305
1 INTRODUCTION
Coursefollowingandkeepinginrestrictedwatersis
usuallyforshipberthingtodockoravoidingcertain
obstacleslikeotherships orconstructions(as shown
inFigure1)inthewaterways.Theprobleminvolves
notonlythecourseplan,butalsothecoursekeeping
performance in confronting the
disturbance of the
ship‐bank interaction. When a ship moves in
proximitytoabank,itwillexperienceasuctionforce
towards the bank and a yawing moment, which is
called ship‐bank interaction or bank effect. Many
publishedstudies presented the way to estimate the
bankinducedforces(Norrbin
1974,Ch’ngetal.1993),
andalsopointedthattheship‐bankinteractioncanbe
quite huge to influence the ship’s course keeping
ability as well as directional stability (Fujino 1968,
Sanoetal.2014,Liuetal.2016).
The under‐actuated nature of the ship
manoeuvringproblems,namely with more
variables
tobecontrolledthanthenumberofcontrolactuators,
makesthecontrolproblemquitechallenging(Fossen
2003). Such problem has received a lot of attention
from the control community. Pettersen & Nijmeijer
(2001) provided a high‐gain, local exponential
tracking result. By applying a cascade approach, a
global tracking
result was obtained in Lefeber et al.
(2003).Pathfollowing approachbasedontheline‐of‐
sight method was proposed by Moreira et al (2007)
and then presented in more researchers’ works
(Børhaugetal.2008,Skjetneetal.2011).However,the
performance of control systems is limited by the
constraints
onthecontrolinputs.Theaforementioned
workshasnottakenthelimitationsintoaccount.
Ship Course Following and Course Keeping in Restricted
Waters Based on Model Predictive Control
H.Liu,N.Ma&X.C.Gu
ShanghaiJiaoTongUniversity,Shanghai,China
ABSTRACT:Shipnavigationsafetyinrestrictedwaterareasisofgreatconcerntocrewmembers,becauseships
sailingincloseproximitytobanksaresignificantlyaffectedbytheso‐calledship‐bankinteraction.Thepurpose
ofthispaperistoapplytheoptimalcontroltheory
tohelphelmsmenadjustships’courseandmaintainthe
targetcourseinrestrictedwaters.To achievethisobjective,themotionofa verylargecrudecarrier (VLCC)
closetoabankismodeledwiththelinearequationsofmanoeuvringandtheinfluenceofbankeffectonthe
ship hydrodynamic force
is considered in the model. State‐space framework is cast in a Multiple‐Input
Multiple‐Output(MIMO)system,wheretheoffset‐freemodelpredictivecontrol(MPC) isdesignedforcourse
followingandthelinearquadraticregulator(LQR)isusedforcoursekeeping.Simulationresultsshowthatthe
controlmethodseffectivelywork
inshipcoursefollowingandcoursekeepingwithvaryingship‐bankdistances
andwaterdepths.Theadvantageofadoptingspeedvariationasthesecondcontrolinputisobvious.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 12
Number 2
June 2018
DOI:10.12716/1001.12.02.11
306
Figure1.Trajectorycontrolforavoidanceinawaterway
The optimal control theory considering physical
constraintswasputforwardtosolvetheproblemsof
ship path planning (Djouani & Hamam 1995).
Researches based on optimal theory have designed
the ship path controller using methods like linear
quadratic regulator (LQR) (Thomas & Sclavounos
2006, Mucha & el Moctar 2013) and evolutionary
algorithms(Szłapczyński2013).Themodelpredictive
control (MPC) approach, which allows multiple
control inputs to response to theshipmotion in the
under‐actuated system, has been used for path
following (Oh & Sun 2009, Li et al. 2010), and
improved with disturbance compensating unit to
apply to
ship heading control (Li & Sun 2012). A
significantworkbyFengetal.(2013)introducedthe
LQR to the MPC scheme to improve the robust
control quality, so that the system can achieve the
offset‐free path following under external
disturbances.
