23
1 INTRODUCTION
Since International Maritime Organization (IMO)
clearly presents standards for the ship
maneuverabilitytoensureshipnavigationsafety,the
predictionofshipmaneuverabilityhasbecomeavital
and attractive issue. The system basedmaneuvering
simulation has been proved as an effective and
economic way to predict the ship maneuverability.
One of the preconditions is the esti
mation of
maneuveringmodels.Toahighdegree,theaccuracy
of the estimation guarantees the effectiveness of
predictionofthemaneuveringmodel.
Themainmethodsforestimatingthemaneuvering
model include towing‐tank experiments, captive
model experiments (Skjetne et al. 2004),
computational fluid dynamics (CFD) and system
i
dentification combined with the full‐scale or free‐
runningmodel(Xuetal.2014).Thelastisbecoming
anattractiveand cost‐effective toolfor estimation of
shipmaneuveringmodels.
System identification is a very broad topic with
different techniques that depend on the character of
models to be esti
mated: linear, nonlinear, hybrid,
nonparametric, etc. (Ljung 2010). Various
Parameter Identification of Ship Maneuvering Models
Using Recursive Least Square Method Based on Support
Vector Machines
M.Zhu&A.Hahn
Carl‐von‐OssietzkyUniversityofOldenburg,Oldenburg,Germany
Y.Q.Wen
SchoolofNavigation,WuhanUniversityofTechnology,Hubei,China
A.Bolles
InstituteofInformationTechnology,Oldenburg,Germany
ABSTRACT:Determinationofshipmaneuvering modelsisa toughtask ofshipmaneuverabilityprediction.
Amongseveralprimeapproachesofestimatingshipmaneuveringmodels,systemidentificationcombinedwith
the full‐scale or free‐ running model test is preferred. In this contribution, real‐time system identification
programsusingrecursivei
dentificationmethod,suchastherecursiveleastsquaremethod(RLS),areexerted
foron‐lineidentificationofshipmaneuveringmodels.However,thismethodseriouslydependsontheobjects
ofstudyandinitialvaluesofidentifiedparameters.Toovercomethis,anintelligenttechnology,i.e.,support
vectormachines(SVM),isfirstlyusedtoesti
mateinitialvaluesoftheidentifiedparameterswithfinitesamples.
As real mea s ured motion data of the Mariner class ship always involve noise from sensors and external
disturbances,thezigzagsimulationtestdataincludeasubstantialquantityofGaussianwhitenoise.Wavelet
methodandempiricalmodedecomposition(EMD)areusedtofilt
erthedatacorruptedbynoise,respectively.
The choice of the sample number for SVM to decide initial values of identified parameters is extensively
discussed and analyzed. With de‐noised motion data as input‐output training samples, parameters of ship
maneuvering models are estimated using RLS and SVM‐RLS, respectively. The comparison between
i
dentification results and true values of parameters demonstrates that both the identified shipmaneuvering
models from RLS and SVM‐RLS have reasonable agreements with simulated motions of the ship, and the
incrementofthesampleforSVMpositivelyaffectstheidentificationresults.Furthermore,SVM‐RLSusingdata
de‐noisedbyEMDshowsthehighestaccuracyandbestconvergence.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 1
March 2017
DOI:10.12716/1001.11.01.01
24
conventional system identification methods, such as
least squares method (LS), maximum likelihood
method(ML)andextendedKalmanfilter(EKF),have
been successfully applied to estimate the ship‐
maneuveringmodel.Forinstance,Xuetal.(Xuetal.
