527
1 BACKGROUND
One of the most important issues in marine navigation is
safe control of the shipʹs movement. As a result of the
relative movementof own ship with the speed V and the
courseandthemetj‐thshipmovingatthespeedVjand
coursej,a
certainsituationatseaisdetermined(Lisowski
2016).
Thevariables characterizingsituationin theform
of distance Dj and bearing Nj to j‐th met ship are
measured by Automatic Radar Plotting Aids ARPA
anti‐collisionsystem(Lebkowski2018).
The standard ARPA system performs automatic
tracking of 20 encountered objects, determination of
their speed and courses as well as
parameter s
approach to the own ship‐Distance of the Closest
Point of Approach
minjj
D
DCPA
and Time to the
ClosestPointofApproach
minjj
TTCPA
(Fig.1).
Figure1.Passingownshipwiththej‐thencounteredship.
TheproperuseoftheARPAanti‐collisionsystem
in order to achieve greater safety of navigation
requires, in addition to the preparation of its
operationanddatainterpretation,supplementingthe
system with appropriate algorithms of computer
navigatormaneuveringdecisionsupport,eliminating
human characteristics and taking into account the
uncertaintyof
the situationand thegame properties
of the control process (Lazarowska 2017, Lisowski
2014,Malecki2013,Mohamed‐Seghir2016).
Thenecessitytotakeintoaccountthestrategiesof
the ships encountered and their kinematics and
dynamics as control objects determines the
The Safe Control Sensitivity Functions in Matrix Game
of Ships
J
.Lisowski
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Thepaperdescribestheuseofmatrixgametheoryforthesynthesisofsafecontrolofashipin
collisionsituations.Ananalysisofthesensitivityoftheshipcontrolalgorithmtotheinaccuracyofprocessstate
information and changes in its parameters was presented.
Sensitivity characteristics were compared on the
exampleofthenavigationalsituationintheKattegatStraitforgoodandrestrictedvisibilityatsea.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 12
Number 3
September 2018
DOI:10.12716/1001.12.03.11
528
applicationofthedifferentialgamemodelforanalysis
(Isaacs1965,Lisowski2012,2014).
Apart from the shipʹs dynamics equation, the
differential game model can be reduced to matrix
gamemodelandj‐thparticipants(Osborne2004).
Taking into account the practical application of
algorithmforcontrollingyourownship
inacollision
situation, it is advisable to conduct the sensitivity
analysis of safe control, on one hand,accuracy of
informationfromARPAanti‐collisionsystem,andon
the other changes of kinematic and dynamic
parametersofthecontrolprocess.
2 GAMESHIPCONTROL
The game matrix R[r
j(j,0)] includes values of the
collisionriskr
jdetermined onbasis ofdata obtained
from the ARPA anti‐collision system for the
acceptable strategies
0 of the own ship and
acceptablestrategies
jofanyparticularnumberofJ
encounteredships(Fig.2).
Figure2.Blockdiagramofthematrixgameofships.
Theriskvalue isdefinedby referringthecurrent
situation of approach, described by parameters D
jmin
andT
jmin,totheassumedevaluationofthesituationas
safe, determined by a safe distance of approach D
s
andasafetimeTs–whicharenecessarytoexecutea
collisionavoidingmanoeuvre(Modarres2006):
1
22
2
min min
123
jj
j
j
sss
DT
D
rc c c
DTD
(1)
Theweightcoefficientsc
1,c2andc3aredepended
on the state visibility at sea, dynamic length L
d and
dynamic beam B
d of the ship, and kind of water
region‐openwatersorfairways(Fig.3).
Figure3.Anexampleofthedependenceofcollisionriskon
the relative values of distance and time of approaching
ships.
In a matrix game player 1 (own ship) has a
possibilityuse
0purevariousstrategies,andplayer2
(encounteredJships)has
jvariouspurestrategies:
0
0
11 10 1
0
0
11 12 1, 1 1
21 22 2, 1 2
12 ,1
0
12 ,1
12 ,1
....
....
.... .... .... .... ....
....
[( , )]
.... .... .... .... ....
....
.... .... .... .... ....
....
n
n
n
jj j jn
mm m mn
jj
rr r r
rr r r
rr r r
Rr
rr r r
rr r r
(2)
The constraints for the choice of a strategy
0
,
j
result from the recommendations of the
COLREGsRules(Szlapczynski&Szlapczynska2016).
Inamatrixgameplayer1has apossibilitytouse
0
pure various strategies, and player 2 has
j various
purestrategies.Constraintslimitingtheselectionofa
strategy result from COLREGs Rules. As most
frequently the game does not have a saddle point,
thereforethebalancestateisnotguaranteed.Inorder
to solve this task, we may use a dual linear
programming(Nisanatal.2007,Kula
2014).
