PRACE NAUKOWE POLITECHNIKI WARSZAWSKIEJ z. 114 Transport 216 Józef Lisowski Gdynia Maritime University, Faculty of Electrical Engineering, Department of Ship Automation OPTIMIZATION OF MULTI-STAGE DECISION-MAKING PROCESS AT THE GAME MARINE ENVIRONMENT The manuscript delivered: March 216 Summary: The paper describes the process of ship game control in collision situations at sea. For the synthesis process control algorithm uses a simplified model of multi-positional game, allowing determination of the safe and optimal game traectory of own ship in passing situations with more ships encountered as a sequence of maneuvers, course and speed. Compared three criteria to optimize decision-making process for determining safe traectory of own ship taking into account the degree of cooperation encountered ships. Considerations exemplified the real situation on the sea registered in the Kattegat Strait. Keywords: maritime transport, safety of navigation, game theory 1. INTRODUCTION Some of the most important issues of the transport processes are optimal and safe control of ships, planes and cars as moving obects. Such processes relate to traffic control multiple obects at the same time, with varying degrees of cooperation, the impact of random factors not known probability distribution and a high proportion of operator subectivity in decisionmaking maneuver. Therefore, control of such processes is done by game control systems that are synthesized by the methods of game theory. The most adequate model of the own ship model control process in situation of m encountered obects is the model of differential game m participants [4, 6, 1]. Model of differential game can be reduced to model of multi-stage positional game, which takes into account the obect dynamics by means of lead time maneuver. The essence of the positional game is the dependence of own obect strategies for position p(t) of encountered obects [3, 12, 13].
218 Józef Lisowski 2. GAME CONTROL PROCESS OF SHIP As a result of the movement of own ship with speed V and in terms of encountered ship moving at a speed V and course is determined a situation at sea. Parameters characterizing the situation as distance D and bearing N for ship are measured by radar anti-collision system ARPA (Automatic Radar Plotting Aids) [5]. The ARPA system enables to track automatically at least 2 encountered ships, determination of their movement parameters (speed V, course ) and elements of approach to the own ship ( Dmin DCPA - Distance of the Closest Point of Approach, Tmin TCPA - Time to the Closest Point of Approach) and also the assessment of the collision risk (Fig. 1). Fig. 1. The situation of own ship passing of encountered ships The proper use of anti-collision system ARPA in order to achieve greater safety of navigation requires, in addition to training in the use and interpretation of the data, supplement the system with appropriate methods of computer-aided manoeuvring decision of navigator in the complex navigational situation in a short time, eliminating the subectivity of man and taking into account the indefiniteness of the situation and the properties game process control [2,8,11]. In practice, there are many possible maneuvers to avoid a collision, from which to select the optimal maneuver, to ensure a minimum the risk of collision or minimum losses of the road for safe passage of encountered ships (Fig. 2).
Optimization of multi-stage decision-making in the game marine environment 219 Fig. 2. Possible traectories of the own ship in case of passing the encountered ships 3. MULTI-STAGE POSITIONAL GAME Taking into account the high complexity of the differential game model, in practice used as simulation model for testing control algorithms, the practical synthesis of control algorithms formulated simplified models, while using chosen methods of artificial intelligence methods [1,7,9]. Individual models can be assigned to safety control ship algorithms in collision situations. For the practical synthesis of control algorithms can be used the positional game model. 3.1. BLOCK DIAGRAM GAME The essence of positional game is an addiction strategy U own ship from the position p(tk) of encountered ships in the current step k. In this way included in the process model possible changes course and speed of encountered ships during the realization of the control (Fig. 3).
