Tackling the Premature Convergence Problem in Monte Carlo Localization Gert Kootstra and Bart de Boer

Transcription

1 Tacklng he Premaure Convergence Problem n Mone Carlo Localzaon Ger Koosra and Bar de Boer Auhors: Ger Koosra (correspondng auhor) Arfcal Inellgence Unvesy of Gronngen, The Neherlands Groe Krussraa 2/ TS Gronngen The Neherlands Fax: +31 (0) G.Koosra@a.rug.nl Bar de Boer Arfcal Inellgence Unvesy of Gronngen, The Neherlands Groe Krussraa 2/ TS Gronngen The Neherlands B.de.Boer@a.rug.nl 1

2 Keywords: Mone Carlo localzaon, parcle fler, premaure convergence, nchng, genec algorhms Absrac. Mone Carlo localzaon uses parcle flerng o esmae he poson of he robo. The mehod s known o suffer from he loss of poenal posons when here s ambguy presen n he envronmen. Snce many ndoor envronmens are hghly symmerc, hs problem of premaure convergence s problemac for ndoor robo navgaon. I s, however, rarely suded n parcle flers. We nroduce a number of so-called nchng mehods used n genec algorhms, and mplemen hem on a parcle fler for Mone Carlo localzaon. The expermens show a sgnfcan mprovemen n he dversy mananng performance of he parcle fler. 1. Inroducon Many pars of our everyday real-world envronmen are more or less dencal. Many offce buldngs and aparmen complexes conss of dencally lookng rooms and hallways,as can for nsance be seen n he Robocs Daa Se Reposory (Radsh) [16]. If we wan our fuure robos o navgae hrough hese envronmens, he robo localzaon echnques need o cope wh hese symmeres and ambguous suaons. Mone Carlo localzaon, based on parcle flerng, s currenly he mos used echnque for robo localzaon (e.g., [28]). A parcle fler however canno deal wh ambguous suaons, snce suffers from he problem of premaure convergence [2, 28]. The nheren properes of he parcle fler make he parcle populaon quckly loose dversy. Because of he loss of dversy, he fler s no able o manan several possble soluons, needed when here s ambguy, bu converges o one soluon ha mgh no represen he robo s real locaon. The problem s frequenly obscured by usng an enormous amoun of parcles. We, on he oher hand, propose an mprovemen upon sandard Mone Carlo localzaon ha deals wh symmerc envronmens n a more fundamenal and moreover a more compuaonally effcen way. 2

3 Premaure convergence s no only a problem n Mone Carlo localzaon, bu also n genec algorhms [4, 11]. Ths s no very surprsng, snce genec algorhms and parcle flers are very smlar echnques. Boh algorhms use a se of ndvduals or parcles o represen an underlyng probably funcon. In Mone Carlo localzaon hs probably funcon s he probably of he robo s poson, n genec algorhms hs s he fness funcon. In boh mehods, a new se of parcles or ndvduals s formed based upon random changes and selecon. As a consequence of he smlary beween genec algorhms and parcle flers, boh echnques face he same problems, bu can also benef from he same soluons. Despe he smlary beween boh algorhms, he ransfer of knowledge from one feld o he oher has been explored n only a few papers. Hguch [14], for nsance, appled he genec algorhm s operaors muaon and crossover o he predcon sep of a parcle fler. Benvenue e al. [2] used fness sharng, a well known nchng algorhm n genec algorhms, n her parcle fler o overcome premaure convergence of he populaon. The sably properes of parcle flers and genec algorhms are dscussed n [24]. Fnally, Kwok e al. [17] am o preserve dversy by applyng crossover and a dversfcaon echnque n a roboc Smulaneous Localzaon and Mappng (SLAM) algorhm. These sudes ndcae ha a sysemac exploraon of hese smlares can be benefcal for boh genec algorhms and parcle flers. We wll specfcally focus upon he problem of premaure convergence. Ths problem s wellsuded n he genec algorhm feld. A number of so-called nchng mehods are developed o ackle he problem [4, 11, 20]. Alhough hese mehods proved o be very effecve, hey are no used n parcle flers, wh he noable excepon of Benvenue e al [2], who use sharng, one of he nchng mehod, n Mone Carlo flerng. In hs paper, we wll sysemacally explore he possbles o use he nchng mehods from genec algorhms n Mone Carlo localzaon. We beleve ha he proposed mehods can be an mporan mprovemen for usng Mone Carlo localzaon n symmercal envronmens. More general, our resuls wll be vald for parcle flers a large. Furhermore, we hope o convnce he reader of he possbly and 3

