Optimal Arragemet of Buoys Observable by Meas of Radar TOMASZ PRACZYK Istitute of Naval Weapo ad Computer Sciece Polish Naval Academy Śmidowicza 69, 8-03 Gdyia POLAND t.praczy@amw.gdyia.pl Abstract: - The paper addresses the problem of buildig a automatic, spare, radar system to coastal avigatio. To fix positio the system uses the iformatio about buoys surroudig the ship. Accuracy of the system depeds o may factors. Oe of them is the way of locatig buoys o the give area of the sea. To mae the tas of the system easier ad to mae the positio fixed by the system more accurate the buoys should be appropriately arraged. The paper suggests the solutio based o geetic algorithms to arrage the buoys. The solutio proposed was tested experimetally ad results of the tests are preseted at the ed of the paper. Key-Words: - optimizatio, geetic algorithms, radar coastal avigatio, spare avigatioal systems, positio approximatio, eural etwors Itroductio There are a few cocepts of automatig radar avigatio [[6],[7],[8],[9]]. Oe of them is based o a system of self buoys, i.e. buoys recogizable by the positioig system (The problem of differig self buoys from other buoys is preseted amog other thigs i [[0]]). To fix positio the system determies a vector of distaces (ad optioally bearigs) to all self buoys surroudig the ship, usig radar for that purpose. The vector of distaces fixed by the system is passed o to appropriately prepared artificial eural etwor. The tas of the etwor is to fix positio of the ship. Geerally, the tas of the etwor ca be viewed as approximatio of a positio fuctio defied as follows: F : V P () D VD R - the set of vectors of size icludig distaces (ad optioally bearigs) to buoys; P - the set of poits i geographical coordiate system, i.e. the set of poits i the followig form ( ϕ, λ), ϕ is the latitude ad λ is the logitude. Accuracy of approximatig the fuctio above strogly depeds o its shape. The more the fuctio is udulated the greater difficulties with its approximatio. To mae the tas of the etwor easier ad to mae the positio fixed by the etwor more accurate the shape of the positio fuctio should be possibly the easiest. Sice, the shape of the positio fuctio depeds o mutual arragemet of buoys, it ca be easily modified through movig buoys ito differet places. However, the problem is how to arrage buoys to obtai flat, easy shape of the fuctio. To solve this problem differet methods ca be used. Oe of them, suggested i the paper, are geetic algorithms (GAs). To test GAs i the problem of arragig the buoys simple experimets were performed. I the experimets, classical Caoical GA (CGA) [,[3],[4],] ad Eugeic Algorithm (EuA) [[],[5],[]] were used. The tas of the algorithms was to arrage te buoys o a virtual area of the sea of the size 0x0. To evaluate arragemets they were used to create positioig eural etwors which, the, were tested i terms of accuracy of positio geerated i testig poits of the cosidered research stretch. I the experimets, arragemets geerated by GAs were compared to radom arragemets. The paper is orgaized as follows: sectio is a short presetatio of the positioig system; sectio 3 is a descriptio of the problem of arragemet of the buoys; sectio 4 is a descriptio of evolutioary techiques used to solve the problem metioed, sectio 5 is a report from the experimets; ad sectio 5 is a summary. ISBN: 978--6804-056-5 00
The cocept of the positioig system There are three ey elemets of the system: radar, the system of self buoys, ad positioig eural etwor. Radar is a commo device o every ship used to measure distace ad bearig to differet obects. The iformatio about obects ad geerally the iformatio about world surroudig a ship is preseted i a radar scree. Additioally, the iformatio about obects is trasmitted outside radar. Most curretly used radars geerate the message formatted accordig to NMEA 083 (the stadard established by Natioal Marie Electroics Associatio) stadard. The message icludes all the iformatio required by the system, i.e. the iformatio about distaces (ad optioally bearigs) to all obects visible by meas of radar. With regard to buoys, they are preset o almost every stretch to facilitate avigatio. I our case, the buoys have to have three basic features. First, they caot chage positio over time (the so-called super-buoys). Secod, they have to be visible by meas of radar (each buoy has to be equipped with the so-called raco, i.e. a device to reflect radar waves). Third, all the buoys have to be recogizable by the system, i.e. they have to be differetiable from other buoys occurrig o the give area of the sea. To differ the self buoys from other buoys ideas from artificial immue systems ca be used [[0]]. The third elemet of the system is approximatig eural etwor. At preset, to approximate positio, two Geeral Regressio Neural Networs (GRNN) are used. f ) Z = 0 σ) Wϕ = 0 σ = Z () ϕ σ) ( x x ) σ ϕ σ) = e (3) fσ) - the value of approximated fuctio i poit x (i our case f is oe of uow coordiates of the ship, i.e. ϕ or λ, as x is the vector of distaces to buoys fixed i the positio (ϕ,λ )) σ - the parameter W - the true value of the fuctio f i the