Enlightened by the previous work, this paper
proposeda
modifiedMPCschemefortheshipcourse
following in close proximity to a bank, which
combines the LQR to overcome the bank induced
forcesandmaintaintheshipcourse.Therationaleof
the MPCscheme aiming atcourseplanning and the
introduced unit of LQR for course keeping and
system
stabilization is elaborated in section 2. The
illustration of bank induced forces and the
corresponding hydrodynamic derivatives is given in
section 3. In section 4, the performance of the
controller is evaluated on the condition of different
ship‐bank distances, andthe conclusion is drawnin
thesection5.
2 NUMERICAL
MODEL
2.1 Maneuveringtheory
Figure2showsthecoordinatesystemsaswellasthe
variables used in the equations of ship motion. The
earth‐fixed coordinate system O
0‐ξηζ and ship fixed
coordinatesystemO‐xyzareright‐handedcoordinate
systems with the positive ζ and z axispointing into
thepageandtheoriginOatthemid‐shippoint.The
shipinitiallymovesinthedirectionoftheξaxiswith
speed, U. Resulting from the bank
effect, the ship’s
motion is defined as the vector [u , v, r], and the
headingangleψaswellasthedriftangleβappears.δ
denotes the rudder angle and y
bank denotes the ship‐
bankdistance.
Figure2.Coordinatesystems
Smalldeviationinswayandyawmotioniscaused
by the rudder deflection and wind/bank forces.
Therefore,the equationof surgemotion isneglected
and t the 2 ‐DoF linear manoeuvring model of ship
horizontalmotionaregivenas:
yxG
mmv mmurmxrY
(1)
zz G
I
Jrmxvur N
(2)
wheremistheshipmassandI
zistheyawmomentof
inertia.m
x,my andJzare theadded massand added
moment of inertia for the surge, sway and yaw
motion. Y and N represent the hydrodynamic sway
force and yaw moment acting on the ship, which
include hydrodynamic inertia terms and will be
expressedbythefollowingpolynomialequations.
=
+
rvr
YYrYvYrY Y Y
0
(3)
=0
+
vvr
NNvNvNrN N N
(4)
Hereintheforceandmomentcausedbythebank
effect are expressed as
0
YY
and
=0
NN
.
Thesubscript“η=0”meanstheconstantbankinduced
forces on the initial lateral position, and
Y
and
N
means the force due to the ship’s lateral
displacementη.Thefouritemsarecalledasymmetric
derivatives.
All variables in the Equations 1 and 2 above are
nondimensionalizedintermsofshiplengthL,draftT,
speedUandwaterdensityρthroughtheequationsas
follows.
, , ,
0.5 0.5
, , , = ,
,
0.5 0.5
z
z
G
G
I
mvLrL
mI vr
LT LT U U
x
uvrL
uvr x
UUUL L
YN
YN
LTU L TU
2
2422
222
(5)
Substituting Equations 3 and 4 into Equations 1
and 2, and nondimensionalizing the equations, the
equationsofshipmanoeuvringis:
307
RB
vv
rr
MNLF F
(6)
where
=
v
r
G
v
Grz
Ym
Ymx
Nmx
NI
M
(7)
v
rv
r
G
Ym
Y
N
mx
N
N (8)
0
0
RB
Y
Y
Y
NNN
LF F
(9)
With variables v, r and η existing in (6), the
heading angle ψ is added and the following
expressionsarededuced.
11
1
0
0
01 0 0
10 1 0
R
B
vv
rr
MN ML
MF F
(10)
Since this dynamic system accepts a small scale
variation of speed that is defined asΔu, the bank
inducedsuctionforceandyawmomentinrelationto
forward speed can be expressed in the following
form.