4 relevance of ransferrng knowledge from he feld of genec algorhms o he feld of parcle flers. In he nex secon, we descrbe he genec algorhm and he parcle fler and gve her smlares. In secon 3, we nroduce he problem of premaure convergence and a number of exsng nchng mehods n he feld of genec algorhms. We demonsrae he applcably of hese mehods o parcle flers for Mone Carlo localzaon n secon 4, and descrbe he resuls n secon 5. Fnally, we dscuss he mplcaons of hese resuls n secon Parcle Flers and Genec Algorhms Parcle flers and genec algorhms are wo sochasc search algorhms ha are based upon he same prncples. These prncples are varaon, selecon, and reproducon. The purpose of varaon s o explore he search space, and selecon and reproducon o explo he good soluons. Alhough he applcaons of he algorhms mgh be dfferen, boh algorhms, n her essence, are dencal. Parcle flers can be raced back o Meropols & Ulam [22], who nvened he Mone Carlo mehod. The parcle fler res o represen a dsrbuon by a se of random samples (parcles) drawn from ha dsrbuon. Through a process of varaon, selecon and resamplng, he parcles are redsrbued n order o beer represen he dsrbuon. Parcle flers are wdely used for sae esmaon, for problems such as rackng, and robo localzaon (e.g. [6]). The goal of a parcle fler s o approxmae a dsrbuon by a se of weghed samples (or parcles), so ha he densy of he samples s proporonal o he probably densy of he dsrbuon. Parcle flerng works n hree seps: a predcon sep, a correcon sep and a resamplng sep. In Mone Carlo localzaon, he dsrbuon we ry o approxmae s ha of he poson of he robo n s envronmen. I can be approxmaed by usng knowledge of he prevous poson of he robo, of he acon ha was performed and of he sensor observaons ha were made. The frs wo seps of he parcle fler are hus mplemened by he ranson 4

5 model (wha happens o he robo afer a gven acon) and he sensor model (wha would we observe, gven he robo s sae and he map of he envronmen). Le = { S x, w = 1,..., n} be he populaon a me, where x s he sae of parcle and w s he mporance wegh of ha parcle. A = 0, he parcles are usually randomly dsrbued over he search space wh equal mporance weghs. Every eraon, he algorhm akes he prevous populaon S 1, he acon performed u 1 and he observaon z. In he frs sep, he nex sae of he parcles s predced by samplng from he ranson model P( x x, u ). The ranson model s ypcally a Gaussan dsrbuon wh a mean defned by 1 1 he former poson of he robo, x 1, and he acon of he robo, u 1 (see fgure 3 for an example). The sandard devaon of he dsrbuon s specfc for he robo and reflecs he expeced error of performng an acon. In he second sep, he mporance weghs are deermned hrough he sensor model w = P( z x ). Ths s ypcally he lkelhood of makng sensory observaon z, gven ha he robo s a poson, x. A model of he sensor nose s used for hs calculaon (see fgure 3 for an example). Fnally, he populaon s resampled o generae a new populaon of n parcles. Each new parcle s drawn from he prevous populaon, where he probably ha a parcular parcle s seleced s proporonal o s mporance wegh. For more nformaon on Mone Carlo localzaon and parcle flers we refer o [6, 28]. Genec algorhms have been nroduced by Holland [15] and a smlar echnque, evoluonary sraeges, by Rechenberg [26]. Inspred by naural evoluon, he genec algorhm mplemens a populaon of ndvduals ha s prone o varaon and (naural) selecon and reproduces generaon afer generaon. Nowadays, dfferen forms of he algorhm are used as effecve search mechansms for a varey of search and opmzaon problems (e.g., [10, 23]). Whn populaon genecs, a subfeld of bology, echnques smlar o genec algorhms are used o sudy he dynamcs of evoluonary processes [12]. 5

6 The close smlary beween parcle flers and genec algorhms has been poned ou by Hguch [13] (as ced n [14]) and Moral, Kallel & Rowe. [25]. As menoned before, boh parcle flers and genec algorhms are based upon on varaon, selecon and reproducon. Varaon s used n boh algorhms for explorng he search space. In parcle flers, hs s aken care of n he predcon sep by randomly samplng from he dsrbuon P( x x 1, u 1). Genec algorhms use muaon o nroduce varaon n he populaon. Secondly, selecon and reproducon accoun for he exploaon of good soluons. In parcle flers, he probably of selecng a sample s defned by he mporance weghs, whch are assgned o he samples n he correcon sep (whch n Mone Carlo-localzaon s based on he sensor model). Smlarly, n genec algorhms ndvduals are seleced wh a probably ha s proporonal o her fness. Fnally, ndvduals n boh algorhms reproduce proporonally o her wegh (or fness). Alhough reproducon s called resamplng n parcle flers, he mplemenaons are analogous. Snce boh algorhms are based upon he prncples of varaon, selecon and reproducon, and he mplemenaon of hese prncples s analogous, we conclude ha parcle flers and genec algorhms are essenally he same. The consequence of hs smlary s ha boh algorhms face smlar problems, and ha smlar soluons can be appled o hese problems. In he nex secon, we wll dscuss he problem of premaure convergence and he exsng soluons for hs problem n genec algorhms. Subsequenly, we wll demonsrae how hese soluons can be used n parcle flers. 3. Premaure Convergence and how o Couner In hs secon we wll frs dscuss he problem of premaure convergence, whch boh parcle flers and genec algorhms suffer from. Then we wll revew a number of nchng mehods ha are used n genec algorhms o couner hs problem. 6

