Internatonal Journal on Marne Navgaton and Safety of Sea Transportaton Volume 6 Number 3 September 2012 Evolutonary Sets of Safe Shp Trajectores: Evaluaton of Indvduals R. Szlapczynsk Gdansk Unversty of Technology, Gdansk, Poland J. Szlapczynska Gdyna Martme Unversty, Gdyna, Poland ABSTRACT: The paper presents a descrpton of the evaluaton phase of the Evolutonary Sets of Safe Shp Trajectores method. In general, the Evolutonary Sets of Safe Shp Trajectores method combnes some of the assumptons of game theory wth evolutonary programmng and fnds an optmal set of cooperatng trajectores of all shps nvolved n an encounter stuaton. Whle developng a new verson of ths method, the authors decded to use real maps nstead of a smplfed polygon modellng and also to focus on better handlng of COLREGS. The upgrade to the method enforced re-desgnng the evaluaton phase of the evolutonary process. The new evaluaton s thoroughly descrbed and t s shown how evaluaton affects fnal solutons returned by the method. 1 INTRODUCTION A desred soluton to a mult-shp encounter stuaton would nclude a set of planned, optmal trajectores for all the shps nvolved n an encounter, such that no collson or doman volatons occur when these shps follow the trajectores. When solvng ths stuaton the key dffculty s that even a sngle course change performed by one shp nvolved n the encounter may force one or even more the other shps to manoeuvre. Thus the optmsaton method utlzed to fnd a soluton to the problem should be flexble enough to effcently look through the vast search space and handle even mnor changes n the shp s behavour e.g. n ts moton parameters. There s a number of approaches to solvng a mult-shp encounter stuaton. Two basc trends are ether utlzaton of dfferental games (Lsowsk 2005) or searchng for a sngle trajectory (for the own shp) by evolutonary algorthms (Smerzchalsk et al. 2000). The former method assumes that the process of steerng a shp n mult-shp encounter stuatons can be modeled as a dfferental game played by all shps nvolved, each havng ther strateges. Unfortunately, hgh computatonal complexty s ts serous drawback. The latter approach s the evolutonary method focused on fndng only a sngle trajectory of the own shp. In short, the evolutonary method uses genetc algorthms, whch, for a gven set of pre-determned nput trajectores fnd a soluton that s optmal accordng to a gven ftness functon. However, the method s lmtaton s that t assumes targets moton parameters not to change and f they do change, the own trajectory has to be recomputed. Ths lmtaton becomes a serous one on restrcted waters. If a target s current course colldes wth a landmass or another target of a hgher prorty, there s no reason to assume that the target would keep such a dsastrous course untl the crash occurs. Consequently, plannng the own trajectory for the unchanged course of a target wll be futle n the majorty of such cases. Also, the evolutonary method does not offer a full support to VTS operators, who mght face the task of synchronzng trajectores of multple shps wth many of these shps manoeuvrng. Therefore, the authors have proposed a new approach, whch combnes some of the advantages of both methods: the low computatonal tme, supportng all doman models and handlng statonary ob- 345
stacles (all typcal for evolutonary method), wth takng nto account the changes of moton parameters (changng strateges of the players nvolved n a game). Instead of fndng the optmal own trajectory (from the own shp s perspectve) for the unchanged courses and speeds of targets, an optmal set of safe trajectores of all shps nvolved s searched for (from the coast, e.g. VTS, perspectve). The method s called evolutonary sets of safe trajectores and ts early verson has been presented by one of the authors n (Szlapczynsk 2010). The newly developed verson of the method uses real maps nstead of smplfed polygon modellng and focuses on COLREGS complance. The upgrade to the method enforced changes n all phases of the evolutonary process ncludng evaluaton. The paper presents a descrpton and a dscusson of the new evaluaton phase. The rest of the paper s organzed as follows. In the next secton a bref descrpton of the problem s gven, ncludng basc constrants of the optmzaton problem as well as the addtonal constrants - the COLREGS rules, whch are taken nto account. Secton 3 covers the ssue of detectng varous constrants volatons. Ths s followed by a Secton 4, where t s shown, how, on the bass of prevous sectons, the ftness functon s formulated. In secton 5 dfferent evaluaton approaches and the consequences of applyng them are compared by means of smulaton experments. Fnally the summary and conclusons are gven n Secton 6. 