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Research ArticleDynamic Games Methods in Synthesis ofSafe Ship
Control Algorithms
Józef Lisowski
Department of Ship Automation, Faculty of Electrical
Engineering, Gdynia Maritime University, 81-225 Gdynia, Poland
Correspondence should be addressed to Józef Lisowski;
[email protected]
Received 16 May 2018; Accepted 16 September 2018; Published 1
October 2018
Academic Editor: Alain Lambert
Copyright © 2018 Józef Lisowski.This is an open access article
distributed under the Creative CommonsAttribution License,
whichpermits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The paper presents application of dynamic games methods,
multistage positional and multistep matrix games, to automate
theprocess control of moving objects, on the example of safe
control of own ship in collision situations when passing many
shipsencountered. Taking into consideration two types of ships
cooperation, for each of the two types of games, positional and
matrix,four control algorithms for determining a safe ship
trajectory supporting the navigator’s maneuvering decision in a
collisionsituation are presented.The considerations are illustrated
by examples of computer simulation in Matlab/Simulink software of
safetrajectories of a ship in a real situation at sea. Taking into
account the smallest final deviation game trajectory from the
referencetrajectory of movement, in good visibility at sea, the
best is trajectory for cooperative matrix game, but in restricted
visibility at sea,the best is trajectory for cooperative positional
game.
1. Introduction
One of the most important transport issues are the processesof
optimal and safe control of ships, airplanes, and cars asmoving
objects [1–4]. Such processes relate to managing themovement of
many objects at the same time, with varyingdegrees of interaction,
the impact of random factors with anunknown probability
distribution, and a large share of oper-ator’s subjective in
maneuvering decisions [5–8]. Therefore,the management of such
processes is accomplished by meansof game control systems, whose
synthesis is carried out withthe methods of game theory [9, 10].
Game theory is a branchof mathematics, covering the theory of
conflict situations andbuilding and analyzing their models [11,
12]. Conflict can beas follows: military, political, social, and
economic, in a socialgame, in the game with nature, and in the
implementationof the control process during interferences of
disturbances orother control objects [13]. A game in the concept of
controltheory is a process consisting of several control
objectsremaining in a conflict situation or a process with
undefineddisturbances or incomplete information. Players as
controlobjects participating in a conflict situation have certain
setsof strategies. Strategy is a set of rules of action, player
control,which cannot change the actions of an opponent or
nature
[14–16]. The strategies are implemented by man,
automaton,regulator, and computer. Strategies can be pure as
elements ofa set of strategies or mixed as a probability
distribution on aset of clean strategies. The result of the game is
the paymentin the form of winning, losing, or the probability of
carryingout a certain action—control [12, 17–19].
The first concept of game theory and the theorem onmini-max was
formulated by E. Borel (1921, 1927). The maincreators of game
theory are John vonNeumann (1928) andO.Morgenstern (1944).
The largest class of games that can be used in the gamecontrol
of dynamic transport processes and among themcontrolling the
movement of ships, planes, and cars representdifferential games,
described by state and output equations,and state and control
constraints [20–24].
The application of the theory of differential games incontrol
theory, including motion control of objects, wasdone by W.H.
Fleming (1957-1964), L.S. Pontriagin (1964-1966), R. Isaacs (1965),
N.N. Krasovski (1965-1974), W.P.Paciukov (1968-1976), A.W. Merz and
J.S. Karmarkar (1976),J. Kazimierczak (1973), T. Miloh and S.D.
Sharma (1977),V. Kudriaszov and J. Lisowski (1979-1980), P.N. Tiep
and J.Lisowski (1993-1997), M. Mohamed-Seghir and J.
Lisowski(1979-2013), and Z. Zwierzewicz (1994-2013).
HindawiJournal of Advanced TransportationVolume 2018, Article ID
7586496, 8 pageshttps://doi.org/10.1155/2018/7586496
http://orcid.org/0000-0001-9281-376Xhttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/7586496
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2 Journal of Advanced Transportation
.D
$D
$M
4DGCH
$DGCH
own shipencountered
ship
j
V
61, 1
6*, *
62, 2
6D, D
Figure 1: The situation of own ship passing of 𝑗 = 1, 2, . . . ,
𝐽encountered ships.