7 Fgure 1. An example of premaure convergence n Mone Carlo localzaon. In each snapsho, he robo s shown as a crcle wh s dsance measuremens. The parcles are depced as black dos. The gray areas are obsacles and he whe areas are free space. The envronmen s hghly symmercal wh he excepon of an objec n he lef hallway. In he frs suaon some dversy s sll presen n he parcle populaon. However, he dversy quckly dsappears as a consequence of random drf n samplng. Ths resuls n he sysem s nably o correcly localze he robo when he dsambguang objec s found Premaure Convergence One of he major problems ha boh genec algorhms and parcle flers suffer from s he problem of premaure convergence (see [5, 20] for genec algorhms and [2] for parcle flers). Premaure convergence relaes o he loss of dversy n he populaon. Gven a problem wh mulple soluons, he enre populaon wll soon converge o jus one of hese soluons. The reason for hs s he presence of randomness n he selecon process along wh a fxed populaon sze. Ths causes random genec drf [4, 12, 18 chaper 2]. Imagne a populaon conssng of 50% A and 50% B, each wh equal fness. Because of he randomness n he 7

8 selecon process, he populaon n he nex generaon may, for example, conss of 52% A and 48% B. In he nex generaon, he resamplng may cause he populaon o drf even furher from a balanced dsrbuon. I can be shown ha random genec drf always resuls n a homogeneous populaon, eher A or B, wh a me o convergence nversely proporonal o he populaon sze [12]. The resulng loss n dversy s undesrable for boh genec algorhms and parcle flers, snce he manenance of all poenal soluons s crucal o fnd he global opmum. As an llusraon of he problem of premaure convergence, consder he suaon n Mone Carlo localzaon as shown n fgure 1, where a loss of dversy resuls n a sub-opmal soluon. As can be seen, he envronmen s hghly symmerc excep for a dscrmnang objec n he lef hallway. In hs suaon, many of he robo s observaons could have been made n more han one place. In oher words, here are many ambguous suaons, and herefore mulple hypoheses of he robo s poson should be mananed. The sandard parcle fler s used o deermne he locaon of he robo. Inally, he parcles are randomly dsrbued. However, he dversy quckly dsappears as a consequence of random genec drf. In he frs suaon shown n he fgure, here s sll some dversy and a number of parcles are close o he acual locaon of he robo. In a shor me, however, hese parcles are los n favor of he oher ambguous locaon. As a consequence, he PF seles a a sub-opmal soluon and wll never be able o fnd he correc soluon. If he PF had mananed more dversy, all ambguous soluons would be mananed and he opmal soluon would be found once he robo s near he dsambguang objec. Ths example clearly shows ha premaure convergence of he parcle fler can cause he algorhm o become suck a a local opmum. Hence, s mporan o manan dversy boh n genec algorhms and parcle flers. The man cause of premaure convergence s ha soluons n dfferen nches (.e. peaks n he fness landscape) compee wh each oher for lmed resources (.e., a lmed number of places n he populaon). In he remander of hs secon we wll dscuss hree mehods from he genec 8

9 algorhm leraure ha ackle he problem of compeon beween dfferen nches. All hree mehods have a dfferen approach. One promoes compeon among ndvduals whn he same nche. The oher gves a fness dsadvanage o ypes of ndvduals ha are frequen n he populaon and an advanage o nfrequen ypes. And he las one uses a dynamcally changng populaon sze n conjuncon wh prevenon of compeon beween dfferen nches Nchng Mehods A number of soluons o he problem of premaure convergence exs n he feld of genec algorhms. These are called nchng mehods [20]. The role of nchng mehods s o fnd and manan mulple soluons durng he whole search process, even f some of hese soluons have lower fness han ohers. Nchng mehods resul n selecon pressure whn a regon (nche), bu no beween dfferen regons. In hs paper we wll dscuss and mplemen hree nchng mehods,.e., crowdng, fness sharng (as well as s varan frequency dependen selecon), and local selecon. The frs wo mehods are ofen used n genec algorhms and domnae he leraure. We ncluded he less well-known echnque of local selecon, because we beleve ha s especally sued for Mone Carlo localzaon Crowdng The basc prncple of all crowdng models s ha new ndvduals ener he populaon by replacng he mos smlar ndvdual. Crowdng models are so called seady-sae GAs, meanng ha only a subse of he populaon reproduces a a me, nsead of he whole populaon a once. De Jong s crowdng facor model was he frs model ha naed hs branch of nchng algorhms [4]. In hs model a proporon, called he generaon gap, of he populaon s seleced for reproducon va fness proporonae selecon. Muaon and possbly crossover are used o generae offsprng. For each offsprng, a proporon, cf, of he populaon, he crowdng facor, s 9

