185
1
INTRODUCTION
AccordingtotheInternationalAssociationofMarine
Aids to Navigation and Lighthouse Authorities
(IALA‐AISM) [1], a domain is defined as the
operational zone around, above or below a vessel
withinwhichanincursionbyanotherfixedormoving
object, or another domain, may trigger reactions or
processes.
Domainsas
atoolforavoidingtheriskofcollision
are classified by Jurdzinski [2] as two‐dimensional
domainsinthe formof circular,elliptical, polygonal
and mixed forms and three‐dimensional domains in
theformofanellipsoidalandirregularsolid.
Goodwin[3], Davisetal. [4]andColleyetal.
[5]
conducted the first domain analyses. Detailed
analyses of domain properties in various applied
conditionswerecarriedoutbySzlapczynskietal.[6],
Starup[7],Chenetal.[8]andWang[9].Pietrzykowski
andWielgosz[10,11]andMarcjanetal.[12]presented
the use of domains for planning anticollision
maneuvers.
Domain
designbasedoninformationfromanAIS
system was carried out byHorteborn et al.[13] and
Zhangetal.[14],Smierzchalski[15]andZhuetal.[16]
made the first attempts to shape domains using
artificialintelligence.
Thus far, no comparison of the possible domain
shapesfortheoptimality
ofaplannedsafetrajectory
ofashiphasbeenperformed.
Thethesisofthispaperistoshowthatthroughthe
experimentalanalysisofdifferentdomainshapes,itis
possible to assess the optimality of cruise routes in
emergencysituations.
Thenewelementsofthispaper,contributingtothe
development
ofmethodsandtoolsforsafemaritime
transport,are:
Adetailedcomparativeanalysisofsafetrajectories
fordifferentdomainshapes,determinedaccording
tothemodifiedoptimallyneuralalgorithm,which
waspreviouslydevelopedbyLisowski[17];
Effect of Ship Neural Domain Shape on Safe and
Optimal Trajectory
J
.Lisowski
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Thisarticlepresentsthetaskofsafelyguidingaship,takingintoaccountthemovementofmany
othermarineunits.Anoptimallyneuralmodifiedalgorithmfordeterminingasafetrajectoryispresented.The
possible shapes of the domains assigned to other ships as traffic restrictions
for the particular ship were
subjectedtoadetailedanalysis.ThecodesforthecomputerprogramNeuro‐Constraintsforgeneratingthese
domains are presented. The results of the simulation tests of the algorithm for a navigational situation are
presented.Thesafetrajectoriesoftheshipwerecomparedatdifferentdistances,
changingthesailingconditions
andshipsizes.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 17
Number 1
March 2023
DOI:10.12716/1001.17.01.20
186
Ananalysisoftheoptimalityasafunctionofsafe
passingdistancesofshipsandthediscretizationof
thecomputercalculations.
This paper’s content is presented as follows: In
Section 1, we introduce a synthetic review of the
literature.Section2presentsamodelofthesafeand
optimal ship control
process. The optimally neural
algorithm for safe ship trajectory is described in
Section 3. Section 4 illustrates the results of the
algorithmsimulationstudies.Section5concludesthis
paperandlooksatfuturedirectionsforimprovement.
2
TASKOFSAFESHIPGUIDING
The description of the task of safely guiding ships
consistsofkinematicsanddynamicsequations,inthe
followingform:
f, , txxu
(1)
where
x are the measurable variables of the object
dynamics;x
1andx2arethecomponentsoftheshipʹs
position; x
3 is the ship’s course; x4 is the angular
velocity; x
5 is the speed; x6 is theacceleration of the
ship’smotion;andtistime.
Thestateandcontrolconstraintsare:
g0x
(2)
h0u
(3)
where
u is the control quantities: u1—rudder angle;
u
2—rotationalspeedorpitchoftheship’spropeller.
Contrarytothestabilizationoftheship’scourseat
small angles of rudder deflection, anticollision
maneuvering, according to Rule 8b COLREGs [17],
requires visibly larger changes of course at greater
rudderdeflections.
Therefore, nonlinear ship dynamics equations
wereusedhere.
On the other
hand, the state constraints contain
kinematic equations of the motion of passing ships.
