79
1 INTRODUCTION
The paper focuses on the situation of dangerous
approach of the ships that requires a maneuver
passing,thenatureofwhichdeterminesthesystemof
coordinationoftheirinteractionandrealizesthearea
ofmutualresponsibilities.Forthepreventionofeach
of the vessels collisions, the coordination system
prescribes a specific type of beha
vior (the
maneuveringorsavingoftheparametersofmotion).
Ithelpstooperatethe ship flexible strategypassing,
which contains a priority maneuver, taking into
accounttheprescribedtypeofbehaviorofinteracting
ships, and reserves the maneuver for the case of
ignoring the partner prescribed duties. To av
oid
collisions,themaneuverofchangingtheshipʹscourse
ispreferable.Thepaperstressesuponthecalculation
of the altering course maneuver parameters,
considering the dynamic characteristics of the
maneuvering of the vessel. It is also given the
dynamic model of rotational motion of the vessel,
which is the most suit
able for calculating the
parameters of the altering course maneuvers of the
vessels.
2 MAINPART
2.1 Settingupthetask
Let’s see the situation dangerous approach of
operating the vessel (c
1) to (c2) when near the
proposed passing area is another ship (c
3). In this
case, the matrix of situational disturbance
is
}
ij
{ωW (Pyatakov2015).SeeEquation1below:
0
32
ω
31
ω
23
ω0
21
ω
13
ω
12
ω0
W
(1)
For the case (1) the values of the situational
disturbances
12
ω
and
21
ω
are not equal to zero.
Therefore, the vessels
1
c
and
2
c
interact. Binary
coordinator
(2)
Coor
, implemented in COLREGs‐72,
determinesthenatureofsuchinteraction.Thevessels
decide to make a coordinated maneuver of passing,
compensating situational disturbance, i.e. ensuring
the circulation of values, situational disturbances
12
ω
and
21
ω
zero.Thestrategyofpassing,formedby
The Choice of the Maneuver of the Vessel’s Passing
Considering the Coordination’s System of the
Interactive Vessels and Their Dynamic Characteristics
Y.L.Volkov,E.N.Pyatakov&Y.V.Kalinichenko
OdessaNationalMaritimeAcademy,Odessa,Ukraine
ABSTRACT:Themaneuverofthealteringcourseofthevesselisamorepreferabletoavoidacollision.Dueto
that the calculation of the parameters of the avoidance maneuver should be done considering the dynamic
characteristicsofthevesselinmaneuvering.
Thepaperanalyzesthedynamicmodelsofthevessel rota
tionmotioninordertoselectmoreappropriateone
forthecalculationofavoidancemaneuverofthevesselapplyingthealteringofthecourse.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 1
March 2017
DOI:10.12716/1001.11.01.08
80
operatingvessel,dependsontheinitialvaluesforthe
situationaldisturbances
31
ω
13
ω
and
32
ω
23
ω
.
2.2 Thecaseofinteracting
In the case
0
13
ω
and
0
23
ω
the presence of the
vessel
3
c
does not cause the appearance of its
interaction with operating vessel and target. In this
situation
(1)
G
there is only the interaction Bz12
between the operating vessel c1 and a dangerous
target c
2and the coordinator Coor(2) that is based on
the relative positions of the vessels
12
S
and their
statuses
1
St
and
2
St
directs interactive vessels
coordinating signals
12
γ
and
21
γ
(Pyatakov 2015;
Burmaka2016).
These signals determine their behavior in the
process of the passing, each of them prescribing
reciprocalduties,allowingtheshipstomakechoiceof
strategiespassing.Whileoneofthevesselsretainsits
motion parameters, the second one executes a
maneuver passing or both vessels perform
coordinatedmaneuversofthepassing.
2.3 Choosingthemaneuver
Therefore,operatingship
1
c
choosesastrategyofthe
passing
)1(
1
D
, grounding on the coordinating signal
γ
12 and realized area of mutual obligations. More
over,ifthecoordinatorinstructstheoperatingvessel
1
c
to give way to the target
2
c
(
1γ
)1(
12
), altering
course maneuver D
1
(1)
is selected by the strategy of
passingD
1
(1)
.

If
0
)1(
12
γ
the operating vessel gets the
instruction from the coordinator to maintain the
motion parameters D
1
(0)
. Then the ship follows the
prescribed strategy D
1
(0)
until the pointing
time
t
~
when performing the maneuver )t
~
(D
~
1
it
should prevent a possible collision because of the
inactiontargetc
2,whichdoesnotgiveawayD2
(0)
In
this situation
(1)
G , strategy D1
(1)
is defined as
follows(2):
.t
~
t),0(
2
D ,0
)1(
12
γif ),t
~
(
1
D
~
;t
~
t),0(
2
D ,0
)1(
12
γif ),0(
1
D
);1(
2
D ,0
)1(
12
γif ),0(
1
D
;1
)1(
12
γif ),1(
1
D
)
1(
1
D
(2)
2.4 Thesecondcaseofcooperating
Let’sseethesituationG
(2)
when
1ω
13
and
0ω
23
.
