251
1 INTRODUCTION
Shipcollisionreferstothephysicaleventthattwoor
more ships occupy the same point or area on the
ocean surface at the same time, it is the overall
embodimentofthetemporalandspatialeffectsofthe
marinetraffic.Andshipcollisionriskindex,whichis
the elementary and import
ant concept related with
ships, is the researching focus of the marine traffic
fieldandisthe keytechnologyforthe realizationof
ship automation (Chen, 2011). To improve the
navigation security level and realize the economic
and green navigation, since the end of 1970s,
researchers have been researching the a
utomatic
collisionavoidancesystemandhaveproposedamass
of related calculation methods. However, there
always have been divergences on the calculation of
collision risk index in the encountering situation. In
thispaper,themodeloftheshipcollisionriskindex
space(generalizedspace)basedontheunificationof
tempora
l and spatial effects is proposed, which can
quickly realize the risk degree calculation in the
encountering situation of two or more ships, and
solvethetime‐seriescausalrelationships in the two‐
dimensional temporal and spatial space. And using
the collision risk index on the two‐dimensional
temporal and spatial space, ship tra
jectories can be
tracked,whichhasobviousphysicalsignificance.
2 REVIEWOFTHERESEARCHINGMETHODSOF
SHIPCOLLISIONRISKINDEX
Thereareseveralquantitativecalculationmethodsof
shipcollisionriskindex,whichareasfollows:
2.1 Macroscopicshipcollisionriskindex
Accordingtothetheoryofmarinet
rafficengineering,
by analyzing the ship encountering probability and
historical collision incidents within certain time
period in specific water area, the ship collision
Modeling of Ship Collision Risk Index Based on
Complex Plane and Its Realization
X.Xu
WuhanUniversityofTechnology,Wuhan,China
X.Geng&Y.Wen
WuhanUniversityofTechnology,Wuhan,China
HubeiKeyLaboratoryofInlandShippingTechnology,Wuhan,China
ABSTRACT: Ship collision risk index is the basic and important concept in the domain of ship collision
avoidance.Inthispaper,theadvantagesanddeficienciesofthevariouscalculationmethodsofshipcollision
riskindexarepointedout.Thentheshipcollisionriskmodelba
sedoncomplexplane,whichcanwellmakeup
forthedeficienciesofthewidely‐usedevaluationmodelproposedbyKearon.JandLiururuisproposed.On
this basis, the calculation method of collision risk index under the encountering situation of multi‐ships is
constructed,thenthethree‐dimensionalimageandspatialcurveoftheriskindexarefiguredout.Finally,single
chipmicrocomputerisusedtorealize themodel. And a
ttaching this singlechipmicrocomputertoARPAis
helpfultothedecision‐makingofthemarinenavigators.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 10
Number 2
June 2016
DOI:10.12716/1001.10.02.07
252
probabilityintheresearchedarea isdetermined and
the collision risk index of ships navigating in the
waterareaisfiguredout.(Wang,1998)
2.2 DCPAandTCPA
With the development of computer technology and
the application of ARPA, the two parameters of
DCPA (Distance to Closest Point of Approach)
and
TCPA (Time to Closest Point of Approach) can be
easilyacquired.(Zhaoetal.,1999)
2.2.1 DirectweightingmethodofDCPAandTCPA
In 1977 foreign scholar Kearon.J firstly proposed
the evaluation criteria and evaluation model of ship
collision risk in the international seminar of
mathematic problems in international
marine
navigation (Kearon, 1979). In this model, the ship
collision risk index is calculated as:
22
()()aDCPA bTCPA
,
a
、
b
are the
weightingcoefficientsofDCPAandTCPA,whenthe
coming ship is from the starboard, then
5a
、
0.5b
;whenthecomingshipisfromtheportside,
then
5a
、
1b
.
represents the collision risk
index in the encountering situation of the own‐ship
and target ship, and smaller
is, it will be more
dangerous.
DCPAandTCPAofthetargetshipcanbeacquired
using current ARPA, and caution circle of the own‐
ship can be established by setting the secure DCPA
(called DCPAs for short) and secure TCPA (called
TCPAs for short). However, determining the danger
alarm
onlybydividingthevaluerangeofDCPAand
TCPAhasgreatlimitations. In theevaluation model
ofshipcollisionriskindexproposedbyKearon.J,the
collision risk index is quantitatively calculated
as:
22
()()aDCPA bTCPA
, but the model has
severe deficiencies, even mistakes, which are as
follows:
1 The most severe deficiency of the model is the
dimension disunity of variables. DCPA is the
dimension of length, TCPA is the dimension of
time,however,weightingsummationofthemonly
involves the value, without consideration of
the
dimensions, which are in accordance with the
actualsituations.