2
0
22
0
0.5
0.5 2
bank
FLdYUu
LdY U U u u
(11)
2
2
0
22 2
0
0.5
0.5 2
bank
FLdMUu
LdM U U u u
(12)
Thelasttermintheequationsaboveis ofsecond
order and negligible, while the second term directly
showsthecontributionoftheperturbationvelocityΔu
to the bank induced forces. Keeping only leading
order terms in the perturbation, the matrix of ship‐
bankhydrodynamicsisrewrittenas:
2
2
2
22
0
0
0
0
0.5
2
0.5
1 2
bank
B
bank
F
LdU
UUu
M
U
LdU
u
Y
N
Y
N
F
(13)
Nowthelinearizedmanoeuvringmodelisputinto
thestateequationwithmatrixform:
xAxΒuE
(14)
where
,
TT
vr u
xu
(15)
11
0
0
01 00
10 01
MN ML
A
(16)
1
1
2
, = 0
00
0
00
B
RB
MF
MF F
BE
(17)
2.2 Coursefollowingcontrol
To simulate the course following process in the
discrete‐time scheme, the continuous‐time model of
Equation14isfirstdiscretizedas:
1kdkdkd
xAxBuE (18)
The standard MPC scheme can be formulated
based onthediscretemodelbut unable to eliminate
the steady cross‐track error due to the existing
disturbance of bank effect. So the MPC scheme
proposedhereisderivedfromthesteadystatesystem
equationsunderthesteadydisturbanceE
d,
ddd
d
xAxBuE
yCx
(19)
whereu
∞isthesteady statecontrol; y∞is the output
that is controlled to approach the desired value.
Consideringthefollowingequation:
1
11
dddddd
II
uCABCAE
(20)
The current error dynamic formulation,in which
A
dhastwoeigenvaluesat1,leadstoasingular(I‐Ad)
matrix.Inthiscasethecontrollercannotgiveprecise
308
response to balance the bank induced forces. To
resolvethisproblem,theoriginalsystemneedstobe
stabilized first through feedback while the MPC
schemedeterminestheoptimalcontrolinputateach
time step.In this study, the stabilizationisachieved
by using LQR controller. The cost function and
weighting
matricesfortheLQRcontrollerarechosen
tobethesameasfortheMPC.Morespecifically,the
control input at each time step is divided into two
parts:
MPC LQR MPC
kk k k LQRk
uu u u Kx (21)
Andthemodifiedsystemdynamicswillbe:
1
MPC
kdkdk LQRk
M
PC MPC
d d LQR k dk dk dk
xAxBu Kx
A BK x Bu Ax Bu
(22)
Withthestablesystem,thecorrespondingcontrol
input to overcome the ship‐bank interaction forces
canbecalculatedby:
1
11
MPC
dddddd
II
uCABCAE(23)
Itcanbeseenfrom Equation 22and Equation23
that the function of LQR includes: 1) to avoid the
singular matrix (I‐ A
d) as the system varies at each
timestep;2)tostabilizethepartofcoursekeepingin
thecontrolsystemsothattheMPCalgorithmcanbe
runinanoffset‐freesituation.
As to the standard MPC scheme, first the cost
functionforminimizationis:
1
0
;
P
N
MPC T
k k kj kj
j
T
MPCMPC MPCMPC
kj kj
J
Ux xQx
uu Ruu
(24)
11
,,,
P
MPC MPC MPC MPC
kkk kN
Uuu u
(25)
N
P is the prediction horizon; Q and R are the
weightingmatricesfor thestatesand controlinputs,
respectively.
M
PC
k
U
istheoptimalcontrolsequence,in
which the
M
PC
k
u
only is remained by the MPC
schemeastheactualcontrolinputtobeimplemented.
Notice that
M
PC
u
is included in the costfunction to
achieve offset‐free path following. The optimal
controlsequenceissubjectto:
max max
LQR MPC LQR
kj kj kj
uu u uu (26)
1max 1
1max 1
MPC LQR LQR LQR
kj kj s kj kj
MPC LQR LQR LQR
kj kj s kj kj
T
T
uu u uu
uu u uu
(27)
where j=0, 1 ,…,N
P; umax is the actuator constraint;
Δu
maxistheconstraintsontherateofactuatorchange;
T
sisthesamplingtime.