2014)incorporatedLSwithintegralsamplestructure
and Euler method to identify the
linear
hydrodynamic model in the horizontal plane of an
underwatervehicleusingsimulateddata.Åstromand
Kållstrom(Åstrom&Kållstrom1976) appliedMLto
determine steering dynamics of a freighter and a
tanker using free steering experiments on full‐scale
ships.Shi et al.(Shi et al. 2009) tackled
identification
ofanon‐linearshipmaneuveringmodel
basedonEKF.ThismethodwasalsousedbyPerera
et al. (Perera et al. 2015) to identify the stochastic
parameters of a nonlinear ocean vessel steering
model. In recent years, a variety of novel methods
basedonthemodernartificialintelligenttechnology,
such
as the artificial neural network (ANN), genetic
algorithm(GA)andsupportvectormachines(SVM),
have been used successfully in the parameter
identification of the ship maneuvering model. ANN
wasusedbyRajeshandBhattacharyya(Rajeshetal.
2008)todealwithsystemidentificationofanonlinear
maneuvering model for large
tankers. Sutulo and
Guedes Soares (Sutulo & Guedes Soares 2014)
developed an identification method based on the
classic genetic algorithm to estimate a mathematical
model describing the ship maneuverability by using
simulation data corrupted by the white noise of
various levels. Comparatively, SVM directs at finite
samples, which requires no initial
estimation of
parametersbuthas goodgeneralizationperformances
andglobaloptimal(Luo&Cai2014).In2009,Luoand
Zou (Luo & Zou 2009) firstly successively applied
SVM to identify hydrodynamic derivatives of
Abkowitz model from the free‐running model test,
and predicted zigzag tests using the regressive
Abkowitzmodel.
Otherstudiescanbefoundfromthe
research group guided by Zou (Zhang et al. 2013 &
Zhangetal.2011&Xuetal.2012&Wangetal.2013)
andreferencestherein.
Insuch a variety of identification methods, some
are developed to on‐line identify time‐varying
coefficients,
for instance, recursive least square
method (RLS) algorithm and least mean squares
(LMS) algorithm (Ljung 2002). Since the change of
current weather and ship loading conditions can
cause parameter variations of ship maneuvering
models, the well‐known RLS with an advantage of
simple construction is used in this paper to
identify
parametersofshipmaneuveringmodels.
As well known, the identification results of RLS
aresensitivetotheinitialvaluesofparameters(Zhang
et al. 2013). Hence, this contribution aims at
conqueringsuchdrawbackofRLS bybenefitingfrom
applying firstly SVM which is a kind of batch
identification technique and
requires no initial
estimation of the parameters, to provide RLS initial
values. Additionally, this paper makes an effort to
analyze the choice of the training sample number
applied for SVM to identify initial values of ship
maneuveringmodels.
The data for learning and validation of
identificationprocedureareobtainedfrom
simulation
ofshipmaneuveringmodelscombinedwithexisting
particulars. For consideration of real navigation
situationinfluencedbydifferentdisturbances,suchas
wind,waveandcurrents,thesimulationtestdataare
corrupted by non‐correlated white noise, i.e.,
Gaussian white noise. Then, two different filters,
namely,Waveletfilters(Barford1992)andEmpirical
Mode Decomposition (EMD) algorithm (Wang et al.
2014)areusedtoomitnegativeinfluenceofexternal
disturbancesonidentificationresults.
Thepaperisorganizedasfollows.Insection2,the
mathematical model of ship maneuvering is
described.The identification methods including RLS
and SVM are introduced in section 3. The
implementation of ship maneuvering models’
identification is conducted and the identification
results are analyzed in section 4. Finally, the
conclusionoftheworkissummarizedinsection5.
2 THEMATHEMATICALMODELOFSHIP
MANEUVERING
Ship dynamics are complex due to nonlinear and
coupling characteristics. At present, three types of
mathematical
model of ship maneuvering are
common. MMG model is modular model separately
describing rudder effects and propeller effects.
