COMPUTATIONAL INTELLIGENCE AND OPTIMIZATION METHODS APPLIED TO SAFE SHIP S CONTROL PROCESS

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1 COMPUTATIONAL INTELLIGENCE AND OPTIMIZATION METHODS APPLIED TO SAFE SHIP S CONTROL PROCESS Józef Lisowski Gdynia Maitime Academy, Poland 81-5 Gdynia, 83 Moska St., lis@wsm.gdynia.pl ABSTRACT The pape descibes the application of selected methods of an optimal contol theoy, game theoy and atificial neal netwoks with the aim to atomatize the pocess of safe handling of the ship's movement. The desciptions contained in this pape show the stcte of a mlti-dimensional steeing system, define the task of a safe steeing and conseqently models of the pocess and methods fo detemination of the ship's stategy in collision sitations have been pesented. The consideations have been illstated with example of calclating safe own ship's taectoies in sitation of passing twenty obects enconteed. INTRODUCTION To asse safety ship navigation is one of the most impotant poblems in the maine navigation. The poblem of collision avoidance has become an gent isse, theefoe it is necessay to descibe the pocess of collision avoidance moe accately. Accoding to the Lloyd s data the volme of tonnage lost as a eslt of collisions sstained by vessels in popotion to all losses sstained as a eslt of sinking, fie, missing o shipweck vaies depending on a yea fom a few p to a dozen o so pecent. It is assmed that appoximately half of the losses may be avoided sing bette methods of comptesppoted safe steeing of the ship's movement, specially with latest methods of a moden theoy of contol engineeing - static and dynamic optimization with state constaints togethe with methods of comptational intelligence diffeential games, fzzy sets, neal netwoks, expet systems, genetic algoithms and evoltionay pogamming. Pete Padfield in his book An agony of collisions stated that the shipmastes, with exceptions eqiing special attention, ae not eckless, impdent maniacs of low sense of esponsibility who dive one into the othe in a totally absent-mindedly attack. The shipmastes ae simply people facing the tasks which ae impossible to be pefomed. They stive to cove the distance fom A to B in a detemined time withot colliding with othe vessels while the latte also stive to avoid the fist. At that citical moment on one knows how the othe intends to behave and, even moe, how that othe is obliged to behave in the pevailing sitations. Few shipmastes ae ganted the sixth sense [17]. This is exactly the eason why the poblem of safe steeing of the ship's movement is of a concen not only to the mastes opeating a ship, bt also to the lectes at maitime academies who pepae candidates fo this maine pofession thogh lectes, classes n on compte simlatos and physical models of ships. These ae followed by appenticeship on boad vaios types of eal ships. This poblem is also of a concen to electonic and contol enginees who constct all kinds of technical aangements to sppot a demanding and esponsible ob to be pefomed by shipmastes on the bidge.

2 SYSTEM OF SHIP SAFE CONTROL The challenge in eseach fo effective methods to pevent ship's collisions has become imminent with inceasing sizes, speeds and nmbe of ships paticipating in sea caiage. An obvios contibtion in inceasing safety of shipping has been fistly the application of adas and then the development of ARPA (Atomatic Rada Plotting Aids) anti-collision system (Fige 1). Fige 1. The mltidimensional system of ship safe contol: ψ and ψ-efeence and eal cose of own ship, V and V- efeence and eal speed of own ship, α and α-efeence and eal dde angle of own ship, n and n-efeence and eal otational speed of scew popelle, D and N -distance and beaing to th enconteed obect, D s-safe distance, T m-time manoeving The ARPA system enables to atomatically tack at least obects enconteed, to detemine thei movement paametes (speed V, and cose ψ ) and elements indicating thei appoach to own ship min (DCPA = D - Distance of the Closest Point of Appoach, TCPA = T min - Time to the Closest Point of Appoach) togethe with the assessment of isk of a collision (Fige ). Fige. The pocess of a safe passing of the own ship with a nmbe of ships enconteed Howeve, the ange of fnctions of a standad ARPA system ends p with a simlation of a manoeve selected by navigato. The poblem of selecting sch a manoeve is vey difficlt as the pocess of contol is vey complex since it is dynamic, non-linea, mlti-dimensional, non-stationay and game making in its nate. In pactice methods of selecting a manoeve assme a fom of appopiate steeing algoithms pogammed in the memoy of a micopocesso-based contolle geneating an option within the ARPA anti-collision system o a taining simlato fond in a laboatoy of a maitime academy.

