MANOEUVRABILITY-BASED CRITICAL TIME FOR PREVENTING CLOSE-QUARTERS SITUATIONS

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1 Journal of Marine Science and Technology, Vol. 25, No. 3, (2017) 249 DOI: /JMST MANOEUVRABILITY-BASED CRITICAL TIME FOR PREVENTING CLOSE-QUARTERS SITUATIONS Chi-Cheng Tai 1, Jiang-Ren Chang 1, and Chih-Li Chen 2 Key word: colliion avoidance, cloe-quarter ituation, COLREGS. ABSTRACT Thi aer deal with the COLREGS-comlaint action and the critical time for reventing cloe-quarter ituation, from the viewoint of a tand-on veel. A model baed on the dynamic game of comlete information (DGCI) i formulated for three colliion ituation, and the COLREGS-comlaint action are obtained. The critical time i then comoed by the alteration time and the hyical time delay. The alteration time i determined by uing analytic geometry with relative motion. Conidering manoeuvrability of the veel, an equation to etimate the hyical time delay i derived from the MSC.137(76) Standard for Shi Manoeuvrability. Then, the critical time can be determined by correcting the hyical time delay. Finally, three real cae are analyed to validate the rooed reult and demontrate that they can rovide criteria to enure navigation afety for ractical alication. I. INTRODUCTION Colliion avoidance can be regarded a a roce wherein the officer of the watch (OOW) take roer action in amle time, baed on the rule of the road, o that two veel can a at a afe ditance. The rule of the roadinternational Regulation for Preventing Colliion at Sea, 1972, or COLREGServe a a guidance for the roce. However, ome critique of the COLREGS, uch a their qualitative nature and unneceary comlication, have been reented (Weber, 1995; Stitt, 2002). Thee diadvantage lead to divergent interretation of the rule by different individual. In uch cae, ubjective judgment baed on the exerience of the OOW are uually adoted to deal with thee ituation, rather than bureaucratically following recritive rule (Belcher, 2002). Uncertaintie in colliion ituation may arie (Wu, 1984; Taylor, 1990; Jame, 1994). Paer ubmitted 10/05/16; revied 12/01/16; acceted 12/07/16. Author for correondence: Chih-Li Chen ( clchen@mail.ntou.edu.tw). 1 Deartment of Sytem Engineering and Naval Architecture, National Taiwan Ocean Univerity, Keelung, Taiwan, R.O.C. 2 Merchant Marine Deartment, National Taiwan Ocean Univerity, Keelung, Taiwan, R.O.C. Therefore, a further te to enure afety would be a formaliation of criteria for ractical alicationwhich i the motivation of thi reearch. Cloe-quarter ituation (CQS) i one of the qualitative term needing clarification, although the objective of the COLREGS i to revent the develoment of uch critical ituation (Zhao, 2008). Conidering the condition of being in ight of one another, the roce of a colliion may be divided into four tage baed on the obligation of the give-way and tand-on veel, a follow: free manoeuvre, action required by the give-way veel, action required by the tand-on veel and the CQS (Cockcroft and Lameijer, 1996). In thi regard, a CQS can be viewed a a ituation where it i imoible for one veel manoeuvring alone to avoid the other by a ubtantial alteration of coure (Mankababy, 1987). The give-way veel i required to manoeuvre in the econd tage, namely action required by the give-way veel, when the rule begin to aly (Zhao, 2008). However, ome critical occaion arie if the give-way veel neglect to kee a roer lookout (Wu, 1984); the tand-on veel then ha to manoeuvre alone under Rule 17(a)(ii) before the lat moment to revent the develoment of a CQS. In uch cae, how and when to manoeuvre by herelf alone become an urgent tak for the tand-on veel. Therefore, analyi of the COLREGS-comlaint action and determination of the critical time for reventing a CQS from the viewoint of the tand-on veel are neceary for maritime afety, and are the main goal of the reent effort. The relevant literature i divided into two categorie: analyi of the COLREGS-comlaint action, and determination of the critical time or ditance. To analye the COLREGS-comlaint action, a tatic game of comlete information wa firt adoted by Cannell (1981); Tai et al. (2014) then rooed a model baed on a dynamic game of comlete information (DGCI) to reflect the dynamic nature of colliion avoidance. Thee qualitative reearche interreted the COLREGS in trategic and formalied manner and obtained the COLREGS-comliant action for the econd tage. However, action and critical time to revent a CQS for the tand-on veel if the give-way veel neglect to fulfill her obligation are not addreed. A for determining the critical ditance, there are three tye of reearch: rule of thumb, decritive model, and recritive model. A a rule of thumb, Cockcroft and Lameijer (1996) uggeted that the boundary of the CQS be conidered one nautical mile (nm) for veel in ight