Abkowitz model is whole‐ship model regarding
influences on the ship as the whole using Taylor
seriesexpansion.Theresponsemodel,particularly,is
theNomotomodels (Fossen2011).In thisstudy, the
problem of determining ship steering dynamics is
focused from the point of view of parameter
identification.Assumingthattheshipforwardspeed
is constant (
0
u
), the steering dynamics of a surface
shipcanbedescribedas(Åstrom&Kållstrom1976)
0
0
001
0
0
010 0
mY mx Y
vGr
v
mx N I N r
Gv zr
YYm Y
vr
v
NNmx r N
vr G
(1)
where
m
isthenon‐dimensionalmassoftheship;
x
G
isthenon‐dimensionallongitudecoordinateofthe
ship’s center of gravity;
I
z
is the non‐dimensional
inertia moment about
z
‐axis;
v
,
r
are non‐
dimensionalsmallperturba tionsrespectively;
v
isthe
non‐dimensionalswaylinearvelocity;
,
r
arenon‐
dimensional yaw rate;
is the non‐dimensional
heading angle;
is the rudder angle;
Y
v
,
Y
r
,
Y
,
Y
v
,
Y
r
arerespectivehydrodynamiccoefficientsof
theswaymotion;
N
v
,
N
r
,
N
,
N
v
,
N
r
arerespective
hydrodynamiccoefficientsoftheyawmotion,and
,
2
vL
v
U
2
,
2
rL
r
U
,
L
U
,
v
v
U
,
rL
r
U
,
22
Uuv
o
.
The normalized equations of motion, i.e., Eq.(1),
areeasilyconverted tostandardstatespacenotation
25
bysolvingforthederivatives
v
and
r
,whichis
givenas
0
11 12 11
0
21 22 21
010 0
vaa vb
raa rb
(2)
wheretheparametersareexpressedrespectivelyby
()( )
11
()( )( )( )
INY mxYN
zrv Grv
a
mY I N mx Y mx N
vz r Gr G v
()()( )( )
12
()( )( )( )
I N Y m mx Y N mx
zrr Grr G
a
mY I N mx Y mx N
vz r Gr G v
()( )
21
()( )( )( )
mY N mx N Y
vv G vv
a
mY I N mx Y mx N
vz r Gr G v
()( )( )()
22
()( )( )( )
mY N mx mx N Y m
vr G Gvr
a
mY I N mx Y mx N
vz r Gr G v
()( )
11
()( )( )( )
INY mxYN
zr Gr
b
mY I N mx Y mx N
vz r Gr G v
()( )
21
()( )( )( )
mY N mx N Y
vGv
b
mY I N mx Y mx N
vz r Gr G v
Rewriting the state variables of Eq.(2) with
dimensionalformat,itcanbegivenas
2
11 12 11
2
21 22 21
22
UU
av arUb
LL
v
UUU
ravarb
L
LL
r
(3)
3 IDENTIFICATIONMETHOD
3.1 LS‐SVMMethod
With several years’ application of SVM, it has been
provedthatitcanalsobedesignedtodealwithsparse
dataintheconditionofmanyvariablesbutfewdata
(Vapnik 2000). LS‐SVM is the one modified form of
SVM, which
has the ability to simultaneously
minimizetheestimationerrorinthetrainingdata(the
empirical risk) and the model complexity (the
structuralrisk)forbothregressionandclassification.
Consideramodelintheprimalweightspace
() () ( , )
Tn
y
xxbxRyR
(4)
where
x
is the input data;
y
is the output data;
b
isabiastermfortheregressionmodel;
isamatrix
of weights; and
(.)
:
n
h
RR
is the mapping to a
high‐dimensional Hilbert space, the
n
h
can be
infinite. The optimization problem in the primal
weightspaceforagiventrainingset
{,}
1
N
s
xy
ii
i
with
N
s
asthenumberofsamplesbecomes
11
2
(,)
22
1
N
s
T
min J e C e
i
w, b,e
i
(5)
subjectto
()
T
yxbe
iii
(6)
where
e
i
is regression error;
C
is the penalty factor
withpositivevalues.
In the case of
becoming infinite dimensional,
the problem in the primal weight space cannot be
solved. The Lagrangian is computed to derive the
dualproblem
(,,, ) (,) ( ( ) )
1
N
s
T
J
be J e x b e y
iii
i
i
(7)
where
(1,, )iN
is
are the Lagrange multipliers.