Ina dualproblem player1 aimsto minimizethe
risk of collision and pla y er 2 aims to minimize the
collisionrisk.The componentsof themixed strategy
expressthedistributionoftheprobabilityofusingby
the playerstheir pure strategies. As aresult, for the
optimalcontrolquality
indexintheform:
0
min min
j
j
Ir
(3)
matrix probability P of applying each one of the
particularpurestrategiesisobtained:
529
0
0
11 10 1
0
0
11 12 1, 1 1
21 22 2, 1 2
12 ,1
0
12 ,1
12 ,1
....
....
.... .... .... .... ....
....
[(,)]
.... .... .... .... ....
....
.... .... .... .... ....
....
n
n
n
jj j jn
mm m mn
jj
pp p p
rr r r
pp p p
Pp
pp p p
pp p p
(4)
Thesolutionforthecontrolproblemisthestrategy
representingthehighestprobability:
0
00
max
,
jj
Iu p
(5)
The safe trajectory of own ship is treated as a
sequenceofsuccessivechangesintimeofhercourse
andspeed.Asafepassingdistanceisdeterminedfor
theprevailingvisibilityconditionsatseaD
s,advance
timetothemanoeuvret
manddurationofonestageof
thetrajectory∆t
kasacalculationstep.
At each one step the most dangerous object
relative to the value of the collision risk r
j is
determined. Then, on the basis of semantic
interpretationofCOLREGsRules,thedirectionofthe
ownshipturnrelative tothemostdangerousshipis
selected. A collision risk matrix R is determined for
theacceptablestrategies ofthe ownship
0andthat
forthej‐thencounteredship
j.
By applying a principle of the dual linear
programming for solving matrix games the optimal
course of the own ship and that of the j‐th ship is
obtained at a level of the smallest deviations from
theirinitialvalues(Zak2013).
Fig. 4 presents the hyper‐surface of the
collision
riskforvalues
0andjofthestrategies.
Figure4. Dependence of the collision risk on the course
strategiesoftheownship
0andthosejoftheencountered
ship.
If,atagivenstep,thereisnosolutionattheown
shipspeedV,thenthecalculationsarerepeatedfora
speed decreased by 25%, until the game has been
solved.
Thecalculationsarerepeatedstepbystepuntilthe
momentwhenallelementsofthematrixRare
equal
to zero and the own ship, after having passed
encountered ships, returns to her initial course and
speed(Tomera2012).
Bydualusingfunctionlinprog–linearprogramming
fromOptimizationToolboxofMATLABsoftwarethe
cooperative multi‐step Matrix Game MG algorithm
wasdevelopedtodeterminethesafegametrajectory
ofshipincollisionsituation.
3 CONTROLSENSITIVITYANALYSIS
Thesensitivitytheorymethodswerewidelyusedfor
solvingvarioustheoreticalandappliedproblemswith
analysis and synthesis, identification, adjustment,
monitoring, testing, tolerance distribution (Eslami
1994,Wierzbicki1977).
At the same distinction is made between the
sensitivity of model control process
for changing its
parametersandprocessoptimalcontrolsensitivityto
changes in its parameters and disturbance influence
(Rosenwasser&Yusupov2000).
In previous papers dealt with sensitivity of
deterministicsystemsdonotgamesystems.
At sea, land and air transport processes occur of
own ship and many encountered ships. Control of
suchprocesses,duetothehighproportionofhuman
subjectivity in the decision‐making maneuver, often
takesthecontrolasgamecharacter(Lisowski2013).
Main investigation method in sensitivity theory
consistsinusingsocalledsensitivityfunctions.
The first‐order sensitivity functions f
x of optimal
controluofgameprocessdescribedbystatevariables
xcanbepresentedasfollowingpartialderivate:
x
I
xu
f
x
(6)
It is also possible consider of the sensitivity
functions r‐th order of optimal control
f
r, x
in the
followingform:
1
,12
1
...
...
m
r
rx m
r
r
n
Ixu
f
rr r r
xx
(7)
Quality game control index I
0,j takes the form of
game payment, consisting of integral payments and
finalpayment:
0
2
K
t
j
KK
t
I
xt r t dt
(8)
Theintegralpaymentrepresentslossofwaybythe
ship while passing the encountered ships and the
final payment determine the final risk of collision
r
j(tK)relativetothej‐thshipandthefinaldeviationof
ownshiptrajectoryd(t
K)fromthereferencetrajectory.
