The dynamic game theory methods applied to ship control with minimum risk of collision

Transcription

1 Risk Analysis V: Siulation and Hazard Mitigation 293 The dynaic gae theory ethods applied to ship control with iu risk of collision J. Lisowski Departent of Ship Autoation, Gdynia Maritie University, Poland Abstract This paper describes the application of eleents of the gae theory for the purpose of optial control of soe dynaic continuous processes. Using as an exaple the process of safe ship s control, the paper presents the proble of applying a positional non-cooperative gae of obects for the description of the process considered as well as for the synthesis of optial strategies. A atheatical odel of differential gae is forulated and its approxiation in the for of a dual linear prograg proble is used for the synthesis of safe ship s traectory as a ultistage process decision. The considerations have been illustrated in an exaple of a coputer siulation (the POSGAME prograe) to detere the safe ship s traectory in a situation of passing any of the obects encountered. Keywords: differential gaes, positional gaes, dual linear prograg, safety navigation, risk analysis, ship control. 1 Introduction In order to ensure safe navigation the ships are obliged to observe legal requireents contained in the International Regulations for Preventing Collision at Sea (COLREG). However, these Rules refer exclusively to two ships under good visibility conditions, in case of restricted visibility the Rules provide only recoendations of general nature and they are unable to consider all necessary conditions of the real process. Therefore the real process of the ships passing exercises occurs under the conditions of indefiniteness and conflict accopanied by an iprecise co-operation aong the ships in the light of the legal regulations. Consequently, it is reasonable - for ship operational purposes - to present this process and to develop and exae ethods for a safe steering of WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press doi:1.2495/risk6281

2 294 Risk Analysis V: Siulation and Hazard Mitigation the ship by applying the rules of the gae theory. A necessity to consider siultaneously the strategies of the encountered obects and the dynaic properties of the ships as the steering obects is a good reason for the application of the differential gae odel - often called the dynaic gae - for the description of the processes [1, 1, 12, 13]. The challenge in research for effective ethods to prevent ship collisions has becoe iportant with the increasing size, speed and nuber of ships participating in sea carriage. An obvious contribution in increasing safety of shipping has been firstly the application of radars and then the developent of ARPA (Autoatic Radar Plotting Aids) anti-collision syste. The ARPA syste enables to track autoatically at least 2 encountered obects, deteration of their oveent paraeters (speed V, course ψ ) and eleents of approach to the own ship ( D DCPA - Distance of the Closest Point of Approach, T TCPA - Tie to the Closest Point of Approach) and also the assessent of the collision risk r, fig. 1. Figure 1: Navigational situation representing the passing of the own ship with the -th obect. However the functional scope of a standard ARPA syste ends with the siulation of the anoeuvre altering the course ± ψ or the ship's speed ± V selected by the navigator. The ost general description of the own control obect passing the nuber of other encountered oving obects is the odel of a dynaic gae of a nuber of obects [2, 3, 4, 5, 11]. WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

3 2 The positional gae odel of process Risk Analysis V: Siulation and Hazard Mitigation 295 The dynaic gae is reduced to a positional ultistage gae of a nuber of participants who do not co-operate aong the [6, 7], fig. 2. The state and control variables are represented by the following values: x X, x Y, x X, x Y, u ψ, u V, u ψ, 1, 2,..., 1 u 2 V (1) Figure 2: Block diagra of a positional odel of the gae. The aking of a continuous positional gae discrete and reducing it to a ultistage positional gae is detered by own ship and depends on: the axiu relative speed of the own ship under the current navigational situation, the range of the situation and the dynaic characteristics of the ship [8]. The essence of the positional gae is to ake the strategies of the own ship dependent on current positions p(t k ) of the obects encountered at the current step k. In this way possible course and speed alterations of the obects encountered are considered in the process odel during the steering perforance. The current state of the process is detered by the co-ordinates for the position of the own ship and of the obects encountered: ( X,Y ), x ( X,Y ), 1,,..., x 2 (2) The syste generates its steering at the oent t k on the basis of the data which are obtained fro the ARPA anti-collision syste concerning the current positions of the ships: x( tk ) p( tk ) 1, 2,..., k 1, 2,..., K (3) x ( tk ) It is assued, according to the general concept of the ultistage positional gae, that at each discrete oent of the tie t k the position of the obects p(t k ) is WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

4 296 Risk Analysis V: Siulation and Hazard Mitigation known on the own ship. The constraints of the state co-ordinates: { ( ), ( ) } x t x t P (4) constitute the navigational constraints, while the steering constraints: u U, u U 1, 2,..., (5) take into consideration the kineatics of the ships oveent, the recoendations of the COLREG Rules and the condition to aintain the safe passing distance D s : D D ( t) Ds (6) The closed sets U and U defined as the sets of the acceptable strategies of the players: [ p( t) ],U [ p( t) ] U (7) are dependent, which eans that the choice of the steering u by the -th obect alter the sets of the acceptable strategies of the other obects. Let refer to the set of the acceptable strategies of the own ship while passing the -th encountered obect at a distance D s. The area, when aintaining stability in tie of the course and speed of the own ship and the ship encountered is static and is coprised within the seicircle of a radius equal to the set reference speed of the own ship V r within the arrangeent of the co-ordinates X Y with the axis X directed to the direction of the set course, fig. 3. Figure 3: Deteration of the acceptable strategies of the own ship and the obect encountered. WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

