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1
INTRODUCTION
Trackingashipʹspositionatseahasbeenoneofthe
most important tasks of sailors since ancient times.
Initially, sailors used natural clues such as the sun,
starsandoceancurrentstodeterminetheirpositionat
sea. As science and technology advanced, sailing
becamemorecomplicated,sailors began
touse tools
and technology that helped them more accurately
determinetheirpositionatseaandplantheirroute s.
In the 15th century, Portuguese sailors such as
BartolomeuDiasandVascodaGamatookadvantage
of ocean winds and currents to cross the Indian
Ocean. Between 1519 and 1522, Portuguese
sailor
FerdinandMagellan made thefirstcircumnavigation
oftheworld,whichbecameamilestoneinthehistory
ofsailingandnavigation.Inthe19thcentury,British
scientistandmathematicianGeorgeAirydevelopeda
system of geodetictriangulationthatallowedsailors
to accurately determine their position at sea using
astronomical observations. In
1837, Airy published
„Mathematical Tracts on the Lunar and Planetary
Theories, the Figure of the Earth, Precession and
Nutation, the Calculus of Variations, and the
UndulatoryTheoryofOptics”,wherehedescribedhis
system of geodetic triangulation. In 1960, the U.S.
militaryʹs TRANSIT satellite system became the first
satellite
system to enable maritime navigation. In
1978, the US military introduced the Global
Positioning System (GPS), which became a widely
used navigation tool at sea and on land. With the
developmentofcomputertechnologyandnavigation
software, routing a shipʹs passage has become
increasingly automated. Modern navigation systems
makeit
possibletoquicklyandaccuratelydeterminea
The Concept of Determining the Ship’s Route Based
on the Capability Plots
J
.Wnorowski&A.Łebkowski
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Everyyear,newvesselsequippedwithdynamicpositioning(DP)systemsarebuiltinshipyards
aroundtheworld.Duetotheincreasingnumberofoffshorevessels,aclienthiringavesselshouldanalysethe
vesselʹspositioningcapabilitychartstodeterminewhichwaterareasthevessel
isdesignedfor.Thesechartsare
representedaspolardiagrams.Inthecentreofthechartisashapesymbolisingtheshipʹsbody,andthevalues
onthechartrepresentthemaximumwindspeedthatcanaffecttheshipatagivenangle,atwhichthevessel
will maintain
its position.Vessel capability charts can also be usedby the crewduringthrustersfailuresto
determineatwhatangletothewinddirectionthevesselshouldstandtominimisetheimpactofwindforces.
Analysesthatdetermineavesselʹsabilitytokeeppositioncanbeperformedbyclassificationsocieties
orother
companies with approval from classification societies. The article presents the concept of a pathfinding
algorithmtodeterminestherouteoftheship’spassagewithminimalenergyconsumption.Thealgorithmuses
theinformationaboutenvironmentalforcesaffectingtheshipandinformationaboutthrustallocationobtained
fromCapabilityPlots
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 17
Number 1
March 2023
DOI:10.12716/1001.17.01.21
194
route based on data on a shipʹs position, weather,
currentsandotherfactorsaffectingnavigation[1–6].
An algorithm that can be used to determine the
pathofashipʹspassageistheA*(A‐star)algorithm.
This is a graph search algorithm that is used to
determinethe
shortestpathbetweentwovertices.In
thecaseofshippassagerouting,theA*algorithmcan
beusedtofindtheshortestpath,takingintoaccount
various constraints, such as the speed of the ship,
obstacles in the way or sea currents. In publications
[7–10],theauthorspresentedanexample
application
ofthealgorithmtodeterminethepassageroute of a
shiptakingintoaccountchangingweatherconditions
along the route. Another algorithm that works on a
similar principle to A* is the D* (D‐star) algorithm.
The difference between the two is that the D*
algorithmisusedwhenthe
obstaclesalongtheroute
are unknown, which means that the information
provided to the algorithm must be updated. An
exampleoftheuseoftheD*algorithmispresentedin
thepublication[11,12],wheretheauthorsshowhow
unmanned units can determine their own passage
routesindependently.
Anotherfamily
ofalgorithms are thoserelatedto
artificial intelligence. These include genetic
algorithms, artificial neural networks and swarm
algorithms [13–15]. A genetic algorithm is an
optimization method that simulates the processes of
population genetics. In the case of determining the
passagerouteofaship,thisalgorithmcanbeusedto
find
thebestrouteundercomplexconditions,taking
intoaccountanumberofcriteria,suchasminimizing
voyage time, minimizing costs, minimizing fuel
consumption, etc. In the publication [16, 17], the
authorspresent awaytodetermineashipʹspassage
route taking into consideration weather changes,
other ships on the
route and minimizing energy
consumption.Swarmalgorithmscan also be usedto
optimize a shipʹs passage route, taking into account
various external factors. These are metaheuristic
optimization methods that mimic the behavior of
flocksandswarmsinnature.Oneofthemorepopular
algorithmsbelongingtothistypeisan
algorithmthat
mimicsthebehaviorofantcolonies[18].