Inthiscase,inadditiontotheinteractionBz
12between
theshipsc
1andc2thereisaninteractionBz13between
the ships c
1 and c3. Consequently, the coordinator
Coor
(2)generates coordination signalsγ12
(2)
,γ21
(2)
,γ13
(2)
andγ13
(2)
andγ31
(2)
. The signalsγ12
(2)
andγ13
(2)
are
addressedtotheoperatingvesselbythecoordinator.
Theycan prescribe itone type ofbehavior(γ
12
(2)
= 1,
γ
13
(2)
= 1or anotherγ12
(2)
= 0,γ13
(2)
= 0), that means to
maneuverormaintainaconstantflowparameters,or
contradict each other (γ
12
(2)
= 1,γ13
(2)
= 0 orγ12
(2)
= 0,
γ
13
(2)
=1).
Inthecaseofsequencecoordinationofsignalsγ
12
(2)
=1andγ12
(2)
=1,theoperatingvesselmustgivewayto
vessels c
2 and c3 by the passing maneuver D1
(2)
(1),
whichcanberealizedinoneoftwopossibleoptions:
common for vessels c
2 and c3 maneuver or two
consecutivemaneuversforeachofthevessels.
2.4.1 Strategyofcooperatingthevesselsinthesecond
case
If the coordinating signalsγ
12
(2)
= 0,γ13
(2)
= 0)
require from the operating vessel to maintain a
constant flow parameters relative to both vessels c
2
and c3 then the operating ship follows a constant
courseandspeedD
1
(2)
(0),providedthattheobjectives
c
2 and c3 carry out altering course maneuvers D2(1)
andD
3(1).Ifoneofthetargetorboth targets donot
givewaytotheoperatingvessel,thelatterfollowsa
constantcourseand speed D
1
(2)
(0)until thepointing
time
t
~
after which the vessel
1
c
is forced to
maneuver
)t
~
(
(2)
1
D
~
topreventapossiblecollision.
In the case when coordinating signals contradict
each other (γ
12
(2)
= 1,γ13
(2)
= 0 orγ12
(2)
= 0,γ13
(2)
= 1),
operatingvesselusesalteringcoursepassingintime
zero
)t
~
(
(2)
1
0D
~
,whichissafeforships c2andc3.Inthis
situationG
(2)
,strategyD1
(2)
isdefinedasfollows(3):
.0 γ,1 γif ),t(D
;1 γ,0 γif ),t(D
;t
~
t,D ,D ,0 γ,0 γif ),t
~
(D
~
;t
~
t,D ,D ,0 γ,0 γif ),0(D
;D ,D ,0
γ,0 γif ),0(D
;1 γ,1 γif ),1(D
D
)2(
13
)2(
12
o
)2(
1
)2(
13
)2(
12
o
)2(
1
)0(
3
)0(
2
)2(
13
)2(
12
(2)
1
)0(
3
)0(
2
)2(
13
)2(
12
)2(
1
)1(
3
)1(
2
)2(
13
)2(
12
)2(
1
)2(
13
)2(
12
)2(
1
)2(
1
(3)
2.5 Thethirdcaseofcooperating
Another situation G
(3)
is characterized by situational
disturbances
0ω
13
and
1 ω
23
,thusinadditionto
the interactions Bz
12 there is an interaction Bz23
between the ships c2 and c3. In this situation, the
coordinatorformedacoordinatingsignalsγ
12
(3)
,γ21
(3)
,
γ
23
(3)
andγ13
(2)
andγ32
(3)
.
Ifthesignalsofcoordinationγ
12
(3)
=1,γ21
(3)
=0,γ23
(3)
=1
andγ32
(3)
=0realized,
theoperatingship
1
c should
give a way to the ship c
2 performing the maneuver
D
1
(3)
(1),rememberingthatc2performsthe maneuver
D
2
(1)
withtheshipc3.
Similarly,astrategyofthepassingoftheoperating
vesselunderthecoordinationofthesignalsγ
12
(3)
=1,
γ
21
(3)
= 0,γ23
(3)
= 0 andγ32
(3)
= 1 is formed.The
maneuverD
3
(1)
ofthepassingofthevesselc3istaking
intoaccount.