10 randomly sampled. The offsprng replaces he mos smlar ndvdual n he sample, where he smlary can be defned eher by he dsance beween he genoypes or he dsance beween he phenoypes. The model s ecologcally nspred. Smlar ndvduals n a naural populaon usually lve n he same envronmenal nche, and hereby compee wh each oher for lmed resources. Dssmlar ndvduals, e.g. dfferen speces, on he oher hand, lve n dfferen nches and do no compee for he same resources. To ge an nuve dea why he algorhm promoes dversy, consder he followng suaon: We have a populaon wh many ndvduals n nche A and jus a few n nche B. In he reproducon sep, a sample of he sze of he crowdng facor s aken for every ndvdual ha reproduces. Snce ndvduals of ype A are more frequen n he populaon, he sample wll conan manly ndvduals of ype A. There s even a far chance ha for a reproducng B, he sample consss of only As. In ha case, an ndvdual A s replaced by he ndvdual B, whle he reverse rarely happens. Ths process resores he balance n he populaon, hus mananng dversy. The crowdng algorhm urns ou o be capable of mananng mulple soluons, bu s of lmed use for fndng and opmzng mulple soluons. The man reason for hs s he fac ha old ndvduals are replaced by new ones rrespecve of he muual fness values. The only fness dependen selecon s up fron when reproducng ndvduals are seleced. Ths means ha f ndvduals are seleced for reproducon, bu hey ypcally wll replace smlar ndvduals, whch ofen are smlarly f. Ths resuls n f ndvduals replacng f ndvduals, hereby reducng he selecon pressure. To deal wh hs problem, Sedbrook, Wrgh & Wrgh, exended he sandard crowdng algorhm by selecng he crowdng facor sample from he wors par of he populaon [27]. Ths mehod, called closes-of-he-wors, only replaces low fness ndvduals, hereby ncreasng he selecon pressure. 10

11 Fness sharng Holland nroduced he concep of fness sharng[15]. The dea behnd s he fac ha each nche has an assocaed oal amoun of fness and ndvduals occupyng he same nche share hs amoun. If he number of ndvduals exceeds he carryng capacy of a nche, he ndvduals are beer off seekng less crowded nches. A populaon s sable f each nche conans a number of ndvduals proporonal o s oal amoun of fness. Fness sharng uses a form of frequency dependen selecon. Ths means ha he fness of a phenoype s dependen on s frequency n he populaon relave o oher phenoypes. Phenoypes ha are rare n he populaon have a fness advanage, whle common phenoypes have a dsadvanage. Ths mechansm s ofen found n naure. Consder, for nsance, a predaor ha can specalze n hunng one of wo prey ypes. I s hen mos advanageous for he predaor o specalze n hunng he mos common prey, hereby reducng he fness of ha prey and gvng a fness advanage o he rare ype. Ths ype of negave frequency dependen selecon resuls n he balancng of he populaon and he manenance of dversy. Goldberg and Rchardson [11] mplemened fness sharng by calculang he shared fness, f '( ), of ndvdual, as he normal fness value, f ( ), dvded by he nche coun: f ( ) f '( ) = (3.1) n s d j j= 1 ( (, )) In he above equaon, d(, j ) s he dsance beween ndvdual and j. Ths can be eher he genoypc or phenoypc dsance. The nche coun s he sum of all sharng funcon values, s( d ), beween ndvdual and all n members of he populaon. The funcon s( d ) s a monoonously decreasng funcon of he dsance d. Sharng s an effecve nchng mehod, whch s capable of fndng and mananng mulple soluons. The man drawback s he compuaonal cos of calculang he nche coun; he 2 mehod as s presened above has Θ ( n ) complexy. The compuaonal complexy can be 11