5 Risk Analysis V: Siulation and Hazard Mitigation 297 The area U is detered with the inequalities (8), (9) and (1): where: a x y u b u c ( ) ( ) x 2 y 2 2 u u Vr + (8) + (9) 2 x y Vr V u( u, u) a χ cos( qr + χ δ ) b χ sin( qr + χ δ ) c χ V sin( q + χ δ ) + Vr cos( qr + χ δ ) 1 for W1 ( starboard ) χ 1 for W2 ( port ) The sybol χ is detered with the use of an appropriate logical function Z which characterises particular COLREG Rules. The for of the function Z depends on the interpretation of these recoendations in order to consider it in the steering algorith, however: (1) 1 Z then then χ 1 χ 1 (11) The interpretation of the COLREG Rules in the for of the anoeuvring diagras developed by A.G. Corbet, S.H. Hollingdale, E.S. Calvert and K.D. Jones akes it possible to forulate the logical function Z as a seantic interpretation of the legal rules of anoeuvring. Each particular ships passing situation is referred to the appropriate logical variable and its value 1 or : A encounter of a ship fro bow or fro other directions, B closing or sailing away of the ship, C passing the ship behind the stern or ahead of the bow, D closing of the ship fro bow or fro astern, E closing of the ship fro the starboard side or the port side. By iising the logical function Z using the ethod of Karnaugh s tables we can obtain: Z A A( B C D E ) (12) The resultant area of the acceptable anoeuvres of the own ship in relation to the obects is: U U, 1 1 2,..., (13) and is detered by an arrangeent of the inequalities (8) and (9). WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

6 298 Risk Analysis V: Siulation and Hazard Mitigation On the other hand, however, the set of the acceptable strategies of the -th obect in relation to the own ship U is detered by the following inequalities: where: x u x 2 a y y 2 + b u c (14) 2 ( u ) ( u ) V + (15) x y V u ( u, u ) a cos( q + ) χ χ δ b χ sin( q + χ δ ) c χ V sin( q + ) χ δ (16) The sybol χ is detered by analogy to the deteration of use of the logical function Z described by the equation (12). 3 Algorith of the ultistage positional gae χ with the The optial steering of the own ship u ( t ), equivalent for the current position p(t) to the optial positional steering u ( p ), is detered in the following way: - fro the relationship (14) and (15) for the easured position p(t k ) of the steering status at the oent t k sets of the acceptable strategies U [ p( t k )] are detered for the encountered obects in relation to the own ship, and fro w the relationship (8) and (9) the output sets U [ p( tk )] of the acceptable strategies of the own ship in relation to each one of the encountered obects, - a pair of vectors u and u, is detered in relation to each -th obect and then the optial positional strategy of the own ship u ( p ) fro the condition: when the encountered obects non co-operate 1 I U 1 u U ax u U S u U (u ) [ x (t ),L ] k S when the encountered obects co-operate (x,l ), U U w (17) WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

7 Risk Analysis V: Siulation and Hazard Mitigation I U 1 u U u U S u U (u ) [ x (t ),L ] k S (x,l ), U U w (18) for the non-gae optial control 3 I U 1 u U { S [ x (t ),L ]} k S (x,l ), U U w, 1, 2,..., (19) where: [ x ( t ), L ] u ( t )dt + r ( t ) + t L S (2) t K d( tk ) refers to the goal control function of the own ship in the for of the payents the integral payent and the final one. The integral payent deteres the distance of the own ship to the nearest turning point L on the assued route of the voyage and the final one deteres the risk of collision and final gae traectory deflection fro reference traectory. The criteria for the selection of the optial traectory of the own ship is reduced to the deteration of her course and speed which ensure the sallest losses of way for the safe passing of the encountered obects at a distance not saller than the assued value D s, having regard to the ship s dynaic in the for of the advance tie to the anoeuvre t. The sallest losses of way are achieved for the axiu proection of the speed vector of the own ship on the direction of the assued course leading to the nearest turning point L. The optial steering of the own ship is calculated at each discrete stage of the ship s oveent by applying the SIMPLEX ethod to solve the proble of the linear prograg, assug the relationship (2) as the goal function and the constraints are obtained by including the arrangeent of the inequalities (8) and (9) or (14) and (15). The above proble is then reduced to the following operations: we have to detere the function of the control goal as the axiu of the proection of the speed vector of the own ship on the assued direction of the oveent: [ F( x, y ) ] ax 1 1 x1 (21) with linear constraints approxiating the oint set of the safe strategies of the own ship U : WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