Nowthederivativeswithrespectto
,,be
i
,and
i
arecomputedandsettobezero,respectively
(,,,)
0()
1
(,,,)
00
1
(,,,)
0
(,,,)
0() 0
N
s
Jbe
x
ii
i
N
s
Jbe
i
b
i
Jbe
Ce
ii
e
i
Jbe
T
xbey
iii
i
(8)
After straightforward computations, variables
and
e are eliminated from Eq.(8). Then the kernel
trickisapplied. Thekerneltrickallowsustoworkin
large dimensional feature spaces without explicit
computations on them. Therefore, the problem
formulationyields
() (, )
1
N
s
yx Kxx b
i
i
i
(9)
where
(, )
K
xx
i
represents the kernel function. For the
problemofparameteridentification,thelinearkernel
function is usually adopted, i.e.,
(, ) ( )
K
xx xx
ii
,
because the identification equation ofthe steering
model is linear with respect to identification
parameters. So the identified parameters
can be
regressed by using linear kernel based on LS‐SVM,
theregressionmodelis
1
N
s
x
i
i
i
(10)
3.2 RLSmethod
Considering the limitation of space, RLS is briefly
introduced. RLS is developed for on‐line parametric
identification based on off‐line method, LS. Given a
systemorganizedwithalinearregressionformusing
a model parameter vector
, a lagged input‐output
26
data vector
() [ (1) (2) ()]
T
TT T
Xk x x x k
, and an
unspecifiednoiseprocess
()vk
asfollows
() () ()
T
y
kXk vk
(11)
Then,parameters
areestimatedusingRLSas
ˆˆ ˆ
() ( 1) ()[() ()( 1)]
1
() (1)()[ ()(1)()]
()[ () ()]( 1)
T
kkKkykxkk
T
Kk Pk xk I x kPk xk
T
Pk I Kkx k Pk
(12)
4 PARAMETRICIDENTIFICATIONBYSVM‐RLS
4.1 ConstructionofSamples
The continuous Eq.(3) is discretized using Euler
forwardmethod.Itsdifferenceformcanbeexpressed
as
11
() () () () ()
12
(1)
(1)
21 22
() () () () ()
2
2
11
() ()
2
21
() ()
2
at
vk vkUk a trkUk
vk
L
at at
rk
rk vkUk rkUk
L
L
bt
kU k
L
bt
kU k
L
(13)
where
1k
and
k
denote two successive sampling
times,
t
is the sampling interval. Then the input‐
outputpairsareusedforSVMandRLStoidentifythe
parametersinEq.(13).
Theinputsareexpressedas
2
[ ( ), ( ) ( ), ( ) ( ), ( ) ( )]
41
T
vk vkUk rkUk kU k
Y
(14a)
2
[(),() (),() (), () ()]
41
T
rk vkUk rkUk kU k
Z
(14b)
Let
[1 ]
12314
bb b
B
,
[1 ]
12314
cc c
C
, then the
outputsare
(1)vkBY
,
(1)rkCZ
.
Once the parameters of
B
and
C
are obtained
through identification algorithms, the parameters of
the state space model (Eq.(2)) can be achieved
immediately,namely,
1
,
11
bL
a
t
2
,
12
b
a
t
2
,
12
b
a
t
3
11
bL
b
t
,
2
1
,
21
cL
a
t
2
,
22
cL
a
t
2
3
21
cL
b
t
.
4.2 DataPreprocessing
The data used for learning and validation of
parametric identification of the ship steering model
are generated by synergistically employing forth‐
order Runge‐Kutta algorithm and Eq.(3) with
parameters extracted from the study in (Åstrom &
Kållstrom 1976). The ship parameters are shown in
Table1.