22 Józef Lisowski Fig. 3. Game system of ships positional control The current state of the process is determined by the position coordinates of the own ship and m encountered ships: x X Y, x X, Y, 1, 2, m...,, (1) The system generates its control at the time tk on the basis of the data it receives from ARPA anti-collision system for the current tracked ships positions: x( tk ) p( tk ) 1, 2,..., m k 1, 2,..., K (2) x ( tk ) 3.2. SETS OF ADMISSIBLE STRATEGIES It is assumed, as a general concept of multi-stage positional game that at any discrete moment of time tk on the own ship is known position p(t) of encountered ships. Constraints of the state process control are navigational limitations: x t, x t P (3) Control constraints include kinematics ship moving, traffic rights recommendations sea route COLREG (COLlision REGulations) and condition of keeping a safe passing distance: where: U admissible set of own ship strategies, U admissible set of encountered ship. u U, u U 1, 2,..., m (4)
Optimization of multi-stage decision-making in the game marine environment 221 Sets of admissible strategies game participants to each other U and U are dependent which means that the choice of control encountered ship varies sets of acceptable strategies of other ships: U [ p( t)], U [ p( t)] The geometric construction of state process constraint associated with safety passing of encountered ship at a safe distance Ds is shown in Figure 4, in the form of two areas acceptable maneuvers to change own ship course - to port side or starboard side U. U PS (5) SS Fig. 4. Determining the set of admissible strategies U U PS USS of own ship in a positional game relative to encountered ship The total area of acceptable maneuvers of the own ship for m encountered ships describes the relationship: m U 1 U 1, 2,..., m (6)
222 Józef Lisowski Example of the total set of admissible maneuvers of own ship for six encountered ships shown in Figure 5. Fig. 5. Set of admissible maneuvers in the passing situation of six encountered ships Similarly, determining a set of acceptable strategies of encountered ship U in relation to the own ship, and the total area admissible maneuvers U. Taking into account the COLREG rules is possible by an appropriate choice of a set of maneuver. Presentation COLREG rules in the form of appropriate diagrams maneuver allows the formulation of a logical function Z as a semantic interpretation of these rules. To determine the optimal maneuver of the own ship, the total for all m encountered ships allowable area U controls, is possible use the principle of linear programming, capable of being formulated as follows: designate values x 1, x 2, corresponding components *,V *, for maximum throw x1 of own ship velocity V on a given course z, in the constraints of admissible own ship strategies: U x, x (7) 1 2
Optimization of multi-stage decision-making in the game marine environment 223 ai 1 x1 ai2 x2 bi, i 1, 2,..., m (8) The first seven constraints due to the linear approximation of the circle with a radius of V with an accuracy of 1%, the next pair of constraints are associated with lines tangent to a circle with a radius Ds in the port side PS and starboard side SS. The value of constraints coefficients are for encountered ship: i i,1 oz o a7 Z cos q Z (9) i, 2 oz o a7 Z sin q Z (1) q Z V cosq Z o b7 Z sin (11) where: Z =-1 for the subset of U PS, andz =1 for the subset of U SS. Angle values are unambiguous function of traffic parameters encountered ship: speed V, course, bearing N and distance D. o oz o 3.3. OPTIMIZATION OF DECISION-MAKING PROCESS Determination of the own ship optimal control, equivalent to the current position p(t) the optimal positional control, is carried out in two stages: determine the admissible sets strategies U pt k of encountered ships relative to own w ship and output admissible sets U pt k of own ship in relation to each of the encountered ship, determination with respect to each encountered ship a pair of vectors u and then the optimal positional strategy u ( p ) of the own ship, according to the optimization quality index taking into account the degree of obects cooperation. Quality index of non-cooperative positional game control for selection of the optimal and safe game traectory of the own ship comes down to determine its course and speed to ensure the smallest loss of the way for safe passing encountered ships at a distance of not less than a predetermined value Ds or the shortest distance S to the point of return Lz, including the own ship dynamics in form lead time maneuver. The smallest losses way reached the maximum proection of the own ship velocity on the direction of the z reference course. I min max min S x( tk ) S, m m U U u U u U ( u ) u 1 1,2,..., m m u (12)