2. Classification of Control Processes ofMoving Objects
As a result of the movement of own ship with speed 𝑉 andcourse 𝜓
in terms of encountered 𝑗 ship moving at a speed𝑉𝑗 and course 𝜓𝑗 a
situation at sea is determined. Parameterscharacterizing the
situation as distance 𝐷𝑗 and bearing 𝑁𝑗for 𝑗 ship are measured by
radar anticollision system ARPA(Automatic Radar Plotting Aids)
[25].
The ARPA system enables us to track automatically atleast 20
encountered 𝐽 ships, determine of their movementparameters
(speed𝑉𝑗, course𝜓𝑗) and elements of approach tothe own ship (𝐷𝑗min
= 𝐷𝐶𝑃𝐴𝑗: Distance of the Closest Pointof Approach, 𝑇𝑗min = 𝑇𝐶𝑃𝐴𝑗:
Time to the Closest Point ofApproach), and also assess the
collision risk (see Figure 1).
The proper use of anticollision system ARPA in orderto achieve
greater safety of navigation requires, in additionto training on
the use and interpretation of the data, sup-plements the system
with appropriate methods of computer-aidedmaneuvering decision of
navigator in the complex nav-igational situation in a short time,
eliminating the subjectivityof man and taking into account the
indefiniteness of thesituation and the properties game process
control [26, 27].
In practice, there are many possible maneuvers to avoida
collision, from which to select the optimal manoeuver, toensure a
minimum the risk of collision or minimum losses ofthe road for safe
passage of encountered ships (see Figure 2).
The movement of objects in time is influenced by
controlvariables 𝑢 from appropriate admissible control sets 𝑈:
𝑢 ∈ 𝑈 (𝑈(𝜃)0 , 𝑈(𝜃)𝑗 ) , (1)where
𝑈(𝜃)0 set of strategies for own object,𝑈(𝜃)𝑗 set of strategy 𝑗
of encountered object fromamong the total number of 𝐽 objects,𝜃 = 0
symbolically means stabilization of the setobject trajectory,
1
23
safe and optimalgame trajectory
ofown ship
dangeroustrajectories
safe and non-optimaltrajectories
own ship
Figure 2: Possible trajectories of the own ship in case of
passing theencountered ships.
𝜃 = 1 symbolically the implementation of an anti-collision
manoeuver to minimize the risk of collision,which in practice is
achieved by meeting inequalities:
𝐷𝑗min = min𝐷𝑗 (𝑡) ≥ 𝐷𝑠, 𝑗 = 1, 2, . . . , 𝐽 (2)𝐷𝑗min: smallest
distance of approaching own object tothe object you are meeting,𝐷𝑗:
current distance to the object 𝑗,𝐷𝑠: safe proximity distance in
given ambient con-ditions, traffic rules, and dynamic properties of
theobject,
𝜃 = −1: symbolic maneuvering the object in order toachieve the
shortest approaching distance, for example, whentransferring the
load.
Following types of motion control objects can be
distin-guished:
(1) Conflict Games:
(i) situations of unilateral dynamic game: 𝑈(𝑈(−1)0 𝑈(0)𝑗 )and
𝑈(𝑈(0)0 𝑈(−1)𝑗 );
(ii) chase situations: 𝑈(𝑈(−1)0 𝑈(1)𝑗 ) and 𝑈(𝑈(1)0 𝑈(−1)𝑗 );(2)
Unilateral Games:
(i) avoiding collisions with
(1) manoeuvers of own ship: 𝑈(𝑈(1)0 𝑈(0)𝑗 );(2) manoeuvers of 𝑗
object encountered:𝑈(𝑈(0)0 𝑈(1)𝑗 );(3) cooperating maneuvers:
𝑈(𝑈(1)0 𝑈(1)𝑗 );
(ii) meeting of objects: 𝑈(𝑈(−1)0 𝑈(−1)𝑗 );(3) Optimal
Control:
(i) stabilization of reference trajectory of object move-ment:
𝑈(𝑈(0)0 𝑈(0)𝑗 ).