3 MODELS OF THE CONTROL PROCESS The manne in which a ship, being a mlti-dimensional and non-linea dynamic obect, is steeed depends on the extent and pecision of the infomation on cent navigational sitation and also on the adopted model fo the pocess. Ding its development it is essential to take into consideation the following: eqations of kinematics and dynamics of own ship, distbances oiginated by the sea motion, the wind and the sea cent, navigational constaints, stategy of the obects enconteed and the fnction of the steeing taget. A wide vaiety of models possible to be adopted diectly inflences the synthesis of vaios contol algoithms and, conseqently, on the effects of safe steeing of own ship's movements. Basic model It was aleady many yeas ago when Pete Padfield, mentioned in the intodction, stated that, ncetainty is a soce of all diect cases of collisions. Uncetainty eslting fom ambigos eglations with espect to collision pevention, ncetainty of a ship enoying the ight of way as to the intentions of the ship giving the way, ncetainty as to which of the two ships shold become the "giving way ship" and which enoys pioity, ncetainty as to the cose of the foeign ship and as to he position in the fog. In each one case of a collision at least one of these ncetainties was a totally obvios case of the accident. The qotation above explains the fact that the pocess of steeing a ship in collision sitations, when enconteing a geate nmbe of obects often develops nde conditions of non-detemination and conflict accompanied by a non-pecise co-opeation of the obects in the light of ecommendations contained in the povisions of the COLREG (Intenational Reglations fo Peventing Collisions at Sea). Theefoe the model of a diffeential game has been adopted as the most adeqate fo this pocess - whee geneally ships ae monitoed as steeed obects (Fige 3). Fige 3. A block diagam of a diffeential game paticipants This is a model of a game since it takes nde consideation the stategies of the obects enconteed in tems of "thinking playes"; it is diffeential as it mst contain the desciption of the dynamics and kinematics of own ship and the enconteed obects. Sch a game model consists of: non-linea and mlti-dimensional state eqations, non-linea and changing in time state and steeing constaints, and a qality contol index in the fom of integal and final payment. The model pefoms a fnction of a eseach simlation tool [9,15,18,19,]. Geneal dynamic feates of the pocess ae descibed by a set of state eqations in the following fom: v v v x 1 f [(x,x,...,x,...,x m 1 ), (,,...,,..., m ),t] i i 1 m 1 m (1) i 1,,...,( ), 1,,...,m whee: x (t) x (t) o (t) (t) dimensional state vecto of the own ship detemined in a time span t [t,t ], dimensional state vecto of the th obect, dimensional contol vecto of the own ship, dimensional contol vecto of the th obect. k

4 The constaints of the contol and the state of the pocess ae connected with the basic condition fo the safe passing of the ships at a safe distance D s in compliance with COLREG Rles, geneally in the following fom: g (x, ) The synthesis of the decision making patten of the ship s handling leads to the detemination of the optimal stategies of the playes who detemine the most favoable, nde given conditions, condct of the pocess. Fo the class of non-coalition games, often sed in the contol techniqes, the most beneficial condct of the own ship as a playe with th obect is the minimization of he goal fnction in the fom of the payments the integal payment and the final one: () I x (t) dt t k min tk t (3) The integal payment detemines the loss of way of the own ship to each a safe passing of the enconteed obects and the final one detemines the isk of collision. Geneally two types of the contol tasks ae taken into consideation - pogammed contol (t) and positional contol [x (t)]. The basis fo the decision making steeing ae the decision making pattens of the positional contol pocesses, the pattens with the feedback aangement epesenting the diffeential games [5,6,8,16]. Appoximated models Having egad to a high complexity of the basic model in the fom of a model of a diffeential game fo the pactical synthesis of safe steeing algoithms vaios appoximated models ae fomlated, sch as fo example: positional game, matix game, dynamic, static and speed tiangle models. The degee of simplification is dependent on a contol method applied. ALGORITHMS OF SAFE CONTROL Each paticla simplified models of the pocess may be assigned espective methods of safe steeing of a vessel, aligning them fom the most complex to the simplest one (Table 1). Table 1. Methods of detemining the ship's stategy in a collision sitation Model of the pocess Method of contol Algoithm of safe contol Positional game Dal linea pogamming Positional game taectoy POSGAME Mlti-step matix game Dal linea pogamming Risk game taectoy RISKGAME Dynamic pogamming Dynamic Maximm pinciple Fzzy contol Dynamic taectoy Atificial neal netwok DYNTRAJ Evoltionay pogamming Static Linea pogamming Fzzy contol Optimal manoeve OPTMAN Speed tiangle Expet system Necessay manoeve NECMAN In this pape the fist thee algoithms of safe ship contol: POSGAME, RISKGAME and DYNTRAJ will be pesented.

