89
1 INTRODUCTION
Ship’sintelligentcontrolandnavigationofunmanned
marinevehicleshasbeenstudiedformanyyears.One
of the most important objectives is finding an
optimumtrajectoryandspeedforashiptokeepaway
from both stationary obstacles and moving ships
whilethedistancetoitsdestinationistheminimum.
Several solutions have been int
roduced to solve this
problem, such as Game Theory, Genetic or
Evolutionary Algorithm and so on. Many studies
have achieved rather satisfactory results. An
evolutionary Planner/Navigator algorithm was
proposed by Smierzchalski early in 1999. In the
model, the problem was reduced to a dynamic
optimization under both dynamic and static
constraints.Themodelcanbeint
egratedinAutomatic
RadarPlottingAids(ARPA)tomakedecisionsupport
forseafarers.Smierzchalski,etal(2000)extendedthe
work and proposed a novelθEP/N++ model. The
main target is trying to reduce calculation and
searching for trajectory very quickly. By using the
algorit
hm, a safe and near optimum trajectory for
each involved ship can be found within1 minute.
When finding an optimum trajectory for ships, the
requirement from International Regulations for
PreventingCollisionatSea(COLREGs)shouldnotbe
neglected.Michael,etal(2006)studiedtheproblemof
unmannedma
rinevehiclesautonomousoperationby
multi‐objective optimization and interval
programming under COLREGs. An in‐field
experiment with two crafts was also carried out to
validatethemodel.Theworkdidnotconsidermulti‐
shipanti‐collisionproblem.Szlapczynski,etal(2011)
considered multi‐ship trajectory planning problems.
Instead of finding optimum tra
jectory for only one
ship,theirstudycanfindsafetrajectoriesforallships
involved to avoid all ship domains violation and
stationary constraints. In all the above studies, the
focuses were trajectory searching. However, ship’s
Ship Trajectory Control Optimization in Anti-collision
Maneuvering
J
.F.Zhang,X.P.Yan&D.Zhang
IntelligentTransportSystemsResearchCenter,WuhanUniversity ofTechnology,Wuhan,China
Engineering Research Center for Transportation Safety (Ministry of Education), Wuhan University of Technology,
Wuhan,China
ResearchandDevelopmentBaseonWaterwayTransportationSafetyandAnti‐pollutionofCJRDCMinistryofTransport,
WuhanUniversityofTechnology,Wuhan,China
S.Haugen
Department of Production and Quality Engineering, Norwegian University of Science and Technology, Trondheim,
Norway
ABSTRACT:Alotofattentionisbeingpaidtoship’sintelligentanti‐collisionbyresearchers.Severalsolutions
havebeenintroducedtofindanoptimumtrajectoryfor ship, suchasGameTheory,Genetic orEvolutionary
Algorithms and so on. However, ship’s maneuverabilit
yshould be taken into consideration before their real
applications.Ship’strajectorycontrolinanti‐collisionmaneuveringisstudiedinthispaper.Atfirst,asimple
linearshipmaneuverabilitymodelisintroducedtosimulateitsmovementunderdifferentspeedandrudder
angle.Afterthat,ship’strajectorycontrolisstudiedbyconsideringthedurationofrudder,operationdistance
to turni
ng points, and maximum angular velocity. The details for algorithm design are also introduced. By
giving some restrictions according to the requirements from COLREGs, the intervals for rudder angle in
differentcircumstancescanbedeterminedbasedonthecurves.Theresultscangiveverymea
ningfulguidance
forseafarerswhenmakingdecisions.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 1
March 2013
DOI:10.12716/1001.07.01.11
90
maneuverability should also be taken into
consideration, especially in turning points. So the
resultscannotbeusedinrealitydirectly.