3 ASYMMETRICHYDRODYNAMICFORCES
Table 1 lists the principal dimensions of the test
model KVLCC2, which is a crude oil tanker as a
benchmark for study of ship hydrodynamics. The
PMMtestwasconductedintheCWCatShanghaiJiao
Tong University. The dimensions of measuring
section are 8.0m×3.0m (water width) ×1.6m (water
depth). In this experiment, the ship was placed
laterallyoffthecenterlineoftheCWCwithdifferent
displacements η=0m‐0.9m, corresponding to
y
bank=2.8B‐0.8B.Theconditionofwaterdepthincludes
h=1.25T,h=1.5Tandh=10.0T.
Table1.PrincipledimensionsoftheKVLCC2
_______________________________________________
ParametersFullscale Model
_______________________________________________
Lengthbetweenperpendiculars L m 320.00 2.4850
BreadthBm 58.00 0.4504
DesigndraftTm 20.80 0.1615
DisplacementΔm
3
312540 0.1464
LCBfromMid‐shipxBm11.04 0.086
Scale128.77:1
_______________________________________________
Figure 3 plots the nondimensional asymmetric
force Y’ and yaw moment N’ versus lateral
displacementinthreewaterdepths.Asthesignofthe
values indicate, the ship experiences a suction force
towardsthebankandabow‐outmomentturning the
bowoffthebankside.Theforcesincreaseas
theship
movestothe bank,andtheamplitudesoftheforces
soar asthewaterdepthdecreases.Eachpoint is the
valueof
0
Y
or
0
N
atthecorrespondingposition.
The asymmetric derivatives
Y
and
N
are
analyzed using the data surrounding the point. The
asymmetric derivatives
Y
and
N
at ybank=1.7B in
thethreewaterdepthsarepresentedinTable2.
Figure3. Asymmetric hydrodynamic forces versusη’ in
differentwaterdepths.
Table2.Asymmetricderivativesaty
bank=1.7Binthreewater
depths
_______________________________________________
h/T=1.25 h/T=1.5 h/T=10
_______________________________________________
Y
0.0608 0.0463 0.0045
N
‐0.0235‐0.0161‐0.0007
_______________________________________________
0.00.10.20.30.4
-0.012
-0.009
-0.006
-0.003
0.000
'
Exp Fit
h=10T
h=1.5T
h=1.25T
N'
0.0 0.1 0.2 0.3 0.4
0.000
0.008
0.016
0.024
0.032
0.040
Exp Fit
h=10T
h=1.5T
h=1.25T
Y'
'
309
4 SIMULATIONOFCOURSECONTROL
The controller focuses on the trajectory following
problemthatisillustratedinFigure1.FirstlytheMPC
schemeistunedintheconditionofh/L=0.5wherethe
bankeffectcanbeignored.Theaimoftheschemeis
tomaketheshipproceed
onaparallelcourseto the
bankwithalateraldistanceoffromtheship’sinitial
course.Theactuatorconstraintsareu
max=[0.5240.08]
T
andΔu
max=[0.210.03]
T
;thesamplingtimeTs=1sec.
4.1 Controllertuning
Thelengthofpredictionhorizonisfirsttunedbythe
simulationsas shownin Figure4.It revealsthat the
simulation results converge to the disired lateral
valuewhenN
Papproaches to20.Andthisachievesa
goodbalancebetweenthecoursecontrolperformance
andthetimeconsumption.Sothelengthofprediction
horizon is set as N
P =20 for further tuning of the
weightingmatrices.
TheweightingmatricesQandRshapetheclosed
loopresponsetoachievethedesired performancein
theformofQ={0,0,q
1,q2}andR={r1,r2}.Thetuningfor
thematricesisactuallythetrial‐and‐errorprogramto
findtheoptimumratiobetweenq
1andq2aswellas
the ratio between r
1 and r2.The ratio q1/q2 sets the
preference for the controller to eliminate the cross‐
trackerrorswhiletheratior
2/r1setsthepreferencefor
the controller to use one actuator over the other.
Firstlyq
1=10andr1=1,r2=2areselectedwhilethevalue
of q
2isvaried to examine the simulationresults. As
showninFigure5,thecontrolschemeeliminatesthe
cross‐trackerroreffectivelywiththerangeofq
1/q2=1
to10,butthepathfollowingperformancedeteriorate
when q
1/q2 reach 100. The amplitude of speed
reduction also increases as the value of q
2 grows.