Theinvestigationofsensitivityofthegamecontrol
makes, for sensitivity analysis of the final payment
d(t
K):
530
K
xi
i
dt
f
x
(9)
Takingintoconsiderationthepractica lapplication
ofthe gamecontrolalgorithmfor theown ship in a
collisionsituation it isrecommended to perform the
analysisofsensitivityofasafecontrolwithregardto
theaccuracydegreeoftheinformationreceivedfrom
the anti‐collision ARPA radar system
in the current
approach situation, from one side and also with
regard to the changes in kinematical and dynamic
parametersofthecontrolprocessfromtheotherside.
Admissibleaverageerrors,itcanbecontributedby
sensors of anti‐collision system can have following
valuesfor:
radar,
bearing:±0,22
o
,
formofcluster:±0,05
o
,
formofimpulse:±20m,
marginofantennadrive:±0,5
o
,
samplingofbearing:±0,01
o
,
samplingofdistance:±0,01nm,
gyrocompass:±0,5
o
,
log:±0,5kn,
GPS:±15m.
The algebraic sum of all errors, influent on
picturingofnavigationalsituation,cannotexceed±5%
or±3
o
.
3.1 SensitivityofSafeShipControltoInaccuracyof
InformationfromARPASystem
Let X represent such a set of state process control
informationonthenavigationalsituationthat:
[, , , , , ]
jjj j
X
VV DN
(10)
Let then X
e represent a set of information from
ARPAsystemcontainingerrorsofmeasurementand
processingparameters:
[, , , , , ]
ejjjjjjjj
X
VV VV D DN N
(11)
Relative measure of sensitivity of the final
paymentinthegamef
xasafinaldeviationoftheship
safetrajectoryd
Kfromthereferencetrajectorywillbe:
() ()
100%
()
Ke K
x
K
dX dX
f
dX
(12)
[, , , , , ]
x
VVjjDjNj
f
ssss s s
(13)
3.2 SensitivityofSafeShipControltoProcessParameters
Alterations
LetPrepresentasetofparametersofthestateprocess
control:
[, , , ]
ms k
P
tD t V
(14)
Let then P
e represent a set of parameters
containing errors of measurement and processing
parameters:
[, ,, ]
em ms sk k
Pt tD Dt tV V
(15)
Relativemeasureofsensitivityoffinalpaymentin
the game f
p as a final deflection of the ship safe
trajectoryd
Kfromtheassumedtrajectorywillbe:
() ()
100%
()
Ke K
p
K
dP dP
f
dP
(16)
,, ,
ptmDstkV
f
sss s
(17)
where:
t
m‐advancetimeofthemanoeuvrewithrespecttothe
dynamicpropertiesoftheownship,
t
k ‐timeofonestageoftheshipʹstrajectory,
D
s–safedistance,
Ts–safetimeofapproach.
4 SENSITIVITYFUNCTIONS
ComputersimulationofMGalgorithm,asacomputer
software supporting the navigator manoeuvring
decision, were carried out on an example of a real
navigational situation of passing J=9 encountered
ships(Fig.5),(Tab.1).
Figure5. Nineteen‐minute speed vectors of the own ship
and nine encountered ships in a situation in the Kattegat
Strait.
Table1.Ownandencounteredshipsmovementparameters.
_______________________________________________
j Dj[nm] Nj[deg] Vj[kn] j[deg]
_______________________________________________
0‐ ‐20.0 00.0
1 08.8 326 14.5 090
2 14.3 006 16.0 180
3 07.5 011 16.0 200
531
4 12.0 340 00.0 000
5 12.0 050 00.0 000
6 06.0 225 15.0 290
7 08.0 290 12.0 300
8 05.0 140 09.0 045
9 14.0 030 06.0 000
_______________________________________________
Thesituationswereregistered inSkagerrakStrait
onboardr/vHORYZONTII,aresearchandtraining
vessel of Gdynia Maritime University, on the radar
screenofARPAanti‐collisionsystemRaytheon.
4.1 SensitivityFunctionsofGameShipControlinGood
VisibilityatSeaforDs=0.5nm
The safe trajectory of own
ship and sensitivity
functions determined by MG algorithm in Matlab
softwarearepresentedinFig.6and7.
Figure6. The safe trajectory of own ship for MG_gv
algorithmingoodvisibilityatseaD
s
=0.5nminsituationof
passingJ=9encounteredships,r(t
K
)=0,d(t
K
)=2.15nm.
Figure7.Sensitivityfunctionsofthematrixgamecontrolof
own ship in good visibility at sea according to MG_gv
algorithm.