In term of ship maneuverability, Proportion
Integration Differentiation (PID) control is widely
used. There are also a lot of ship control
methodologies under different kind of desaturations
suchaswind,wave,andcurrentandsoon.Zha
ng,et
al (2002) proposed a trajectory control system called
closed‐loop gain shaping. Sliding model, including
back‐stepping (Lin, et al, 2000) and fuzzy sliding‐
mode (Yuan, et al, 2011) is another course control
algorithm. Soda et al (2012) studied numerical
simulationofship’snavigationundertheinfluenceof
windandwave.Inthosemodels,thecharact
eristicsof
disturbance signals are unknown. A series of fuzzy
logic are used to make decision inferences. The
models perform well even in nonlinear systems and
prioriknowledgeisnotneeded.Intheabovestudies,
a lot of attentions are pa
id to course control. But in
anti‐collisionmaneuvering,shipshouldnotonlyalter
to its target course, but also should keep close to its
planned tra jectory. By taking the problem into
consideration,apathcontrollingsystemwasdesigned
by Fossen, et al (2003) by minimizing the difference
between designed and act
ual speeds and the track
error simultaneously. The operations are totally on
automaticways.
It must be admitted that most anti‐collision
operations are still carried on by seafarers. So it is
necessary to find some regular patterns when ships
are altering their courses by rudder angle operation.
By doing so, seafarers decision ma
king could be
supported, which would help them make better
performance. This paper will focus on ship’s
trajectory control in turning points during anti‐
collision maneuvering. At first, a ship
maneuverability model is built to simulate its
movementunderdifferentvelocityandrudderangle.
After that, an algorit
hm is introduced to find the
relationships between rudder angle and other
parameters such as course alteration and time and
soon.
The rest of paper is organized as follows: In
section2,ashipmaneuverabilitymodelisintroduced
in detail. Section 3 will give algorithm design for
ruddercontrolduringcoursea
lteration.Casestudies
are carried out in section 4. Conclusions and future
works are summarized in section 5 and
acknowledgmentsaregiveninsection6.
2 SHIP’SMANEUVERABILITYMODEL
Asimplelinearshipmaneuverabilitymodelproposed
byNomoto(1957)wasusedinthispaper.Themodel
gives the relat
ionship between rudder angle and
angularvelocitybyusingthefollowingequation:
Tr r K
(1)
whererisship’syawrate,δisrudderangle.TandK
aretimeconstantandruddergainrespectively,which
should be obtained by field experiment for a typical
ship. Solving the above inhomogeneous differential
equation, the angular velocity at time t can be
obtainedbythefollowingequation:
-
0
() (1- )
t
T
rt K e
(2)
where δ
0 is ship’s initial rudder angle. Furthermore,
the course alteration during the period can be
computedby integratingangular velocityduring the
timeasfollow:
-
0
0
-
0
() (1- )
(- )
t
t
T
t
T
tKedt
KtTTe
(3)
where
()t
isship’scoursealterationafterperiodt.
Duringanti‐collisionoperation,therudderwillreturn
to midship aftersome period, so that the ship could
reach the target course gradually under its inertia.
Under this situation, ship maneuvering equation
changesintothefollowingversion:
0Tr r
(4)
By solving the above equation, the course
alteration after the rudder returning to midship as
follow:
--
00
0
() (1- )
tt
t
TT
tredtrTe
(5)
wherer
0isship’s angularvelocitywhenrudderbegan
to return to midship. It can be conveyed that when
ship’s course became stable (t→∞), course alteration
fromreturningtomidshipwouldber
0T.
3 ALGORITHMDESIGNANDCASESTUDY
3.1 Phasesforcoursealteration
A course alteration operation can be divided into
three phases. As can be seen in figure 1, ship can
changeitscourseby steering. In the firstphase until
t
1, the rudder is altered from midship to a certain
angle. In the second phase, from t
1 to t3 the rudder
angleremainsconstant.Inthelastphasefromt
3tot4
therudderanglereturntomidship.
Figure1.Demonstrationforship’scoursealteration
91
During the three phases, ship’s course alteration
canstillbedividedintothreesteps.Accordingtothe
course alteration curve, the course changes quickly
andthecurveisconcaveuntilt
2.Thecurvecomesto
beastraightlinebetweent
2andt3,whichmeansthere
is equilibrium between the torque from rudder and
flowresistance.Thecurvebetweent
3andt4turnout
to be convex, which means that course is altering
slowly. The angular velocity curve comes from the
differential of the curve course alteration. It is still
dividedintothreesteps.Itshouldbementionedthat
thereisatleastamaximumvalueforangularvelocity
accordingtoitscurve.Ofcourse,thethreepha
sesdo
not absolutely exist simultaneously in reality. It is
possiblethattheruddermayreturntomidshipbefore
ship’sangularvelocitybecomesstable.Thecurvesin
figure 1 are just conceptual demonstrations and
qualitative study will be carried out in next
subsection.