2
CAPABILITYPLOTS
In order to determine the ability of a DP‐equipped
vesseltokeepapositionundervariousenvironmental
conditions, Capability Plots are made. These charts
are presented in the form of polar diagrams. In the
centerofthechartisasymbolrepresentingtheshape
ofthevessel,while
thevaluesontheedgerepresent
the angle of the environmental forces acting on the
vessel. Figure 1 shows an example of this kind of
chart:
Figure1.TheexampleofCapabilityPlot
In the graph above, the green line indicates the
maximumvalueofwindspeed(inknots)thatcanact
on the vessel at a given angle to make it keep its
presetposition.Tocountertheenvironmentalforces,
thevesselusesthethrustersitisequippedwith.
Inordertomake
thediagraminFigure1,itisfirst
necessary to develop a mathematical model of the
environmentalforcesthataffecttheship.
2.1
Mathematicalmodeloftheenvironmentalforces
The mathematical model of environmental forces
consistsofthefollowingcomponents:
Windforceaffectingthevessel
Seacurrentforceaffectingthevessel
Seawavesforceaffectingthevessel
2.1.1
Thewindforceandtorque
In calculating the wind force on a vessel, it is
assumedthatthewindonlyaffectstheLPP/2pointof
thevessel(LPP‐LengthBetweenPerpendiculars).By
substitutingthewindspeedandwindangleintothe
following functions, the value of the wind force is
obtained,
actingontheX‐axisandY‐axisofthevessel
andthetorque.
wind w x w FW
X
qC A
(1)
wind w Y w LW
YqC A
(2)
wind w M w FW FW
MqC AH
(3)
where::q
w–windpressureontheship’shullfactor;
C
X–hullshapecoefficientforX‐axis;αw–windangle
relative to LPP/2 point [⁰]; A
FW – frontal projected
windarea[m
2
];CY‐hullshapecoefficientforY‐axis;
A
LW–longitudinalprojectedwindarea[m
2
];CM‐hull
shape coefficient for torque; H
FW – longitudinal
positionoftheareacenterofA
LW[m]
Theseacurrentforceandtorque
The force of the sea current and its torque are
determinedanalogouslytothewindforce(equations
1‐3).Thedifferencebetweenthemisthatinthecaseof
windforce,theabove‐waterpartoftheshipistaken
into account, and
in the case of the sea current, the
underwaterpart:
195
current c x C FC
X
qC A
(4)
current c Y C LC
YqCA
(5)
current c M C FC FC
M
qC A H
(6)
where: q
c – sea current pressure on the ship’s hull
factor;C
X–hullshapecoefficientforX‐axis;αc–sea
currentanglerelative toLPP/2point[⁰];A
FC–frontal
projected sea current area [m
2
]; CY‐hull shape
coefficientforY‐axis;A
LC–longitudinalprojectedsub‐
mergedcurrentarea[m
2
];CM‐hullshapecoefficient
for torque; H
FC – longitudinal position of the area
centerofALC[m]
Seawaveforceandtorque
Themathematicalmodelofseawavesdependson
the wave spectrum used. The most commonly used
spectra are JONSWAP [19] and Pierson Moskowitz
spectrum [20]. Classification society‐DNV [21]
recommendstheuse of theMoskowitz
spectrumfor
analyses of a shipʹs ability to keep its position. The
following is a mathematical model for determining
thecomponentsofseawaveforces:
,, '
wave c temp a WL surge
XqBhdirbowcfT
(7)
12
,, 0.09 ,,
temp a WL temp a WL temp
h dir bow C h dir bow C h dir
(8)
.45
1
0.8
Aa a
hbow bow
(9)
2
1
0.7 , 0.85,1.15
BWL WLWL
hC CC
(10)
, 0
2, 2
temp temp
temp
temp temp
dir dir
dir dir
dir dir
(11)
11 11
,,
temp
temp a WL A a B WL A a
dir dir
hdir bowC hbow hC hbow
(12)
1
2
0.05 0.95 tan 1.45 1.75
temp temp
h dir dir dir
(13)
3
'
31
1, 1
, 1
T
if T
fT
Te ifT
(14)
'
0.09 sin
wave c OS temp sway
YqL dir fT
(15)
( 0.05 0.14
temp
wave wave Los OS
dir dir
M
YX L
(16)
'
0.33
0.9
z
surge
pp
T
T
L
(17)
'
0.5
0.75
z
sway
T
T
B
(18)
0.3125 0.62
sw
HV
(19)
0.741 0.536
zw
TV
(20)
where:B–maximumbreadthatwaterline[m];
bow
a – angle between the vessel x‐axis and a line
drawnforemostpointinthewaterlinetothepointat
y=B/4 on the water line [°]; C
WL – water plane area
coefficient;dir
temp – waves coming from direction[°];
L
os–longitudinaldistancebetweentheforemostand
aft most point under water [m]; X
Los – longitudinal
position of Los/2 [m]; L
pp – length between
perpendiculars[m];H
s–significantwaveheight[m];
V
w–windspeed[m/s]
Tounderstandtherelationshipbetweenmathematical
formulas 7‐20, it is necessary to read scientific
description prepared by classification society DNV
[21].