2 250 Journal of Marine Science and Technology, Vol. 25, No. 3 (2017) of one another and 2 to 3 nm in retricted viibility. However, thi imle concet fail to take into account the variou ize, characteritic and eed of the veel, a tated by Mr. Jutice Willmer in the cae of Grea/Verna (Cockcroft and Lameijer, 1996). Decritive model are contructed baed on tatitical analye. Davi et al. (1980) rooed a concet of the arena, in which the OOW would tart to take action to avoid a CQS, baed on the quetionnaire and analyi of the average attern. Habberly and Taylor (1989) deigned a imulation and analyed the alteration ditance made by the ubject. Taylor (1990) and Jame (1994) ubequently rovided robability ditribution of the alteration ditance baed on imulation data collected by Habberly and Taylor (1989). Thee decritive model can calculate the alteration ditance within which manoeuvre are initiated, but are unable to take into account the delay ditance or time a affected by the manoeuvrabilitie of the veel. Finally, recritive model are contructed by uing mathematical concet. Hilgert (1983) defined the critical ditance for reventing a CQS a that ditance at which a crahing to i made by both veel. However, thi model ubequently roved ineffective in avoiding a CQS if the manoeuvre i made by one veel alone (George, 1984). Colley et al. (1983) and Wu (1984) both develoed equation for the ditance of lat-minute action (LMA). However, the former equation can only determine the ditance in a head-on ituation, and the latter i too riky due to it aumtion that the initial ditance to cloet oint of aroach (DCPA) and the final aing ditance of the veel both equal zero. Baed on the definition of a CQS, thi aer attemt to formalie the afety criteria of how and when to manoeuvre for the tand-on veel if the give-way veel neglect to fulfill her obligation. The COLREGS-comlaint action in overtaking, head-on and croing ituation are firt analyed by a DGCI, which deal with equential rocee in which thinking eole interact with each other (Fudenberg and Tirole, 1991; Gibbon, 1992; Montet and Serra, 2003; Ramuen, 2005). Then, the critical time for the tand-on veel to revent a CQS i conidered. The alteration time i determined by the analytic geometry with relative motion. Since relative motion i unable to take into account the effect of manoeuvrability, the arameter of MSC.137(76) Standard for Shi Manoeuvrability (IMO, 2002) are adoted to erve a maximum limit and to derive an etimated equation of the hyical time delay. So the alteration time can be corrected and the critical time for reventing the CQS can be determined, and the uroe of thi reearch can be met. Thi aer i organized a follow: Section 2 decribe the DGCI and the analyi of the COLREGS-comlaint action. The theoretic background and rocedure required to obtain the critical time for a CQS are reented in Section 3. In Section 4, three real accident are analyed to demontrate the feaibility of the rooed aroach. Finally, concluion are reented in Section 5. OOW 1 /o OOW 2 /o /o /o (u 1(, ), u 2(, )) (u 1(, /o), u 2(, /o)) (u 1(, ), u 2(, )) (u 1(/o, ), u 2(/o, )) (u 1(/o, /o), u 2(/o, /o)) (u 1(/o, ), u 2(/o, )) (u 1(, ), u 2(, )) (u 1(, /o), u 2(, /o)) (u 1(, ), u 2(, )) Fig. 1. Extenive-form rereentation of the colliion game. II. COLREGS-COMPLAINT ACTIONS FOR PREVENTING A CLOSE-QUARTERS SITUATION In thi ection, a brief introduction of the DGCI i reented. Then, three colliion ituation are formalied and analyed by a DGCI, and the COLREGS-comlaint action are obtained. Finally, the action required by the tand-on veel in the cae of negligence of the give-way veel are dicued. 1. Dynamic Game of Comlete Information Game theory i a trategic-thinking aroach that deal with real-life ituation where thinking eole interact with each other (Montet and Serra, 2003). That i, each individual choice deend on what the other may do (Fudenberg and Tirole, 1991). The DGCI articularly concern ituation where individual, who have full undertanding of the ituation, make choice in equence (Gibbon, 1992; Ramuen, 2005). It i adoted here to analye the roce of colliion avoidance. When one conider two veel in a direct colliion ituation in the oen ea with good viibility, alteration of coure alone may be the mot effective action to avoid colliion. Therefore, the eed of the veel are aumed to be maintained at a afe level. Two OOW of the veel equentially take action in two tage before a CQS can develo. The framework of the colliion game can be decribed by an extenive-form rereentation, a hown in Fig. 1. The baic element