83
1 INTRODUCTION
Alargepartinincreasingthe safetyofnavigationis
the use of ARPA anti‐collision system, which
enablestotrackautomaticallyatleast20encountered
jships,determinationoftheirmovementparameters:
speed V
j, courseψj and elements of approach: Dj –
distance,N
j–bearing,
jj
DCPAD
min,
‐Distance
oftheClosestPointofApproach,
jj
TCPAT
min,
‐
TimetotheClosestPointofApproach(Fig.1).
Thefunctionalscopeofa standardARPA system
ends with manoeuvre simulation to achieve the safe
passing distance D
s by altering course
or
speed
V selected by the navigator (Bist 2000,
Cockroft&Lameijer2006,Cahill2000).
The most general description of the own control
object passing the j number of other encountered
objects is the model of a differential game of a j
number of objects (Basar & Olsder 1982, Engwerda
2005,Isaacs1965,Mesterton‐Gibbons2001).
Figure1. The own ship moving with speed V and course
duringofpassingjencounteredships.
Thismodelconsistsbothofthekinematicsandthe
dynamics of the ship’s movement, the disturbances,
Cooperative and Non-Cooperative Game Control
Strategies of the Ship in Collision Situation
J
.Lisowski
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Thepaperintroducesthepositionalcooperativeandnon‐cooperativegameofagreaternumberof
metshipsforthedescriptionoftheprocessconsideredaswellasforthesynthesisofoptimalcontrolstrategies
oftheownshipincollisionsituation.Theapproximatedmathematicalmodelofdifferent
ialgameintheformof
triple linear programming problem is used for the synthesis of safe ship trajectory as a multistage process
decision.Theconsiderationshavebeenillustratedanexampleofprogramcomputersimulationtodetermine
thesafeshiptrajectoriesinsituationofpassingamanyoftheshipsencountered.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 12
Number 1
March 2018
DOI:10.12716/1001.12.01.09
84
thestrategiesoftheownshipandencounteredships
andthequalitycontrolindex(Clarke2003,Kula2015,
Osborne2004).
The diversity of possible models directly affects
the synthesis of ship control algorithms which are
afterwards affected by the ship control device,
directlylinkedtotheARPAsystemandconsequently
determines effects of safe and optimal control
(Fletcher1987,Lisowski2011).
2 MODELOFGAMESHIPCONTROLPROCESS
2.1 Stateandcontrolvariables
Thedifferentialgamedescribedbystateequation:
,,
x
fxut
(1)
is reduced to a positional multistage game of a j
numberofparticipants(Fig.2).
Figure2. Block diagram of the positional game model of
passingtheownshipandencounteredjships.
Thestatexandcontroluvariablesarerepresented
by(Lisowski&Lazarowska2013):
0,1 0,2 ,1 ,2
0,1 0,2 ,1 ,2
,, ,
,, ,
1, 2, ...,
oojjjj
j
jj j
x
Xx Yx Xx Y
uuVu uV
jm
(2)
The making of a continuous positional game
discrete and reducing it to a multistage positional
gameisdeterminedbyownshipanddependson:the
maximum relative speed of the own ship under the
current navigational situation, the range of the
situationandthedynamic characteristicsofthe own
ship(Gluwer&Olsen1998,Isil&Koditschek2001).
Theessenceofthepositionalgameistomakethe
strategies of the own ship dependent on current
positionsp(t
k)oftheships encounteredatthecurrent
step k. In this way possible course and speed
alterations ofthe objects encountered are considered
intheprocessmodelduringthesteeringperformance
(Lazarowska2012,Luus2000).
Thecurrentstate of theprocess is determinedby
theco‐ordinatesforthe
positionoftheownshipand
oftheshipsencountered:
0
,
,
1, 2, ...,
oo
j
jj
x
XY
x
XY
j
m
(3)
Thesystemgeneratesitssteeringatthemomentt
k
onthebasisofthedatawhichareobtainedfromthe
ARPA anti‐collision system concerning the current
positionsoftheownandencounteredships:
0
()
()
()
1, 2,...,
1, 2,...,
k
k
j
k
x
t
pt
x
t
j
m
kK
(4)
Itisassumed,accordingtothegeneralconceptof
the multistage positional game, that at each discrete
momentofthetimet
kthepositionoftheencountered
ships p (t
k) is known on the own ship (Gierusz &
Lebkowski 2012, Lisowski 2012, Malecki 2013,
Mohamed‐Seghir2016,Zak2013).