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Journal of Advanced Transportation 3
O11 OM11 O
1D O
MDD O
1*
OM**
object 1 object j object J
· · ·
· · ·· · · · · ·
· · ·· · ·
· · ·· · ·
...... own ship
R11R
z11 R
1D R
TDD R
1* R
T**
O10
OM00
R10
RT00
Figure 3: Diagram of the differential gamemodel of moving
objectscontrol process.
3. Dynamic Games Models of Moving ObjectsControl Processes
3.1. Differential Game Model. The most adequate model ofthe
process of controlling own object in the situation with𝐽
encountered objects is the differential game model of 𝐽participants
(Figure 3).
The dynamic properties of control process are describedby the
state equation:
�̇�𝑖 = 𝑓𝑖 [(𝑥𝑧00 , 𝑥𝑧11 , . . . , 𝑥𝑧𝑗𝑗 , . . . , 𝑥𝑧𝐽𝐽 ) ,(𝑢𝑠00 ,
𝑢𝑠11 , . . . , 𝑢𝑠𝑗𝑗 , . . . 𝑢𝑠𝐽𝐽 ) , 𝑡] ;
𝑖 = 1, 2, . . . , (𝑛 𝑧𝑗 + 𝑧0) ; (𝑗 = 1, 2, . . . , 𝐽) ,(3)
where→𝑥 𝑧00 (𝑡) − 𝑧0 dimensional state vector of own object,→𝑥
𝑧𝑗𝑗 (𝑡) − 𝑧𝑗dimensional state vector of 𝑗th object,→𝑢 𝑠00 (𝑡) − 𝑠0
dimensional control vector of own object,→𝑢 𝑠𝑗𝑗 (𝑡) − 𝑠𝑗
dimensional control vector of 𝑗th object.
For example, the equations of the state of the own shipcontrol
process in collision situations, taking into account theown ship’s
hydromechanics equations and the kinematics ofrelative motion of
own ship and ship encountered, will takethe form (4).
�̇�10 = 𝑥20,�̇�20 = 𝑎1𝑥20𝑥30 + 𝑎2𝑥30 𝑥30 𝑥40 + 𝑏1𝑥30 𝑥30
𝑢10,�̇�30 = 𝑎4𝑥30 𝑥30 𝑥40 𝑥40 (1 + 𝑥40) + 𝑎5𝑥20𝑥30𝑥40 𝑥40
+ 𝑎6𝑥20𝑥30𝑥40 + 𝑎7𝑥30 𝑥30 + 𝑎8𝑥50 𝑥50 𝑥60+ 𝑏2𝑥30𝑥40 𝑥30 𝑢10,
�̇�40 = 𝑎3𝑥30𝑥40 + 𝑎4𝑥30𝑥40 𝑥40 + 𝑎5𝑥20𝑥40 + 𝑎9𝑥20 +
𝑏2𝑥30𝑢10,
�̇�50 = 𝑎10𝑥50 + 𝑏3𝑢20,�̇�60 = 𝑎11𝑥60 + 𝑏4𝑢30,�̇�1𝑗 = −𝑥30 +
𝑥2𝑗𝑥20 + 𝑥3𝑗 cos𝑥3𝑗 ,�̇�2𝑗 = −𝑥20𝑥1𝑗 + 𝑥3𝑗 sin𝑥3𝑗 ,�̇�3𝑗 = −𝑥20 +
𝑏4+𝑗𝑥3𝑗𝑢1𝑗 ,�̇�4𝑗 = 𝑎11+𝑗𝑥4𝑗 𝑥4𝑗 + 𝑏5+𝑗𝑢2𝑗 .