8 3 Risk Analysis V: Siulation and Hazard Mitigation a x 1 + b y 1 c a + x1 b y1 c a x1 + b y1 c (22) a x1 + b y1 c + l+ s + l+ s + l+ s a x1 + b y1 c The above constraints are detered by the obects onitored by the ARPA syste; this refers to the first constraints, the next l describe the coastline and the reaining inequalities s approxiate the circle of a radius equal to the speed of the own ship. After the interval of tie t k the current fixing of the ship s position is carried out and then coes the solving of the proble using the algorith for the positional steering. Using the function of lp linear prograg fro the Optiization Toolbox contained in the Matlab, the positional ultistage gae anoeuvring progras: POSGAME 1 for criterion (17), POSGAME 2 for criterion (18) and POSGAME 3 for criterion (19) has been designed for the deteration of the safe ship s traectory in a collision situation [9]. 4 Coputer siulation Siulation tests of the POSGAME 1, POSGAME 2 and POSGAME 3 progras have been carried out with reference to real situation recorded on the basis of the radar screen. Fig. 4 shows the coputer siulation, perfored on the POSGAME progras, for the own ship s traectory in a situation of passing 42 encountered obects in the English Channel region. 5 Conclusion The application of the odel of a ultistage positional gae for the synthesis of an optial anoeuvring akes it possible to detere the safe gae traectory of the own ship in situations when she passes a greater nuber of the encountered obects. The traectory has been described as a certain sequence of anoeuvres with the course and speed. The positional gae anoeuvring POSGAME coputer progra designed in the Matlab language also takes into consideration the following: Rules of the Convention on the International Regulations for Preventing Collisions at Sea COLREG, advance tie for a anoeuvre t calculated with regard to the ship s dynaic features and the assessent of the final deflection d(t K ) between the real and reference traectories. The big influence for deteration of the safe traectory has the values of tie digitization tk and advance tie to the anoeuvre t. The presented odel and ethod of steering a ship ay be used in practice both to iprove the shipboard anti-collision syste and to construct appropriate ARPA training siulator at Officer Training Centre. WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

9 Risk Analysis V: Siulation and Hazard Mitigation 31 Good visibility, D s.6 n Restricted visibility, D s 3. n r (t K ), d(t K )4.67 n POSGAME 1 r (t K ), d(t K )12.11 n r (t K ), d(t K )1.67 n POSGAME 2 r (t K ), d(t K )3.16 n r (t K ), d(t K )1.97 n POSGAME 3 r (t K ), d(t K )2.25 n Figure 4: The results of the coputer siulation for the safe anoeuvring of the own ship in a situation of passing 42 encountered obects. WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press

10 32 Risk Analysis V: Siulation and Hazard Mitigation References [1] Isaacs R., Differential gaes, John Wiley and Sons: New York, [2] Lisowski J., Mohaed-Seghir M., The safe ship control with iu risk of collision, Risk Analysis I, Wit Press, Coputational Mechanics Publications: Southapton and Boston, pp , [3] Lisowski J., Siulation odels of the safe ship s steering process in collision situations, Risk Analysis II, Wit Press, Coputational Mechanics Publications: Southapton and Boston, pp , 2. [4] Lisowski J., Gae control of oving obects. Proc. of the 15 th IFAC World Congress: Barcelona, pp.1-6, 22. [5] Lisowski J., Multi-step atrix gae with the risk of ship collision, Risk Analysis IV, Wit Press, Coputational Mechanics Publications: Southapton and Boston, pp , 24. [6] Lisowski J., The ultistage positional gae of process decision in arine collision situations. Proc. of the XV International Conference on Systes Science: Wroclaw, pp. 1-1, 24. [7] Lisowski J., Matheatical odeling of a safe ship optial control process. Polish Journal of Environental Studies, 14(1), pp , 25. [8] Lisowski J., Gae and coputational intelligence decision aking algoriths for avoiding collision at sea. Proc. of the IEEE Int. Conf. on Technologies for Hoeland Security and Safety: Gdansk, pp , 25. [9] Łebkowski A., Study and coputer siulation of gae positional control algorith with consideration ship s speed changes in Matlab: M.Sc. thesis, Gdynia Maritie Acadey, 21 (in Polish). [1] Merz A.W. & Kararkar J.S., Collision avoidance systes and optial turn anoeuvres. Journal of Navigation, 29(2), pp , [11] Miloh T. & Shara S.D., Maritie collision avoidance as a differential gae. Schifftechnik, 24(1), pp , [12] Olsder G.J. & Walter J.L. A differential gae of approach to collision avoidance of ships. Proc. of the 8 th IFIP Syposiu on Optiization Techniques, pp , [13] Vincent T.L., Collision avoidance at sea. Proc. of the Workshop Enschede Differential Gaes and Applications, WIT Transactions on Ecology and the Environent, Vol 91, 26 WIT Press