2 The determination of the values of
a
、
b
only
considered which side the coming ship is from,
however, once the direction of the target ship is
determined, the value of
a
、
b
is determined.
Therefore, different weighting relationships of
TCPAandDCPA,thatis,thecontributionofTCPA
and DCPA to the value of
, under different
conditionsofthesamedirectioncan’tbereflected
3 The evaluation criteria is that the smaller
, the
more dangerous, which will easily cause
confusion.Becauseitisusuallyconsideredthatthe
larger the risk value is, the more dangerous will
be.
4 Wrongconclusionswillbeacquiredbythismodel
insome situations. Supposing that forshipA,its
DCPAiszeronmileandTCPA
is11min;forship
B,itsDCPAis2nmileandTCPAis1min(Zheng,
2010). And for both ship A and B,
5a
,
1b
.
Then according to the evaluation criteria of
Kearon,
121
A
,
101
B
, and
AB
,
which indicates that ship B is more dangerous
thanshipA.Whileinthenavigationpractice,ship
AismuchmoredangerousthanshipB.Therefore,
this evaluation model has its deficiencies, and
mistakes,andneedstobeimproved.
2.2.2 WeightingofDCPA,TCPAandsubjectivefactors
A new equation
was proposed to calculate the
collision risk index of the own‐ship, which is as
follows(LiuandHu,2012):
00
0
00
D
CPA TCPA
RS Tt
RT
(1)
InEq.1,
0
R
istheradiusofthesafedomainofthe
own‐ship;
0
T
depends on the last opportunity of
steeringruddertoavoidcollision.
Using the product of
0
0
D
CPA
Rs
R
and
0
0
D
CPA
Tt
T
to
definethecollisionriskindexisnotreasonable.Foran
example, when
0
D
CPA
RS
,
0
D
CPA
Tt
, the risk index is
zero;when
0
0,
DCPA TCPA
s
tT
,theriskindexiszeroas
well,thoughthevaluesoftheriskindexarethesame,
thesetwoconditionsaretotallydifferent.Atthesame
time, when
00
,
DCPA TCPA
SRtT
, the risk index
0
is
negative;and when
00
,
DCPA TCPA
SRtT
,the risk index
isnegativeaswell,however,boththesetwocasesare
wrong.
D
CPA
S
reflects the spatial urgency degree,
TCPA
t
reflects the temporal urgency degree, and
smaller
TCPA
t
is, the more important this factor will
be, therefore, there should be a changing weighting
coefficient.
In general, when
s
D
CPA DCPA
or
DCPA MINDCPA
,theshipissecure,justasthe
districtIinfig.1shows.
When
D
CPA DCPAs
(or
D
CPA MINDCPA
) and
TCPA TCPAs
(or
TCPA MINTCPA
), the ship is in danger, just as
the district Ⅱ in fig.1 shows. At this moment, the
ships are in close‐quarters situation. When
D
CPA DCPAs
(or
DCPA MINDCPA
) and
TCPA TCPAs
(or
TCPA MINTCPA
), just as
the district Ⅲ in fig.1 shows, then the ships are in
immediate danger, and actions need to be taken at
oncebynavigators.
When DCPA is in the range of 2‐3n miles, and
TCPAisintherangeof15‐30min,thenthechanging
ofcollisiondanger
includesthreeprocesses,theyare
security, close‐quarters situation and immediate
danger.Onlybymakingfullpreparationintheclose‐
quarterssituation,theprobabilityoftheoccurrenceof
immediate danger can be effectively reduced.
Multiplying
0
0
D
CPA
RS
R
with
0
0
D
CPA
Tt
T
can’t
253
fullyreflectthetwo‐dimensionaltemporalandspatial
physicalproperties.
D
CPA
TCPA
D
CPAs
TCPAs
I
II
III
Figure1. Ships in security, in danger and in immediate
danger
2.3 Fuzzymathematicalmethod
Whenusingfuzzymathematicalmethodtodetermine
theshipcollisionriskindex,factorset,evaluationset
and evaluation indexes needs to be established, and
the subordinating degree function of all the
parameters in the evaluation set needs to be
established.Alltheseshouldbedescribedusingfuzzy
language, for the factors considered are not
comprehensive enough, and human factors such as
thedeterminationofevaluationvalueandselectionof
subordinate function will affect the results. (Yan,
2002)
2.4 Methodofartificialneuralwork
This method extensively connects simple artificial
neurons to simulate human brain behavior and
function.