(4)
State variables 𝑥𝑧00 of own ship are represented by 𝑥10:
course,𝑥20: exchange rate, 𝑥30: linear velocity, 𝑥40: drift angle,
𝑥50: rota-tional speed, and 𝑥60: pitch of main propeller. The
variables ofencountered ships are determined by the following
values: 𝑥1𝑗 :distance, 𝑥2𝑗 : bearing and 𝑥3𝑗 : course, and 𝑥4𝑗 :
speed.
The control values 𝑢𝑠00 for movement of own ship are𝑢10: rudder
deflection angle, 𝑢20: reference value of rotationalspeed, and 𝑢30:
reference value of the main propeller pitchstroke, and control
values 𝑢𝑠𝑗𝑗 of encountered ship are 𝑢1𝑗 :heading and 𝑢2𝑗 : linear
velocity.
For example, for the passing situation of own ship with𝐽 = 20
ships encountered, the differential game model of thisprocess is
represented by 𝑖 = 86 state variables [28].
The constraints of state and control variables result fromthe
fact that ship maintains a safe passing distance 𝐷𝑠 inaccordance
with the COLREGs (Collision Regulations) rulesof maneuvering with
each ship encountered [29, 30]:
𝑔𝑗 (𝑥𝑧𝑗𝑗 , 𝑢𝑠𝑗𝑗 ) ≤ 0; (𝑗 = 1, 2, . . . , 𝐽) , (5)The synthesis
of game control object consists in minimizingthe control quality
index in form of integral and finalpayment:
𝐼𝑗0 = ∫𝑡𝑘𝑡0
[𝑥𝑧00 (𝑡)]2 𝑑𝑡 + 𝑟𝑗 (𝑡𝑘) + 𝑑 (𝑡𝑘) → min, (6)If the velocity of
own object variable is assumed, then theintegral payment will show
the length of the trajectory of itsown object during the passing of
objects encountered. Thefinal payment determines the final risk of
collision of ownship to the 𝑗th object and the final deviation of
own objecttrajectory from the previously reference trajectory of
traffic[2, 6, 8]. The advantage of the game model is an
accuratedescription of the kinematic and dynamic properties of
theprocess of controlling objects.The disadvantage of this modelis
the large number of state variables and the complexityof
mathematical dependencies. Therefore, this model is bestused as a
simulation model for testing practical algorithmsfor controlling
the safe movement of objects [31–36].
For the purposes of synthesis of real control
algorithms,simplified game models in the form of positional and
matrixgames are used.
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4 Journal of Advanced Transportation
3.2. Positional Game Model. The differential game modelcomes
down to a multistage positional gamemodel, in whichthe object’s
dynamics are taken into account by the timeof the manoeuver ahead
of time. The essence of positionalgame is dependence of own
object’s strategy on the positionsof the objects 𝑝(𝑡) taught. In
this way, possible changes inthe course and speed of the objects
encountered during thecontrol implementation are taken into account
in the processmodel. The current state of the process at the moment
𝑡𝑘 isdetermined by the coordinates of own ship position 𝑥0
andencountered objects 𝑥𝑗:𝑝 (𝑡𝑘) = [𝑥0 (𝑡𝑘)𝑥𝑗 (𝑡𝑘)] ;
𝑥0 = (𝑋𝑗, 𝑌𝑗) ;𝑥𝑗 = (𝑋𝑗, 𝑌𝑗) ;
(𝑗 = 1, 2, . . . , 𝐽) ; (𝑘 = 1, 2, . . . , 𝐾) ,
(7)
It is assumed that at each discrete moment of time theown ship