5 Positional game taectoy The diffeential game is edced to a positional mlti-stage game of a nmbe of paticipants who do not co-opeate among them (Fige 4). Fige 4. Block diagam of a positional game paticipants The state and contol vaiables ae epesented by the following vales: x 1 X, x Y, x 1 X, x Y, 1, V, 1, V 1,,..., m (4) In this way possible cose and speed alteations of the enconteed obects ae taken into accont as the steeing pocess is in pogess. The aim of the own ship in the pocess of pefomance of the ship's sailing pogamme is to each a cetain set L k (a pot o anothe way point on the ote), which may be shaed by a few obects, fo example at the appoaches of the vessel to the pot, howeve, only L k set is known on boad own ship. The essence of the positional game is to make the stategies of the own ship dependent on cent positions p(t k) of the obects enconteed at the cent step k. The system geneates its steeing at the moment t k on the basis of the data which ae obtained fom the ARPA anticollision system concening the cent positions of the ships: The constaints of the state co-odinates: x(t k ) (X, Y,t k ) p(t k ) (5) x (t k ) (X, Y, t k ) = 1,,, m, k = 1,,..., K x t, x t P constitte the navigational constaints, while the steeing constaints: (6) U U 1,,..., m (7) take into consideation the kinematics of the ships movement, the ecommendations of the COLREG Rles and the condition to maintain the safe passing distance D s : Dmin tds min D (8) The closed sets U and U defined as the sets of the acceptable stategies of the playes ae dependent, which means that the choice of the steeing by the th obect alte the sets of the acceptable stategies of the othe obects.

6 The optimal steeing of the own ship (t), eqivalent fo the cent position p(t) to the optimal positional steeing (p), is detemined in the following way: - fo the meased position p(t k) of the steeing stats at the moment t k sets of the acceptable stategies p p t k U ae detemined fo the enconteed obects in elation to the own ship, and the otpt sets w t k U of the acceptable stategies of the own ship in elation to each one of the enconteed obects, - a pai of vectos m and positional stategy of the own ship (p), is detemined in elation to each -th obect and then the optimal fom the condition: I min m U U 1 max m U min S x (t k ),Lk S (x,lk ) U( ) w U U 1,,...,m (9) t Lk x(t), Lk (t) dt t S (1) whee S efes to the continos fnction of the manoeving goal of the own ship, chaacteising the distance of the ship at the moment t to the neaest tning point L k on the assmed ote of the voyage. The optimal steeing of the own ship is calclated at each discete stage of the ship s movement by applying the SIMPLEX method to solve the poblem of the linea pogamming, assming the elationship (9) as the goal fnction and the contol constaints (7). Using the fnction of lp linea pogamming fom the Optimization Toolbox contained in the Matlab 5.3, the positional mlti-stage game manoeving POSGAME pogam has been designed fo the detemination of the safe ship s taectoy in a collision sitation [7,14]. Risk game taectoy The diffeential game is edced to a matix game of a nmbe of paticipants who do not co-opeate among them (Fige 5). Fige 5. Block diagam of a model matix game paticipants The state and contol vaiables ae epesented by the following vales: x D, x N,, V,, V 1,,..., m (11) The matix game incldes the vales detemined peviosly on the basis of data taken fom an anticollision system ARPA the vale a collision isk with egad to the detemined stategies of the own ship and those of the -th enconteed obects. The fom of sch a game is epesented by the isk matix R containing the same nmbe of colmns as the nmbe of paticipant I (own ship) stategies.