Theconstraintsofthestateco‐ordinates:
0
,
j
x
txt P
(5)
constitute the navigational constraints, while the
steeringconstraints:
0
1, 2,...,
jj
uUo
uU
jm
(6)
take into consideration the kinematics of the ship
movement, the recommendations of the COLREGS
Rulesandtheconditiontomaintainthesafepassing
distanceD
s:
,min
min
jjs
D
Dt D
(7)
2.2 Setsofacceptablestrategies
TheclosedsetsU
o,jandUj,odefinedasthesetsofthe
acceptablestrategiesoftheshipsasplayers:
,1,2,
,1,2,
oj o j o j
jo j o j o
UptS S
UptS S
(8)
aredependedonthepositionp(t),whichmeansthat
the choice of the steering u
j by the j‐th encountered
ship alter the sets of the acceptable strategies of the
otherships(Fig.3).
85
Figure3. Determination of the acceptable safe strategies
areas of the own ship
,1, 2,oj o j o j
US S
and the
encounteredjship
,1, 2,
j
ojojo
US S
.
Let refer to the set of the acceptable strategies of
theownshipwhilepassingthej‐thencounteredship
at a safe distance D
s (Jaworski, Kuczkowski,
Smierzchalski 2012; Lebkowski 2015, Nisan et al.
2007,Straffin2001,Tomera2015,Lisowski2014a).
Thearea,whenmaintainingstabilityintimeofthe
course and speed of the own ship and the ship
encountered is static and is comprised within the
semicircleofaradiusequaltothe
setreferencespeed
oftheownshipVwithinthearrangementoftheco‐
ordinates 0X’Y’ with the axis X’ directed to the
directionofthereferencecourse(Millington&Funge
2009,Modarres2006,Szlapczynski2012).
ThesetU
o,jisdeterminedwiththeinequalities:
,,' ,,' ,
22 2
,' ,'
oj ox oj oy oj
ox oy
au bu c
uuV
(9)
where:
,' ,'
,, ,,,
,, ,,,
,,,
,,
,,,
1,
,
2,
(, )
cos( )
sin( )
sin( )
cos( )
1()
1()
oox oy
oj oj oj oj oj
oj oj oj oj oj
jjoojoj
oj oj
oj oj oj
oj
oj
oj
Vuu u
aq
bq
Vq
c
Vq
for S Port side
f
or S Starboard side
(10)
The value
jo,
is determined by using an
appropriate logical function F
j characterising any
particular recommendation referring to right of way
containedinCOLREGSRules(Lisowski2014b).
2.3 SemanticinterpretationofCOLREGS Rules
TheformoffunctionF
jdependsoftheinterpretation
oftheaboverecommendationsforthepurposetouse
theminthecontrolalgorithm,when:
,
,
11
01
oj
j
oj
then
F
then
(11)
InterpretationoftheCOLREGSRulesintheform
of appropriate manoeuvring diagrams enables to
formulate a certain logical function F
j as a semantic
interpretationoflegalregulations formanoeuvring.
Eachpa rticulartypeofthesituationinvolvingthe
approachoftheshipsisassignedthelogicalvariable
valueequaltooneorzero(Lisowski2015a,2015b):
A–encounteroftheshipfromboworfromanyother
direction,
B–approaching
ormovingawayoftheship,
C–passingtheshipasternorahead,
D–approachingoftheshipfromtheboworfromthe
stern,
E–approachingoftheshipfromthestarboardorport
side(Lisowski2015c,2015d).
By minimizing logical function F
j by using a
method of the Karnaughʹs Tables the following is
obtained:
_
___ ____
()
j
F
AABCDE
(12)
Theresultantarea ofacceptablemanoeuvresofthe
ownshipinrelationtothemencounteredshipsis:
,
1
1, 2, ...,
m
ooj
j
UU
jm
(13)
isdeterminedbyanarrangementofinequalities(9).