2 SOLVING MULTI-SHIP ENCOUNTER SITUATIONS AS AN OPTIMIZATION PROBLEM It s assumed that we are gven the followng data: statonary constrants (such as landmasses and other obstacles), postons, courses and speeds of all shps nvolved, shp domans, tmes necessary for acceptng and executng the proposed manoeuvres. Shp postons and shp moton parameters are provded by ARPA (Automatc Radar Plottng Ad), or, f there s no relable dentfcaton assured, AIS (Automatc Identfcaton System) systems. A shp doman can be determned based on the shp s length, ts moton parameters and the type of water regon. Snce the shape of a doman s dependent on the type of water regon, the authors have assumed and used a shp doman model by Davs (Davs et al. 1982), whch updated Goodwn model (Goodwn 1975), for open waters and to use a shp doman model by Coldwell (Coldwell 1982), whch updated Fuj model (Fuj et al. 1971), for restrcted waters. As for the last parameter the necessary tme, t s computed on the bass of navgatonal decson tme and the shp s manoeuvrng abltes. By default an assumed 6-mnute value s used here. Knowng all the abovementoned parameters, the goal s to fnd a set of trajectores, whch mnmzes the average way loss spent on manoeuvrng, whle fulfllng the followng condtons: none of the statonary constrants are volated, none of the shp domans are volated, the mnmal acceptable course alteraton s not lesser than 15 degrees (assumed to elmnate slow and nsgnfcant turns), the maxmal acceptable course alteraton s not to be larger than assumed 60 degrees, speed alteraton are not to be appled unless necessary (collson cannot be avoded by course alteraton up to 60 degrees), a shp manoeuvres, f and only f she s oblged to, t s assumed that manoeuvres to starboard are favoured over manoeuvres to port board. The frst two condtons are obvous: all obstacles have to be avoded and the shp doman s an area that should not be volated by defnton. All the other condtons are ether mposed by COLREGS (IMO 1977) and good marne practce or by the economcs. In partcular, the course alteratons lesser than 15 degrees mght be msleadng for the ARPA systems (and therefore may lead to collsons) and the course alteratons larger than 60 degrees are not recommended due to effcency reasons. Also, shps should only manoeuvre when necessary, snce each manoeuvre of a shp makes t harder to track ts moton parameters for the other shps ARPA systems (Wawruch 2002). Apart from these man constrants, addtonal constrants selected COLREGS rules have to be drectly handled. The COLREGS rules, whch are of nterest here are: Rule 13 overtakng: an overtakng vessel must keep well clear of the vessel beng overtaken. Rule 14 - head-on stuatons: when two powerdrven vessels are meetng head-on both must alter course to starboard so that they pass on the port sde of the other. Rule 15 - crossng stuatons: when two powerdrven vessels are crossng, the vessel, whch has the other on the starboard sde must gve way. Rule 16 - the gve-way vessel: the gve-way vessel must take early and substantal acton to keep well clear. Rule 17 - the stand-on vessel: the stand-on vessel may take acton to avod collson f t becomes 346
clear that the gve-way vessel s not takng approprate acton. There are also some addtonal COLREGS-related assumptons, namely: there are always good vsblty condtons, all consdered shps are equally prvleged, all consdered shps have motor engne (no salng shps taken nto account), no narrow passages are taken nto account no port board manoeuvres are assumed when overtakng, no manoeuvres to bypass navgatonal sgns are taken nto account. In the followng sectons t wll be analysed how these constrants volatons can be detected, n what order should they be taken nto account and how severely should they be penalzed durng the evaluaton phase by the ftness functon of the evolutonary method. However, the current verson of the method uses a vector map of a gven area. Whle vector maps also uses polygons defned by coordnates of ther vertces, the number of vertces and thus the edges rses drastcally, when compared to the smplfcaton used before. Even after lmtng the map to a certan area, the numbers of the edges that have to be checked for possble crossngs are stll much larger. Ths s shown n Fgure 2. Fgure 