józeflisowski · 2019. 7. 30. · distance of own ship to the nearest point of return on a given...

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

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 ).

Journal of Advanced Transportation 3

Ｏ11 ＯＭ11 Ｏ

1ＤＯ

ＭＤＤＯ

1＊

ＯＭ＊＊

object 1 object j object J

· · ·

· · ·· · · · · ·

· · ·· · ·

· · ·· · ·

...... own ship

Ｒ11Ｒ

z11 Ｒ

1ＤＲ

ＴＤＤＲ

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.


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


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.


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.


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.

References

[1] X. Jiang, Y. Ji, M. Du, and W. Deng, “A Study of Driver’sRoute Choice Behavior Based on Evolutionary Game Theory,”Computational Intelligence and Neuroscience, vol. 2014, ArticleID 124716, 10 pages, 2014.

[2] D.-W. Song and P. M. Panayides, “A conceptual application ofcooperative game theory to liner shipping strategic alliances,”Maritime Policy&Management, vol. 29, no. 3, pp. 285–301, 2002.

[3] M. Bongini and G. Buttazzo, “Optimal control problems intransport dynamics,” Mathematical Models and Methods inApplied Sciences, vol. 27, no. 3, pp. 427–451, 2017.

[4] J. C. Engwerda, “Stabilization of an uncertain simple fisherymanagement game,” Fischeries Research, vol. 31, pp. 1–21, 2017.


[5] X. Chen, X. Zhu, Q. Zhou, and Y. D. Wong, “Game-theoreticcomparison approach for intercontinental container trans-portation: a case between China and Europe with the BRinitiative,” Journal of Advanced Transportation, vol. 2018, 15pages, 2018.

[6] C. S. Fisk, “Game theory and transportation systems mod-elling,” Transportation Research Part B: Methodological, vol. 18,no. 4-5, pp. 301–313, 1984.

[7] M. G. H. Bell, “A game theory approach to measuring theperformance reliability of transport networks,” TransportationResearch Part B: Methodological, vol. 34, no. 6, pp. 533–545,2000.

[8] M. Ishii, P. T.-W. Lee, K. Tezuka, and Y.-T. Chang, “A game the-oretical analysis of port competition,” Transportation ResearchPart E: Logistics and Transportation Review, vol. 49, no. 1, pp.92–106, 2013.

[9] J. C. Engwerda, LQ dynamic optimization and differential games,John Wiley & Sons, West Sussex, UK, 2005.

[10] M. J. Osborne, An introduction to game theory, Oxford Univer-sity Press, New York, NY, USA, 2004.

[11] D. H. Sadler, “The Mathematics of Collision Avoidance at Sea,”Journal of Navigation, vol. 10, no. 4, pp. 306–319, 1957.

[12] O. Glass and L. Rosier, “On the control of the motion of a boat,”Mathematical Models and Methods in Applied Sciences, vol. 23,no. 4, pp. 617–670, 2013.

[13] A. S. Nowak and K. Szajowski, Advances in dynamic games,applications to economics, finance, optimization and stochasticcontrol, Birkhauser, Berlin, Germany, 2000.

[14] T. Radzik, “Characterization of optimal strategies in matrixgames with convexity properties,” International Journal of GameTheory, vol. 29, no. 2, pp. 211–227, 2000.

[15] M. Leng and M. Parlar, “Game theoretic applications in supplychain management: a review,” INFOR. Information Systems andOperational Research, vol. 43, no. 3, pp. 187–220, 2005.

[16] W. Qi, H. Wen, C. Fu, and M. Song, “Game Theory Model ofTraffic Participants within Amber Time at Signalized Intersec-tion,” Computational Intelligence and Neuroscience, vol. 2014,Article ID 756235, 7 pages, 2014.

[17] A. Haurie and J. B. G. Kun, Stabilizity, controllability, andoptimal strategies of linear and nonlinear dynamical games[Ph.D. thesis], RWTH-Aachen, 2001.

[18] J. Sanchez-Soriano, “An overview on game theory applicationsto engineering,” International Game Theory Review, vol. 15, no.3, 1340019, 18 pages, 2013.

[19] D. Wells, Gamesandmathematics, Cambridge University Press,London, 2013.

[20] T. Miloh, Determination of critical manoeuvres for collisionavoidance using the theory of differentialgames, Institut fürSchiffbau, Hamburg, Germany, 1974.

[21] A. Bressan and K. T. Nguyen, “Stability of feedback solutions forinfinite horizon noncooperative differential games,” DynamicGames and Applications, vol. 8, no. 1, pp. 42–78, 2018.

[22] M. Breton and K. Szajowski, Advances in dynamic games:theory, applications, and numerical methods fordifferential andstochastic games, Birkhauser, Boston, Mass, USA, 2010.

[23] W. A. van den Broek, J. C. Engwerda, and J. M. Schumacher,“Robust equilibria in indefinite linear-quadratic differentialgames,” Journal of Optimization Theory and Applications, vol.119, no. 3, pp. 565–595, 2003.