Theseconstraints,intheirsimplestform,takeafixed
or variable shape generated by an artificial neural
network[18,19].
3
OPTIMALLYNEURALALGORITHMFORSAFE
SHIPTRAJECTORY
Dynamic programming with the following control
quality index can be used to solve the task of
calculatingthe optimal time to complete a safe ship
trajectory:
t
*
5
u
0
min x dt t
(4)
This leads to the determination of the ship’s
optimaltrajectory,safelyavoidingencounteredships
as the neural constraints of this process states, as
showninFigure1.
Figure1. Movement of various shapes of ship neural
domains in the shape of an ellipse, circle, hexagon and
parabola, illustrating the mapping of subjectivity of the
navigator assessing the risk of collision when passing
anothership.
Fourtypesofshipdomainshapes,inthe formof
an ellipse, circle, hexagon and parabola, were
examined and compared in this study. A pretrained
three‐layer neural network with hyperbolic tangent
andlogisticactivationfunctionswasusedtogenerate
thesedomains.
The MATLAB Neural Network Toolbox was
utilizedtosynthesize
theneuralnetwork,andanerror
propagation method with an adaptive learning rate
andmomentumwasemployedforitsadaptation.The
networkwastestedagainstvariouspracticalscenarios
and its expected response from approximately 350
experiencednavigators.
For each variant of the scenarios, a navigator
subjectively decided, in accordance with
their
previousseapractice, the approximatelyappropriate
change of course orship speed.The neural network
prepared in this way represented the average
experienceofalargegroupofnavigators.
The final synthesis of the optimally neural
algorithm combined dynamic programming as a
discrete op timization multistep method with the
neuralnetwork
mappingreal stateconstraintsofthe
collisionavoidanceprocesswithotherships.
Figure2illustratesthefunctioningoftheoptimally
neuralcontrolalgorithm,developedbytheauthorof
this paper, for determining safe route of further
voyagewhileavoidingmanyships.
Aninnovativesolutiontothetaskofdetermining
the
optimalandsafetrajectoryofashipwhenpassing
a larger number of other ships is the design of an
artificialneuralnetworkgeneratingshipdomainsand
theuseoftheBellmandynamicprogrammingmethod
tosynthesisthecontrolalgorithm[20‐22].
187
Figure2.Optimallyneuralalgorithmfordeterminingasafe
trajectory: V – own ship speed, s – number of stage, dd –
number of nodes in one stage, ii‐node number in the
previous stage, jj‐node number at the current stage, t*‐
optimal,shortestsafetrajectorytime.
A computer program in the MATLAB/Simulink
software representing the Neuro‐Constraints
procedure developed by the author of this work
(Algorithm 1) is shown on the right. The learning
material this neuralnetwork consisted of navigational
scenarios, 300 responses were recorded from experienced
navigatorsinARPAsystemtrainingcourses.
4
ALGORITHMSIMULATIONSTUDIES
The algorithm was subjected to simulation tests in
MATLAB/Simulink software for the scenario of
passingtenships(Table1andFigure3).
Table1.Navigationscenariomeasurementdata.
________________________________________________
Ship,Speed,Vk Course,k Distance,Dk Bearing,Nk
k (kn) (deg) (nm) (deg)
________________________________________________
0 16.2 0 00
1 15.8 261 7.945
2 10.5 120 4.2320
3 14.2 180 7.9349
4 12.8 351 4.562
5 9.3358 4.224
6 4.10 8.130
7 6.7180 6.7336
8 3.9180 3.8301
9 11.9 330 7.618
10
8.10 3.610
________________________________________________
First, Figures 4 and 5 show a comparison of the
safe trajectories in situations of good and restricted
visibility at sea for different shapesof ship domains
andwithoutdomains.
Figure3.Vesseltrafficnavigationscenario.
188
Figure4.Comparisonofsafetrajectoriesinthesituationof
goodvisibilityatseaford
s=0.5nm,fordifferentshapesof
shipdomains:(a)–ellipse,(b)–hexagon,(c)‐parabolaand
withoutdomains–(d).
Figure5.Comparisonofsafetrajectoriesinthesituationof
restrictedvisibilityatseaford
s=2.0nm,fordifferentshapes
ofshipdomains:(a)–ellipse,(b)–hexagon,(c)‐parabola
and(d)‐withoutdomains.