2.5.1 Strategyofcooperatingthevesselsinthethird case
Inthecaseofsignalcoordinationγ
12
(3)
=0,γ21
(3)
=1,
γ
23
(3)
=1γ32
(3)
=0
andγ32
(3)
=1,whentheoperatingship
81
shouldmaintainitsmotionparameters,itsstrategyis
to follow with constant course and speed D
1
(3)
(0), if
thetargetc
2performsamaneuverpassingD2
(1).
A similar strategy D
1
(3)
(0) is used in the
implementationofsignalcoordinationγ
12
(3)
=0,γ21
(3)
=
1,γ
23
(3)
=1andγ32
(3)
=0.If, however,forbothcasesthe
target c
2 in contrary to the requirements of the
coordinator does not give a way to the operating
vesselD
2(0)thoughitgotthecoordinatingsignals,the
latter implement a strategy D
1
(3)
(0) of point‐in‐time
t
~
, and then the operating ship
1
c takes its own
)t
~
(
(3)
1
D
~
maneuverofthepassing.Thus,inasituation
G
(3)
the strategy D1
(3)
is calculated according to the
Equation4:
.t
~
t,D ,0 γif ),t
~
(D
~
;t
~
t,D ,0 γif ),0(D
;D ,0 γif ),0(D
;1 γif ),1(D
D
)0(
2
)3(
12
(3)
1
)0(
2
)3(
12
)3(
1
)1(
2
)3(
12
)3(
1
)3(
12
)3(
1
)3(
1
(4)
2.6 Interactioncase
The final situation G
(4)
realizes under the situational
disturbances
1ω
13
and
1 ω
23
inwhichbesidesthe
interactionsofBz
12theremaybealsointeractionsBz13
and Bz23 accordingly between vessels
1
c
and c3, and
between vessels c
2 and c3. In this situation, the
coordinator forms coordinating signalsγ
12
(4)
,γ21
(4)
,
γ
13
(4)
,γ31
(4)
,

γ23
(4)
andγ32
(4)
. Therefore, the operating
behavioroftheship
1
c
inthefirstplaceisdetermined
bytheratioofthecoordinationsignalsγ
12
(4)
andγ13
(4)
.
Ifthesesignals areagreed uponγ
12
(4)
=1andγ13
(4)
=1,
then the operating ship
1
c
should perform a passing
maneuver
)1(
)4(
1
D
, which can be common to ships c2
andc3andconsistoftwoconsecutivemaneuvers,and
givewaytovesselsc
2andc3.
2.6.1 Strategyforinteractioninthefourthcase
Inthecaseofthecoherentcoordinationofsignals
γ
12
(4)
=0 andγ13
(4)
=0 if the ships c2 and c3 carry out
altering course maneuvers D
2
(1) and D3
(1) the
operating vessel must maintain a constant flow
parameters, implementing strategies
)0(
)4(
1
D
. In the
casewhenoneofthetargetsdoesnotgiveawayto
the operating vessel, the latter follows a constant
course and speed
)0(
)4(
1
D
until the pointing time
t
~
after which vessel
1
c performs the maneuver
)t
~
(
(4)
1
D
~
toavoidofapossiblecollision.
In the case when coordinating signals contradict
each other (γ
12
(4)
= 1,γ13
(4)
= 0 orγ12
(4)
= 0γ13
(4)
= 1),
operatingvesselistakingpassingmaneuverata zero
pointoftime
)
o
t(
)4(
1
D
,whichissafetyforbothc2andc3
vessels.Thus,inasituation
(4)
G
strategy
)4(
1
D
isshown
inEquation5:
.0
)4(
13
γ,1
)4(
12
γесли ),
o
t(
)4(
1
D
;1
)4(
13
γ,0
)4(
12
γесли ),
o
t(
)4(
1
D
;t
~
t),0(
3
D ),0(
2
D ,0
)4(
13
γ,0
)4(
12
γесли ),t
~
(
(4)
1
D
~
;t
~
t),0(
3
D ),0(
2
D ,0
)4(
13
γ,0
)4(
12
γесли ),0(
)4(
1
D
);1(
3
D ),1(
2
D ,0
)4(
13
γ,0
)4(
12
γесли ),0(
)4(
1
D
;1
)4(
13
γ
,1
)4(
12
γесли ),1(
)4(
1
D
)4(
1
D
(5)
2.7 Generalsituationsofvesselsinteraction
The situation
(i)
G
(
41i
) provides for binary
interaction of the vessels with the opposite
coordinating signals when one of the ships should
takeanalteringcoursepassing,andthesecond ship
must not change the parameters of its motion. The
coordinatorCOLREG‐72mentionsonlyonesituation
(Rule14)whenthestandardinteractionof
theshipsis
in the simultaneous maneuvering of the vessel and
operatingtarget.Inthissituation,the operatingship
chooses the maneuver of altering course to the
starboardindependentlyfromthetargetandthethird
vesselbehavior.