[24] E. Dockner, G. Feichtinger, and A. Mehlmann, “Noncoopera-tive solutions for a differential game model of fishery,” Journalof Economic Dynamics & Control, vol. 13, no. 1, pp. 1–20, 1989.

[25] J. Lisowski and A. Lazarowska, “The radar data transmission tocomputer support system of ship safety,” Solid State Phenomena,vol. 196, pp. 95–101, 2013.

[26] D. S. Bist, Safety and security at sea, Butter Heinemann, Berlin,Germany, 2000.

[27] H. Gluver and D. Olsen, Ship Collision Analysis, CRC Press,CRC Press Taylor & Francis Group 6000 Broken Sound Park-way NW, Suite 300 Boca Raton, FL 33487-2742, 2018.

[28] T. Perez, Ship motion control, Springer, Berlin, Germany, 2005.[29] G. J. Olsder and J. L. Walter, “A differential game approach to

collision avoidance of ships,” in VIII Symp. IFIP On Optim.Techn, vol. 6, 1977.

[30] L. P. Perera, J. P. Carvalho, and C. G. Soares, “Decision makingsystem for the collision avoidance of marine vessel navigationbased on COLREGs rules and regulations,” in Proceedings of13th Congress of International Maritime Association of Mediter-ranean, pp. 1121–1128, Istanbul, Turkey, 2009.

[31] J. Lisowski, “Game control methods in avoidance of shipscollisions,” Polish Maritime Research, vol. 19, pp. 3–10, 2012.

[32] J. Lisowski, “Sensitivity of computer support game algorithms ofsafe ship control,” International Journal of Applied Mathematicsand Computer Science, vol. 23, no. 2, pp. 439–446, 2013.

[33] J. Lisowski, “Analysis of Methods of Determining the SafeShip Trajectory,” TransNav, the International Journal on MarineNavigation and Safety of Sea Transportation, vol. 10, no. 2, pp.223–228, 2016.

[34] I. Millington and J. Funge, Artificial Intelligence for Games,Elsevier, Amsterdam, Netherlands, 2009.

[35] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani,Algorithmic Game Theory, Cambridge University Press, NewYork, NY, USA, 2007.

[36] M. Modarres, Risk Analysis in Engineering, Francis Group,London, UK, 2006.

[37] P. Krawczyk and G. Zaccour, Games and Dynamic Games,World Scientific, New York, NY, USA, 2012.

[38] T. Basar and P. Bernhard, H-Infinity Optimal Control andRelatedMini-MaxDesign Problems: ADynamicGameApproach,Springer, Berlin, Germany, 2008.

[39] M. Piattelli, “Anticollisione e teoria dei giochi,” L’AutomazioneNavale, vol. 2, pp. 5–9, 1971.

[40] P. V. Reddy and G. Zaccour, “Feedback Nash equilibria inlinear-quadratic difference games with constraints,” Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl, vol. 62, no. 2, pp. 590–604, 2017.

[41] T. Başar and G. J. Olsder, Dynamic Noncooperative GameTheory, SIAM, 2013.

[42] E. V. Gromova and L. A. Petrosyan, “On an approach toconstructing a characteristic function in cooperative differentialgames,”Automation andRemote Control, vol. 78, no. 9, pp. 1680–1692, 2017.

[43] M. Mesterton-Gibbons, An Introduction to Game TheoreticModeling, AmericanMathematical Society,NewYork,NY,USA,2001.

[44] I. Millington and J. Funge, Artificial Intelligence for Games,Elsevier, London, UK, 2009.

International Journal of

AerospaceEngineeringHindawiwww.hindawi.com Volume 2018

RoboticsJournal of

Hindawiwww.hindawi.com Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwww.hindawi.com

Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of


Hindawiwww.hindawi.com

Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling &Simulationin EngineeringHindawiwww.hindawi.com Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

www.hindawi.com Volume 2018

Advances in

Multimedia

Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/ijae/https://www.hindawi.com/journals/jr/https://www.hindawi.com/journals/apec/https://www.hindawi.com/journals/vlsi/https://www.hindawi.com/journals/sv/https://www.hindawi.com/journals/ace/https://www.hindawi.com/journals/aav/https://www.hindawi.com/journals/jece/https://www.hindawi.com/journals/aoe/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/jcse/https://www.hindawi.com/journals/je/https://www.hindawi.com/journals/js/https://www.hindawi.com/journals/ijrm/https://www.hindawi.com/journals/mse/https://www.hindawi.com/journals/ijce/https://www.hindawi.com/journals/ijap/https://www.hindawi.com/journals/ijno/https://www.hindawi.com/journals/am/https://www.hindawi.com/https://www.hindawi.com/

józeflisowski · 2019. 7. 30. · distance of own ship to the nearest point of return on a given...

Documents