Compared totheratio q
1/q2=10, the system succeeds
in yielding a smaller fluctuation of the speed
reductionduring thewholeprocessoftheshipcourse
adjustment and stabilization when q
1/q2=1. So
q
1=q2=10ischosenasthevalueoftheQmatrix.
Finally,thetuningofr
2/r1isconductedbyvarying
r
2 with the value of r1 kept at 1. The results are
presented in Figure 6. In the range from r
2/r1=0.5 to
r
2/r1=10,the simulation results of ship trajectory and
therudderreflectiondonotchangequitelarge,which
proves the finding by Feng et al. (2013) that the R
matrix has less influence on the course changing
performancethantheQmatrix.Buttoosmallr2will
putlargerweightontheusage
ofshipspeedchange
and cause the failure of the speed variation
constraints. As the ship approaches to the desired
pathwiththeleastovershootrudderangleattheratio
r
2/r1=2, the value r1=1and r2=2 is selected for the R
matrixinthefollowingsimulations.
Figure4. Simulations of the course following for different
predictionhorizons
Figure5. Simulations of the course following for different
ratiosbetweenq
1andq2
0 10203040
0.00
0.05
0.10
0.15
0.20
0.25
/L
/L
N
P
=10
N
P
=20
N
P
=40
desired position
0 20406080100120
-0.10
-0.05
0.00
0.05
0.10
u'
time (s)
N
P
=10
N
P
=20
N
P
=40
0 20406080100120
-2
0
2
4
6
8
10
(°)
time (s)
N
P
=10
N
P
=20
N
P
=40
0 20 40 60 80 100 120
-0.20
-0.15
-0.10
-0.05
0.00
u'
time (s)
q
1
/q
2
=0.1
q
1
/q
2
=1
q
1
/q
2
=10
q
1
/q
2
=100
0 10203040
0.00
0.05
0.10
0.15
0.20
/L
/L
q
1
/q
2
=0.1
q
1
/q
2
=1
q
1
/q
2
=10
q
1
/q
2
=100
desired position
0 20 40 60 80 100 120
-2
0
2
4
(°)
time (s)
q
1
/q
2
=0.1
q
1
/q
2
=1
q
1
/q
2
=10
q
1
/q
2
=100
310
Figure6. Simulations of the course following for different
ratiosbetweenr
1andr2.
4.2 Simulationincloseproximitytobanks
The tuned scheme is further used for the course
following of KVLCC2 ship from the initial course,
which is y
bank=1.7B and ybank=1.35B from the bank, to
thetargetcoursethatis0.16Llaterallyfromtheinitial
coursein thewater depth ofh=10T. The simulations
areshowninFigure7andFigure8,respectively.The
combinationofrudderdeflectionandspeedvariation
as Multiple‐Input Multiple‐Output control (denoted
as MIMO in the figures) is compared with the
performance of rudder‐control only (denoted as
Rudder‐only). The results show that the favourable
predictionhorizonandweightingmatriceschosenfor
multipleinputsystemalsoworkforthesinglerudder
case. The steady state error of η/L for two control
schemesindicatesthatMIMOcaneleminatethecross‐
trackerrormoreeffectivelythantheschemethatuses
the rudder only. This is because that the speed
variation as the second input improves the course
controlperformance.It isalso noticeablethatdueto
the offset‐free scheme, there will be
nonzero steady
rudder angle as well as a certain value of speed
reductiontoovercomethebankinducedforces.
Next, the effect of MIMO combined with offset‐
freeMPConcoursefollowingandkeepingischecked
for different water depths, as presentedin Figure 9.
