The safe ships trajectory in a restricted area
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1 Scienific Journals Mariime Universiy of Szczecin Zeszyy Naukowe Akademia Morska w Szczecinie 214, 39(111) pp , 39(111) s ISSN The safe ships rajecory in a resriced area Zbigniew Pierzykowski, Janusz Magaj Mariime Universiy of Szczecin 7-5 Szczecin, ul. Wały Chrobrego 1 2, {z.pierzykowski; j.magaj}@am.szczecin.pl Key words: safey of navigaion, ship rajecory, resriced area, ship domain, opimizaion Absrac This paper presens he problem of deermining a safe ship rajecory in a resriced area. The roue choice ask is defined as a dynamic opimizaion problem. The roue choice algorihm and crieria are presened and discussed. The ship fuzzy domains are used as a safey crierion. The crierion of he number of manoeuvres o be performed by he own ship is added o he algorihm. The ship encouners simulaed siuaions. Calculaed safe ship rajecories were presened and discussed. iscusses he implemenaion of he presen process, in real erms. Inroducion Ensure he safey of passengers, cargo, and he environmen is one of he basic asks navigaor. This involves he need o deermine a safe ship rajecory and is modificaions if he siuaion is changing. The mos imporan crieria for rajecory choice are safey and cos-effeciveness. The basic consrains for he rajecory choice in an open sea area are ship encouners, hydrological and meeorological condiions and own ship manoeuvring abiliy. Navigaion in resriced areas should ake ino accoun waer area parameers lengh, widh and deph. As in resriced waers raffic is ofen dense, he issue of safe rajecory of he vessel is essenial. I seems ha soon he auomaic deerminaion of a safe ship rajecory will become an available basic funcionaliy in ship navigaional decision suppor sysems. I is imporan o apply crieria similar o he crieria used by navigaors a he sea. This will allow o generae soluions in accordance wih navigaors expecaions and use he generaed soluions (rajecories) in conducing he ship. One of he possibiliies is he use of he fuzzy ses heory mehods and ools for he formulaion of safe ship rajecory deerminaion crieria. This paper coninues he auhors previous work in his field [1]. Ship rajecory choice in resriced waers Resriced waer areas are characerized by he lack of free roue choice and he need o use safey principles aking ino accoun local condiions and regulaions. The choice of a specific rajecory requires mosly a greaer number of consrains, in paricular he shoreline, shallows and oher navigaional dangers. The choice crieria, in paricular safey crieria (passing anoher ship, navigaional hazards or offshore / por srucures) should be in general modified. Planning a ship s rajecory refers o solving siuaions of encounering oher vessels (arges) and saionary objecs in order o pass hem safely. The navigaor, planning a manoeuvre, bears in mind he consrains (navigable area limis, ship s manoeuvrabiliy) and uses cerain crieria [1, 2]: safey crieria: safe disance of passing, overaking or crossing arge s course; safe disance of passing saionary objecs: land, navigaional dangers; crieria derived from regulaions in force and good sea pracice [3]: 122 Scienific Journals 39(111)
2 The safe ships rajecory in a resriced area course aleraion, readily apparen o anoher vessel; sufficienly early performance of he manoeuvre; manoeuvre recommended by regulaions; economic crieria: loss of ime, disance covered, fuel consumpion ec. The basic crieria are safey crieria. The closes poin of approach (CPA) is he common crierion used in open sea areas. However, he use of he menioned crierion on he resriced area is ofen impossible. The reason may be he widh of he waerway. The crierion of ship domain is an alernaive o CPA. The auhors propose wo- and hree-dimensional domains. The wo-dimensional domain is an area around he ship ha is navigaor should keep clear of oher vessels and objecs [4, 5]. The ship fuzzy domain is an exended domain concep [6]. This is an area around he ship ha he navigaor should keep clear of oher vessels and objecs, and he shape and size of his area depend on he assumed level of navigaional safey, undersood as a degree of membership of a navigaional siuaion o he fuzzy se dangerous navigaion. The shape and size of ship domain and ship fuzzy domain depend on many facors, which makes is deerminaion difficul. This problem is discussed in a number of publicaions, for example [4, 5, 6, 7, 8, 9]. In mos cases a domain is assigned o he objec for which a safe rajecory is being deermined own ship. In an alernaive approach domains can be