Table1.TheparametersofMerchantshipMarinerclass
_______________________________________________
Mariner
_______________________________________________
Length
Lm
161 Speed
(/)
o
ums
7.7
11
a
‐0.693
21
a
‐3.41
12
a
‐0.304
22
a
‐2.17
11
b
0.207
21
b
‐1.63
_______________________________________________
TheRunge‐Kuttaalgorithmisrepresentedas
(, )xfx
(15)
1
211/2
321/2
431
(, )
(,)
2
(,)
2
(,)
nn
nn
nn
nn
kfx
h
kfx k
h
kfx k
kfxhk
(16)
(2 2 )
11234
6
h
x
xkkkk
nn
(17)
where
n donatestime series;
( 1,2,3,4)ki
i
represents
the intermediate variable;
h
is the sampling
interval;
[,, ]
x
vr
is the state vector; and
()/2
1/ 2 1nnn
.
Two groups of zigzag simulation tests, that are
20 /20
for identifying parameters and
10 /10
for
validation, are derived with initial states including
theforwardspeedof
7.7 /ms
,therudderangleof
0
,theheadingangleof
0
,theyawrateof
/
0
s
,and
theswayspeedof
0/ms
.Thesamplingtimeis1000s,
andtheintervalis0.5s.2000measurementpairsof
v
,
r ,
, and resultant speed
U
are recorded for
parameter identification of the steering model. The
simulationresultsareillustratedinFig.1.
Figure1. Comparison of training data of the
20 / 20
zigzagtest
As the real data of ship maneuvering will be
inevitably corrupted by measurement noise and
environmental disturbances which are generally
considered as Gaussian white noise assumed to be
independent one with zero means, the original
simulation data are corrupted by Gaussian white
noise.Then the data are de‐noised by filters.
In
order to effectively analyze the influence from de‐
noised data by different filters on the accuracy of
27
identification results, wavelet filters and EMD are
resorted, respectively. The comparison between the
original simulation data, the simulation data
corrupted by Gaussian white noise, and de‐noised
databyrespectivewaveletandEMDarepresentedin
Fig.1.
4.3 SelectionofSampleNumberforSVM
LS‐SVMasabatchtechnique
avoidslengthyiteration
and needs no initial estimation of parameters.
However,itcanbeseenthattheproblemofapplying
LS‐SVMisthechoiceofthenumberofsamples.The
solution of such problem is proposed by analyzing
the convergence of LS‐SVM used with different
numbers of samples.
The samples are selected from
originalsimulationtestdata,thenumbervariesfrom
10with the interval of10 to 2000.The identification
resultsofdifferentnumbersofsamplesareshownin
Fig.2 where the upper‐right one is identified
parametersofthesteeringmodelandthedown‐right
one is
a partially enlarged view. Additionally, the
upper‐left one indicates the relative error of each
parameter between identification results and true
values, and the down‐left one represents a partially
enlargedview.Obviously,allparametersmatchwell
withtrue values while the sample numberincreases
to around 80. Considering that
the training data are
corruptedby noise, the samples used for LS‐SVM is
160thatmeansthe80s.
Figure2. Theidentification results of different numbers of
samples
Table2.Identifiedvaluesofsteeringmodel
__________________________________________________________________________________________________
Parameters Truevalue RLS(Waveletalgorithm) RLS(EMD)LSSVM‐RLS(EMD&160)
__________________________________________________________________________________________________
Identified Relative Identified RelativeIdentified Relative
valueerror value errorvalue error
__________________________________________________________________________________________________
11
a
‐0.693‐2.5808 2.2738‐0.7665 0.1061‐0.7140 0.0303
12
a
‐0.304‐0.9821 2.2306‐0.2835‐0.0674‐0.2949‐0.0299
11
b
0.2070.1972‐0.0473 0.1987‐0.04010.1904‐0.0802
21
a
‐3.410.3409‐4.0691‐3.1485‐0.0767‐3.2232‐0.0548
22
a
‐2.17‐3.5498 0.6359‐2.2424 0.0334‐2.1846 0.0067
21
b
‐1.63‐1.4722‐0.0968‐1.4996‐0.08‐1.5159‐0.07
__________________________________________________________________________________________________
Figure3.Comparisonbetweentrueandidentifiedparameters
28
4.4 IdentificationResults
Forthepurposeofclearlyshowingtheidentification
resultsofthesteeringmodel,theformer500sresults
are selected and presented in Fig.3, because most
parameters converge well after 150s except for
11
b
.