12 reduced by esmang he nche coun from a fxed szed sample of he populaon nsead of usng he enre populaon [11]. Ths resuls n Θ( k n) complexy, where k s he sze of he sample. In he exreme case, a sample sze of 1 can be used. Alhough hs opmzaon yelds good resuls for he requred amoun of compuaon, comes wh he cos of loosng some of he dversy mananng powers. Ths s caused by he nroducon of addonal sample varance, whch s a source for exra random genec drf. In fac, for every dversy preservng scheme, here s a radeoff beween compuaonal complexy and he dversy preservng capables. Anoher drawback of he sharng algorhm, as s presened, s he need o defne he nche radus, r. Pror knowledge abou he fness landscape s needed o deermne hs value. In our expermens, we wll exclude he nche radus from he equaons. Ths adapaon wll be dscussed n secon Local selecon The sandard genec algorhm and parcle fler have a fxed populaon sze. Ths s one of he causes of premaure convergence, snce nherenly makes ndvduals n dfferen nches compee for he lmed avalable posons n he populaon. The above-menoned nchng mehods compensae for hs phenomenon n dfferen ways, bu keep he populaon sze fxed. The local selecon algorhm [21], on he oher hand, has a flexble populaon sze. Over me, an ndvdual accumulaes fness. The rae of accumulaon depends on he ndvdual s qualy and on he number of nearby oher ndvduals. The accumulaed fness s compared o a hreshold o decde wheher he ndvdual wll reproduce or de. The fxed hreshold and he flexble populaon sze elmnae he nheren compeon beween ndvduals of dfferen nches. In hs way, local selecon mnmzes he nfluence of premaure convergence. Menczer, Degerau & Sree mplemen hs as he evoluonary local selecon algorhm (ELSA) [21]. In hs algorhm, each ndvdual accumulaes energy over me. Each me sep, explores a canddae soluon, ', smlar o self. The ndvdual looses a consan amoun of 12

13 energy, E ou. Furhermore, collecs energy from he envronmen, E n, proporonal o he fness of canddae soluon '. Indvduals n he same neghborhood have o share her fness. To do hs n an effcen way, he envronmen s subdvded no separae bns of equal sze. Each me sep, he number of ndvduals per bn s couned. The energy consumpon of he ndvdual s hen se o s fness dvded by he number of ndvduals n he same bn. When he oal amoun of energy accumulaed exceeds a hreshold θ, he ndvdual reproduces, meanng ha a new copy of canddae soluon, ' eners he populaon. If he energy level drops below zero, he ndvdual des. Oherwse, he ndvdual says n he populaon. I s neresng o noe ha smlar algorhms are used n he feld of ndvdual-based modelng, for nsance n he Sugarscape smulaon [8]. Local selecon changes he populaon sze dynamcally. The sze depends on he carryng capacy of he fness landscape, as he number of ndvduals per soluon s proporonal o he fness of ha soluon. An adapve populaon sze mgh be a useful propery for Mone Carlo localzaon, because MCL has o deal wh dynamc fness landscapes, snce he poson of he robo consanly changes. Dependng on he complexy of he suaon, more or fewer members n he populaon are needed. Dynamc populaon sze n Mone Carlo localzaon s also suded by Fox [9]. The populaon sze and he brh and deah raes n he populaon are conrolled by wo parameers, θ and E ou. Boh parameers ogeher nfluence he populaon sze. The rae of generang new ndvduals, s se by θ. E ou conrols he rae wh whch ndvduals de. I s mporan o se hese parameers correcly, such ha Mone Carlo localzaon can adap properly o changng envronmens. In conras wh he oher menoned nchng mehods, local selecon does no selec ndvduals for reproducon by comparng her fness wh he res of he populaon n a sochasc selecon process. Insead, reproducon and deah of an ndvdual only depend on he 13

14 Fgure 2. The map wh square symmery used n he expermens. ndvdual s local envronmen. Indvduals survve as long as he envronmen s capable of supporng hem. Therefore, hs nchng mehod s less prone o random genec drf. Furhermore, he algorhm has a compuaonal complexy of Θ ( n). However, snce n s adapve, expermens need o show wheher hs really s a compuaonal mprovemen over he oher nchng mehods. A dsadvanage of local selecon s he fac ha parameers θ and need o be se n order o conrol he magnude of he populaon sze and he brh and deah rae. E ou In hs subsecon, we have descrbed hree nchng mehods ha are used n genec algorhms. The frs wo mehods, crowdng and sharng, have proven o be successful nchng mehods n many sudes. The dsadvanage s ha hey have hgher compuaonal complexy. The oher nroduced mehod, local selecon, has been less well suded, bu has he advanage of lnear complexy wh respec o he populaon sze. Moreover, local selecon has he propery of beng able o adap he populaon sze o he suaon, whch mgh be profable for Mone Carlo localzaon. 14

15 4. Expermens We mplemened he nchng mehods gven n secon 3.2 and esed hem on her dversy mananng capables. We wll frs descrbe he robo smulaon ha we used. Then we wll descrbe he used algorhms. Fnally, we wll descrbe he expermens The Robo Smulaon The Mone Carlo localzaon expermens were run n a roboc smulaon. The map we used has square symmery (see fgure 2). In hs symmerc envronmen here are always four ambguous suaons for he robo. The sze of he map s 150 by 150. The robo n he smulaon has 16 dsance sensors, whose values are deermned by ray racng n he map, wh a maxmum dsance of 20. Gaussan nose s added o he sensor readngs wh sandard devaon of 1.0. Furhermore, he robo has dfferenal drve, wh whch can navgae hrough he envronmen. The robo provdes odomery nformaon, conssng of he ranslaonal and roaonal speed of he robo measured n uns per me sep and radans per me sep respecvely. There s Gaussan nose on he odomery, wh sandard devaon of 1.0 and 0.04 respecvely. The robo s conrolled by a smple obsacle avodance behavor, smlar o Braenberg Vehcle 3b, he Explorer [3]. The defaul ranslaonal speed s 8 uns per me sep. 15