assigned o arges [8, 1]. In his case our (own) ship is considered as a poin. This also requires ha domains be assigned o oher objecs navigaional dangers. An imporan supplemen o he menioned safey crieria are crieria resuling from he regulaions in force and principles of good sea pracice: subsanial and visible course aleraion, sufficienly early performed manoeuvre, recommended urn o sarboard. The subsanial course aleraion is undersood as such aleraion ha will be noiced by he navigaors on ships in viciniy. When a dangerous siuaion occurs (risk of collision), he navigaor should ake prevenive acions early enough o solve he collision siuaion. The regulaions recommend urn-o-sarboard manoeuvre o solve a collision siuaion. Economic crieria are also very imporan, being mainly requiremens imposed by he shipowner. These are mosly formulaed as loss of ime, exra disance covered, fuel consumpion ec. Accepable values of hese losses in some cases may be deermined. Ship rajecory deerminaion as an opimizaion ask The roue choice is associaed wih he deerminaion of manoeuvre / manoeuvres and is / heir parameers o ensure he safe passing of encounered objecs [9]. This ask can be formulaed as he ship s course and/or speed deerminaion. The movemen conrol of a ship which represens a mulidimensional nonlinear dynamic objec, requires muliple decision making. ecisions are dynamic and involve he seings of conrols (rudder, machine) in order o execue an effecive manoeuvre. The problem of ship rajecory deerminaion can be formulaed as a dynamic opimizaion ask. The problem leads o he formulaion of mulicrieria opimizaion when addiional crieria, such as e.g. loss of way, fuel consumpion are aken ino accoun. One of he sandard mehods of dynamic opimizaion is dynamic programming, used in problems of mulisage decision making and conrol. Opimal ship conrol in erms of prese conrol qualiy indicaor can be deermined by using he Bellman s principle of opimaliy. The opimizaion problem is o find such conrol funcion u(), defining he opimal rajecory x() ha he qualiy funcional J will assume a minimum value: J (ˆ( x ), uˆ( ), ) min k u( ) U, x( ) X f ( x( ), u( ), )d (1) where: X{x 1,... x n } n-dimensional space of saes X; U = {u 1,... u m } m-dimensional space of conrols U; f funcion of insananeous losses; u()u se of allowable conrols; x()x allowable (maximal) rajecory space; ime;, k sar and sop ime. The conrol sraegy, deermining he opimal rajecory, consiss of a series of conrols: u ˆ ˆ ˆ ˆ (2) u, u,..., o u 1 k1 where: û conrol in ime i, i =,1,, k. i Zeszyy Naukowe 39(111) 123
3 Zbigniew Pierzykowski, Janusz Magaj Uncerainies (imprecisions) of goals and consrains in he rajecory choice can be accouned for by using sysems of fuzzy inference, including sysems employing mehods of mulisage conrol in a fuzzy environmen [11, 12]. Mulisage conrol in a fuzzy environmen The fuzzy environmen can be presened as an ordered four G, C,, U (G fuzzy goal, C fuzzy consrains, fuzzy decision, U se of decisions). For a given n-dimensional space of saes X = {x 1,... x n } and m-dimensional space of conrols U = {u 1,... u m } he fuzzy goal is defined as a fuzzy se G U wih he membership funcion G : μ : X U [,1] R (3) G and he fuzzy consrain as a fuzzy se C U wih he membership funcion C : μ : X U [,1] R (4) C If a decision is made in a fuzzy environmen, i.e. wih a consrain C and goal G, described, respecively, by membership funcions C (x) and G (x), he fuzzy decision is deermined from his relaionship: μ ( x) min( μ ( x), μ ( x)) (5) xx The conrol process consiss in selecing conrols u wih imposed consrains C (x), wih goals G (x) imposed on he saes x in subsequen conrol sages. As a qualiy indicaor of he mulisage decision making (conrol) process for k conrol sages, his fuzzy decision is adoped: k1 k x C G C G C G described by he membership funcions: μ u,..., u x k μc u μg x μ u μ x G Ck C 1 k1 Gk k (6)... (7) The problem of mulisage conrol in a fuzzy environmen is hen formulaed as follows: μ u..., u x max μ u, u x,..., k1 k1 (8) The opimal sraegy, consequenly, has he form of his series of conrols u * : * * * * u u, u,..., u 1 (9) k 1 The fuzzy goal and fuzzy consrains of fuzzy ses described by heir respecive membership funcions are proposed: goal: safe disance of passing an objec (anoher ship, navigaional danger); consrain 1: possibly small losses of disance (shif of he original rajecory); consrain 2: noiceably manoeuvre (i.e. visible course aleraion). To avoid he deerminaion of rajecories, where muliple manoeuvres occur, he addiional fuzzy consrain is formulaed: consrain 3 small number of