Theinitialcourseislocatedat
ybank=2.8Bandthetarget
course is 0.16L laterally from the initial course. The
steady‐state error in the lateral position becomes
larger as the water depth decreases, owing to the
rapidly increasing bank induced forces in shallow
water. In the subplots of the control output, the
amplitudeoftherudder
angleresponsesignificantly
increases, and the speed reduction reaches the
saturation point ofΔu’
max=0.08, indicating that the
demand of speed reduction is more than the upper
limit of the actuatorconstraint. The possible way to
eliminate steady‐state errors is re‐tuning the
weighting matrix Q and R and turn up the limit of
actuatorconstraint.
Figure7. Simulations of the course following in close
proximitytoabank(y
bank=1.7B)
Figure8. Simulations of the course following in close
proximitytoabank(ybank=1.35B).
0 10203040
0.00
0.04
0.08
0.12
0.16
0.20
/L
/L
r
2
/r
1
=0.5
r
2
/r
1
=1
r
2
/r
1
=2
r
2
/r
1
=10
desired position
0 20406080100120
-2
0
2
4
(°)
time (s)
r
2
/r
1
=0.5
r
2
/r
1
=1
r
2
/r
1
=2
r
2
/r
1
=10
0 20 40 60 80 100 120
-0.20
-0.15
-0.10
-0.05
0.00
u'
time (s)
r
2
/r
1
=0.5
r
2
/r
1
=1
r
2
/r
1
=2
r
2
/r
1
=10
0 10203040
0.00
0.06
0.12
0.18
/L
/L
y
bank
=1.7B, MIMO
y
bank
=1.7B, Rudder-only
desired position
0 20406080100120
-4
0
4
8
Rudder angle (°)
time (s)
y
bank
=1.7B, MIMO
y
bank
=1.7B, Rudder-only
0 20 40 60 80 100 120
-0.12
-0.08
-0.04
0.00
u/U
time (s)
y
wall
=1.7B, MIMO
0 10203040
0.00
0.06
0.12
0.18
/L
/L
y
bank
=1.35B, MIMO
y
bank
=1.35B, Rudder-only
desired position
0 20406080100120
-4
0
4
8
Rudder angle (°)
time (s)
y
bank
=1.35B, MIMO
y
bank
=1.35B, Rudder-only
0 20 40 60 80 100 120
-0.12
-0.08
-0.04
0.00
u/U
time (s)
y
wall
=1.35B, MIMO
311
Figure9. Simulations of the course following in
differentwaterdepths.
5 CONCLUSIONS
The paper proposes the modified MPC scheme to
solveshipcoursefollowingincloseproximitytothe
bank as well as counteracting bank effect. A linear
manoeuvring model including the hydrodynamic
componentsofbankinducedforces
isintroducedand
transformed to state equations for the design of
control system. To stabilize the error dynamics
formulation and to get a reliable solution of the
control response to bank effect, the LQR unit is
introduced into the system. And then the steady
control input to overcome the disturbance
due to
bankeffectisimplementedintotheMPCscheme.The
controller is tuned first to obtain the suitable
prediction horizon and weighting matrices for
improvingperformance.Then,simulationsofvarying
ship‐bankdistances areconducted.The feasibilityof
the offset‐free MPC scheme in course following for
ship proceeding close
to a bank is proved. The
advantageofadoptingspeedvariationasthesecond
control input is that the steady state errors under
ship‐bankinteractionsarereduced.Inshallowwaters,
thesteady‐stateerrorincreasesunderthesignificantly
enhancing ship‐bank interaction. Therefore, the
weighting matrix and the actuator
constraint are
suggested to be adjusted to provide higher control
inputs.
ACKNOWLEDGEMENT
TheresearchissupportedbytheNationalKeyBasic
Research Program of China: No. 2014CB046804 and
theChinaMinistryofEducationKeyResearchProject
“KSHIP‐II Project” (Knowledge‐based Ship Design
Hyper‐IntegratedPlatform):No.GKZY010004.
REFERENCES
Børhaug,E.,Pavlov,A.,Pettersen,K. Y.2008.IntegralLOS
control for path following of underactuated marine
surface vessels in the presence of constant ocean
currents. Proc. 47th IEEE Conference on Decision and
Control:4984‐4991.