traiig poit x (i our case W is the value of either ϕ or λ ) x - th traiig poit (i our case the sample vector of distaces to buoys for which we ow accurate positio of the ship) Z the umber of traiig poits. x W x x 3 W Σ W3 / f(x) x - x- x W4 Fig. Structure of a sigle GRNN The first oe is used to approximate the latitude while the secod oe to approximate the logitude. Both GRNNs implemet the followig fuctio: Σ Fig. The cocept of the system To prepare the etwors to wor we should have at our disposal the set of traiig poits i the form ( v, ϕ, λ ), =..Z, v is the vector recorded i the poit ( ϕ, λ ) icludig distaces to buoys. All the traiig poits ca be produced i a laboratory, o a persoal computer. There is o ecessity to perform ay measuremets at sea what is the great advatage of the preseted system. ISBN: 978--6804-056-5 0
The system wors as follows. At the begiig, self buoys are selected from amog all buoys visible o a stretch. Next, all self buoys are ordered accordig to North-South directio. I the followig step, the vector of size (at each poit i time at most buoys ca be visible by meas of radar; the field of visio of radar is restricted to the area determied by the rage of radar observatio R) icludig distaces to self buoys is created. At the begiig of the vector distaces to North-most buoys are stored. Distaces to South-most buoys are at the ed of the vector. Whe fewer tha self buoys are visible o a stretch, the ed of the vector is filled i with zeros. The last activity of the system is activatio of approximatig eural etwors. The etwors fix positio of a ship based o the vector of distaces prepared beforehad. Illustratio of the system is preseted i Fig.. The greatest advatages of the system described above are simplicity i creatig ad autoomy, i.e. idepedece o outside sources of iformatio. The latter feature is particularly importat i the case of military ships. 3 Optimizatio of locatig the buoys The most importat evaluatig criterio of a positioig system is its accuracy. Accuracy of approximatig the positio fuctio by meas of GRNN depeds o the value of σ ad o traiig poits used to create the etwor. The first step to adust a iitially created GRNN to a tas is to tue the value of σ. The oly method to improve performace of the etwor whe the value of σ is fixed ad results of the etwor are still usatisfactory is to add ext traiig poits. The ewly itroduced poits should come from areas the greatest errors of the positio occur. The additioal poits o the oe had ehace performace of the etwor but o the other had exted the calculatig time ecessary to fix a positio (i our case we deal with software, sequetial implemetatio of GRNNs). Accordigly, we caot simply add ext traiig poits to mae the system more accurate. Too may traiig poits used to create positioig GRNNs ca mae them too slow ad thereby upractical. The solutio to this problem is to facilitate the positio fuctio. The simpler the shape of a fuctio the fewer traiig poits ecessary to accurately approximate it. I the case of the positio fuctio to mae it flat ad easy to approximate similar vectors of distaces should correspod to similar positios. Moreover, differet vectors should represet differet positios. The oly method to achieve such effect is to appropriately arrage the buoys (each chage i locatig the buoys iduces a chage of the shape of the positio fuctio). To accomplish the flat shape of the positio fuctio the buoys should be arraged so as to miimize the followig evaluatio fuctio: ( ) ( ( )) * d d a m F e( a) = i i (4) m m i< a= ϕ λ, ϕ, λ,, ϕ, λ the vector, icludig positios of self buoys (the arragemet of buoys); m the umber of poits uiformly distributed o the stretch ad used to evaluate arragemets of buoys; d the ormalized distace betwee i th ad th i poit of the stretch; d the ormalized Euclidea distace betwee i th * i vector (icludig distaces to buoys) represetig i th poit of the stretch ad th vector represetig th poit of the stretch. I the experimets reported further, to create arragemets of buoys, we used two types of GAs, i.e. CGA ad EuA. To evaluate created arragemets both algorithms used the followig fitess fuctio: F f( a) = (5) F a 4 Experimets e ( ) I the experimets, the tas of GAs was to arrage te buoys (=0) o the virtual stretch of the size 0x0. To evaluate arragemets geerated durig the evolutioary process, GAs used the fitess fuctio (5) built based o hudred poits (m=00) uiformly distributed o the research stretch. The best arragemets produced by GAs were tested as a elemet of the positioig system. Each selected arragemet was tested te times, i.e. te differet positioig systems (a sigle positioig system cosisted of two GRNNs) were produced for each selected arragemet. The positioig systems created to test a sigle arragemet differed i traiig sets used to produce them. The traiig sets used i the experimets icluded te, twety, fifty or hudred traiig poits ad represetig them ISBN: 978--6804-056-5 0