7 She has; e.g. a constant cose and speed, alteation of the cose o to staboad, o to pot etc., and contains a nmbe of lines which coespond to a oint nmbe of paticipant II ( obects) stategies: , 1 1n R [(, )] (1) m1 1 m 1, 1, 1, 1 m, 1 1n n n mn The constaints affecting the choice of stategies (ν, ν ) ae a eslt of the ecommendations of the way pioity at sea. Playe I (own ship) may se of vaios pe stategies in a matix game and playe II (enconteed obects) has of vaios pe stategies. As the game, most feqently, does not have saddle point the state of balance is not gaanteed thee is a lack of pe stategies fo both playes in the game. The poblem of detemining an optimal stategy may be edced to the task of solving dal linea pogamming poblem. Mixed stategy components expess the distibtion of pobability (, ) of sing pe stategies by the playes. As a eslt of sing the following fom fo the P steeing citeion: I min max (13) the pobability matix P=[P (ν, ν )] of sing paticla pe stategies may be obtained. The soltion fo the steeing goal is the stategy of the highest pobability. This will also be the optimal vale appoximated to the pe stategy: o [p (, max )] Using the fnction of lp linea pogamming fom the Optimization Toolbox contained in the MATLAB, the RISKGAME pogam has been designed fo the detemination of the safe ship s taectoy in a collision sitation [11]. Dynamic taectoy It is assmes that the movement of the enconteed obects, geneally the -th one is ectilinea and egla with a speed V and a cose ψ, with egad to the dynamic feates of own ship (Fige 6). (14) Fige 6. Ship as an obect of steeing nde collision sitations: α efeence dde angle, n efeence otational speed of scew popelle, ψ cose, - angla tning speed, V speed, V - acceleation, X, Y position, O - constaints vecto (enconteed obects) The desciption of the own ship dynamic allows fo the following epesentation of the state eqations in a discete fom: xi,k 1 xi,k x i,k (xi, 1, ) i 1,,...,7 (15) whee: x 1 = X, x = Y, x 3 =, x 4 =, x 5 = V, x 6 = V, x 7 = t, 1 = / max, = n / nmax max

8 The basic citeim fo the ship's contol is to ense safe passing of the obects, which is consideed in the state constaints: g (X,Y, t) (16) This dependence is detemined by the aea of the collision hazad the so-called ship's domain and which assmes the fom of a cicle, paable, ellipse, hexagon, etc. [,3]. The ships' domains may have a pemanent o vaiable shapes geneated, fo example, by an atificial neal netwok [4,1,1,13]. The netwok was modelled with the se of Neal Netwok Toolbox fom the MATLAB package. Moeove, a citeim of optimization is taken into consideation in the fom of smallest possible way loss fo safe passing of the obects, which, at a constant speed of the own ship, leads to the timeoptimal contol: t k t k I(1,) x5 dt x5 dt min (17) Detemination of the optimal contol of the ship in tems of an adopted index of the contol qality may be pefomed by applying Bellman's pinciple of optimization [1].The optimal time fo the ship to go thogh k stages is as follows: t k min, 1,k,k [t k 1(x1,k, x,k, x3,k1, x 4,k1, x5,k1, x6,k1 ) t k (x1,k, x,k, x1,k 1,x,k1,x 5, k k 3, 4,...,K )] (18) The optimal time fo the ship to go thogh the k stages is a fnction of the system state at the end of the k-1 stage and contol ( 1,k,,k ) at the k- stage (Fige 7). Fige 7. The detemination of safe ship s optimal taectoy by means dynamic pogamming method By going fom the fist stage to the last one the fomla (18) detemines the Bellman's fnctional eqation fo the pocess of the ship's contol by the alteation of the dde angle and the otational speed of the scew popelle. The constaints fo the state vaiables and the contol vales geneate the neoconst pocede in the algoithm DYNTRAJ fo the detemination of the safe ship's taectoy. The consideation of the constaints eslting fom maintaining safe appoaching distance and the ecommendations of the way pioity law COLREG is pefomed by checking whethe the state vaiables have not exceeded constaints in each of the intesections consideed and by eecting the intesections in which the excess has been discoveed.