2. A shp s trajectory crossng a landmass on a btmap 3 DETECTING CONSTRAINTS VIOLATIONS Below t s descrbed how the constrants volatons can be detected and, n case of varous possble approaches, whch one has been chosen by the authors and why. 3.1 Detectng statc constrants volatons (collsons wth landmasses and safety sobate) In the frst verson of the method (Szlapczynsk 2009) smplfed polygon modellng of the statc constrants have been appled, nstead of usng real maps. Therefore t was natural to fnd collsons by detectng all crossngs of the shps trajectores wth polygons edges. Ths s shown n Fgure 1. A number of operatons that the algorthm has to perform to fnd collsons n such stuaton s proportonal to the number of the edges of all polygons n a gven area. Therefore the authors have decded not to process vector map drectly for crossng detecton, but to use t for generatng btmap of an area. Although t s a tme-takng operaton, fortunately, t s enough to generate such btmaps offlne and only once for each area. Then, when the method s runnng n real tme, nstead of checkng the edges for geometrcal crossngs, each btmap cell, whch the trajectory of a shp traverses, s read and checked f t belongs to landmass, water or safety sobate. For a btmap, whose detal level reflects ths of a gven vector map, the computatonal tme would be much shorter: proportonal to the number of traversed cells, nstead of a number of all vertces. Ths approach s also more flexble n terms of future mplementaton of bathymetry: f every cell contaned nformaton on the water depth, t would be easy to check, whether a cell s passable or not for a partcular shp. Fgure 1. A shp s trajectory crossng a landmass modeled as a polygon. The geometrcal crossngs of the trajectory and polygon edges are marked n black 347
3.2 Detectng collsons wth other shps 3.3 Detectng COLREGS volatons Detectng COLREGS volatons s much more dffcult than volatons descrbed n the prevous two sub-sectons. In general, there may be three types of COLREGS volatons: a shp does not gve way, when t should, a shp gves way, when t should not, because t s a stand-on shp, a shp manoeuvres to port-board when t should manoeuvre to starboard. Fgure 3. Algorthm for shp-to-shp collson detecton The algorthm for detectng shp-to-shp collsons (Fgure 3) s as follows. Each shp s trajectory s checked aganst all other shps. For each par of shps, the start tme and end tme of each trajectory s segments are computed. If two segments of the two trajectores overlap n tme, they are checked for geometrcal crossng. In case of a crossng, the approach factor value s computed. Then, f the approach factor value ndcates collson, the type of an encounter (head-on, crossng or overtakng) s determned on the bass of the shps courses and t s decded, whch shp s to gve way (both shps n case of head-on). The collson s only regstered for the gve way shp and the nformaton on the collson are stored n the trajectory data structure. 348 Each of these three stuatons may happen on ether open or restrcted waters, whch gves us a total of sx cases to handle. The dffculty wth decdng, whether a shp has acted lawfully, or not, les n the nature of evolutonary algorthms as well as n the nature of the problem tself: COLREGS specfy only the procedures for shp-to-shp encounters. Lookng at a set of shp trajectores for a mult-target encounter t s sometmes mpossble to tell, what was the reason for a partcular manoeuvre: whch shp was gven way ntentonally, and whch one benefted from t only as a sde effect. A partal soluton to ths problem s storng n the trajectory data the nformaton on the reasons of the manoeuvres. The possble reasons mght be: landmass avodance or other statc constrant volaton avodance, gvng way to a prvleged shp, any other, e.g. due to the shp s passage plan. However, the course alteratons that each trajectory contans may be made ntentonally as a result of applyng a collson avodance operator or unntentonally as a result of crossng or mutaton. A manoeuvre whch resulted accdentally from crossng or mutaton may be just as good as the one beng the effect of a specalsed operator s more conscous work. Therefore the any other manoeuvre s reason cannot always be regstered as COLREGS volaton. All ths consdered, the authors have decded to lmt the used types on the manoeuvre s reasons to: obstacle avodance and any other. The fnal COLREGS volatons detecton rules are: 1 On open waters: a) f a shp s not oblged to gve way to any other shp, any manoeuvre (other than the manoeuvres gven by the passage plan) t performs s regstered as COLREGS volaton, b) f a shp s oblged to gve way, and does not perform a manoeuvre t s regstered as COLREGS volaton, c) all manoeuvres to port board are regstered as COLREGS volatons. The c) pont may rase some doubts, but t must be emphaszed that COLREGS volatons regstraton s done for the sake of future penalzng of