The identified values of the steering model by
differentidentificationmethodsarelistedinTable2.
Itisobviousthattheidentificationresultsfromusing
de‐noised data by EMD are more precise than the
ones from Wavelet algorithm. Both the estimated
valuesofthesteeringmodelbyRLSand
LSSVM‐RLS
convergewellintothetruevalues.
Comparatively,theidentifiedvaluesfromLSSVM‐
RLS have higher accuracy, in particular
,,
12 21 21
aab
,
because the initial values of those parameters
provided by LSSVM are close to true values.
Additionally, LSSVM‐RLS shows better convergence
performance.Itisdeservingtonotethattheidentified
valueof
11
b
byLSSVM‐RLSisworsethanRLS.This
may be attributed to two aspects. Firstly, under
conditions of training data corrupted by noise even
filtered, LSSVM still needs more data samples to
achieveaccuratevaluesofparameters.Secondly, the
differencebetweentheinitialvalueof
11
b
appliedto
identification algorithms and true value has an
impact.Theinitialvalueof
11
b
setforRLSiscloserto
thetruevaluethantheoneobtainedfromLSSVM.
4.5 PredictionandVerification
Verification of identification results is the essential
procedure for parameter identification. Hence, a
10
o
/10
o
zigzagtestispredictedbyusingtheidentified
steering model.As presented in Fig.4, the
comparison between predicted data and original
simulation data indicates that the identified steering
modelhasasatisfiedagreementwiththerealmodel,
which illustrates that the identification method
preformsgoodgeneralization.
Figure4.Predictionandcomparisonof the
10 /10
zigzag
test
5 CONCLUSION
In this paper, we have developed a solution to
overcome the problem of initial value definition for
parameter identification in linear ship dynamic
modelsusingrecursiveleastsquares(RLS)approach.
For the definition of the parameters, we combined
LSSVMwithanRLSalgorithm.Toshowthebenefitof
this
approach,wehaveexecutedazig‐zagsimulation
basedevaluation,inwhichweaddedGausssiannoise
calculated by signal ration proportional approach to
generate realistic training and validation data. To
filterthe noise we used a wavelet algorithm and an
empirical mode decomposition (EMD) for the RLS
approach, and EMD
for the LSSVM approach. We
have shown that our LSSVM‐RLS approach for
parameter identification is suitable and for most
parameters even better than the RLS‐only approach
with predefined initial values. We also have shown
that EMD filtering provides better results for de‐
noisingdata.
Forthcoming work will focus on
expanding the
application of the proposed parameter identification
methodtothenonlinearidentificationalgorithm,such
as ExtendKalman filter algorithm, for the nonlinear
shipmaneuvering model.The furtherpointsworthy
of attention will be data acquisition through
extractingfromrealshipnavigationmotionsrecorded
bynavigationdevicesmountedinship
body,andthe
datafiltering.
ACKNOWLEDGE
ThepaperissupportedbytheMinistryofScienceand
CultureofLowerSaxonyfortheGraduateSchoolSafe
AutomationofMaritimeSystems(SAMS).
REFERENCES
Skjetne, S., Smogeli, V., & Fossen, T. I. 2004. Modeling,
identification, and adaptive maneuvering of Cybership
II:Acompletedesignwithexperiments.IFACConference
on Control & Application in Marine System (pp.203 ‐208).
Ancona,Italy.
Xu,F.,Xiao,T.,etal.2014.IdentificationofNomotomodels
withintegralsamplestructure
foridentification.The33
rd
Chinese Control Conference (pp.6721‐6725). Nanjing,
China.
