevectoras:
T
,, , , ,
NE
VVCOGSOG
x , (19)
where
‐latitude,
‐longitude,
,
NE
VV‐ projections of the vector of speed over
groundontheparallelandthemeridian,
COG ‐courseoverground,
SOG
‐speedoverground.
Thequantitiesmeasured(measurementmodel)are
thefollowingparameters: positioncoordinatesfroma
GPSsystem
,
GPS GPS
,projectionsofthevectorof
speed over ground on the parallel and meridian
(
,
NE
VV),courseoverground(COG)andspeedover
ground (SOG). Hence, the vector of measurements
willtakethisform:
, , , , ,
GPS GPS N E
VVCOGSOG
T
z . (20)
Each of the system matrices was appropriately
selected for vectors (19) and (20) [4], [5], [6]. DR
navigationinthisalgorithmisanelementofamodel
(trend) of ship movement, but also‐as is apparent
from(19)‐someofitsparametersareestimated.
5 ANEXAMPLE
For a comparison of
the two methods of
measurements fusion, simulation tests have been
carriedout.The same measurement conditions were
adoptedforthetwomethodsofdatafusionwithDR
results:LS‐DRandKF‐DR.Therefore,thetestswere
basedonthesameseriesofmeasurementsofposition
coordinatesfromaGPS,
gyrocompasscourseandlog
speed.Inthiscase,thefollowingvaluesofvarianceof
specificmeasurementsweretaken[5],[6]:
DGPS system:
2, 0
m, 1, 5
m, the
coordinatesarenon‐correlated;
courseoverground:
0
1, 5
COG
;
speedoverground:
0,5
SOG
knot.
For these data the DR position errors for one
second interval were equal to
1, 414
m.
Thesevalueshavebeenadoptedinbothcasesofthe
estimatedpositioncalculations.Fig.2showsthatin
694
the macroscopic sense both methods of coordinates
estimationyieldthesameresult,i.e.bothtrajectories
coincide.
Figure2.Macroscopiccompatibilityofthetwotrajectories.
The same initial position in both cases was
adopted (Figure 3). Only in the next steps the
particular positions, KF, LS start to differ
significantly. These differences become more visible
wherelargerchangesofthevelocityvectoraremade,
inthiscaseduringtheturningcircle(Figure4).This
is because
the KF algorithm identifies the velocity
vectortrend,whiletheLSalgorithmdoesnot.
Figure3.Theinitialsectionofthetwotrajectories.
Figure4.Thetwotrajectoriesatapointofintersectionwith
theturningcircle.
Figure5depictsmagnifieddifferencesbetweenKF
and LS trajectories on a steady course (uniform
rectilinearmovement).
Figure5. Example differences between the two trajectories
ofashiponsteadycourse(magnified).
Figure 6 presents distances between KF and LS
positionsalongatrajectory(insubsequentiterations).
Themaximumdeviationis6metres,theminimum0.5
meter. The average distance between estimated
positions(forthewholeset)fromKFandLSis2.747
meters.
Figure6.ThedistancesbetweenKFandLSpositions.
6 SUMMARY
Thepresentedmodelsandalgorithmsillustratetwoof
many possibilities of the navigational application of
themethodofleastsquaresandtheKalmanfilterfor
the integration of navigational data. In the Kalman
filterthestatevectorreproducesthesystemevolution
(movement trend) on the basis of dead reckoning
navigation.ThemainadvantagesofKalmanfilter,in
this case, are its recurrence, which is a natural
necessityincaseofshipnavigationandthepossibility
ofusingtheshipmovementdata(itstrend).
The fusion of position parameters with DR
navigationbytheLSmethodgivessimilarresultsto
thoseobtainedbytheKFmethod,inwhichdatafrom
DR are also used. The differences between the
position coordinates obtained from using these
695
methodsarealmostinsignificant(onaveragelessthan
3 meters). One advantage of the KF over the LS
method,owingtothestructureoftheformer,isthatit
offers a possibility to identify a trend of ship
movement parameters. On the other hand, the KF
algorithm requires more complex
computations.
However,toachievethesameleveloffunctionalityof
theLS method, theestimationmodelwouldhaveto
be expanded to include the identification, which in
turn would increase significantly the computational
complexityofthealgorithm.
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