position and the met objects positions are known.Constraints of
state variables are navigational constraints onthe surrounding of
encountered objects:
[𝑥0 (𝑡)𝑥𝑗 (𝑡)] ∈ 𝑃, (8)Constraints of control variables take
into account the motionkinematics of objects, legal recommendations
for traffic reg-ulations (maritime traffic law, air traffic law,
and road code),and the condition of maintaining a safe passing
distance:
𝑢0 ∈ 𝑈0;𝑢𝑗 ∈ 𝑈𝑗 (𝑗 = 1, 2, . . . , 𝐽) , (9)
The sets of acceptable strategies of the players in relation
toeach other are dependent, which means that the choice ofcontrol
by the 𝑗th of the object changes the sets of acceptablestrategies
of other objects:
{𝑈𝑗0 [𝑝 (𝑡)] , 𝑈0𝑗 [𝑝 (𝑡)]} , (10)The resultant area of
acceptable manoeuvers of own object inrelation to 𝐽 objects is
𝑈0 = 𝐽⋂𝑗=1
𝑈𝑗0 ; (𝑗 = 1, 2, . . . , 𝐽) , (11)Optimal control of the own
object, ensuring minimal roadloss on safe passing of the objects
encountered, is determinedby the static optimizationmethod from the
set of permissiblecontrols 𝑈0:
𝑢∗0 ∈ 𝑈0, (12)
3.3. Matrix GameModel. The differential gamemodel comesdown to
amultistepmatrix gamemodel, in which the object’sdynamics are
accounted for by the time of manoeuver ahead.The game matrix
[𝑟𝑗(𝑠0, 𝑠𝑗)] contains collision risk values 𝑟𝑗determined for
permissible strategies 𝑠0 of own object andacceptable strategies 𝑠𝑗
of encountered 𝑗th object. Collisionrisk value is defined as a
reference to the current approxi-mation situation, described by
objects close-up parameters𝐷𝑗min and 𝑇𝑗min to the assumed
assessment of the situation assafe, determined by the safe distance
proximity 𝐷𝑠 and safetime𝑇𝑠, necessary for the collision avoidance
manoeuver anddistance 𝐷𝑗:
𝑟𝑗 = [𝜁1(𝐷𝑗min𝐷𝑠 ) + 𝜁2(
𝑇𝑗min𝑇𝑠 ) + 𝜁3 (𝐷𝑗𝐷𝑠 )]
−0.5
, (13)
where𝜁1, 𝜁2, and 𝜁3 are coefficients depending on the state
ofobject movement environment.
In a matrix game, own object as player A has the abilityto use
𝑠0 different pure strategies, and 𝐽 objects representingplayer B
have 𝑠𝐽 different pure strategies:
𝑅 = 𝑟𝑗 (𝑠0, 𝑠𝑗)
=
𝑟1,1 𝑟1,2 . . . 𝑟1,𝑠𝑗 . . . 𝑟1,𝑠𝐽𝑟2,1 𝑟2,2 . . . 𝑟2,𝑠𝑗 . . .
𝑟2,𝑠𝐽. . . . . . . . . . . . . . . . . .𝑟𝑠0−1,1 𝑟𝑠0−1 ,2 . . .
𝑟𝑠0−1,𝑠𝑗 . . . 𝑟𝑠0−1,𝑠𝐽𝑟0,1 𝑟𝑠0 ,2 . . . 𝑟𝑠0 ,𝑠𝑗 . . . 𝑟𝑠0 ,𝑠𝐽
, (14)
Constraints on the choice of strategy (𝑠0, 𝑠𝑗) result from
legalrecommendations of traffic COLREGs regulations. Becauseusually
the game has no saddle point, so there is no guaran-teed balance
[37, 38].
4. Game Ship Control Algorithms
The synthesis of algorithms for the control of moving objectswas
carried out on the example of the safe motion controlprocess of
one’s own ship during the meeting of other ships.Individual models
of the process can be assigned the appro-priate algorithms of
computer-aided navigatingmaneuveringdecisions in collision
situations [39, 40].
The exact but complex model of differential game servesas a
simulation model to check the correctness of controlalgorithms
based on approximate positional andmatrix gamemodels.