AsCOLREGs‐72pointsout,undertheconditionof
sufficient water space more
preferable maneuver of
thepassingistochangetheshipʹscourse.Therefore,
afteridentifyingsituationaldisturbancetheoperating
ship should realize the situation
(i)
G
and calculate
theparametersofthepassingstrategy
)i(
1
D
,including
somealternativealteringcoursemaneuvers,andthen,
basedonthetargetsandthevessel
3
c
behavior,and
perform the appropriate option of altering course
maneuver.
2.8 Methodsofaccountingthedynamicsofvesselswhile
choosingthemaneuver
To perform the altering course maneuver it is
necessaryto calculatethe alteringcourse K
yand the
timet
yofthe beginning ofthe turn accordingtothe
Equation6(Tsimbal2007):
)]
oty
K
c
sin(K
1
ρarcsin[
oty
K
y
K
y
L
d
L
arcsin
y
α
oty
K
(6)
α
y ‐ the value of the bearing to the target at the
beginningofthealteringcourse;
L
y ‐ distance to the target at the starting point of
alteringcourse;
L
d‐thedistanceoftheclosestpositionapproach.
Next step is calculating the meaning of ty (see
Equation7):
82
)
oty
K
otn
Ksin(
otn
V
)
c
K
oty
τsin(K
c
V
oty
Ksiny
oty
Kcosx
y
t
ˆ
y
t
(7)
V
c,Kc‐thespeedandheadingofthetarget;
V
otn,Kotn‐theinitialvalueoftherelativespeedand
course;
x , y ‐ coordinates the operating of the vessel
duringthemaneuver;
τ ‐durationofrotationoftheoperatingofavessel;
y
t ‐thetimeofstartingthemaneuver,
y
t is calculated without taking into account the
dynamicsofthevessel(Equation8):
)
oty
K
otn
Ksin(
otn
V
)
oty
K
n
αsin(
n
L
d
L
y
t
(8)
Thevalues
x , y andintheexpressiontyare
definedforthedynamicmodeldescribingthemotion
oftheship.Themostadequateprocessofrealturnof
the vessel is described by the course changes of the
vessel depending on the angle of the rudder
(Equation9):
)
2
T
1
T(/}])
2
T/t(-exp1[
2
2
T])
1
T/t(-exp1[
2
1
T{)
r
ω
o
ω(t
r
ω
o
KK
(9)
k
β
ω
k
r
ω
ω
k
–efficiencyoftherudder;
k
β
–theangleoftherudder;
o
K
,
o
ω
‐the initial rate and the angle speed of the
operatingvessel;
T
1, T2‐ the time constant depending on the vessel
dynamics.
Duration of rotation
τ is calculated in Equation
10:
t
k
tτ (10)
where,
)
2
T
1
T(/}])
2
T/
k
t(exp1[
2
2
T])
1
T/
k
t(exp1[
2
1
T{t
k
t
‐
‐
})
2
T
1
T(/])
2
T/
k
t(exp
2
T)
1
T/
k
t(exp
1
T[2{
х
)TT(/}])T/t(-exp1[T])T/t(-exp1[T{
21
2
2
2
1
2
1
+К/
ω
a ,
}
-1
L]
1
)/T
2
T-
1
T[()
2
t/Texp()
1
T/
2
Tln{(
1
Tt
)
2
T
1
T(/])
2
T/
k
t(exp
2
T)
1
T/
k
t(exp
1
T[2L
Thevalues∆t
kand∆tarecalculatedbythemethod
ofsimpleiterations.
Thecoordinatesoftheoperatingvesselduringthe
turningwecalculateinEquation11and11a:
,td](t)K
o
K[sin
τ
0
o
VxΔ
(11)
td](t)K
o
K[cos
τ
0
o
VyΔ
(11a)
3 CONCLUSION
Theobtainedvalue
x
, y
andallowtocalculate
the time t
y of the beginning of the turn, taking into
accountthedynamicsofthevessel.
REFERENCES
Burmaka, I.A., Pyatakov, E.N. & Bulgakov, A.Yu. 2016.
Upravleniye sudami v situatsii opasnogo sblizheniya.
Germany:LAPLambertAcademicPublishing.
COLREG–72.1972.Odessa:Phenix.
Pyatakov, E.N., Buzhbetskiy, R.Y., Burmaka, I.A. &
Bulgakov, A.Y. 2015. Vzaimodeistvie sudov
priraskhozhdeniidlyapreduprezhdeniyastolknoveniya.
Kherson:Grin.
Tsimbal, N.N., Burmaka, I.A. & Tupikov, E.E.
2007.
Gibkiestrategiirashozhdeniyasudov.Odessa:KPOGT.