16 A. The moon model B. Samplng from he moon model p(x +1 x,u ) u x C. The sensor model p(z k x ) z k* Fgure 3. Dagram A shows he dsrbuon of he moon model p( x + 1 x, u ). The dsrbuon has a banana shape, caused by he Gaussan dsrbuon on he ranslaon and roaon of he robo. Dagram B shows 1000 samples of hs dsrbuon. The sensor model s k shown n dagram C. The probably p( z x ) s defned by a Gaussan funcon wh he robo s sensor readng, z, as varable, he parcle s readng, z k k*, as mean and 2 σ s as varance Algorhms In secon 3.2 we gave an overvew of four nchng mehods. We have mplemened he nchng mehods and some varaons on hese mehods for Mone Carlo Localzaon. We wll frs descrbe he moon and sensor model as well as he used resamplng scheme, whch all form he bass of he parcle flers. We wll hen descrbe he ndvdual nchng algorhms. 16

17 Table 1. Funcons for he moon model, he sensor model and sochasc unform samplng. Algorhm: SAMPLE_MOTION_MODEL( u, x ) rans ro u = u, u : robo s moon x = x, y, α : poson n polar coordnaes 2 1 δ sample_from( Ν( u rans, σ )) δ rans T sample_from( Ν( u, σ )) ro 2 2 ro γ 3 xˆ x + δ cos( α + δ ) rans ro 4 yˆ y + δ sn( α + δ ) rans 5 ˆ α α + δro 6 reurn x ˆ ˆ ˆ 1 x, y, α + = ro Algorhm: SENSOR_MODEL( z, x ) 1 q 0 * 2 z smulae_observaon( x) 3 for all sensors k k k* ( z ) 2 z σ 2 s 4 p e 2 2πσ s 5 q q p 6 end 7 reurn q Algorhm: SAMPLE( S, N ) Sochasc Unversal Samplng (Baker, 1987) {, 1,, } S = p = x w = n :populaon a me. 1 S c w 1 4 r rand(0, N ) 5 for n 1 o N 6 whle r > c c c + w 9 end S S p r r + N 1 11 end 12 reurn S Moon model, sensor model and resamplng 17

18 In secon 2, we descrbed he basc parcle fler. I consss of hree seps. Frs, he moon model reposons he parcles. Then, he sensor model deermnes he wegh of he parcles. And fnally, he se of parcles s resampled, or n oher words, some parcles are seleced for reproducon. In he followng paragraphs, he mplemenaon of hese seps s descrbed n deal. The pseudo-code of he algorhms s gven n able 1. Moon Model. Every me sep, he parcles are reposoned based on he acon of he robo. As we only wan o nvesgae he dynamcs of he parcle fler, we use a relavely smple model. In hs model, he new poson has a mean value equal o he sum of he old poson and he acon of he robo. Addonal Gaussan nose s added o he ranslaon and roaon of he robo. Ths resuls n a dsrbuon, P( x + 1 x, u ), smlar o ha depced n fgure 3A. The nose n he moon model reflecs he expeced nose of he robo s acon. I s, however, mporan ha he moon model overesmaes hs nose, so ha he parcle fler effecvely explores he search space. In he expermens we used sandard devaons of σ T = 2 and σ γ = 0.2 for ranslaon and roaon respecvely. The nex poson of he parcles s deermned by samplng from he moon model (see also fgure 3B). Sensor Model. The sensor model deermnes he wegh of each parcle by calculang P( z x ), he probably ha he robo makes observaon z a poson an esmae s made of he value of each sensor k f he robo were a locaon ( ) k* RT x. Usng ray racng, x. Ths s gven by z = f x. The wegh of each parcle s hen calculaed by he produc of he ndvdual lkelhoods, P z z, ha he robo s sensor value k k* ( ) k z resembles he parcle s sensor value k* z. Ths resemblance s calculaed usng he densy funcon of he Gaussan dsrbuon, wh k z as varable, k* z as mean and P z x = P z z = N z z σ k k* k k* 2 ( ) ( ) ( ;, s ) k sensors k sensors 2 σ s as he sensor model varance (see fgure 3C). Ths resuls n: (4.1) 18