On the other hand, however, the set of the
acceptablestrategiesofthej‐thobjectinrelationtothe
ownshipisdeterminedbythefollowinginequalities:
,,' ,,' ,
22 2
,' ,'
jo jx jo jy jo
jx jy j
au bu c
uuV
(14)
where:
,' ,'
,, ,,,
,, ,,,
,, ,,,
1,
,
2,
(, )
cos( )
sin( )
sin( )
1()
1( )
jjjxjy
jo jo jo jo jo
jo jo jo jo jo
jo jo o j jo jo
jo
jo
jo
Vuu u
aq
bq
cVq
for S Port side
f
or S Starboard side
(15)
Thesymbol
oj,
isdeterminedbyanalogytothe
determination of
jo,
with the use of the logical
functionF
jdescribedbytheequation(11).
86
Consideration of the navigational constraints, as
shallow waters and coastline, generate additional
constraintstothesetofacceptablestrategies:
,,' ,,' ,nk ox nk ox nk
au bu c
(16)
where:
k–isthenearestpointofintersectionofthestraight
linesapproximatingthecoastline.
3 TYPESOFGAMEANDSAFESHIPCONTROL
STRATEGIES
3.1 Gameoptimalcontrolrules
The optimal control
*
o
ut of the own ship,
equivalentforthecurrentpositionp(t)totheoptimal
positional control
*
o
up, is determined in the
followingway:
from the relationship (14) for the measured
position p(t
k), the control status at the moment tk
sets of the acceptable strategies
,jo k
Upt
are
determinedfortheencounteredships inrelationto
the own ship, and from the relationship (9) the
output sets
,oj k
Upt
of the acceptable
strategiesoftheownshipinrelationtoeachoneof
theencounteredships,
a pair of vectors u
j,o and uo,j, are determined in
relation to each j encountered ship and then the
optimal positional strategy of the own ship
)( pu
o
from the condition of optimum value I *
qualityindexcontrol:
whentheencounteredshipsnon‐cooperate:
mj
LLtxLI
ncokko
uUu
Uu
UUu
nc
jjojo
ojoj
m
j
jooo
...,,2,1
),(
min
max
min
,
)(
,,
,,
1
,
(17)
whentheencounteredshipscooperate:
mj
LLtxLI
cokko
uUuUu
UUu
c
jjojoojoj
m
j
jooo
...,,2,1
),(
minminmin
,
)(
,,,,
1
,
(18)
forthenon‐gameoptimalcontrol:
mj
LLtxLI
ocokko
UUu
oc
m
j
jooo
...,,2,1
),(
min
,
1
,
(19)
where:
0
(), () ( ) ( )
K
t
ok k oK K
t
L
xt L Vtdt rt dt
(20)
referstothegoalcontrolfunctionoftheownshipin
theformofthepayments–theintegralpaymentand
thefinalone(Lisowski2016a).
The integral payment determines the distance of
theownshiptothenearestturningpointL
konthe
assumed route of the voyage and the final one
determines:r
o(tK)‐thefinalriskofcollisionandd(tK)‐
final game trajectory deflection from reference
trajectory.
The criteria for the selection of the optimal
trajectory of the own ship is reduced to the
determinationofhercourseandspeed,whichensure
thesmallestlossesofwayforthesafepassingofthe
encounteredships at adistance notsmaller
than the
assumed safe value D
s, having regard to the ship’s
dynamic in the form of the advance time t
m to the
manoeuvre(Lisowski2016b).
3.2 Parametersofshipdynamics
Atthetimeadvancemaneuvert
mconsistsofelement
m
t during course manoeuvre
or element
V
m
t
duringspeedmanoeuvre V .
Thedynamicfeaturesoftheshipduringthecourse
alteration by an angle
is described in a
simplifiedmannerwiththeuseoftransferfunction:
() ()
()
()
() (1 )
o
Ts
kke
s
Gs
ssTs s
(21)
where:
TT
o
‐ manoeuvre delay time which is
approximatelyequaltothetimeconstantoftheship
asacoursecontrolobject,
)(
k ‐gaincoefficientthevalueofwhichresults
fromthenon‐linearstaticcharacteristicsoftherudder
steering.