4.2 SensitivityFunctionsofGameShipControlwith
RestrictedVisibilityatSeaforDs=1.5nm
The safe trajectory of own ship and sensitivity
functions determined by MG algorithm in Matlab
softwarearepresentedinFig.8and9.
Figure8. The safe trajectory of own ship for MG_rv
algorithm in restricted visibility D
s
=1.5 nm in situation of
passingJ=9encounteredships,r(t
K
)=0,d(t
K
)=4.54nm.
532
Figure9.Sensitivityfunctionsofthematrixgamecontrolof
ownshipinrestrictedvisibility atsea accordingto MG_rv
algorithm.
5 CONCLUSIONS
Thealgorithm ofmultistep cooperativematrix game
takes into consideration the Rules of the COLREGs
Rules and the advance time of the manoeuvre
approximating shipʹs dynamic properties and
evaluates the final deviation of the real trajectory
fromreferencevalue.
Sensitivityofthefinalgamepayment:
isthe
leastforchangesofthedurationofonestage
tis least relative to the sampling period of the
trajectoryandadvancetimemanoeuvre,
most is relative to changes of the own and met
shipsspeedandcourse,
it grows with the degree of playing character of
the control process and with the quantity of
admissiblestrategies.
trajectory and for changes of the advance time
manoeuvre.
Thematrixgamecontrolalgorithmis,inacertain
sense, formal model of the thinking process of a
navigator steeringthe ship’s movement and making
upmanoeuvringdecisions.
Thereforethey
maybeappliedintheconstruction
of a new model of ARPA system containing a
computer supporting the navigator’s decision
making.
REFERENCES
EslamiM. 1994.Theory of sensitivityin dynamic systems.
Berlin:Springer‐Verlag.
Isaacs R. 1965. Differential games. New York: John Wiley
andSons.
KulaK.2014.Cascadecontrolsystemoffinstabilizers.Proc.
XIX Conference Methods and Models in Automation
andRobotics,MMAR,Miedzyzdroje:868‐873.
Lazarowska A. 2017. A new
deterministic approach in a
decision support system for ship’s trajectory planning.
ExpertSystemswithApplications,Vol.71:469‐478.
Lebkowski A. 2018. Design of an autonomous transport
system for coastal areas. TransNav – International
JournalonMarineNavigationandSafetyatSea,Vol.12,
Issue1:117‐124.
Lisowski J.
2012. Game control methods in avoidance of
shipscollisions.PolishMaritimeResearchVol.19,No.1:
3‐10.
Lisowski J. 2013. Sensitivity of computer support game
algorithmsofsafeshipcontrol.InternationalJournalof
Applied Mathematics and Computer Science. Vol. 23,
Issue2:439‐446.
Lisowski J. 2014. Comparison of dynamic games
in
application to safe ship control. Polish Maritime
Research,Vol.21,No.3:3‐12.
Lisowski J. 2014. Optimization‐supported decision‐making
in the marine game environment. Solid State
Phenomena,Vol.210:215‐22.
Lisowski J. 2016. Analysis of methods of determining the
safeshiptrajectory.TransNav–InternationalJournal
on
Marine Navigation and Safety at Sea, Vol. 10, Issue 2:
223‐228.
Malecki J. 2013. Fuzzy track‐keeping steering designfor a
precisecontroloftheship.SolidStatePhenomena,Vol.
197:140‐147.
ModarresM.2006.Riskanalysisinengineering.BocaRaton:
TaylorandFrancisGroup.
Mohamed‐Seghir M.
2016. Computational intelligence
method for ship trajectory planning. Proc. XXI
Conference Methods and Models in Automation and
Robotics,MMAR,Miedzyzdroje:636‐640.
NisanN.,RoughgardenT.,TardosE.&VaziraniV.V.2007.
Algorithmic game theory. New York: Cambridge
UniversityPress.
Osborne M.J. 2004. An introduction to game theory. New
York:
OxfordUniversityPress.
RosenwasserE.&YusupovR.2000.Sensitivityofautomatic
controlsystems.BocaRaton:CRCPress.
Szlapczynski R. & Szlapczynska J. 2016. An analysis of
domain‐based ship collision risk parameters. Ocean
Engineering,Vol.126:47‐56.
Tomera M. 2012. Nonlinear observers design for
multivariable ship motion control. Polish
Maritime
Research,Vol.19,Issue1:50‐56.
Wierzbicki A. 1977. Models and sensitivity of control
systems(inPolish).Warszawa:WNT.
Zak A. 2013. Trajectory tracking control of underwater
vehicles.SolidStatePhenomena,Vol.196:156‐166.
