http://www.transnav.eu the International Journal on Marine Navigation and Safety of Sea Transportation Volue 12 Nuber 1 March 2018 DOI: 10.12716/1001.12.01.09 Cooperative and Non-Cooperative Gae Control Strategies of the Ship in Collision Situation J. Lisowski Gdynia Maritie University, Gdynia, Poland ABSTRACT: The paper introduces the positional cooperative and non cooperative gae of a greater nuber of et ships for the description of the process considered as well as for the synthesis of optial control strategies of the own ship in collision situation. The approxiated atheatical odel of differential gae in the for of triple linear prograing proble is used for the synthesis of safe ship traectory as a ultistage process decision. The considerations have been illustrated an exaple of progra coputer siulation to deterine the safe ship traectories in situation of passing a any of the ships encountered. 1 INTRODUCTION A large part in increasing the safety of navigation is the use of ARPA anti collision syste, which enables to track autoatically at least 20 encountered ships, deterination of their oveent paraeters: speed V, course ψ and eleents of approach: D distance, N bearing, D, in DCPA Distance of the Closest Point of Approach, T TCPA, in Tie to the Closest Point of Approach (Fig. 1). The functional scope of a standard ARPA syste ends with anoeuvre siulation to achieve the safe passing distance Ds by altering course or speed V selected by the navigator (Bist 2000, Cockroft & Laeier 2006, Cahill 2000). The ost general description of the own control obect passing the nuber of other encountered obects is the odel of a differential gae of a nuber of obects (Basar & Olsder 1982, Engwerda 2005, Isaacs 1965, Mesterton Gibbons 2001). Figure 1. The own ship oving with speed V and course during of passing encountered ships. This odel consists both of the kineatics and the dynaics of the ship s oveent, the disturbances, 83
the strategies of the own ship and encountered ships and the quality control index (Clarke 2003, Kula 2015, Osborne 2004). The diversity of possible odels directly affects the synthesis of ship control algoriths which are afterwards affected by the ship control device, directly linked to the ARPA syste and consequently deterines effects of safe and optial control (Fletcher 1987, Lisowski 2011). 2 MODEL OF GAME SHIP CONTROL PROCESS The current state of the process is deterined by the co ordinates for the position of the own ship and of the ships encountered: 0 o, x X Yo x X, Y 1, 2,..., The syste generates its steering at the oent tk on the basis of the data which are obtained fro the ARPA anti collision syste concerning the current positions of the own and encountered ships: (3) 2.1 State and control variables The differential gae described by state equation: x f x, u, t (1) x0 ( ) ( tk ) ptk x ( tk ) 1, 2,..., k 1, 2,..., K (4) is reduced to a positional ultistage gae of a nuber of participants (Fig. 2). It is assued, according to the general concept of the ultistage positional gae, that at each discrete oent of the tie tk the position of the encountered ships p(tk) is known on the own ship (Gierusz & Lebkowski 2012, Lisowski 2012, Malecki 2013, Mohaed Seghir 2016, Zak 2013). The constraints of the state co ordinates: 0, x t x t P (5) constitute the navigational constraints, while the steering constraints: u u 0 Uo U 1, 2,..., (6) Figure 2. Block diagra of the positional gae odel of passing the own ship and encountered ships. The state x and control u variables are represented by (Lisowski & Lazarowska 2013): x0,1 Xo, x0,2 Yo, x,1 X, x,2 Y u0,1, u0,2 V, u,1, u,2 V 1, 2,..., The aking of a continuous positional gae discrete and reducing it to a ultistage positional gae is deterined 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 own ship (Gluwer & Olsen 1998, Isil & Koditschek 2001). The essence of the positional gae is to ake the strategies of the own ship dependent on current positions p(tk) of the ships 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 (Lazarowska 2012, Luus 2000). (2) take into consideration the kineatics of the ship oveent, the recoendations of the COLREGS Rules and the condition to aintain the safe passing distance Ds: in (7) D,in D t Ds 2.2 Sets of acceptable strategies The closed sets Uo, and U,o defined as the sets of the acceptable strategies of the ships as players: Uo, p t So1, So2, U o, p t S1, os2, o are depended on the position p(t), which eans that the choice of the steering u by the th encountered ship alter the sets of the acceptable strategies of the other ships (Fig. 3). (8) 84