Thecoursemanoeuvredelaytimeisasfollows:
mo
tT
(22)
Inpractice,dependingonthesizeandtypeofvessel
advancetimetotheanti‐collisionmanoeuvrethrough
achangeofcourseis:
st
m
72060
.
Differential equation of the second order
describingtheshipʹsbehaviourduringthechangeof
the speed by
V
is approximated with the use of
theinertiaofthefirstorderwithatimedelay:
87
()
()
() 1
ov
Ts
v
V
v
ke
Vs
Gs
ns Ts
(23)
where:
T
ov‐time of delay equal approximately to the time
constant for the propulsion system: main engine‐
propellershaft‐screwpropeller,
T
v‐ thetimeconstantoftheshipʹshullandthemass
oftheaccompanyingwater.
Thespeedmanoeuvredelaytimeisasfollows:
3
V
movv
tTT
(24)
In practice, depending on the size and type of
vessel advance time to the anti‐collision manoeuvre
throughachangeofspeedis:
st
V
m
900120
.
3.3 Computerprogramsofpositionalmultistagegame
shipcontrol
The smallest losses of way are achieved for the
maximum projection of the speed vector of the own
shiponthedirectionoftheassumedcourseleading
tothenearestturningL
kpoint.
Theoptimalcontrolofthe ownshipiscalculated
at each discrete stage of the ship’s movement by
applying triple linear programming SIMPLEX
method, assuming the relationship (20) as the goal
functionandtheconstraintsareobtainedbyincluding
thearrangementoftheinequalities(8),(14)and(16).
The
above problem is then reduced to the
determination the function of control goal as the
maximum of the projection of the own ship speed
vectoronreferencedirectionofthemovement:
,' ,' ,'
min max ,
ox oy ox
LVuuu
(25)
withlinearconstraintsapproximatingthejointsetof
thesafestrategiesoftheownshipU
o,j:
,1 , ' ,1 , ' ,1
,,' ,,' ,
,,' ,,' ,
,1,' ,1,' ,1
,,',,',
oox ooy o
oj ox oj oy oj
om ox om oy om
om ox om oy om
omkpox omkpoy omkp
au bu c
au bu c
au bu c
au bu c
aubuc
(26)
where:
m–numberonencounteredships,
k–numberofconstraintsapproximatingcoastline,
p‐number of segments approximating a semi‐circle
witharadiusequaltoownshipspeed.
Aftertheintervaloftimet
kthecurrentfixingofthe
ship position is carried out and then comes the
solving of the problem using the algorithm for the
positionalcontrol.
Using the function of lp – linear programming
from the Optimization Toolbox contained in the
Matlab/Simulink the positional multistage game
manoeuvringprograms:mpgame_ncforcriterion
(17),
mpgame_c for criterion (18) and mpngame_oc for
criterion(19)hasbeendesignedfordeterminationof
thesafeshiptrajectoryinacollisionsituation.
4 COMPUTERSIMULATION
4.1 Navigationalsituation
Computersimulationofmpgame_nc,mpgame_cand
mpngame_oc algorithms was carried out in
Matlab/Simulinksoftware on an example of the real
navigational
situation of passing j=19 encountered
shipsintheSkagerrakStraitingoodvisibilityDs=0.5
1.0nmandrestrictedvisibilityDs=1.52.5nm(nautical
miles),(Fig.4andTab.1).
Figure4. The place of identification of navigational
situationsinSkagerrakandKattegatStraits.
The situation was registered on board r/v
HORYZONTII,aresearchandtrainingvesselofthe
Gdynia Maritime University, on the radar screen of
the ARPA anti‐collision system Raytheon (Fig. 5
and6).
88
Table1. Movement parameters of the own ship and
encountered19ships.