9 COMPUTER SIMULATION Simlation tests of the POSGAME, RISKGAME and DYNTRAJ compte pogams have been caied ot with efeence to eal sitations at sea. The sitations have been ecoded on the basis of the ada sceen. The compte simlation eslts in Matlab/Simlink of the safe own ship s taectoy in the sitation twenty enconteed obects is pesented in Figes 8a-8d own ship Fige 8a. The photo of a ada sceen showing a sitation at the moth of the Rive Thames Fige 8b. Positional game taectoy in good visibility fo D s=.5 nm, T m=1 min own ship own ship V[kn] dpsi[deg] t[min] Fige 8c. Risk game taectoy, speed and cose changes of own ship in good visibility fo D s=.5 nm, T m=1 min - -4 Fige 8d. Dynamic taectoy, otational speed and dde angle in good visibility fo D s=.5 nm, T m=1 min

10 CONCLUSION To sm p it may be stated that the methods of steeing consideed in this stdy ae, in a cetain sense, fomal models fo the thinking pocesses of a navigating office condcting own ship and making p decisions on manoeves. Theefoe they may be sed in pactice to constct both appopiate taining simlatos at maitime academies and vaios options of the ARPA anti-collision system. May I have a hope that the analysis of the poblem of atomatization of safe steeing of a ship will contibte in the neaest fte to change the meaning of Pete Padfield's thesis that "navigating by sea is neithe as safe as it is deemed to be, no as safe as it cold be". Acknowledgement This wok was financially sppoted by the state Committee fo Scientific Reseach of the Govenment of Poland nde statte eseach of the Gdynia Maitime Academy No DS/155/1 implemented in 1. Refeences [1] Bellman R.E.: Dynamic pogamming, Pinceton Univesity Pess, New Yok, 1957 [] Colley B.A., Ctis R.G., Stockel C.T.: Manoeving times, domains and aenas, Jonal of Navigation, No 36, 1983, p [3] Davie P.V., Dove M.J., Stockel C.T.: A compte simlation of maine taffic sing domains and aenas, Jonal of Navigation, No 33, 198, p.15- [4] Hnt K.J., Iwin G.R., Wawick K.: Neal netwok engineeing in dynamic contol systems, Advances in Indstial Contol, Spinge, 1998 [5] Isaacs R.: Diffeential games, John Wiley and Sons, New Yok, 1965 [6] Kasovski N.N., Sbbotin A.I.: Game theoetical contol poblems, Spinge Velag, Belin, 1988 [7] Kdiaszov W., Lisowski J.: Model of positional game applied fo the synthesis of safe ship contol, Achiwm Atomatyki i Telemechaniki, No 4, Vol.4, 1979, p (in Polish) [8] Leitmann G., Skowoński J.: Avoidance contol, Jonal of Optimization Theoy and Application, No 4, Vol.3, 1977, p [9] Lisowski J.: The analysis of diffeential game models of safe ship s contol pocess, Jonal of Shanghai Maitime Institte, No 1, 1985, p.5-38 [1] Lisowski J., Rak A., Czechowicz W.: Neal netwok classifie fo ship domain assessment, Jonal of Mathematics and Comptes in Simlation, No 3-4, Vol.51,, p [11] Lisowski J.: The ship s safe contol as a mlti-step matix game with the isk of collision, 9 th Intenational Symposim on Dynamic Games and Applications, Adelaide,, p [1] Lisowski J.: Comptational intelligence and optimization methods applied to safe ship s contolthe dynamic pogamming and neal netwok methods, Jonal of Shanghai Maitime Univesity, No 3, Vol.1,, p [13] Lisowski J.: Comptational intelligence methods in the safe ship contol pocess, Polish Maitime Reseach, No, Vol.8, 1, p.18-4 [14] Lisowski J.: A positional game model of safe ship contol pocess, 5 th IFAC Confeence on Contol Applications in Maine Systems, Glasgow, 1, p.1-8 [15] Mez A.W., Kamaka J.S.: Collision avoidance systems and optimal tn manoeves, Jonal of Navigation, No, Vol.9, 1976, p [16] Miloh T., Shama S.D.: Maitime collision avoidance as a diffeential game, Schifftechnik, No 4, 1977, p [17] Padfield P.: An agony of collisions, London, 1966 [18] Si-Ying Z.: A new appoach of solving qalitative diffeential games and detemining the bonday of contollable egion, Poc. of IFAC 8 th Tiennal Wold Congess, Kyoto, 1981 [19] Vincent T.L.: Collision avoidance at sea, Poc. of Wokshop Enschede Diffeential Games and Applications, 1977 [] Zwiezewicz Z.: Methods of mathematical contol theoy and thei applications to some optimization poblems of moden maine navigation, Scientific Poc. of Szczecin Maitime Univesity, Szczecin, 1994, p.1-15