Afterdeterminingtheenvironmentalforcesthataffect
the ship, the next step is to find the distribution of
thrust forces on the individual thrusters.
The total
value of the thrust force coming from all the
propellers should balance external forces. To model
the propellers, a standard from the classification
societyDNVcanbeused,markedasDNV‐ST‐0111.
3
DETERMININGTHEROUTEOFTHESHIP
Capability Plots are made to determine in which
waters a ship can carry out positioning operations.
When simulations are carried out, data are obtained
ontheeffectsofenvironmentalforcesontheshipand
the distribution of thrust values to individual
thrusters.Thisdatacan
beusedtodeveloptheshipʹs
passage route taking into account additional
constraints,suchaslimitingenergyconsumption.
Thefirststageofdefiningthepassagerouteisto
specify the start and end points. Then divide the
selected area into points with the chosen resolution.
Figure 2 shows an
example of dividing area into
pointswiththestartandendpointsmarked.
196
Figure2. Example of division of the area into points with
prohibitedareashighlightedinred
After declaring the control point matrix, the
capability plot for each point should be determined.
Sinceasmallrangeofareaistakenintoaccount,the
environmental conditions at each point will be the
same,makingthecapabilityplotsthesameaswell.
Aftercalculatingthechartsoftheshipʹs
abilityto
keep its position, it is necessary to determine the
weights of the point‐to‐point transition. Since the
environmental force acting on the ship depends on
theangleofincidenceofwind,seacurrentandwaves
onthehull,thetransitionweightscanbedetermined
according to the
expected course at the next point.
Knowing the maximum value of the environmental
forces acting on the shipʹs hull, the weights can be
normalized.Figure3showsanexampleoftransition
weightsbetweenpoints.
Figure3. Weightsof transitionsbetween points depending
ontheroute
Figure 3 shows the first three waypoints for 3
differentshiproutesEachroutehasdifferentvaluesof
transitionweightstothenextpoint.Theydependon
theangleofincidenceofthewindonthehull.From
Figure1,itcanbeseenthatshipsholdpositionmost
poorly when
environmental forces act at about 90
degreesonthehull.Thisisbecausethelateralareais
the largest,which translates intothe greatest impact
ofenvironmentalforces.Forthisreason,theweights
inFigure3weredeterminedasfollows.Thecloserto
the 90‐degree angle the environmental forces
act on
theship,thehighertheweight.
Oncetheindividual transitionweightshave been
determined, it is necessary to find a route such that
thetransitionvalues,whenaddedtogether,comeout
aslowaspossible.InFigure3itcanbeseenthatthe
greenshipgotthesmallest
sum,whichtranslatesinto
the fact that on a given route, environmental forces
willaffecttheshiptheleast.
4
CONCLUSIONS
The article presents the concept of using Capability
Plotsto design a passage route between the starting
and ending points. The algorithm uses information
about the impact of environmental forces on the
vesselʹshulldependingonthecourseanddirectionof
wind, currents and wave action. Using the above
information, the weights of transitions between
successive waypoints can be determined. The route
with the smallest sum of weights will be the most
optimal in terms of the impact of environmental
forces, which translates in theory into the lowest
energyconsumption.
The method presented in the article is based on
graph search methods, so it can be used in
conjunction with other algorithms such as A* when
weather conditions are invariant. When weather
conditions are variable, new graphs of the abilityto
keep a position at fixed time intervals must be
calculated,sointhiscasetheD*algorithmwouldbe
a
bettercombination.
Currently,thealgorithmfordeterminingthethrust
of individual thrusters uses an element of linear
programmingtofindtheminimumthrustdefinedby
theconstraintfunctions.
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