F 1 then o, 1 0 then o, 1 (11) Figure 3. Deterination of the acceptable safe strategies areas of the own ship Uo, So1, So2, and the encountered ship U, o S1, o S2, o. Let refer to the set of the acceptable strategies of the own ship while passing the th encountered ship at a safe distance Ds (Jaworski, Kuczkowski, Sierzchalski 2012; Lebkowski 2015, Nisan et al. 2007, Straffin 2001, Toera 2015, Lisowski 2014a). 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 within the arrangeent of the coordinates 0X Y with the axis X directed to the direction of the reference course (Millington & Funge 2009, Modarres 2006, Szlapczynski 2012). The set Uo, is deterined with the inequalities: a u b u c o, o, x' o, o, y' o, 2 2 2 ox, ' oy, ' u u V V uo( uo, x', uo, y' ) ao, o, cos( qo, o, o, ) b sin( q ) c o, o, o, o, o, o, o, o, V sin( q, o o, o, ) V cos( qo, o, o, ) 1 for So1, ( Port side) 1 for So2, ( Starboard side) (9) (10) The value o, is deterined by using an appropriate logical function F characterising any particular recoendation referring to right of way contained in COLREGS Rules (Lisowski 2014b). Interpretation of the COLREGS Rules in the for of appropriate anoeuvring diagras enables to forulate a certain logical function F as a seantic interpretation of legal regulations for anoeuvring. Each particular type of the situation involving the approach of the ships is assigned the logical variable value equal to one or zero (Lisowski 2015a, 2015b): A encounter of the ship fro bow or fro any other direction, B approaching or oving away of the ship, C passing the ship astern or ahead, D approaching of the ship fro the bow or fro the stern, E approaching of the ship fro the starboard or port side (Lisowski 2015c, 2015d). By iniizing logical function F by using a ethod of the Karnaughʹs Tables the following is obtained: F A A( B C D E) (12) The resultant area of acceptable anoeuvres of the own ship in relation to the encountered ships is: U o 1 U o, 1, 2,..., (13) is deterined by an arrangeent of inequalities (9). On the other hand, however, the set of the acceptable strategies of the th obect in relation to the own ship is deterined by the following inequalities: a u b u c o, x, ' o, y, ' o, 2 2 2 x, ' y, ' u u V V u ( u, x', u, y' ) ao, o, cos( qo, o, o, ) bo, o, sin( qo, o, o, ) c V sin( q ) o, o, o, o, o, o, 1 for S 1, o ( Port side) 1 for S 2, o ( Starboard side) (14) (15) 2.3 Seantic interpretation of COLREGS Rules The for of function F depends of the interpretation of the above recoendations for the purpose to use the in the control algorith, when: The sybol, o is deterined by analogy to the deterination of o, with the use of the logical function F described by the equation (11). 85
Consideration of the navigational constraints, as shallow waters and coastline, generate additional constraints to the set of acceptable strategies: a u b u c (16) nk, ox, ' nk, ox, ' nk, k is the nearest point of intersection of the straight lines approxiating the coastline. 3 TYPES OF GAME AND SAFE SHIP CONTROL STRATEGIES 3.1 Gae optial control rules * The optial control uo t of the own ship, equivalent for the current position p(t) to the optial * positional control uo p, is deterined in the following way: fro the relationship (14) for the easured position p(tk), the control status at the oent tk sets of the acceptable strategies U pt o, k are deterined for the encountered ships in relation to the own ship, and fro the relationship (9) the output sets Uo, ptk of the acceptable strategies of the own ship in relation to each one of the encountered ships, a pair of vectors u,o and uo,, are deterined in relation to each encountered ship and then the optial positional strategy of the own ship nc I u o ( p) fro