19 The sensor model ha we descrbe here consss of a smple Gaussan dsrbuon for a sensor h. Thrun, Burgard & Fox. [28, chaper 5] descrbe a more elaborae sensor model whch beer models a real dsance sensor. Ths sophscaed sensor model, however, s unnecessarly complcaed for our purposes. Resamplng. Genec algorhms and parcle flers srongly depend on samplng he populaon. The loss of dversy s parly a consequence of he randomness n samplng. Furhermore, he speed of convergence s proporonal o he varance of he samplng mehod [20]. Therefore, we used a samplng mehod wh low varance hroughou hs paper. As a subsue for roulee wheel selecon, whch has a relavely hgh sample varance, we used sochasc unversal selecon (SUS), proposed by Baker [1]. SUS dffers from ordnary roulee wheel selecon n ha one does no choose a random locaon on he roulee wheel for every new selecon, bu nally chooses only one random pon on he roulee wheel, and hen moves around wh a number of fxed sze seps ha s equal o he number of selecons ha one wans o make. Boh selecon mehods have he same expeced values, bu he varance of SUS s reduced. The sochasc unversal selecon algorhm s gven n able 1. An addonal advanage of SUS s ha s complexy for a populaon of sze n s Θ ( n), whereas he complexy of roulee wheel selecon s 2 Θ ( n ). The presened moon and sensor model are used n all varans of he Mone Carlo localzaon algorhms dscussed below. The presened samplng mehod s used for resamplng n he sandard parcle fler, he sharng algorhm and boh frequency dependen selecon algorhms. Furhermore, s used for fness proporonal selecon n he crowdng and closesof-he-wors algorhms. 19

20 Table 2. Pseudocode of he sandard Parcle Fler and he addonal code for sharng and frequency dependen selecon. Algorhm Sandard parcle fler {, 1,, } S = p = x w = n : populaon a me. x : poson of parcle a me. w : wegh of parcle a me. u : moon of he robo a me. z : observaon of he robo a me. 1 S0 randomzed_populaon 2 repea 3 for all parcles SAMPLE_MOTION_MODEL( u, x ) 4 x w + 1 SENSOR_MODEL( z 1, x 1 6 end 7 S + 1 SAMPLE( S, n) 8 unl(fnshed) + + ) Algorhm Sharng 20% 5.1 w % n w j= 0 1 ds(,rand_parcle) Algorhm Frequency Dependen Selecon 20% % n w + 1 w + 1 ds(, rand_parcle) j= 0 Algorhm Frequency Dependen Selecon w 1 w 1 ds(,rand_parcle) Algorhm 1: Sandard Parcle Fler As a reference for he nchng mehods, we used he sandard parcle fler for Mone Carlo localzaon as descrbed n secon 2. Inally, he parcles are posoned n he map wh a unform random dsrbuon. The erave algorhm consss of hree seps. Frs, he new 20

21 Table 3. Pseudocode for crowdng and he addonal code for closes-of-he-wors. In he expermens, we used gg = 0.2 and cf = For he samplng n lne 3, we used sochasc unversal samplng. Crowdng 1 S0 randomzed_populaon 2 repea 3 for all parcles SAMPLE_MOTION_MODEL( u, x ) 4 x w + MOTION_MODEL( z 1, x 1 6 end 7 G SAMPLE( S, gg n) 8 for all parcles n G 9 C un_sample( S,sze cf n) 10 j argmn ( ds(, k) k C + + ) j 11 p p // replace j by 12 end 13 unl(end) Closes-of-he-Wors 9 Sˆ wors_parcles( S,sze n/3) 9.1 C unform_sample( Sˆ,sze cf n) poson of all parcles s deermned by samplng from he moon model, usng he odomery of he robo. Nex, he weghs of he parcles s deermnng accordng o he sensor model. Fnally, he se of parcles s resampled usng SUS. The pseudo-code of he sandard parcle fler s gven n able 2. The complexy s Θ ( n) Algorhm 2: Crowdng Where he sandard parcle fler uses sochasc unversal samplng for selecon and reproducon, he crowdng algorhm uses a raher dfferen approach. Afer he moon and sensor model are appled o all parcles, only a proporon of he populaon s seleced for reproducon. A proporon of gg parcles s seleced wh fness proporonae selecon. For 21

22 each seleced parcle, a proporon, he crowdng facor( cf ) of he populaon s unformly sampled. The seleced parcle replaces he parcle n he cf-sample wh he shores dsance n Eucldean space. In he expermens, we used a generaon gap of gg = 0.2 and a crowdng facor cf = The pseudo-code s gven n able 3. The complexy of crowdng s quadrac, 2 Θ( gg cf n ). However, snce gg and cf are small proporons of he populaon, s no very drasc for small populaon szes. In pracce, he algorhm was even faser hen he sandard parcle fler for he maxmum populaon sze ha we used ( n = 2500 ). Ths s caused by he fac ha only gg n copy funcons need o be appled every me sep Algorhm 3: Closes-of-he-Wors The sandard crowdng algorhm has he man dsadvanage ha has lle selecon pressure. The closes-of-he-wors algorhm provdes a soluon for hs. Insead of a random unform selecon of he cf-sample over he whole populaon, he cf-sample s aken from he wors hrd of he populaon. Ths ncreases he complexy of he algorhm, because he populaon needs o be sored on s weghs, addng n log n seps. The algorhm s gven n able Algorhm 4: Sharng Ths algorhm s an exenson of he sandard parcle fler. I nroduces he sharng of weghs beween parcles ha are close ogeher. In he orgnal sharng algorhm [11], a parameer defnng he nche radus, r, s used. Inspecon of he algorhm on some plo expermens showed he bes resul wh hgh values of r. Wh a small radus, many rrelevan clusers appear. Moreover, s desrable o reduce he number of free parameers n he models. We herefore decded o elmnae he nche radus and le he parcles share her weghs wh all oher parcles, bu more wh nearby parcle and less wh dsan parcles. Ths resuls n he followng sharng funcon: 22