This method proposed that the artificial
neural model of ship collision risk index can be
established using multi indexes. Usually, three
indexes are used, they are
D
CPA
S
,
TCPA
t
and whether
thecomingshipisfromtheportsideorstarboard,and
whenthe comingship isfrom theportside,then the
inputvalueis0.1;whenthecomingshipisfromthe
starboard,thentheinputva lueis0.9.However,ifthe
collision risk index model established using this
methodisinappropriate,then the results are lack of
credibility.(ZhouandWu,2004)
3 NEWMODELOFSHIPCOLLISIONRISKINDEX
BASEDONCOMPLEXPLANE
3.1 Securityvector,securitylevelandshipcollisionrisk
degree
DCPArepresentsthedistance anditsunit is nmile,
TCPA represents the time and
its unit is minute.
DCPAreflectsthespatialeffectsandurgencydegree
of collision, and TCPA reflects the temporal effects
and urgency degree of collision. Therefore, security
vector
S
isintroducedinthetwo‐dimensionalspatial
andtemporalspace,and
0
SDCPAiVTCPA
,
i
is
the imaginary unit,
0
V
is the speed, and
0
/V DCPAs TCPAs
,itisrelatedwithfactorssuch
assize,maneuveringperformance,loadingcondition
of the ships and the navigating water area and
atmospheric condition.
is the weighting
coefficient of DCPA and
0
VTCPA
, it has different
values under different conditions of the same
direction of the ship coming. And
is in inverse
proportiontotherelationspeedoftwoships,thatis,
larger the relative speed is, smaller
will be, then
thesecurityvector
S
willbesmallerandriskdegree
willbehigher. Atthistime,theweightingcoefficient
oftemporaleffecttospatialeffectwillincrease,which
can make up for the deficiency of the evaluation
criteria proposed by Kearon that the weighting
coefficientisinvariable.
Themoduleofsecurityvectorisdefinedas:
222 2
00
()/ ( )
ss
s
DCPA V TCPA DCPA V TCPA
(2)
Thecollisionriskdegreeisdefinedas:
2222
00
1/ ( ) / ( )
ss
d s DCPA V TCPA DCPA V TCPA
(3)
We can see from the definition that smaller the
securitycoefficient
S
is,larger
d
willbe,anditwill
be more dangerous, which corrected the cognitive
trap of the evaluation criteria proposed by Kearon
thatsmaller
is,itwillbemoredangerous.
3.2 Shipcollisionriskmodelbasedoncomplexplane
DCPAandTCPAofthetargetshipareacquiredusing
theARPAoftheownship,thenthesecurityvectorof
the target ship can be represented as:
0
S DCPA i V TCPA
.
is the weighting
coefficient between DCPA and
0
VTCPA
, and it is
relatedwith therelativespeedoftwoships.
also
varieswiththeencountersituation,whentheshipis
coming from the starboard,
then
0.5
TCPA TCPAs
TCPAs
;andwhentheshipis
coming from the portside, then
0.55
TCPA TCPAs
TCPAs
.Generallyspeaking, the
conditionwhentheshipiscomingfromthestarboard
ismoredangerousthantheconditionwhentheshipis
comingfromtheportside.Therefore,whentheshipis
comingfromthe portside,
will increase, then the
security vector will increase and it will be less
dangerous.
Shipcollisionriskindexisdefinedas:
2222
00
1/ ( ) / ( )d s DCPAs V TCPAs DCPA V TCPA
(4)
254
In Eq.4,
s
is the security degree,
is the
weighting coefficient between DCPA and
0
VTCPA
, and
0
/V DCPAs TCPAs
,
D
CPAs
is the secure approaching distance,
TCPAs
is the
secure approaching time. When
TCPA TCPAs
and
D
CPA DCPAs
, it is under urgent risk
condition,whichisnotdiscussedhere.
4 COMPARISONOFTWOEVALUATION
CRITERIAANDMODELS
4.1 ComparisonofKearon modelandthenewevaluation
model
ForshipA,DCPA=0.1nmile,TCPA=11min;forship
B, DCPA=2.1 n mile, TCPA=1min. Let a=5, b=1,
22
()()aDCPA bTCPA
, and smaller
is, it
will be more dangerous, then
121.25
A
,
111.25
B
, and
BA
, therefore, risk degree
betweenshipBandthe ownshipislarger thanthat
betweenshipA andtheownship.Whileinfact,the
risk degree of ship A to the own ship is higher.