the condition of optiu value I* quality index control: when the encountered ships non cooperate: 1,2,..., c I in ax in Lxo ( tk), Lk Lo, u, ou, o uo, U o, ( u) uou ouo, 1 when the encountered ships cooperate: 1,2,..., oc I nc (17) in in in Lxo ( tk), Lk Lo, c u U u U u U, o, o o, o, ( ) (18) uou o o, 1 for the non gae optial control: 1,2,..., in L xo( tk), Lk Lo, oc uou o Uo, (19) 1 t K Lxo( tk), Lk V( t) dt ro( tk) d( tk) (20) t 0 refers to the goal control function of the own ship in the for of the payents the integral payent and the final one (Lisowski 2016a). The integral payent deterines the distance of the own ship to the nearest turning point Lk on the assued route of the voyage and the final one deterines: ro(tk) the final risk of collision and d(tk) final gae traectory deflection fro reference traectory. The criteria for the selection of the optial traectory of the own ship is reduced to the deterination of her course and speed, which ensure the sallest losses of way for the safe passing of the encountered ships at a distance not saller than the assued safe value Ds, having regard to the ship s dynaic in the for of the advance tie t to the anoeuvre (Lisowski 2016b). 3.2 Paraeters of ship dynaics At the tie advance aneuver t consists of eleent t during course anoeuvre or eleent V t during speed anoeuvre V. The dynaic features of the ship during the course alteration by an angle is described in a siplified anner with the use of transfer function: () s k( ) k( ) e G () s () s s(1 T s) s T s o (21) T o T anoeuvre delay tie which is approxiately equal to the tie constant of the ship as a course control obect, k ( ) gain coefficient the value of which results fro the non linear static characteristics of the rudder steering. The course anoeuvre delay tie is as follows: t To (22) In practice, depending on the size and type of vessel advance tie to the anti collision anoeuvre through a change of course is: t 60720 s. Differential equation of the second order describing the shipʹs behaviour during the change of the speed by V is approxiated with the use of the inertia of the first order with a tie delay: 86
T s V() s ke ov G () v V s ns () 1Ts v (23) Tov tie of delay equal approxiately to the tie constant for the propulsion syste: ain enginepropeller shaft screw propeller, Tv the tie constant of the shipʹs hull and the ass of the accopanying water. The speed anoeuvre delay tie is as follows: V t T 3T (24) ov v In practice, depending on the size and type of vessel advance tie to the anti collision anoeuvre through a change of speed is: t V 120900 s. Using the function of lp linear prograing fro the Optiization Toolbox contained in the Matlab/Siulink the positional ultistage gae anoeuvring progras: pgae_nc for criterion (17), pgae_c for criterion (18) and pngae_oc for criterion (19) has been designed for deterination of the safe ship traectory in a collision situation. 4 COMPUTER SIMULATION 4.1 Navigational situation Coputer siulation of pgae_nc, pgae_c and pngae_oc algoriths was carried out in Matlab/Siulink software on an exaple of the real navigational situation of passing =19 encountered ships in the Skagerrak Strait in good visibility Ds=0.5 1.0 n and restricted visibility Ds=1.5 2.5 n (nautical iles), (Fig. 4 and Tab. 1). 3.3 Coputer progras of positional ultistage gae ship control 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 Lk point. The optial control of the own ship is calculated at each discrete stage of the ship s oveent by applying triple linear prograing SIMPLEX ethod, assuing the relationship (20) as the goal function and the constraints are obtained by including the arrangeent of the inequalities (8), (14) and (16). The above proble is then reduced to the deterination the function of control goal as the axiu of the proection of the own ship speed vector on reference direction of the oveent: in L ax V u, u u ox, ' oy, ' ox, ' (25) with linear constraints approxiating the oint set of the safe strategies of the own ship Uo,: ao,1 uo, x' bo,1 uo, y' co,1 ao, uo, x' bo, uo, y' co, ao, uox, ' bo, uoy, ' co, ao, 1 uox, ' bo, 1 uoy, ' co, 1 a u b u c o, kp o, x' o, kp o, y' o, kp (26) nuber on encountered ships, k nuber of constraints approxiating coastline, p nuber of segents approxiating a sei circle with a radius equal to own ship speed. After the interval of tie tk the current fixing of the ship position is carried out and then coes the solving of the proble using the algorith for the positional control. Figure 4. The place of identification of navigational situations in Skagerrak and Kattegat Straits. The situation was registered on board r/v HORYZONT II, a research and training vessel of the Gdynia Maritie University, on the radar screen of the ARPA anti collision syste Raytheon (Fig. 5 and 6). 87