contituting a DGCI, uch a layer, action, ayoff, information and trategie, are defined a follow: (1) The layer who take action to maximize their own ayoff are two OOW for thi cae. They are rereented by deciion node from left to right in order of time, and denoted by OOW i, i = 1, 2. It i noted that each deciion node contitute a ubgame. (2) An action, which i rereented by a branch, i a choice the layer can make. When we conider uual ractice in oen ea with good viibility, there are 3 action can be choen, including of altering coure to ort (), tanding on (/o), and altering coure to tarboard (). They are denoted by a i = {, /o, } referring to OOW i. (3) The ayoff are utilitie that reond to action choen by all OOW and are denoted by u i (a 1, a 2 ). In thi cae, the

3 C.-C. Tai et al.: Critical Time for Cloe-Quarter Situation 251 ayoff for each OOW conit of two value. One i a tate utility that correond to the effect of the action, while the other i a enalty utility if either action violate the COLREGS. (4) The information refer to the knowledge of thi game. In articular, comlete information mean that the knowledge i oeed by each OOW. There i no robability ditribution in the framework and the node are ingleton. (5) The trategie are full lan for each OOW reacting to each ubgame at the time he make a choice. In thi regard, OOW 1 ha only one deciion node with three trategie a the action he can chooe. On the other hand, OOW 2 ha three deciion node o he ha 3 3 = 27 trategie denoted by (, /o,, ). That i, for, OOW 2 chooe no matter what the OOW 1 chooe; for /o, OOW 2 chooe if OOW 1 chooe or /o, and chooe /o if OOW 1 chooe ; and o on. The olution of the DGCI i named the ubgame erfect Nah equilibrium (SPNE), which i comoed of the bet trategie for each OOW. It can be obtained by earching through all oible trategie; however, thi i a comlicated tak. A convenient method referred to a backward induction i adoted to achieve the reult. Backward induction rereent a attern of reaoning that tart from the time when OOW 2 make a choice. In thi regard, the quetion each OOW face can be viewed a an otimization roblem (Gibbon, 1992). For each ubgame, OOW 2 face an otimization roblem a follow, a2a2 max u a, a A et of bet action choen by OOW 2 that would maximize hi ayoff in each ubgame, i denoted by a 2 *, alo referred to a the bet trategy for him. Working backward to the time when OOW 1 make a deciion, he anticiate that OOW 2 will chooe the et of bet action and face the other otimization roblem a follow, a1 A1 max u a, a. * OOW 1 would ick an action from a 1 baed on the et of a 2 * to maximize hi ayoff. A mentioned above, thi bet action i alo the bet trategy for him. Thu the combination of thee bet trategie, the SPNE, from which the layer have no incentive to deviate, i obtained. 2. COLREGS-Comlaint Action Colliion ituation can be categorized into overtaking, head-on, and croing ituation, under Rule 13. Overtaking, Rule 14. Head-on Situation, and Rule 15. Croing Situation, in the COLREGS. Auming there i a veel at the centre, the (1) (2) C B A Fig. 2. Region for determining colliion ituation. OOW 1 /o OOW 2 D ( 0, 0 ε ) /o ( 1, 1 ) ( 1, 1 ε ) /o /o ( 1 ε, 1 ) ( 0 ε, 0 ε ) ( 1 ε, 1 ) ( 1, 1 ε ) ( 1, 1 ) ( 0, 0 ε ) Fig. 3. The framework and the SPNE of the game for overtaking ituation. ituation can be determined baed on the region where the other veel i coming u. Region A correond to an overtaking ituation, region B to a head-on ituation and region C and D to croing ituation, a illutrated in Fig. 2. It i noted that the boundarie of region B, namely head-on ituation, are aumed to be 005 and 355 a categoried by Cockcroft (1982). The COLREGS-comlaint action for thee ituation can be analyed by colliion game a follow: 1) Game for Overtaking Situation A hown in Fig. 2, the veel coming from region A i the give-way veel, which take action firt and i referred to a OOW 1, while the centre veel i the tand-on veel that take it own action after oberving give-way veel action, and i referred to OOW 2. The tate utility i deignated 0, indicating danger in the condition that both veel chooe the ame action. Otherwie, it i deignated 1, connoting afety. The rule related to action in thi ituation are Rule 8. Action to Avoid Colliion, Rule 13. Overtaking, Rule 16. Action by the Give-way Veel and Rule 17. Action by the Stand-on Veel. The enalty utility, denoted by with a oitive value between 0 and 1, correond to the action that violate the Rule. Thee violation are: firt, OOW 1 chooe the action /o; econd, OOW 2 chooe action other than /o if OOW 1 ha given way; and third, OOW 2 chooe the action /o if OOW 1 ha choen /o. The framework and the SPNE are illutrated a Fig. 3.