____________________________
j Dj Nj Vjψj
nm deg kn deg
____________________________
0‐‐20 0
1 9 320 14 90
2 15 10 16 180
3 8 10 15 200
4 12 35 17 275
5 7 270 14 50
6 8 100 8 6
7 11 315 10 90
8 13 325 7 45
9 7 45 19 10
10 15 23
6 275
11 15 23 7 270
12 4 175 4 130
13 13 40 0 0
14 7 60 16 20
15 8 120 12 30
16 9 150 10 25
17 8 310 12 135
18 10 330 10 140
19 9 340 8 150
____________________________
Figure5. The research‐training ship of Gdynia Maritime
Universityr/vHORYZONTII.
Figure6. The screen of anti‐collision system ARPA
Raytheon,installedontheresearch‐trainingshipofGdynia
MaritimeUniversityr/vHORYZONTII.
Examinedthenavigationalsituation,illustratedin
the form of navigation velocity vectors of own ship
and19metshipsisshowninFigure7.
Figure7. The 12 minute speed vectors of own ship 0 and
j=19 encountered ships in navigational situation in
SkagerrakStrait.
4.2 Simulationofthemulti‐stagenon‐cooperative
positionalgame
Fig.8 and 9show the safeand optimal trajectory of
the own ship in collision situation, which is
determined using the algorithms of non‐cooperative
positional game in good and restricted visibility at
sea.
Figure8. Computer simulation ofmulti‐stage non‐
cooperative positional game algorithm mpgame_nc for safe
own ship control in situation of passing 19 encountered
ships in good visibility at sea, D
s=1.0 nm, d(tk)=3.34 nm
(nauticalmile).
89
Figure9. Computer simulation ofmulti‐stage non‐
cooperative positional game algorithm mpgame_nc for safe
own ship control in situation of passing 19 encountered
shipsinrestrictedvisibilityatsea,D
s=2.5nm,d(tk)=7.34nm
(nauticalmile).
4.3 Simulationofthemulti‐stagecooperativepositional
game
Fig.10and11showthesafeandoptimaltrajectoryof
the own ship in collision situation, which is
determined using the algorithms of cooperative
positional game in good and restricted visibility at
sea.
Figure10.Computersimulationofmulti‐stagecooperative
positional game algorithm mpgame_c for safe own ship
controlinsituationofpassing19encounteredshipsingood
visibilityatsea,D
s=1.0nm,d(tk)=2.94nm(nauticalmile).
Figure11.Computersimulationofmulti‐stagecooperative
positional game algorithm mpgame_c for safe own ship
control in situation of passing 19 encountered ships in
restrictedvisibilityatsea,D
s=2.5nm,d(tk)=7.06nm(nautical
mile).
4.4 Simulationofthenon‐gameoptimalcontrol
Fig.12and13showthesafeandoptimaltrajectoryof
the own ship in collision situation, which is
determined using the algorithms of non‐game
optimalcontrolingoodandrestrictedvisibilityatsea.
Figure12. Computer simulation ofnon‐game optimal
controlalgorithmmpngame_ocforsafeown shipcontrolin
situationofpassing19encounteredshipsingoodvisibility
atsea,D
s=1.0nm,d(tk)=0.72nm(nauticalmile).
90
Figure13. Computer simulation ofnon‐game optimal
controlalgorithmmpngame_ocforsafeown shipcontrolin
situation of passing 19 encountered ships in restricted
visibilityatsea,D
s=2.5nm,d(tk)=3.76nm(nauticalmile).
5 CONCLUSIONS
The synthesis of an optimal on‐line control on the
model of a multi‐stage positional game makes it
possibletodeterminethesafegametrajectoryof the
own ship in situations when she passes a greater j
numberoftheencounteredobjects.
The trajectory has been described as
a certain
sequenceofmanoeuvreswiththecourseandspeed.
The computer programs designed in the Matlab
also takes into consideration the following:
regulations of the Convention on the International
RegulationsforPreventingCollisionsatSea,advance
time for a manoeuvre calculated with regard to the
ship’s dynamic features and
the assessment of the
final deflection between the real traj ectory and its
assumedvalues.
Theessentialinfluencetoformofsafeandoptimal
trajectoryandvalueofdeflectionbetweengameand
reference trajectories has a degree of cooperation
betweenownandencounteredships.
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