volatons, when the fnal ftness functon values s beng computed. Therefore, the only effect of penalzng the manoeuvres to port board wll be addtonal favourng of manoeuvres to starboard, whch are already favoured by doman models. In no way does penalzng make t mpossble to choose a manoeuvre to port board. It s only less proftable for most cases. 2 On restrcted waters: here, as explaned before every trajectory node, whch s a part of a manoeuvre, contans specal nformaton on the reason why ths partcular node has been nserted or shfted: land or other statonary obstacle avodance, target avodance or accdental manoeuvre generated by evolutonary mechansms. Based on ths, COLREGS volatons are regstered as follows: a) f a shp does not ntally have to gve way to any target and ts frst manoeuvre has reason other than statc constrant volaton avodance, t s regstered as COLREGS volaton, b) any manoeuvre to port board of reason other than statc constrant volaton avodance s regstered as COLREGS volaton. Pont b) means that occasonally the correct manoeuvres ntroduced by crossng or mutaton and avodng statc constrant volaton wll be penalzed unjustly. However, t s not a problem, as long as penaltes for statc constrant volatons wll be larger and trajectores avodng them wll stll be selected for next generatons. After all, we are nterested n the fnal sets of trajectores themselves much more than n ther slghtly mprecse ftness functon values. 4 FORMULATING FITNESS FUNCTION In the evolutonary method all ndvduals (sets of trajectores) are evaluated by the specally desgned ftness functon, whch should reflect optmsaton crtera and constrants (Mchalewcz et al. 2004). In ths secton t s shown, how, on the bass of prevous sectons, ths ftness functon s formulated. 4.1 Basc crteron mnmzng way loss The basc crteron s the economc one mnmzng way losses of trajectores n a set. For each of the trajectores, a trajectory_economy_factor s computed accordng to the formula (1). trajectory _ length way _ loss trajectory _ economy _ factor =, _ (1) trajectory length the ndex of the current shp [/], trajectory_length the total length of the -th shp s trajectory [nautcal mles], way_loss the total way loss of the -th shp s trajectory [nautcal mles] computed as a dfference between the trajectory length and length of a segment jonng trajectory s start pont and endpont. As can be seen, the trajectory_economy_factor s always a number from a (0,1] range. 4.2 Penalzng statc constrant volaton After the trajectory economy factor has been computed the statc constrants are handled by ntroducng penaltes for volatng them. For each trajectory ts statc constrant factor scf s computed. The statc constrants are always vald and ther volatons must be avoded at all cost, therefore penaltes appled here are the most severe hence the square n the formula (2). trajectory _ length trajectory _ cross _ length scf =, _ (2) trajectory length trajectory_cross_length the total length of the parts of the -th shp s trajectory, whch volate statonary constrants [nautcal mles]. The statc constrant factor s a number from a [0,1] range, where 1 value means no statc constrant volaton (no landmasses or other obstacles are crossed) and 0 value s for trajectores crossng landmasses on ther whole length. 4.3 Penalzng collsons wth other shps Analogcally to the statc constrant factor, collson avodance factor caf s computed to reflect the shp s collsons wth all other prvleged shps as shown by (3). n (, j,1) ) caf = mn( fmn (3) j= 1, j n the number of shps [/], j the ndex of a target shp [/], fmn,j the approach factor value for an encounter of shps and j, f -th shp s the prvleged one, the potental collson s gnored and the approach factor value s equal to 1 by defnton. [/]. The collson avodance factor s a number from a [0,1] range, where 1 value means no shp doman volaton and 0 means a crash wth at least one of the targets. 2 349