224 Józef Lisowski The optimal control of own ship is calculated at each discrete stage of the ship movement by applying the Simplex method to solve the problem of triple linear programming. Quality index of cooperative positional game control has following form: I min min min S x( tk ) S, m m U U u U u U ( u ) u 1 Quality index of non-game control is described by formula: 1,2,..., m (13) I m u U U 1 S min x( t ) k S, 1,2,..., m (14) 4. COMPUTER SIMULATION OF POSITIONAL GAME The algorithm multi-stage positional game mgp with the determination of a safe and game traectory of the ship in collision situation was developed using the function lp -linear programming with Optimization Toolbox software Matlab/Simulink. The game traectories of own ship in situation m = 12 encountered ships in the Kattegat Strait, in conditions of good visibility at sea with Ds =.5 nm (nautical miles) and restricted visibility at sea with Ds = 2.5 nm, determined by an algorithm mgp for non-cooperative positional game shown in Figure 6, the cooperative positional game in Figure 7, and for the non-game control in Figure 8. Fig. 6. The traectories in a positional non-cooperative game of the own ship and m =12 encountered ships: left - in conditions of good visibility at sea with D s=.5 nm and final payment
Optimization of multi-stage decision-making in the game marine environment 225 d(t k)=5.42 nm; right - in conditions of restricted visibility at sea with D s=2.5 nm and final payment d(t k)=6.81 nm Fig. 7. The traectories in a positional cooperative game of the own ship and m = 12 encountered ships: left - in conditions of good visibility at sea with D s =.5 nm and final payment d(t k) = 4.36 nm; right - in conditions of restricted visibility at sea with D s = 2.5 nm and final payment d(t k) = 4.87 nm Fig. 8. The traectories in a non-game control of the own ship and m = 12 encountered ships: left - in conditions of good visibility at sea with D s =.5 nm and final payment d(t k) = 2.37 nm; right - in conditions of restricted visibility at sea with D s = 2.5 nm and final payment d(t k) = 4.54 nm The final game payment as a final deviation of safe traectory own ship from the reference traectory is bigger than the non-cooperative game for cooperative game, in conditions of good visibility by 24%, and in conditions of restricted visibility by 4%. The final game payment is less, from 1%to 8%, for non-game ship control, but does not include the risk of collision resulting from maneuvering of met ships.
226 Józef Lisowski Taking into account the visibility conditions on the sea, by increasing the passing safe distance from.5 nautical miles to 2.5 nautical miles, leads to an increased game final payment of 12-9%. 5. CONCLUSION Application to the synthesis of the control algorithm using model of multi-stage positional game allows to determine the optimal safe and game traectory of own ship in situations passing more encountered ships as a sequence of maneuvers, course and speed. Developed control algorithm takes into account the COLREG rules of international law of the sea route of vessel traffic and lead time maneuver, approximating the dynamic properties of the own ship, and evaluates the final traectory deviation from the reference traectory. The control algorithm provides a formal model of the actual decision-making process leading ship navigator and can be used with the system of computer-aided navigator when deciding to maneuver in case of collision at sea. Bibliography 1. Basar T., Olsder G.J.: Dynamic non-cooperative game theory. SIAM, Philadelphia 213. 2. Bist D.S.: Safety and security at sea. Butter Heinemann, Oxford-New Delhi 2. 3. Engwerda J.C.: LQ dynamic optimization and differential games. John Wiley & Sons, West Sussex 25. 4. Isaacs R.: Differential games. John Wiley & Sons, New York 1965. 5. Kouemou G.: Radar technology. Chapter 4 by Józef Lisowski: Sensitivity of safe game ship control on base information from ARPA radar, In-tech, Croatia 29, p. 61-86. 6. Mesterton-Gibbons M.: An introduction to game theoretic modeling. American Mathematical Society, Providence 21. 7. Millington I., Funge J.: Artificial intelligence for games. Elsevier, Amsterdam-Tokyo 29. 8. Modarres M.: Risk analysis in engineering. Taylor & Francis Group, Boca Raton 26. 9. Nisan N., Roughgarden T., Tardos E., Vazirani V.V.: Algorithmic game theory. Cambridge University Press, New York 27. 1. Osborne,M.J.: An introduction to game theory. Oxford University Press, New York 24. 11. Perez T.: Ship motion control. Springer, London 25. 12. Straffin P.D.: Game theory and strategy. Scholar, Warszawa 21. 13. Wells D.: Games and mathematics. Cambridge University Press, Cambridge 213. OPTYMALIZACJA WIELOETAPOWEGO PROCESU DECZYJNEGO Streszczenie: morzu. Do syntezy algorytmu sterowania procesem wykorzystano model wieloetapowe gry pozycyne, statków, ako :