Accordingto thenewevaluationcriteriaandmodel,
thefollowingequationcanbe
acquired:
0
(* / )S DCPA i V TCPA DCPA i DCPAs TCPAs TCPA
(5)
According to Eq.5, let DCPAs=2 n
mile,
15minTCPAs
, Then
0.05
A
S
,
1.05
B
S
,
20
A
d
,
0.9524
B
d
,
AB
dd
,larger
d
is, it will
bemoredangerous,therefore,theriskdegreeofship
Atotheown‐shipislargerthanthatofshipB,which
isinaccordancewiththefact.
4.2 Comparisonofthesetwomodesunderdifferent
conditions
1 When
AB
DCPA DCPA
and
AB
TCPA TCPA
,largerTCPAis,itwillbemore
secure, and the results of these two evaluation
modelsareinaccordancewiththefact.
2 When
AB
TCPA TCPA
and
AB
DCPA DCPA
, larger DCPA is , it will be
more secure, and the results of these two
evaluationmodelsareinaccordancewiththefact.
3 When
AB
DCPA DCPA
and
AB
TCPA TCPA
,largerTCPAandDCPAare,it
will be more secure, and the results of these two
evaluationmodelsareinaccordancewiththefact.
4 TheconditionisdifferentwhentheDCPAofone
ship is smaller than that of the other ship, while
TCPA of it is larger than
that of the other ship.
Table1showsthecountering conditionofshipA,B
andtheown‐ship.
Table1.CounteringconditionofshipA,Bandtheown‐ship
_______________________________________________
ShipA ShipB
_______________________________________________
DCPA 0.5nmile1.5nmile
TCPA
18min 15min
_______________________________________________
Theriskdegreeofwhichshipis higher cannot be
directlyjudgedunderthiscondition,thereforeitmust
berealizedbycomputer,usingJAVAtocomparethe
risk degree of one ship whose DCPA is larger and
TCPA is smaller and another ship whose DCPA is
smallerandTCPAislarger.
The calculated results of the evaluation model
proposed by Kearon is
330.25
A
,
281.25
B
;
the calculated results of the evaluation model
proposed in this paper is
2
A
d
,
0.67
B
d
.
According to the traditional model proposed by
Kearon,
AB
, therefore, ship B is more
dangerous than ship A; while according to the new
evaluation criteria,
AB
dd
, so ship A is more
dangerousthanshipB.ThefactisthatshipAismore
dangerous than ship B, so the evaluation criteria
proposed in this paper is more reasonable and
effective.
5 DISCUSSION
1 TheshipcollisionriskindexproposedbyKearon.J
is represented as:
22
()()aDCPA bTCPA
,
as this model considers DCPA and TCPA
respectively, without establishing the temporal
and spatial relationship between them, therefore,
the temporal causal relationship and the
precedence order cannot be described. The
mathematical model proposed in this paper, of
whichtheresultsareinaccordancewiththefact,is
established on the
generalized space of the two‐
dimensional time and space, and has great
physical significance. In fig.2, the equation of the
two lines are:
00 0
()DCPA DCPA V TCPA TCPA
, and
theshadearearepresentstheabsolutesignificance
ofthegeneralized space of two‐dimensional time
and space.Point A represents the earlier
moment, point
0
P
represents the current
moment,andpointBrepresentsthelatermoment.
Different
and
0
V
are corresponding to two
different intersecting lines, then system of
intersecting straight lines is generated. To certain
and
0
V
, the shade area constructed by their
corresponding intersecting lines is call as the
causalconeintimeseries.Asfig.2shows,eventA
isabsolutelyearlierthanevent
0
P
,becauseeventA
can have an effect on event
0
P
; event B is
absolutelylaterthanevent
0
P
,becauseeventBcan
beinfluencedbyevent
0
P
.
D
CPA
TCPA
000
(,)
P
DCPA TCPA
A
B
Figure2.Spatialandtemporalprecedenceorder
255
2 The contributions of DCPA and TCPA to risk
degree should not be invariable. The traditional
evaluationmodelproposedbyKearonconsidered
thattheweightedvaluesareinvariable,whenthe
coming ship is from the starboard, then
5a
,
0.5b
; when the coming ship is from the
portside, then
5a
,
1b
, which is not in
accordance with the actual situations. The
weightingcoefficientinthenewevaluationmodel
varies with the magnitude relationship of TCPA
andTCPAs,whichislogicalandcorrespondingto
theactualsituations.
