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
solvethetimeseriescausalrelationships in the two
dimensional temporal and spatial space. And using
the collision risk index on the twodimensional
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
forthedeficienciesofthewidelyusedevaluationmodelproposedbyKearon.JandLiururuisproposed.On
this basis, the calculation method of collision risk index under the encountering situation of multiships is
constructed,thenthethreedimensionalimageandspatialcurveoftheriskindexarefiguredout.Finally,single
chipmicrocomputerisusedtorealize themodel. And a
ttaching this singlechipmicrocomputertoARPAis
helpfultothedecisionmakingofthemarinenavigators.
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
mathematicproblemsininternational
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 ownship
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 ownship, which is as
follows(LiuandHu,2012):
00
0
00
D
CPA TCPA
RS Tt
RT

(1)
InEq.1,
0
istheradiusofthesafedomainofthe
ownship;
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 in fig.1 shows. At this moment, the
ships are in closequarters situation. When
D
CPA DCPAs
(or
DCPA MINDCPA
) and
TCPA TCPAs
(or
TCPA MINTCPA
), just as
the district 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 23n miles, and
TCPAisintherangeof1530min,thenthechanging
ofcollisiondanger
includesthreeprocesses,theyare
security, closequarters 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
fullyreflectthetwodimensionaltemporalandspatial
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
isintroducedinthetwodimensionalspatial
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