4.4 Penalzng COLREGS volatons The COLREGS volatons are secondary to statc constrant volatons and to collsons wth other shps and therefore the authors have decded to penalze t moderately, to make sure that constrants from the prevous two ponts are met frst. COLREGS complance factor ccf s computed accordng to the followng formula (4). m ccf = 1 [ COLREGS _ volaton _ penalty k ], (4) k = 1 m the number of COLREGS volaton regstered for the current shp as has descrbed n secton 3.3 [/], k the ndex of a regstered volaton [/], COLREGS_volaton_penalty k the penalty for the k-th of the regstered COLREGS volaton [/]. trajectory _ ftness = = trajectory _ economy _ factor * scf * caf * ccf, (6) The fnal ftness functon value assgned to an ndvdual s an arthmetcal average of ftness functon values computed for all trajectores. It s dscussable, whether all trajectores should have the same mpact on fnal ftness functon value (as t s done here), or should the trajectory ftness functon values be taken wth weghts proportonal to the trajectory lengths. When combned wth the formula for trajectory economy factor, the current approach means that we are tryng to mnmze average relatve way loss computed over all trajectores, nstead of total absolute way loss (wth weghts beng used). However, experments have shown, that mnmzng total absolute way loss leads to dscrmnaton of shps, whose basc trajectores are shorter and to ther large relatve way losses (secton 5.2). The penalty values for all regstered COLREGS volatons descrbed n secton 3.3 by ponts 1. a) - c) and 2. a) - b) are confgurable n the method and are set to 0.05 by default. 4.5 Ftness functon value Once all aforementoned factors have been computed, the ftness functon value s calculated. The authors wanted the ftness functon to be normalzed, whch s convenent for further evolutonary operatons, mostly for selecton purposes. When ftness functon values are normalzed, we do not need any addtonal operatons on them and they can drectly be used for random proportonal and modfed random proportonal selecton n the reproducton and successon phases of the evolutonary algorthm. We can also easly measure and see progress we make wth each generaton. However, normalzed ftness functon s harder to obtan, because we have to make sure that we keep the hgh resoluton of evaluatng the ndvduals, namely that we dffer between varous levels of penaltes: statonary constrants, beng more mportant than collson avodance and collson avodance beng more mportant than COLREGS complance. Here, we succeeded n formulatng a normalzed ftness functon, whle keepng relatvely hgh resoluton of evaluaton: mnor statonary constrants volatons are penalzed smlarly as major collsons wth other shps and mnor collsons wth other shps are penalzed smlarly as multple COLREGS volatons. The fnal ftness functon s as follows: n trajectory _ ftness ftness =, (5) n = 1 5 COMPARING DIFFERENT EVALUATION APPROACHES In the followng subsectons dfferent evaluaton approaches and the consequences of applyng them are compared. 5.1 Penalzng COLREGS volatons: how t affects solutons returned by the method Even when a doman model, whch favours COLREGS s appled, t s possble to fnd an encounter stuaton, where addtonal COLREGS volatons penaltes must be used, as has been descrbed n secton 4.4 or otherwse the method wll return ncorrect soluton. A smple example s a head-on encounter of two shps, whose parameters are shown n Fgure 4. In ths scenaro, followng the Rule 14 of COLREGS for head-on stuatons, t s requred that: ( ) both (vessels) must alter course to starboard so that they pass on the port sde of the other. Fgure 4. Parameters of two shps n a head-on encounter Because the method tends to propose manoeuvres no lesser than 15 degrees, a manoeuvre from one shp only would be enough to avod collson. From the way loss mnmzaton pont of vew, the extra manoeuvre from the second shp s redundant. Consequently, ndvduals contanng trajectores wth manoeuvres from both shps would be ranked lower 350