4.1. Algorithm of Positional Noncooperative Game. The opti-mal
control of your own ship is calculated by determiningthe sets of
acceptable strategies for the ships you meet withrespect to own
ship and the sets of acceptable own shipstrategy for each of the
ships you meet. Then the optimal
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Journal of Advanced Transportation 5
positional strategy of the own ship is determined from
thecondition:
𝐼∗ = min𝑢0
max𝑢𝑗
min𝑢𝑗0
𝐼 [𝑥0, 𝑃𝑘] = 𝑆∗0 , (15)The goal control function of own ship 𝑆0
characterizes thedistance of own ship to the nearest point of
return 𝑃𝑘on a given voyage route. The criterion for choosing
theoptimal trajectory of own ship is to determine its courseand
speeds ensuring the smallest loss of the path for safepassing of
encountered ships, at a distance not lower thanthe assumed value of
𝐷𝑠, taking into account the dynamicsof own ship in the form of
advance time of manoeuver.First, the control of own ship is
determined to ensure theshortest trajectory of the flight, the
smallest loss of the road(min condition) for noncooperating control
of every shipencountered, contributing to the largest extension of
thetrajectory of the own ship (max condition). At the end, fromthe
set of controls of own ship to particular 𝑗 placed ships,the
control of own ship is selected in relation to all 𝐽
shipsencountered, ensuring the smallest loss of the road
(conditionmin). According to the optimization three conditions
(minmax min), the linear programming method is used to solvethe
game, obtaining the optimal values of the course and thespeed of
own ship. The smallest road losses are achieved
forthemaximumprojection of the ship’s own speed vector on thecourse
direction. Optimal control is calculated many timesat each discrete
stage of motion using the SIMPLEX methodto solve the linear
programming problem for variables in theform of components of the
ship’s own speed vector [41].
4.2. Algorithm of Positional Cooperative Game. For a
cooper-ative game, the control criterion (15) will take the
followingform:
𝐼∗ = min𝑢0
min𝑢𝑗
min𝑢𝑗0
𝐼 [𝑥0, 𝑃𝑘] = 𝑆∗0 . (16)The difference in relation to the
previous algorithm resultsfrom the cooperation in avoiding
collision by all objectsencountered 𝐽 and replacing the second
condition max formin [42].
4.3. Algorithm of Matrix Noncooperative Game. The duallinear
programming method can be used to determine theoptimal control. In
the dual issue, player A seeks to minimizethe risk of collision,
while player B in the noncooperativegame aims to maximize the risk
of collision [43, 44]. Thecomponents of the mixed strategy express
the probabilitydistribution of players using their pure strategies.
As a result,for the control criterion in the form
𝐼∗ = min𝑢0
max𝑢𝑗
𝑟𝑗, (17)a matrix of probabilities of using individual pure
strategies isobtained.
The most secure probability of 𝑝𝑗 is the solution to thetask of
safe control of own ship:
𝑢∗0 = 𝑢𝑠00 {[𝑝𝑗 (𝑠0, 𝑠𝑗)]max} , (18)
Applying dual linear programming to matrix game solution,the
optimal values of own ship course and 𝑗th met shipare obtained,
with the smallest deviations from their initialvalues.
4.4. Algorithm of Matrix Cooperative Game. For a coopera-tive
game, the control criterion (17) will take the followingform:
𝐼∗ = min𝑢0
min𝑢𝑗
𝑟𝑗, (19)The difference in relation to the previous algorithm
resultsfrom the cooperation in avoiding collision by all
objectsencountered 𝐽 and replacing the second condition max
formin.
5. Computer Simulation of Game ShipControl Algorithms
Figures 4, 5, 6, and 7 show the own ship safe
trajectoriesdetermined by four algorithms previously in the
MAT-LAB/SIMULINK software, in the situation of 𝐽 = 34encountered
ships in the Kattegat Strait, in conditions of (a)good visibility
at sea for𝐷𝑠 = 0.3 nm (nautical miles) and (b)restricted visibility
at sea for 𝐷𝑠 = 1.5 nm.
The game ends at the moment 𝑡𝑘, when the risk of ownship 𝑟𝑗 in
relation to each 𝑗 ship will reach the value of zero𝑟𝑗(𝑡𝑘) = 0 and
then the final deviation of the trajectory of ownship from
reference trajectory 𝑑(𝑡𝑘) is assessed.