vectors of distaces (Z=0, 0, 50, or 00). Traiig sets geerated for differet arragemets differed oly i vectors of distaces to buoys. Poits i which the vectors were recorded were the same for all tested arragemets. All the positioig systems created durig the experimets were tested i hudred testig poits, the same for all arragemets. To ultimately evaluate the arragemets the followig fuctio was used: P T i i E( a) = ( ϕ ϕ ) + ( λ λ ) (6) TP i T the umber of testig poits (T=00); P the umber of positioig systems created for a sigle arragemet (P=0); ϕ, - the accurate positio i th testig poit; ( λ ) i i ( λ ) ϕ, - the positio geerated by i th positioig system i th testig poit. I the experimets, two GAs, i.e. CGA ad EuA, were used to arrage the buoys. Both GAs processed arragemets i the form of * size iteger vectors ecoded as biary strigs. Each iteger from the vector was ecoded as biary strig of size 7 (this meas that chromosomes processed by GAs were of size 40 bits). Sice, the size of the research stretch was 0x0, to create arragemets, itegers from the vectors were scaled to the rage <0,0>. The remaiig parameters of the evolutioary process are preseted below: the umber of idividuals i populatio icludig arragemets: 00; the umber of evolutioary geeratios: 50 000; Parameters of CGA: crossover probability: 0.7; per-bit mutatio probability: 0.03; the size of touramet: ; Parameters of EuA: selectio oise: 0.0, 0.; creatio rate: 0.0, 0.; restrictio operator: o. I the experimets, arragemets prepared by GAs were compared to radom arragemets. Geerally, thirty arragemets prepared by meas of CGA, thirty arragemets geerated by EuA ad thirty radom arragemets were tested. All the results preseted i the further part of the paper are averaged. a) b) c) d) Fig. 3 Example arragemets of buoys, (a) radom arragemet; (b),(c) arragemet geerated by CGA; (d) arragemet geerated by EuA The experimets showed that appropriate arragemet of buoys improves accuracy of the positioig system. However, as to be expected, the improvemet is oly oticeable for a slight umber of traiig poits used to create the system. The more traiig poits the less ifluece of the arragemet of buoys o the accuracy of positio fixed by the system. With regard to arragemets, it seems that there exists a oticeable differece betwee the radom arragemets ad the arragemets produced by meas of GAs. While buoys from the radom arragemets are usually spread o the whole research area, buoys from the arragemets produced by GAs form somethig lie a circle. The buoys are located aroud the research area without ay buoys i the ceter. Tab. The results of the experimets 0 traiig poits 0 traiig poits 50 traiig poits 00 traiig poits 0 4 F e E CGA.7.4 0.98 0.0 9.8 EuA.6.45.7 0.03 35.9 radom 3.86 3.84.7 0.0 40,6 ISBN: 978--6804-056-5 03
5 Summary The paper presets the problem of appropriate arragemet of buoys beig a elemet of the coastal, radar, positioig system. The system fixes positio of a ship usig for that purpose the iformatio about buoys visible o the give stretch. The positio of a ship is fixed by appropriately traied, approximatig eural etwor. To mae the tas of the etwor easier the buoys should be appropriately arraged. I the paper the experimets are reported i which to arrage the buoys GAs were used. The experimets showed that i the case of iumerous traiig sets used to trai the positioig etwor, appropriate arragemet of buoys ca improve the accuracy of the positio fixed. I the case of traiig sets icludig may traiig samples the effect of appropriate arragemet of buoys is less ad less oticeable. However, the large traiig sets ca also cosiderably slow dow the positioig system maig it useless. coast, Iteratioal Joural of Applied Mathematics ad Computer Sciece, Vol. 7, No., 007, pp. 87-98. [0] Praczy T., Detectio of self avigatioal aids o radar image usig ideas from immue systems, Archives of Cotrol Sciece, Vol. 7, No. 3, 007, pp. 4-59 [] Prior J. W. 998. Eugeic Evolutio for Combiatorial Optimizatio, Master s thesis, The Uiversity of Texas at Austi. TR AI98-68. [] Whitley D. A Geetic Algorithm Tutorial. http://citeseer.ist.psu.edu Refereces: [] Alde M., Va Kestere A. & Miiulaie R, Eugeic Evolutio Utilizig a Domai Model, I Proceedigs of the Geetic ad Evolutioary Computatio Coferece (GECCO-00), Sa Fracisco, CA, Morga Kaufma, 00 [] Arabas J. Lectures o evolutioary algorithms. WNT. Warsaw, 00. [3] Goldberg D. E., Geetic algorithms i search, optimizatio ad machie learig, Addiso Wesley, Readig, Massachusetts, 989. [4] Hollad J. H. Adaptatio i Natural ad Artificial Systems, Uiversity of Michiga Press, A Arbor, Michiga, 975. [5] Polai D. & Miiulaie R., Eugeic Neuro- Evolutio for Reiforcemet Learig. I Proceedigs of the Geetic ad Evolutioary Computatio Coferece (GECCO 000), Las Vegas, NV, 000 [6] Praczy T., Applicatio of eural etwors ad radar avigatioal aids of shore area to positioig, Computatioal Methods i Sciece ad Techology, Vol., No., 006, pp 53-59 [7] Praczy T., Artificial eural etwors applicatio i maritime, coastal, spare positioig system, Theoretical ad Applied Iformatics, Vol. 8, No.3, 006, pp. 75-88 [8] Praczy T., Automatic radar avigatioal system, Theoretical ad Applied Iformatics, Vol. 8, 006, pp. 9-08 [9] Praczy T., Applicatio of bearig ad distace trees to the idetificatio of ladmars of the ISBN: 978--6804-056-5 04