4 252 Journal of Marine Science and Technology, Vol. 25, No. 3 (2017) Situation Region Table 1. COLREGS-comlaint action for three colliion ituation. Obligation Stage of give-way veel Stage of tand-on veel The other Centre veel The other Centre veel Centre veel Overtaking A Give-way Stand-on or /o or Head-on B Both required to manoeuvre Croing C Give-way Stand-on /o OOW 1 /o OOW 2 /o /o /o ( 0 ε, 0 ) ( 0 ε, 0 ε ) ( 1 ε, 1 ) ( 0, 0 ε ) ( 1, 1 ε ) ( 1, 1 ) Fig. 4. The framework and the SPNE of the game for head-on ituation. OOW 1 /o OOW 2 /o /o ( 1 ε, 1 ) ( 0 ε, 0 ε ) ( 1 ε, 1 ) ( 0, 0 ε ) /o ( 1, 1 ) ( 1, 1 ε ) Fig. 5. The framework and the SPNE of the game for croing ituation. The bet trategie are () and () for OOW 1 and (/o /o) and (/o /o) for OOW 2. The SPNE of the overtaking game are (, (/o /o)), (, (/o /o)), (, (/o /o)) and (, (/o /o)). The OOW receive the ame ayoff of 1 which imlie the action are not only afe but comly with the COLREGS. That i, the COLREGS-comlaint action are altering coure to ort or tarboard for the overtaking veel and tanding on for the overtaken veel. 2) Game for Head-On Situation Thi ituation aear when one of the two veel i coming from region B a illutrated by Fig. 2. The tate utility i deignated 0, indicating danger in the condition that both veel chooe /o, or if they take ooite direction, i.e., and. Otherwie, it i deignated 1, connoting afety. The only alicable rule in thi ituation i Rule. 14 Head-on Situation, which indicate that both veel hall alter coure to tarboard. That i, the ayoff would contain a enalty utility if either of them doe not chooe. The framework and the SPNE are illutrated in Fig. 4. The bet trategie are () for the firt mover and ( ) and (/o ) for the other. The SPNE are (, ( )) and (, (/o )). The olution i recie in that both veel have to alter coure to tarboard to enure afety and comly with the COLREGS. 3) Game for Croing Situation A croing ituation occur if the other veel coming from region C or D in Fig. 2. For region C, it i a tand-on cae for the centre veel, but for region D it i a give-way cae. From the viewoint of the tand-on veel with the COLREGS, the former cae i conidered here. The order of lay i that the other veel, i.e., the give-way veel referred to a OOW 1, make a choice firt. Then, the centre veel, i.e., the tand-on veel referred to a OOW 2, chooe an action. The tate utility i deignated 0, indicating danger if OOW 1 chooe and then OOW 2 chooe, or if both veel chooe the action of tanding on. Otherwie, it i deignated 1, which connote afety. Conidering Rule 8. Action to Avoid Colliion, Rule 15. Croing Situation, Rule 16. Action by the Give-way Veel and Rule 17. Action by the Stand-on Veel, the enalty utility i determined a follow: firt, OOW 1 chooe or /o; econd, OOW 2 chooe or /o if OOW 1 chooe or /o; and third, OOW 2 chooe or if OOW 1 chooe. In thi way, the game can be formalied, and the framework a well a the SPNE outcome are illutrated in Fig. 5. The bet trategie are () for the OOW of the give-way veel and ( /o) for the OOW of the tand-on veel. The SPNE i (, ( /o)) and both OOW obtain a ayoff of 1. The COLREGScomlaint action are altering coure to tarboard for the giveway veel and tanding on for the tand-on veel. 3. Dicuion on the COLREGS-Comliant Action for Stand-On Veel The COLREGS-comlaint action in the econd tage, namely action required by the give-way veel, are ummaried a follow. Firt, the give-way veel hall alter coure to either ide while the tand-on veel tand on in an overtaking ituation. Second, both veel have to alter coure to tarboard in a headon ituation. Third, the give-way veel hall alter coure to tarboard while the tand-on veel hall tand on in a croing ituation. However, ome critical occaion arie in overtaking and croing ituation becaue the give-way veel may neglect to kee a roer lookout. The roce of a colliion enter the next tage, namely action required by the tand-on veel. The