manoeuvres, which is also described by he corresponding membership funcion. For he safe ship rajecory deerminaion he ijksra algorihm has been used [13]. The research Our research has been aimed a comparing he effeciveness of seleced algorihms for ship s safe rajecory deerminaion in encouner siuaions in a resriced area. The scenario includes moving objecs (vessels), nearby land and oher saionary navigaional dangers (Fig. 1). Fig. 1. A scenario of a ship encouner siuaion in a resriced area We have assumed ha own ship and wo arges included in he scenario are of he same ype (Table 1). The algorihm mulisage conrol in a fuzzy environmen has been examined for a number of varians of goals and consrains and he corresponding crieria, divided ino wo main varians: 1) own ship described by a fuzzy domain; arge ships described by heir conours (ship waerline); 2) own ship described by is conour (ship waerline); arges described by heir fuzzy domains. 124 Scienific Journals 39(111)
4 The safe ships rajecory in a resriced area I is also assumed ha he ship fuzzy domain shapes are ellipical. Each domain is described by heir minimum and maximum boundary (Table 2). Table 1. Ship model (based on [1]) ship: 1) wihou resricions; 2) o 2 (maximum); 3) o 7 (maximum). Ship model LO-RO ship Lengh overall (L c ) [m] 174. Breadh (B) [m] 23. raf forward ( A ) 7.5 Speed over waer (SOW) [w] 16.3 Table 2. omain boundary dimensions of LO-RO ship (based on [1]) omain boundary Lengh Widh minimum (fuzzy) 12 [m] 675 [m] maximum (fuzzy) 367 [m] 1595 [m] For he simulaed coaslines and navigaional dangers, he consan safey zones (domains) adoped are respecively: an area up o 5 m along he land and an area up o 2 m around a navigaional danger. For he deerminaion of fuzzy consrains, we have assumed he maximum shif of he original rajecory (consrain 1), he minimum and maximum values of visible course aleraion (consrain 2), and admissible (maximum) number of manoeuvres for rajecory planning (consrain 3). The resuls The simulaion research has been carried ou for safe rajecory deerminaion wih he use of mulisep conrol mehod. The research has aimed a comparing he effeciveness of seleced algorihms for ship s safe rajecory deerminaion in encouner siuaions in a resriced area. The worked ou soluions have been analyzed in respec o safey and he ime of finding hese soluions, essenial for heir applicaion in he real condiions. The research has made use of compuer simulaion, based on he developed scenario of a navigaional siuaion. The scenario includes moving objecs (vessels), nearby land and oher saionary navigaional dangers (Fig. 1). Varian 1: The own ship fuzzy domain is assumed as he safey crierion. The geomeric dimensions of arge ships (conours) are considered. For he coasline and navigaional dangers he consan safey zones (domains) adoped are respecively: an area up o 5 m along he land and an area up o 2 m around he navigaional danger. Figure 2 shows he deermined safe rajecories wihou and wih resricions (consrains) on he number of own ship manoeuvres fuzzy crierion: small number of manoeuvres performed by own Fig. 2. Own ship rajecories in encouner siuaions varian 1: 1) wihou resricions on he number of manoeuvres performed by own ship; 2) o 2 (maximum); 3) o 7 (maximum) Varian 2: In his case he fuzzy domains of arge ships are assumed as he safey crierion. Own ship is represened by is conour. For he coasline and navigaional dangers he consan safey zones (domains) are deermined as in varian 1. Figure 3 shows he deermined safe rajecories wihou and wih resricion (consrains) on he number of own ship manoeuvres fuzzy crierion: small number of manoeuvres performed by own ship: 1) wihou resricions; 2) o 2 (maximum); 3) o 7 (maximum). Fig. 3. Own ship rajecories in ship encouner siuaions varian 2: 1) wihou resricions on he number of manoeuvres performed by own ship; 2) o 2 (maximum); 3) o 7 (maximum) Such an approach means he definiion of fuzzy domains for all encounered ships should be defined. In his case, he soluion migh be he assignmen of he ships domains, proposed in he Zeszyy Naukowe 39(111) 125