Ch’ng,P.W.,Doctors,L.J.,Renilson,M.R.1993.Amethod
of calculating the
ship‐bank interaction forces and
moments in restricted water. International Shipbuilding
Progress40(421):7−23.
Djouani, K. & Hamam, Y. 1995. Minimum time‐energy
trajectory planning for automatic ship berthing. IEEE
JournalofOceanicEngineering20(1):4‐12.
Feng,P.Y., Sun, J.,Ma,N.2013.Pathfollowingofmarine
surface
vessels using bow and aft rudders in wave
fields. Control Applications in Marine Systems 9(1): 120‐
125.
Fossen,T.I.,Breivik,M.,Skjetne,R.2003.Line ‐of‐sightpath
following of underactuated marine craft. Proceedings of
6thIFACconferenceonmanoeuvring andcontrolofmarine
craft.Girona,Spain.
Fujino, M.
1968. Studies on manoeuvrability of ships in
restrictedwaters.SelectedPapersJournalofSocietyofNaval
ArchitectureofJapan124:51‐72.
Lefeber, E., Pettersen, K. Y., Nijmeijer, H. 2003. Tracking
control of an underactuated ship. IEEE Transactions on
ControlSystemsTechnology11:52–61.
Li,Z.,Sun,J.,Oh,S.2010.
Handlingrollconstraintsforpath
following of marine surface vessels using coordinated
rudder and propulsion control. American Control
Conference6010‐6015.
Li, Z. & Sun, J. 2012. Disturbance compensating model
predictive control with application to ship heading
control. Control Systems Technology IEEE Transactions
20(1):257‐265.
Liu, H., Ma, N.,
Gu, X. C. 2016. Numerical simulation of
PMMtestsforashipincloseproximitytosidewalland
maneuveringstabilityanalysis.ChinaOceanEngineering
30(6):884‐897.
Moreira, L, Fossen, T I, Soares, C G. 2007. Path following
control system for a tanker ship model. Ocean
Engineering34(14‐15):2074‐2085.
Mucha,
P., el Moctar, O. 2013. Ship‐Bank interaction of a
large tanker and related control problems. Proc. of the
32nd ASME International Conference on Ocean, Offshore
andArcticEngineering(OMAE2013).Nantes,France.
Norrbin,N.H.1974.Bankeffectsonashipmovingthrough
a short dredged channel. Proceedings of
the 10th
Symposium on Naval Hydrodynamics: 71−87. Cambridge,
USA.
Oh, S. R. & Sun, J. 2010. Path following of underactuated
marine surface vessels using line‐of‐sight based model
predictivecontrol.OceanEngineering37(2):289‐295.
Pettersen, K. Y. & Nijmeijer, H. 2001. Underactuated ship
tracking control: Theory and experiments.
International
JournalofControl74(14):1435–1446.
Sano,M.,Yasukawa,H.,Hata,H.2014.Directionalstability
of a ship in close proximity to channel wall. Journal of
MarineScienceandTechnology19(4):376‐393.
Skjetne, R., Jørgensen, U., Teel, A. R. 2011. Line‐of‐sight
path‐following along regularly parametrized curves
solved as a generic maneuvering problem. Proc. 50th
IEEEConferenceonDecisionandControl:2467‐2474.
0 20406080100120
0
5
10
15
(°)
t (s)
h = 10T
h = 1.5T
h = 1.2T
0 20406080100120
-0.10
-0.05
0.00
u'
t (s)
h = 10T
h = 1.5T
h = 1.2T
9 121518
0.158
0.159
0.160
0.161
/L
/L
0 6 12 18 24 30 36
0.00
0.05
0.10
0.15
0.20
/L
/L
h = 10T
h = 1.5T
h = 1.25T
desired position
312
Szłapczyński, R. 2013. Evolutionary sets of safe ship
trajectories with speed reduction manoeuvres within
trafficseparationschemes.PolishMaritimeResearch21(1):
20‐27.
Thomas, B.S. & Sclavounos, P.D. 2007. Optimal control
theoryappliedtoshipmaneuveringinrestrictedwaters.
JournalofEngineeringMathematics58(1):301‐315.