Figure 8 compares the trajectories calculated by individ-ual
four algorithms.
In Figure 8(a), showing the safe trajectories of own shipin
conditions of good visibility at sea, the best is trajectory4 for a
cooperative matrix game, providing the smallest finaldeviation from
the reference trajectory of movement, 𝑑(𝑡𝑘) =0.57 nm.
In Figure 8(b), showing the safe trajectories of own ship
inconditions of restricted visibility at sea, the best is
trajectory 2for a cooperative positional game, providing the
smallest finaldeviation from the reference trajectory of movement,
𝑑(𝑡𝑘) =1.56 nm.6. Conclusions
The use of simplified differential game models of the
controlprocess of moving objects, in the form of a
multistagepositional game and multistep matrix game, for the
synthesisof control algorithms allows us to determine the safe
optimaland game trajectory of own object in passing situations
withmore objects as a sequence of manoeuvers at a course andspeed.
The developed control algorithms take into accountthe legal rules
of object movement and manoeuver advancetime, approximating the
dynamic properties of the ownobject and assessing the final
deviation of the actual trajectoryfrom the reference one. The
presented control algorithmsconstitute formal models of the actual
decision-makingprocesses of the ship’s navigator and can be used in
thecomputer navigator support systemwhenmakingmanoeuverdecisions in
collision situation.
-
6 Journal of Advanced Transportation
14
12
10
8
6
4
2
0
−2
−4
−6
−8
−10
−12−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14
(a)
14
12
10
8
6
4
2
0
−2
−4
−6
−8
−10
−12−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14
(b)
Figure 4: Safe trajectory of own ship in the situation of
passing with 𝐽 = 34 met ships, determined by the positional
noncooperative gamealgorithm.
14
12
10
8
6
4
2
0
−2
−4
−6
−8
−10
−12−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14
(a)
14
12
10
8
6
4
2
0
−2
−4
−6
−8
−10
−12−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14
(b)
Figure 5: Safe trajectory of own ship in the situation of
passing with 𝐽 = 34 met ships, determined by the positional
cooperative gamealgorithm.
10
5
0
−5
−10
1050−5−10
(a)
10
5
0
−5
−10
1050−5−10
(b)
Figure 6: Safe trajectory of own ship in the situation of
passing with 𝐽 = 34 met ships, determined by the matrix
noncooperative gamealgorithm.
-
Journal of Advanced Transportation 7
10
5
0
−5
−10
1050−5−10
(a)
10
5
0
−5
−10
1050−5−10
(b)
Figure 7: Safe trajectory of own ship in the situation of
passing with 𝐽 = 34met ships, determined by the matrix cooperative
game algorithm.4
3
2
1
0
−1
−2
[nm]
[nm]0 1 2 3 4 5 6 7 8
X
Y
reference trajectory
4
21
3
(a)
4
3
2
1
0
−1
−2
[nm]
[nm]0 1 2 3 4 5 6 7 8X
Y
reference trajectory
4
2
1
3
(b)
Figure 8: Comparison of safe trajectories of own ship in the
situation of passing by with 𝐽 = 34 met ships, determined by
individual fouralgorithms: (1) positional noncooperative game, (2)
positional cooperative game, (3) matrix noncooperative game, and
(4)matrix cooperativegame; (a) conditions of good visibility at sea
for 𝐷𝑠 = 0.3 nm and (b) conditions of restricted visibility at sea
for 𝐷𝑠 = 1.5 nm.
Data Availability
The description of navigational situations used for
computercalculations and their resulting data used to support
thefindings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of
interestregarding the publication of this paper.
Acknowledgments
The research was supported by the Ministry of Scienceand Higher
Education, within the framework of funds forstatutory activities
(Grant no. 446//DS/2018: Design andComputer Simulation Tests of
Marine Automation Systems in
the Matlab/Simulink and LabVIEW software) of the Electri-cal
Engineering Faculty of Gdynia Maritime University inPoland.
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