5 C.-C. Tai et al.: Critical Time for Cloe-Quarter Situation 253 O O G (x G, y G ) A W O' G (x G, y G ) W O' T A P S (x S, y S ) P' P'' A T A P S (x S, y S ) Fig. 6. The alteration time i determined by Scenario 1. tand-on veel mut take action under Rule 17(a)(ii) before the lat moment at which he can revent a CQS by her manoeuvring alone. The action comlying with the COLREGS for tand-on veel can be found in Fig. 3 and 5 and ummaried in Table 1, which illutrate overtaking and croing ituation, reectively. In thee cae, the tand-on veel hall alter coure to either ide in an overtaking ituation, or alter to tarboard in a croing ituation. Thereafter, the mot imortant iue turn to the quetion of when to manoeuvre, deending on the veel manoeuvrability. Thi i the critical time for reventing a CQS, that i to be reolved in the next ection. III. CRITICAL TIME FOR PREVENTING A CLOSE-QUARTERS SITUATION A mentioned above, a CQS can be viewed a a ituation wherein it i imoible for two veel to a at a afe ditance by a ubtantial alteration of one veel alone. Baed on the viewoint of the tand-on veel, the COLREGS-comlaint action have been analyed in Section 2. Here the critical time, which conit of the alteration time and the hyical time delay, i conidered. The alteration time i calculated by uing equation of analytic geometry with relative motion. To eliminate the diadvantage of relative motion being unable to take into account the manoeuvrability of the veel, the MSC.137(76) Standard for Shi Manoeuvrability i adoted to derive an etimation equation of hyical time delay. Through the correction of the hyical time delay, the critical time can be obtained. In thi rocedure, a afe ditance of 0.85 nm adoted from Goodwin (1975) in the oen ea i aumed to enure afety. Of coure, a ubtantial alteration of coure i aumed to be a coure change of 90. Beide, all the ymbol ued in the following are lited in Nomenclature for a quick reference. Fig. 7. The alteration time i determined by Scenario Alteration Time Given that the oition, coure and eed of the veel are known, equation of analytic geometry with relative motion are adoted. The oition of the tand-on veel i conidered the reference oint for relative motion. The alteration time i determined by an initial relative motion line () and an altered. By earching for a oint of alteration along the initial to yield a afe ditance between the centre veel and the altered, the alteration time can be determined. In articular, there are two cenario for obtaining the alteration time deending on whether the direction difference between the initial and new i greater than 90 a hown in Fig. 6 and 7. The rocedure are conducted a follow: 1) Obtaining the Ditance, True Bearing and Relative Bearing Given the oition of two veel, (x S, y S ) for the tand-on veel and (x G, y G ) for the give-way veel, the difference between the x- and y-coordinate of the oition can be determined firt, i.e., x and y. The ditance between two veel and the true bearing from one veel to the other are conequently determined a follow: D x y 2 2 SG, (3) tan 1 x, if y 0, x0 y tan 1 x 180, if y 0 y tan 1 x TBSG 360, if y 0, x 0, y 90, if y 0, x0 270, if y 0, x0 TB GS TBSG 180, if TBSG TBSG, if TBSG 180 (4a) (4b) Subequently, the relative bearing for both veel can be yielded a:

6 254 Journal of Marine Science and Technology, Vol. 25, No. 3 (2017) RB RB SG GS RB TB C, (5a) TBSG CS, if TBSG CS 0 TBSG CS 360, iftbsg CS 0, TBGS CG, if TBGS CG 0 TBGS CG 360, if TBGS CG 0. (5b) (5c) 2) Initial Relative Motion The relative motion can be derived by vector analyi. The x- and y-comonent of the initial relative motion i yielded a: R V in C V in C, (6a) x G G S S R V co C V co C. (6b) y G G S S By uing ΔR x and ΔRy, the loe of the initial can be obtained a: m Ry. (7) R Baed on the oint-loe form of a line equation, the equation of the initial i derived a: x 0. m x y m x y (8) G G On the other hand, the magnitude and direction of the initial relative motion, i.e., V RM and C RM, can be olved by ubtituting ΔR x and ΔRy for Δx and Δy in Eq. (3) and (4a) where the time cale of V RM i further tranformed to be minute. Uing the formula for the ditance from a oint to a line, the DCPA i yielded a: D CPA m xg yg m 1 2 The time to cloet oint of aroach, TCPA, can be ubequently yielded by dividing the ditance from oint G to oint P by magnitude of the relation motion can be illutrated a: T CPA 3) Altered Relative Motion. (9) 2 2 D D SG CPA. (10) V By a ubtantial alteration of coure the difference of the x- and y-comonent of the altered relative motion are exreed a following equation: RM R ' V inc V in C 90, (11a) x G G S S R ' V coc V co C 90. (11b) y G G S S Here the term i ued to indicate direction of the alteration: for a tarboard turn or for a ort turn. The coure of the altered relative motion, i.e., C ARM, can be obtained by ubtituting Δx in Eq. (4a) with ΔR x ' and Δy with ΔRy '. Likewie, the loe m A can be determined by ubtituting ΔR x in Eq. (7) with ΔR x ' and ΔRy with ΔRy '. The line equation of the altered can alo be formalied by ubtituting m A for m in Eq. (8). 