3.2 Algorithmdesign
It can be seen from figure 1 tha
t course alteration
curveisamonotonicfunctionwithrespecttotime.So
theideaofsqueezeruleisusedinthissection.
Table1showsthepseudocodeforthecalculation
oftimeneededinatypicalcoursealteration.Suppose
the rudder angle is δ and the course a
lteration in a
turning point is Ψ, then the time needed for the
ruddertokeeponthisanglecanbecomputedinthe
followingshownprocedure.
Table1.Pseudocodeforthecalculationoftimeneededina
typicalcoursealteration
_______________________________________________
function: Time=CourseAlteration(Ψ,δ)
_______________________________________________
Initialization:Tmin,Tmax,Tthre
whileT
max‐Tmin>Tthre
T
temp=(Tmax+Tmin)/2
Findcoursealterationbykeepingrudder
angleδforT
temp:Ψtemp
ifΨ
temp>Ψ
T
max=Ttemp
else
T
min=Ttemp
endif
endwhile
return(T
max+Tmin)/2
_______________________________________________
Intheabovepseudocode,theparametersT
minand
T
maxaredeterminedintuitivelytomakesurethatTime
is within them. T
thre is used to define precision. The
smallertheparameteris,thehighertheprecisionwill
be.EitherT
minorTmaxwillbeupdatedineachiteration.
After determining the parameter Time, another
parameter,whichiscalledoperationdistance, should
alsobedetermined.Theoperationdistanceisdefined
as the distance between turning point and the point
thattherudderoperationbegins.Asshowninfigure
2, if the operation distance is too large, ship’s route
willnotreachit
splannedroutewhenshipturnedto
the target course. If operation distance is too small,
ship’sroutewillsurpasstheplannedroute.
In order to determine the operation distance,
another pseudo code is also designed based on
squeeze rule, which is shown in ta
ble 2. The
parameter δ means rudder angle and Trajectory
includes all parameters that can explain the planned
route.
Figure2. Illustration of operation distance when
turningtoplannedroute
The procedure is quite similar with the pseudo
code in Table 1. It should be mentioned that the
parameterD
temp1willbepositiveifship’sactualroute
surpassthe plannedtrajectory. Or elseit isnegative.
EitherD
minorDmaxwillbeupdatedineachiteration.
Table2.Pseudocodeforthecalculationofdistanceneeded
inatypicalcoursealteration
_______________________________________________
function:D=Distance(δ,Trajectory)
_______________________________________________
Initialization:Dmin,Dmax,Dhre
whileD
max‐Dmin>Dthre
D
temp=(Dmax+Dmin)/2
Findthedistancebetweenship’spositionand
Trajectorywhencoursealterationisoverunderthe
circumstancethatrudderangleisδandoperation
distanceisD
temp:Dtemp1
ifD
temp1>0
D
max=Dtemp
else
D
min=Dtemp
endif
endwhile
return(D
max+Dmin)/2
_______________________________________________
4 CASESTUDIES
In this section, case studies by using the above
algorithmsarecarriedout.Withoutlossofgenerality,
theparametersinYuanetal(2011)areused.Intheir
studies, the time constant T is set to be 63.69, the
ruddergainKissettobe0.114andship’svelocityis
set to be 7.2m
/s. The model proposed in this paper
canbeextendedtoothertypesofvesselsbychanging
relevantparameters.AccordingtoCOLREGs,thegive
way ship should make early and substantial actions
so that other ship could notice. On the other hand,
althoughCOLREGsdonothavesimilarrequirements,
ship should also try to av
oid too large course
alteration because it needs to return to target course
after anti‐collision operation is over. When a large
coursealterationisinevitableincertainsituations,the
ship can consider avoiding collision by speed
alterationinstead.Ingeneral,thecoursea
lterationfor
a ship during anti‐collision operation is between 30°
and60°atmosttimes.
At first, the duration of rudder needed for
differentcoursealterationisobtained,whichisshown
in figure 3. The rudder angle varies from 2° to 20°
withanintervalof2°.Thema
ximumrudderangleis
usually 35° for many ships. However, full rudder
92
shouldbe avoided as far as possible in anti‐collision
operations. So the maximum rudder angle is
supposedtobe20°.
