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