4) Obtaining Alteration Time Given that the coure of the altered may be toward or away from the reference oint, the alteration time can be obtained in two different way. The criterion i whether the difference of direction between the two i greater than 90. Scenario 1. The coure of altered relative motion i toward the centre veel. A hown in Fig. 6, in thi cae the difference of direction between the two i le than or equal to 90. Adoting the formula for the ditance from a oint to a line, the new DCPA reult from a ubtantial alteration of coure i obtained a: D ACPA ma xg yg m 1 2 A. (12) Since the ditance to alteration ( GT A ) i the rojection of the difference between the new D CPA ( SP ' ) and the afe ditance ( SP " ) on the initial, the alteration time can be calculated by dividing GT A by the magnitude of initial relative motion a determined in following equation: T A DACPA DS V C C in. RM RM ARM (13) Scenario 2. The coure of altered relative motion i away from the centre veel. In thi cae, the oint of alteration, T A, i the interection of the initial and the circle of afe ditance a illutrated in Fig. 7. Becaue the T A PS i a right-angled triangle, the ditance of TP A can be eaily obtained by the afety ditance ( TS) A and DCPA ( SP ). Once the TCPA and time required for TP A are obtained, the alteration time can conequently be determined a: 2 2 DS DCPA TA TCPA. (14) V RM

7 C.-C. Tai et al.: Critical Time for Cloe-Quarter Situation 255 H 90 Hanjin Singaore O' W O ADVANCE K H 0 K' T PD 1.8 min T A T CQS Fig. 8. The hyical time delay. 2. Phyical Time Delay Since relative motion ha a major diadvantage of failing to account for manoeuvrability, the hyical time delay i introduced to reolve thi roblem. According to Turning ability of the Reolution MSC.137(76) Standard for Shi Manoeuvrability, it i aumed that the veel make a turning circle manoeuvre at a oint, i.e., H 0, around a circle a illutrated in Fig. 8. The track of the manoeuvre arrive at the oint H 90 when the coure change i 90. Conidering that the hyical time delay occur along the arc from H 0 to H 90, a diameter of 5 hi length (L) and an advance of 4.5 L i adoted to aroach the reult. Here KH 0 and KH 90 rereent the radiu of 2.5 L while K ' H 90 rereent the advance of 4.5 L. The angle H 0 KH 90 of can be calculated by: K' H KH co 180 H KH KH. (15) Then H 0H 90 of L which rereent the delay ditance for a veel making a coure change by 90 can be determined accordingly. Thu, the hyical time delay can be obtained by H 0H 90 and eed of the veel. After converting the unit of hi length and eed, the equation of the hyical time delay can be yielded a: T PD L (16) VS 3. Critical Time for Preventing a CQS By conidering the veel manoeuvrability, the critical time for reventing a CQS can be determined by the alteration time and hyical time delay a hown in following equation: TCQS TA TPD. (17) Poition at Coure ( ) Seed (kt) LBP (m) P Situation Obligation Preventing a CQS Phyical time delay (min) Alteration time (min) Critical Time (min) Critical Time at GMT Kocierzyna P'' A Kocierzyna (0, 0) P' Hanjin Singaore (2.4, 5.5) overtaking give-way tand-on A/C to tarbord and ort are ermitted (/b)/29.3 () 15.5 (/b)/27.4 () (/b)/12 52 () Fig. 9. COLREGS-comlaint action and critical time for Kocierzyna. IV. CASE STUDIES Three real colliion cae are dicued to validate the rooed aroach. In the cae of the Kocierzyna and Hanjin Singaore (LLR, 1996), an overtaking ituation where the alteration time i determined by Scenario 1 i demontrated. Second, a croing ituation where the alteration time i determined by Scenario 1 i revealed in the cae of the Sirit and Sitarem (LLR, 2001). Finally, the cae of the Hyundai Dominion and Sky Hoe (MAIB, 2004) i adoted to demontrate the critical time for a croing ituation with Scenario The Kocierzyna and Hanjin Singaore The colliion occurred on Set. 15, 1991, at 1300 GMT in the oen ea where the viibility wa clear for at leat 4 mile. The Kocierzyna wa 95 metre in length. She wa ailing on a coure of 209 (T) and making a eed of 10.5 kt. The Hanjin Singaore wa 242 metre in length. She wa teering a coure of 209 (T) and making a eed of 21 kt. The Kocierzyna wa overtaken by the Hanjin Singaore. The analyi i grounded by