0 2 4 6 8 10 12 14 16 18 20 22
0
50
100
150
200
250
300
Rudder angle(°)
Duration of rudder(s)
Course alteration:30°
Course alteration:40°
Course alteration:50°
Course alteration:60°
Figure3. Relationship between rudder angle and the
durationfortheruddertokeepforcoursealteration
It can be seen from figure 3 that the duration of
rudderisdecreasingasrudderangleincreases,which
isconsistentwithintuition.Inreality,thedurationof
rudder should be large enough for seafarers to
respond. For instance, when course alteration is 30°,
therudderangleshouldnotbelargertha
n12°ifthe
responsetimeissettobe30s.That’stosay,thecurves
can provide an upper bound for rudder angle for
seafarersinanti‐collisionoperations.
0 2 4 6 8 10 12 14 16 18 20 22
400
500
600
700
800
900
1000
Rudder angle(°)
Distance to turning point(m)
Course alteration:30°
Course alteration:40°
Course alteration:50°
Course alteration:60°
Figure4. Relationship between rudder angle and the
distancebetweenoperationstartpointandturningpoint
Figure4showstherelationshipbetweenoperation
distance and rudder angle when course alteration
variesfrom30°to60°withanintervalof10°.
The curves also share a declining trend as well.
COLREGs require that early action is desirable in
anti‐collisionoperations,whichmeansthatoperation
distance should not be too small. Take course
a
lteration of 30° as an illustration, the rudder angle
shouldnotbelargerthan10°iftheoperationdistance
issettobe500m.Consequently,theabovecurves can
also provide an upper bound for rudder angle for
seafarersinanti‐collisionoperations.
COLREGs also require tha
t ship should take
substantial action to make other ships to be able to
notice its intension. The maximum angular velocity
can reflect ship’s action to a large extend. Figure 5
gives the curves of maximum angular velocity for
differentcoursealteration.
0 2 4 6 8 10 12 14 16 18 20 22
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Rudder angle(°)
Maximum angular velocity during turning(°/s)
Course alteration:30°
Course alteration:40°
Course alteration:50°
Course alteration:60°
Figure5. Relationship between rudder angle and the
maximumangularvelocityduringcoursealteration
It can be seen from the figure 5 that the curves
sharearisingtrend.Inreality,ship’sangularvelocity
shouldbelargeenoughforothershipstonotice.The
course alteration is also supposed to be 30°. The
rudder angle should be at least 8° if the maximum
angular velocity is no smaller tha
n 0.35°/s.
Consequently the above curves give a lower bound
forrudderangle. Bycombiningthe aboveevidences,
ship’s rudder angle should be between 8° and 10°
whencoursealterationis30°.Thisisjustanexample.
Rudder angle decision making under other
circumstancescanbemadeinsimilarways.
5 CONCLUSIONSANDFUTUREWORK
Inthi
spaper,ship’strajectorycontrolinanti‐collision
operation is studied. Due to the fact that most
operations during anti‐collision are carried out by
seafarers in most time, rather than totally
automatically, this paper tries to make a decision
supportsystemforant
i‐collisionoperations.Notonly
coursealterationisconsidered,butalsothetrajectory
deviation is taken into consideration. What’s more,
notice that COLREGs require that ship should take
early and substantial actions, the maximum angular
velocityisalsoconsideredtodecidetherudderangle.
Finally, an interval for rudder angle in course
a
lteration can be obtained, which can give guidance
forseafarers.
Ship’s movement is influenced by wind, current
andwave.Inthefuture,wewillfurtherstudy ship’s
trajectory control decision making support system
under uncertainty. What’s more, ship’s trajectory
controlisusuallydonebymakingrudderrectification
graduallyandsmallrudderoperationsareneededto
ma
ke course and route calibrations. Consequently,
further studies are still needed before its real
application.
93
ACKNOWLEDGMENTS
The work was supported by National Science
Foundation of China (NSFC) under grant Nos.
51209165 and 51179146, Marie Curie Actions (FP7‐
PEOPLE‐2012‐IRSES), the China Scholarship Council
(CSC), and Science and Technology project of
transportation from Ministry of Transport (MOT) of
ChinaundergrandNo.2011328201
90.
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