3 In the traditional evaluation model proposed by
Kearon, when
is a constant, the risk degree
curveisanellipse in the first quadrant. While in
the new evaluation model, the risk degree curve
has different shapes when the values of
and
0
V
aredifferent,whichcanprovidemorephysical
connotationandanalysisoftheriskindexspace.
4 Calculation of risk index under the collision
avoidancesituationofmulti‐ships
Assumingthattheencounteringsituationinvolves
6 ships, and DCPA and TCPA of them at the same
moment are measured by ARPA,
the results are
showedintable2.
Table2.DCPAandTCPAofdifferentships
_______________________________________________
Ship1 Ship2 Ship3 Ship4 Ship5 Ship6
_______________________________________________
DCPA 1.61.71.71.61.41.5
(nmile)
TCPA 24 28 18 19 29 22
(min)
_______________________________________________
UsingJAVAtocalculatethecollisionriskdegree,
the results are as follows: d
1=0.5802， d2=0.4663，
d
3=0.5882，d4=0.6173，d5=0.5135，d6=0.6542, and the
sequencing result is
643152
dddddd
,
which shows that the ship 6 is the most dangerous
one,andtheship2isthemostsecureone.
This function is further realized in single chip
microcomputer (STM32F107) and extended TFT
displayscreen.IfitisinstalledtoARPA,thefunction
of APRA will be greatly improved. Under
the
encountering condition of multi‐ships, the
quantitativevalueofthecollisionriskindexbetween
the own‐ship and other ships can be acquired using
thecalculationmodelestablishedinthispaper,which
can help navigators to have a more accurate and
deeperunderstandingoftheencounteringsituationin
advance,
and has great significance to the collision
avoidance and ship security. Different weighting
coefficients are given under different situations, the
weighting coefficient under the head‐on situation is
the largest, that under the crossing encounter
situation is the second largest, and that under the
overtaking situation is the smallest. It can
be
predicted that this model will have promising
application is the intelligent collision avoidance
system,andcanbeextensivelyappliedtothecivilian
ships,militaryaircraft,navalvessel,andetc.
0.7
0.65
0.6
0.55
0.5
0.4
0.45
15
20
25
30
1.8
1.7
1.6
1.5
1.4
1.3
(min)TCPA
()DCPA n mile
d
Figure3.Three‐dimensional imageoftheriskdegreeofship
1to6
Fig.4showsthatwhenTCPAisintherangeof25
to30minutes,theincreasingofcollisionriskdegreeis
slow;whenTCPAisintherangeof20to25minutes,
the increasing of collision risk degree is fast; and
whenTCPAisin the range of 15 to
20 minutes, the
collision risk degree increases rapidly. When
TCPA TCPAs
, the collision risk degree is the
largest; when
TCPA TCPAs
, then the ship is in
immediatedanger,asdistrict Ⅲ infig.1shows.
30 25 20
15
/minTCPA
0
0.1
0.2
0.5
0.3
0.4
0.7
0.6
Figure4. Variationcurve ofrisk degreeof ship1 withthe
riskdegree
6 CONCLUSIONS
TheevaluationmodelproposedbyKearon, Liururu
andetal.hasgreatlimitationsanddeficiencies,even
wrong conclusions. In this paper,
0
iVTCPA
is
takenasacoordinateofthespaceofshipcollisionrisk
index, and security vector is introduced in the two‐
dimensionalspatialandtemporalspace,whichcanbe
expressed in the complex plane
as:
0
SDCPAiVTCPA
. In this new model, the
reasonablepartoftheevaluationmodelproposedby
Kearon is maintained, and the unreasonable part,
evenwrongpartiscorrected.Asthesametime,ithas
effectively made up for the limitations of current
ARPAthatonlyDCPAisconsideredwhengivingthe
collision risk
alarm, and the quantitative value of
256
collision risk index can’t be acquired. In the new
evaluation model of collision risk index, single chip
microcomputer is used to calculate the risk index
undertheencounteringconditionofmulti‐ships,and
sequencing is made, which can help navigators to
better understand the encountering situation and
made reasonable decisions
of collision avoidance.
Therefore, the model proposed in this paper has
promising application in the field of marine
navigation.
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