8 256 Journal of Marine Science and Technology, Vol. 25, No. 3 (2017) A P' O P P'' W Sky Hoe Sirit Poition at Coure ( ) Seed (kt) LBP (m) Situation Obligation Preventing a CQS T A Phyical time delay (min) Alteration time (min) Critical Time (min) Critical Time at ZT T PD 4.3 min T CQS Sitarem W O' Sirit Sitarem (0, 0) (2.9, -0.6) croing give-way tand-on A/C to tarbord i ermitted (/b) 2.1 (/b) (/b) Fig. 10. COLREGS-comlaint action and critical time for Sirit. the fact that Kocierzyna wa at about 5 on the ort bow of Hanjin Singaore at a ditance of about 6 mile at 1225 GMT. Baed on thi information and from the erective of the Kocierzyna, the COLREGS-comlaint action and the critical time for reventing a CQS can be determined and are hown in Fig. 9. A a reult, a ubtantial alteration either to tarboard or to ort of the Kocierzyna i ermitted. The alteration time are 17.3 and 29.3 minute for ubtantial alteration to tarboard and ort, reectively. Conidering her manoeuvrability, the etimation of hyical time delay i 1.8 minute. In thi cenario, the critical time for reventing a CQS i 15.5 minute after 1225 GMT, i.e., 1240 GMT, if a ubtantial alteration to tarboard i made. Otherwie the critical time i 27.4 minute after 1225 GMT, i.e., 1252 GMT, if a ubtantial alteration to ort i made. 2. The Sirit and Sitarem On March 12, 1996 at 2245 zone time (ZT), a colliion occurred between the VLCC Sirit and the coater Sitarem in oen ea with good viibility. The Sirit wa 253 metre in length. She wa the tand-on veel roceeding on a coure of 126 (T) at a eed of 12 kt. The Sitarem wa metre in length. She wa the give-way veel ailing on a coure of 268 (T) and making a eed of 10 kt. The available ground for thi determination are that the Sitarem wa firt een at a ditance of about 3 mile, bearing about 25 on the ort bow of Sirit. It i aumed that the obervation wa made at 2236 ZT, according to the time of colliion and the TCPA. Baed on thi information, the determination i conducted a hown in Fig. 10. A ubtantial alteration to tarboard by the Sirit i ermitted. The alteration time i 6.4 minute for ubtantial alteration to O Poition at Coure ( ) Seed (kt) LBP (m) Situation Obligation Preventing a CQS Phyical time delay (min) Alteration time (min) Critical Time (min) Critical Time at ZT A O' P T CQS T PD 2.8 min T A Hyundai Dominion (0, 0) overtaking tand-on Hyundai Dominion A/C to tarbord i ermitted (/b) 10.9 (/b) (/b) Sky Hoe (-0.8, 4.9) give-way Fig. 11. COLREGS-comlaint action and critical time for Hyundai Dominion. tarboard. Conidering her manoeuvrability, the etimation of hyical time delay i 4.3 minute. Thu, the critical time for reventing a CQS i 2.1 minute after 2236 ZT, i.e., 2238 ZT, if a ubtantial alteration to tarboard i made. 3. The Hyundai Dominion and Sky Hoe The colliion cae haened on June 21, 2004, at 0738 ZT in oen ea with good viibility. The Hyundai Dominion wa metre in length. She wa the tand-on veel teering a coure of 036 (T) and making a eed of 22 kt.the Sky Hoe wa metre in length. She wa the give-way veel ailing on a coure of 091 (T) with a eed of 15.3 kt. The two veel encountered each other in a croing ituation. The available ground to be taken into account are that the Sky Hoe wa firt een by OOW on board Hyundai Dominion on the ort bow at 45 and at a ditance of 5 nm at about Baed on thi information, the determination i conducted a hown in Fig. 11. A ubtantial alteration to tarboard of the Hyundai Dominion i ermitted. The alteration time i 13.7 minute for the ubtantial alteration to tarboard. Conidering her manoeuvrability, the etimation of hyical time delay i 2.8 minute. Thu, the cri-