5 Zbigniew Pierzykowski, Janusz Magaj Table 3. Characerisics of deermined ships rajecories varian 1 Trajecory No. of manoeuvres arge 1 [m] arge 2 [m] danger 1 [m] danger 2 [m] o land [m] Shif [m] Compuaion ime [s] Table 4. Characerisics of deermined ships rajecories varian 2 Trajecory No. of manoeuvres arge 1 [m] arge 2 [m] danger 1 [m] danger 2 [m] o land [m] Shif [m] Compuaion ime [s] lieraure. This mehod, even hough raising some doubs, gives more possibiliies for he own ship rajecory deerminaion. Analysis of he resuls eailed resuls of he simulaions are given in ables 3 (varian 1) and 4 (varian 2). Varian 1. The inroducion of an addiional rierion for a small number of manoeuvres resuls in a significan reducion in he number of manoeuvres o perform, and hus affecs he rouing. I should be recognised as imporan from he viewpoin of he navigaor seering he ship. In he analysed ship encouner siuaion no change of passing disance is observed which seems o be a paricular case. The inroducion of an addiional crierion has no resuled in a significan exension of he calculaion ime. Varian 2. Similarly o varian 1, he inroducion of an addiional crierion for a small number of manoeuvres has resuled in a reducion in he number of manoeuvres in he calculaed rajecories. Minor changes in passing disances of ship No. 1 have been observed. The inroducion of an addiional crierion has no resuled in a significan exension of he calculaion ime (as in opion 1). In opion 2 rajecory calculaions have been observed shorer compuaion imes. The values of compuaion imes for each varian allow o draw conclusions on heir possible use in real condiions (on-line) for solving more complex navigaional siuaions as well. Conclusions This paper presens he problem of deermining a safe rajecory for he ship in case of a meeing in he resriced area. ifferen varians and opions are considered o opimize he ship rajecory wih he use of ship fuzzy domain crierion. In order o reduce he number of manoeuvres, an addiional fuzzy crierion was inroduced crierion of a small number of manoeuvres. Resriced area characerisics land and navigaional dangers are aken ino accoun. Simulaion research has been carried ou for he seleced ship encouner scenario. The inroducion of an addiional crierion has resuled in a significan reducion in he number of manoeuvres under navigaional safey condiions (in compliance wih he safe navigaion condiions). The recorded calculaion imes confirm he applicabiliy of he proposed mehods for deermining a safe rajecory in real condiions (on-line) on ships for solving more complex navigaion siuaions. Boh varian 1 own ship fuzzy domain and conours of arge ships and varian 2 he own ship conour and domains of arge ships are possible o use. Simpler and more reasonable seems o be he defining of own ship fuzzy domain. The second varian gives more flexibiliy for own ship rajecory planning (calculaions), bu i requires he defining of arge ship domains. We are planning o carry ou research for oher, more complex encouner scenarios in resriced waers, as well as aking ino accoun changes of pah predicion parameers, for insance he deerminaion and locaion of graph nodes or subsequen graph nodes in he pah. References 1. PIETRZYKOWSKI Z., MAGAJ J.: The problem of roue deerminaion in ship movemen in a resriced area. Annual of Navigaion 19, par 2, 212, PIETRZYKOWSKI Z.: Fuzzy Conrol in Solving collision Siuaions a Sea. Compuaional Inelligence: Mehods and Applicaions, Eds. L. Rukowski, R. Tadeusiewicz, L.A. Zadeh, J. Żurada, Akademicka Oficyna Wydawnicza EXIT, Warszawa 28, COLREGs 1972, Convenion on he inernaional regulaions for prevening collisions a sea, Inernaional Mariime Organizaion. 4. FUJII Y., TANAKA K.: Traffic capaciy. Journal of Navigaion 24, 1971, Scienific Journals 39(111)
6 The safe ships rajecory in a resriced area 5. GOOWIN E.M.: A saisical sudy of ship domain. Journal of Navigaion 28, 1975, PIETRZYKOWSKI Z.: Ship s fuzzy domain a crierion for navigaional safey in narrow fairways. The Journal of Navigaion 61, The Royal Insiue of Navigaion, Cambridge, 28, pp PIETRZYKOWSKI Z., WIELGOSZ M., SIEMIANOWICZ M.: Simulaion research on he ship domain in he resriced area. In: Proc. of 14 h Inernaional Marine Traffic Engineering, Świnoujście, 211, ŚMIERZCHALSKI R., WEINTRIT A.: omains of navigaional objecs as an aid o roue planing in collision siuaion a sea. In: Proc. of 3 rd Navigaional Symposium, Gdynia, I, 1999, (in Polish). 9. WANG N., MENG X., XU Q., WANG Z.: A unified analyical framework for ship domains. Journal of Navigaion 24, Vol. 62, ŚMIERZCHALSKI R., MICHALEWICZ Z.: Modelling of a ship rajecpry in collision siuaions a sea by evoluionary algorihm. IEEE Transacion on Evoluionay Compuaion, Vol. 4, No. 3, 2, BELLMAN R.E., ZAEH L.A.: ecision making in a fuzzy environmen. Managemen Science 17, KACPRZYK J.: Muli-sage fuzzy conrol, WNT, Warszawa 21 (in Polish). 13. EO N.: The Theory of Graphs and is Applicaion in Technology and Compuer Science. PWN, Warszawa 198 (in Polish). Zeszyy Naukowe 39(111) 127
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