than those wth only one shp manoeuvrng and the fnal soluton wll have only one shp manoeuvrng, whch s shown n Fgure 5. loss. The default penaltes of 0.05 are suffcent for the correct soluton to be chosen. Ths s shown n Fgure 6. 5.2 Optmzaton crteron: total absolute way loss or average relatve way loss Another queston already rased before (secton 4.5) s whether we should mnmze total absolute way loss or average relatve way loss. An example scenaro of an encounter of 6 shps on restrcted waters s presented below. Shp parameters are gathered n Fgure 7. The results of mnmzng total absolute way loss are shown n Fgure 8, the results for the mnmzaton of average relatve way loss n Fgure 9. Fgure 5. An ncorrect soluton to a head-on encounter stuaton returned by the method wthout COLREGS volatons penaltes Fgure 7. Parameters of sx shps n an encounter on restrcted waters Fgure 6. A correct soluton to a head-on encounter stuaton returned by the method wth COLREGS volatons penaltes appled Thus we need to addtonally penalze the ndvduals for COLREGS volatons to favour the ndvduals wth both shps manoeuvrng and larger way As can be seen below, mnmzng average relatve way loss (Fgure 9) results n smoother trajectores for shp 1 and shp 5. Shp 5 also has consderably lesser way loss because t passes the sland on ts left sde (Fgure 9), nstead of rght sde (Fgure 8). Other trajectores (the longer ones) have no major vsual dfferences between them n Fgures 8 and 9, though ftness functon values of some shps are slghtly larger for Fgure 8, because of ther (nsgnfcantly) lesser way losses. Unfortunately, t s mpossble to formally compare the solutons returned by the two varants of the method, whch use dfferent formulas for global ftness functon and thus am at dfferent goals. However, after a seres of smulaton experments, the authors are of the opnon that n general the mnmzaton of average relatve way loss brngs more balanced and ntutve results for most cases and therefore have chosen t to be the default opton of the current verson of the Evolutonary Sets of Safe Trajectores method. 351
Fgure 8. A soluton to a mult-shp encounter stuaton returned by the method wth mnmzaton of total absolute way loss Fgure 9. A soluton to a mult-shp encounter stuaton returned by the method wth mnmzaton of average relatve way loss 6 SUMMARY AND CONCLUSIONS The paper documents the research on the evaluaton phase of the Evolutonary Sets of Safe Shp Trajectores method. For some of the optmsaton constrants, gatherng the data on ther volatons for evaluaton purposes s tme consumng (collsons wth other shps and statc obstacles), whle for others t s dscussable n some cases, whether a constrant has been met or not (COLREGS rules), whch serously lmts detecton possbltes. Even such a seemngly smple ssue as man optmsaton crteron (way loss mnmsaton) becomes a problem, when a partcular ftness functon value s to be formulated. The authors have explored varous possbltes of gatherng the data on constran volaton, as well as usng them n the ftness functons and have presented n the paper ther conclusons: the technques and formulas that, n the course of the research, occurred to be most useful for evaluaton of the sets of shp trajectores. The chosen elements of the method have been llustrated by smulaton examples showng how a change n the evaluaton phase affects the fnal solutons returned by the method. The authors search for the optmal evaluaton s beng contnued, as the whole method s functonal scope s expandng. The current works are focused on handlng Traffc Separaton Schemes drectly n the Evolutonary Sets of Safe Trajectores method, whch brngs new evaluaton ssues. 352
ACKNOWLEDGEMENTS The authors thank the Polsh Mnstry of Scence and Hgher Educaton for fundng ths research under grant no. N N516 186737. REFERENCES Coldwell T.G. 1982. Marne Traffc Behavour n restrcted Waters, The Journal of Navgaton, 36, 431-444. Cambrdge: Cambrdge Journals. Davs P.V. & Dove M.J. & Stockel C.T. 1982. A Computer Smulaton of mult-shp Encounters. The Journal of Navgaton, 35, 347-352. Cambrdge: Cambrdge Journals. Fuj J., Tanaka K. (1971). Traffc Capacty. The Journal of Navgaton, 24, 543-552. Cambrdge: Cambrdge Journals. Goodwn E.M. 1975. A Statstcal Study of Shp Domans. The Journal of Navgaton, 28, 329-341. Cambrdge: Cambrdge Journals. IMO. 1977. Conventon on the Internatonal Regulatons for Preventng Collsons at Sea. The Internatonal Martme Organzaton. Lsowsk J. 2005. Dynamc games methods n navgator decson support system for safety navgaton, Proceedngs of the European Safety and Relablty Conference, vol. 2, 1285 1292. Mchalewcz Z. & Fogel D.B. 2004. How To Solve It: Modern Heurstcs, Sprnger-Verlag. Smerzchalsk, R., Mchalewcz, Z. 2000. Modelng of a Shp Trajectory n Collson Stuatons at Sea by Evolutonary Algorthm, IEEE Transactons on Evolutonary Computaton No. 3 Vol. 4, pp. 227-241. Szłapczynsk R. 2010: Solvng Mult-Shp Encounter Stuatons by Evolutonary Sets of Cooperatng Trajectores. TransNav - Internatonal Journal on Marne Navgaton and Safety of Sea Transportaton, Vol. 4, No. 2, pp. 185-190 Wawruch R. 2002. ARPA zasada dzałana wykorzystana, Gdyna: WSM Gdyna 353