Then,inFigures6and7,acomparisonofthesafe
trajectorieswithcoursemaneuveringonlyandwhen
maneuvering the course and speed for different
shapes of ship domains and without domains was
carriedout.
189
Figure6. Comparison of safe trajectories in a situation of
restricted visibility at sea for d
s = 1.0 nm with course
maneuveringonlywithdV=0,fordifferentshapesofship
domains:(a)–ellipse,(b)–hexagon,(c)‐parabolaand(d)‐
withoutdomains.
Figure7. Comparison of safe trajectories in a situation of
restricted visibility at sea d
s = 1.0 nm when maneuvering
course and speed with dV = 25 %, for different shapes of
shipdomains:(a)–ellipse,(b)–hexagon,(c)‐parabola.
190
Figure8illustratesthetwo‐dimensionaloptimality
ofthesafetrajectorydependingonthediscretization
of the calculations and the required safe passing
distancefordomainformsasanellipse,hexagonand
parabola.
Figure8. The optimal time t* for the execution of a safe
trajectory depending on discretization s of calculations at
different safe distances ds for domains: (a) – ellipse, (b) –
hexagon,(c)‐parabola.
Finally,thedependenceoftheoptimaltimet*for
theexecutionofasafetrajectoryattheadvancetimet
a
of the overtaking maneuver, which was assigned to
the ship’s deadweight, is shown in Figure 9. The
knowledgefromthetheoryofautomaticcontrolwas
usedhere,andtheshipdynamicsarepresentedinthe
formofthemaneuveradvancetime.
Figure9. The dependence of the safe trajectory execution
time on the ship’s dynamics for various shapes of its
domains.
Theadvancetimetaconsistsofthecorrectiontime
ofthesetcoursechange,approximatelyequaltothree
timeconstantsoftheship’sdynamicsasthesubjectof
automaticcontrol.
The sensitivity of theoptimal time t*for thesafe
trajectory of the ship to changes in the safe passing
distance d
s and to the density s of the dynamic
programminggridrangesfrom38%to50%andisthe
lowestfordomainsintheformofaparabola.
On the other hand, the sensitivity of the optimal
time t* to complete the safe trajectory of the ship,
dependingonitssize
DWT,rangesfrom10%to35%
andisthehighestforhexagonaldomains.
Thealgorithm,inordertotakeintoaccount Rule
19COLREGstokeepoutofthewayofanothership
approaching from starboard, assigns port ships
circulardomainsofconstantradiusequaltothesafety
distanced
s,andstarboardshipsellipticalorhexagonal
orparaboladomainsofvariablesize,generatedbyan
artificialneuralnetwork.
5
CONCLUSIONS
This study explored the scientific and research
considerationsoftheoptimallyneuralalgorithmfora
ship’s safe path synthesis, taking into account the
various shapes of the passing ship domains, which
allowedforthefollowingconclusionstobedrawn:
Intermsofvisibilityatsea,ellipse‐shapeddomains
are the least sensitive in terms of extending the
safe trajectory when visibility conditions at sea
deteriorate;
In termsof the abilityto maneuver a ship, when
maneuvering only the ship’s course, the most
191
effective domainsare in the shape of a parabola,
andwhenmaneuveringthecourseandspeed,they
aredomainsintheshapeofanellipse;
Intermsoftakingintoaccountthesizeofaship,
thegreatestoptimalityisprovidedbyaparabola‐
shapeddomain;
In terms of the discretization of calculations,
hexagonaldomainsshowthegre atestsensitivity.
Futureresearchshouldconsider:
Testingthesensitivityofdeterminingasafepathto
inaccuracy and uncertainty of information
regarding the state of the process from various
measurementsources;
Modificationof thealgorithmtakingintoaccount
unforeseenmaneuveringofothervessels,contrary
to COLREGs, during the execution of a safe
trajectory;
Developing a version of the algorithm that takes
into account the specificity of ship navigation in
confinedwatersandinopenwaters.
ACKNOWLEDGMENT
This research was funded by a research project of the
GdyniaMaritimeUniversityinPoland,no.WE/2023/PZ/02,
“Controlalgorithmssynthesisofautonomousobjects”.
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