Åstrom, K., & Kållstrom, C. 1976. Identification of ship
steeringdynamics.Automatica,9‐22.
Barford, L. A., Fazzio, R. S., & Smith, D. R. 1992. An
introduction to wavelets. Hewlett‐Packard Labs, Bristol,
UK,Tech.Rep.HPL‐92‐124,1–29.
Fossen, T.
I. 2011. Handbook of Marine Craft Hydrodynamics
andMotionControl.Wiley.
Jiang,Z.,Yan,W.,Jin,X.,&Gao,J.2012.Identificationofan
Underactuated Unmanned Surface Vehicle. JOURNAL
OFNORTHWESTERNPOLYTECHNICALUNIVERSITY,
699‐705.
Ljung,L.2002.Recursiveidentificationalgorithms.Circuits,
Systems,andSignalProcessing,57‐68.
Ljung,
L.2010.Perspectivesonsystemidentification.Annual
ReviewsinControl,1‐12.
Luo, W. L., & Zou, Z. J. 2009. Parametric Identification of
Ship Maneuvering Models by Using Support Vector
Machines.JournalofShipResearch,19‐30.
Luo, W., & Cai, W. 2014. Modeling of ship maneuvering
motion using optimized
support vector machines.
Intelligent Control and Information Processing (ICICIP),
2014 Fifth International Conference on (pp. 476‐478).
Dalian:IEEE.
Perera,L.P.,Oliveira,P.,&GuedesSoares,C.2015.System
Identification of Nonlinear Vessel Steering. Journal of
OffshoreMechanicsandArcticEngineering,31302‐31307.
Rajesh, G., & Bhattacharyya, S. K.
2008. System
identification for nonlinear maneuvering of large
tankers using artificial neural network. Applied Ocean
Research,256‐263.
29
Sutulo, S., & Guedes, Soares, C. 2014. An algorithm for
offlineidentificationofshipmanoeuvringmathematical
modelsfromfree‐runningtests.OceanEngineering, 10‐25.
Shi,C.,Zhao,D.,Peng,J.,&Shen,C.2009.Identificationof
Ship Maneuvering Model Using Extended Kalman
Filters. TransNav, the International Journal on Marine
NavigationandSafetyofSeaTransportation,105‐110.
Vapnik,V.2000.Thenatureofstatisticallearningtheory.New
York:Springer.
Wang, X., Zou, Z. J., & Yin, J. 2013. Modular Parameter
IdentificationforShipManoeuvringPredictionBasedon
Support Vector Machines. The Twenty‐third (2013)
InternationalOffshoreandPolarEngineering
(pp.834‐839).
Anchorage, Alaska, USA: the International Society of
OffshoreandPolarEngineers.
Wang,Y.‐H.,Yeh,C.‐H.,Young,H.‐W.V.,Hu,K.,&Lo,M. ‐
T. 2014.On the computational complexity of the
empirical mode decomposition algorithm. Physical A:
StatisticalMechanicsanditsApplications,159‐167.
XU,
F.,CHEN,Q.,ZOU,Z.‐J.,&YIN,J.‐C.2014.Modeling
of underwater vehicles’ maneuvering motion by using
integral sample structure for identification. JOURNAL
OFSHIPMECHANICS,211‐220.
Xu, F., Zou, Z. J., Yin, J. C., & Cao, J. 2012. Parametric
identification and sensitivity analysis for Autonomous
Underwater
Vehicles in diving plane. Journal of
Hydrodynamics,744‐751.
Zhang,X.G.,&Zou,Z.J.2011.IdentificationofAbkowitz
model for ship manoeuvring motion usingε‐support
vectorregression.JournalofHydrodynamics,353‐360.
Zhang, X.‐G., & Zou, Z.‐J. 2013. Estimation of the
hydrodynamic coefficients from captive
model test
results by using support vector machines. Ocean
Engineering,25‐31.