9 C.-C. Tai et al.: Critical Time for Cloe-Quarter Situation 257 tical time for reventing a CQS i 10.9 minute after 0710 ZT, i.e., 0720 ZT, if a ubtantial alteration to tarboard i made. V. CONCLUSIONS Thi aer ha uccefully rooed the COLREGScomlaint action and determined the critical time for reventing a CQS baed on the viewoint of a tand-on veel. Aimed at to get rid of the ambiguity of the COLREGS, a DGCI i adoted to analye three colliion ituation uch that the recie COLREGS-comlaint action can be obtained. To take a ubtantial action in amle time for reventing a CQS, the alteration time i firt determined by analytic geometry with relative motion and then i corrected by the hyical time delay with conideration of manoeuvrability of the tand-on veel. Three real cae are conducted a well a analyed to demontrate that the rooed aroach i effective for ractical alication. ACKNOWLEDGMENTS The author would like to exre the gratitude to the National Science Council of Taiwan for financial uort under contract number: NSC E NOMENCLATURE S the oition of the tand-on veel G the oition of the give-way veel C S the coure of the tand-on veel C G the coure of the give-way veel C RM the coure of the initial relative motion C ARM the coure of the altered relative motion V S the eed of the tand-on veel V G the eed of the give-way veel V RM the magnitude of the relative motion L the length of the veel x S the x-coordinate of the tand-on veel x G the x-coordinate of the give-way veel y S the y-coordinate of the tand-on veel y G the y-coordinate of the give-way veel x the difference between x-coordinate of the oint y the difference between y-coordinate of the oint TB SG the bearing of give-way veel from the view of tand-on veel TB GS the bearing of tand-on veel from the view of give-way veel RB SG the relative bearing of give-way veel from view of tandon veel RB GS the relative bearing of tand-on veel from view of giveway veel R x the x-comonent of the initial relative motion R x ' the x-comonent of the altered relative motion Ry the y-comonent of the initial relative motion Ry ' the y-comonent of the altered relative motion m the loe of the initial relative motion line m A the loe of the altered relative motion line D SG the ditance between the give-way and the tand-on veel D S the afety ditance of 0.85 nm D CPA the initial DCPA D ACPA the new DCPA reult from a ubtantial alteration of coure T A the alteration time T PD the hyical time delay TCQS the critical time for reventing a cloe-quarter ituation K the centre of turning circle H 0 the tarting oint of turning circle manoeuvring H 10 the oint of coure changed by 10 the oint of coure changed by 90 H 90 REFERENCES Belcher, P. (2002). A ociological interretation of the COLREGS. The Journal of Navigation 55, Cannel, W. P. (1981). Colliion avoidance a a game of co-ordination. The Journal of Navigation 34, Cockcroft, A. N. (1982). The circumtance of ea colliion. The Journal of Navigation 35, Cockcroft, A. N. and J. N. F. Lameijer (1996). A guide to the colliion avoidance rule: International Regulation for Preventing Colliion at Sea. Fifth edition, Oxford: Butterworth-Heinemann. Colley, B. A., R. G. Curti and C. T. Stockel (1983). Manoeuvring time, domain and arena. The Journal of Navigation 36, Davi, P.V., M. J. Dove and C. T. Stockel (1980). A comuter imulation of marine traffic uing domain and arena. The Journal of Navigation 33, Fudenberg, D. and J. Tirole (1991). Game Theory. The MIT Exre. George, B. D. (1984). Forum: A grahical aement of the cloe-quarter ituation. The Journal of Navigation 37, Gibbon, R. (1992). A Primer in Game Theory. Harveter Wheatheaf. Goodwin, E. M. (1975). A tatitic tudy of hi domain. The Journal of Navigation 28, Habberly, J. S. and D. H. Taylor (1989). Simulated colliion avoidance manoeuvre: a arametric tudy. The Journal of Navigation 42, Hilgert, H. (1983). Defining the cloe-quarter ituation at ea. The Journal of Navigation 36, International Maritime Organization. Maritime Safety Committee. (2002). RESO- LUTION MSC.137(76) Standard for Shi Manoeuvrability. Jame, M. K. (1994). The timing of colliion avoidance manoeuvre: decritive mathematical model. The Journal of Navigation 47, Maritime Accident Invetigation Branch. (2005). Reort on the Invetigation of the Colliion between Hyundai Dominion and Sky Hoe. No.17/2005. Mankababy, S. (1987). The Law of Colliion at Sea. Elevier Science Publiher B.V. Montet, C. and D. Serra (2003). Game Theory and Economic. Palgrave Macmillam. Ramuen, E. (2006). Game and Information: An Introduction to Game Theory, Fourth Edition. Blackwell Publiher. Stitt, I. P. A. (2002). The COLREGS - time for a rewrite? The Journal of Navigation 55, Taylor, D. H. (1990). Uncertainty in colliion avoidance manoeuvring. The Journal of Navigation 43, The Kocierzyna and Hanjin Singaore. (1996) 2 Lloyd Re COURT OF APPEAL. The Sitarem and Sirit. (2001) 2 Lloyd Re ADMIRALTY COURT.

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