Table 1. Moveent paraeters of the own ship and encountered 19 ships. D N V ψ n deg kn deg 0 20 0 1 9 320 14 90 2 15 10 16 180 3 8 10 15 200 4 12 35 17 275 5 7 270 14 50 6 8 100 8 6 7 11 315 10 90 8 13 325 7 45 9 7 45 19 10 10 15 23 6 275 11 15 23 7 270 12 4 175 4 130 13 13 40 0 0 14 7 60 16 20 15 8 120 12 30 16 9 150 10 25 17 8 310 12 135 18 10 330 10 140 19 9 340 8 150 Figure 7. The 12 inute speed vectors of own ship 0 and =19 encountered ships in navigational situation in Skagerrak Strait. 4.2 Siulation of the ulti stage non cooperative positional gae Fig. 8 and 9 show the safe and optial traectory of the own ship in collision situation, which is deterined using the algoriths of non cooperative positional gae in good and restricted visibility at sea. Figure 5. The research training ship of Gdynia Maritie University r/v HORYZONT II. Figure 8. Coputer siulation of ulti stage noncooperative positional gae algorith pgae_nc for safe own ship control in situation of passing 19 encountered ships in good visibility at sea, Ds=1.0 n, d(tk)=3.34 n (nautical ile). Figure 6. The screen of anti collision syste ARPA Raytheon, installed on the research training ship of Gdynia Maritie University r/v HORYZONT II. Exained the navigational situation, illustrated in the for of navigation velocity vectors of own ship and 19 et ships is shown in Figure 7. 88
Figure 9. Coputer siulation of ulti stage noncooperative positional gae algorith pgae_nc for safe own ship control in situation of passing 19 encountered ships in restricted visibility at sea, Ds=2.5 n, d(tk)=7.34 n (nautical ile). Figure 11. Coputer siulation of ulti stage cooperative positional gae algorith pgae_c for safe own ship control in situation of passing 19 encountered ships in restricted visibility at sea, Ds=2.5 n, d(tk)=7.06 n (nautical ile). 4.3 Siulation of the ulti stage cooperative positional gae Fig. 10 and 11 show the safe and optial traectory of the own ship in collision situation, which is deterined using the algoriths of cooperative positional gae in good and restricted visibility at sea. 4.4 Siulation of the non gae optial control Fig. 12 and 13 show the safe and optial traectory of the own ship in collision situation, which is deterined using the algoriths of non gae optial control in good and restricted visibility at sea. Figure 10. Coputer siulation of ulti stage cooperative positional gae algorith pgae_c for safe own ship control in situation of passing 19 encountered ships in good visibility at sea, Ds=1.0 n, d(tk)=2.94 n (nautical ile). Figure 12. Coputer siulation of non gae optial control algorith pngae_oc for safe own ship control in situation of passing 19 encountered ships in good visibility at sea, Ds=1.0 n, d(tk)=0.72 n (nautical ile). 89
Figure 13. Coputer siulation of non gae optial control algorith pngae_oc for safe own ship control in situation of passing 19 encountered ships in restricted visibility at sea, Ds=2.5 n, d(tk)=3.76 n (nautical ile). 5 CONCLUSIONS The synthesis of an optial on line control on the odel of a ulti stage positional gae akes it possible to deterine 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 coputer progras designed in the Matlab also takes into consideration the following: regulations of the Convention on the International Regulations for Preventing Collisions at Sea, advance tie for a anoeuvre calculated with regard to the ship s dynaic features and the assessent of the final deflection between the real traectory and its assued values. 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