23 1 s( d(, j)) = d(, j) (4.2) where d(, j ) s he dsance beween parcle and parcle j and s () s he sharng funcon. Ths resuls n a srong wegh sharng f he dsance s small and a weak sharng f he wo parcles are more dsan. Orgnally, he wegh of parcle s adjused by dvdng by he sum of s( d(, j )) over all parcles j n he populaon. To reduce he compuaonal coss, we choose o calculae he nche coun by summaon over a fracon χ of randomly seleced parcles, nsead of over all parcles: w wˆ = χ n s d j = 0 ( (,rand(0, n) )) (4.3) where w and w ˆ are, respecvely, he wegh and adjused wegh of parcle a me. In our expermens, we used χ = 0.2. Alhough hs reduces he compuaonal demands, he complexy s sll quadrac. Table 2 shows he code for sharng n addon o he sandard parcle fler Algorhm 5: Frequency Dependen Selecon The raonale behnd frequency dependen selecon s ha nfrequen parcles have a fness advanage, whereas frequen parcles do no have hs advanage. Sharng, as descrbed above, s a possble mplemenaon of frequency dependen selecon. However, can also be mplemened dfferenly. Insead of dvdng a wegh by he nche coun, he wegh can be recalculaed by mulplyng wh he dsance from o he res of he populaon. Lke n he sharng algorhm, we used he dsances owards a fracon χ of randomly seleced parcles o reduce he amoun of compuaon. We used χ = 0.2 n our expermens. Ths gves: χ n j= 0 ( ) w ˆ = w d,rand(0, n ) (4.4) The complexy of he algorhm, lke n sharng, remans quadrac, 2 Θ( χ n ). The code for frequency dependen selecon s gven n able 2. 23

24 Algorhm 6: Frequency Dependen Selecon, Sample Sze=1 In he prevous algorhm, he frequency of a parcle s deermned by akng 20 percen of he populaon no accoun. Alhough hs reduces he amoun of compuaon, he algorhm sll has a quadrac complexy. To reduce he complexy o Θ ( n), we need o measure frequency by samplng a fxed sze from he populaon. In hs algorhm, we chose o have a sample sze of one parcle. Ths means ha he wegh of he parcles s mulpled by he dsance owards one randomly seleced parcle. The algorhm s descrbed n able 2. We should menon ha a larger sample sze s expeced o perform beer. However, as s no our am o fnd he bes radeoff beween accuracy and compuaon, we chose o show he smples possble soluon for frequency dependen selecon. The sharng algorhm could be adjused n a smlar way o also have a sample sze of one. Alhough hs has no been esed, one can expec a smlar change n performance Algorhm 7: Local Selecon The las algorhm ha we used n our expermens, local selecon, s very dfferen from he oher algorhms. The above-descrbed algorhms all have a fxed populaon sze, whch causes nheren compeon beween parcles n dfferen nches. As dscussed before, hs s a source of premaure convergence. The nchng algorhms based upon crowdng and sharng, algorhms 2 6, use dfferen mehods o compensae for he nheren compeon, bu do no elmnae he source. In local selecon, on he oher hand, reproducon and deah only depend on he accumulaed fness of a parcle and local sharng of hs fness. As a consequence, he populaon sze adaps o he carryng capacy of he envronmen. 24

25 Table 4. Pseudocode for he local selecon algorhm. In he expermens, θ was he varable and E ou = 0.2 θ. The used value of E ou was derved from a number of plo expermens. Local Selecon {, 1,, } S = p = x E = n :populaon a me. E : energy of parcle a me. S0 randomzed_populaon for all parcles E θ repea for all parcles x + 1 SAMPLE_MOTION_MODEL( u, x ) w + 1 MOTION_MODEL( z 1, + x + 1) b ge_world_bn() wbn[ b] wbn[ b] + 1 end for all parcles E /wbn[ ] n w + 1 b ( ) E E + E E n ou f( E + 1 > θ ) ˆ copy( ) ˆ / E E E ˆ {, } S S p p else f( E + 1 > 0) S + 1 S + 1 p + 1 else // parcle des end end unl(fnshed) The pseudocode of local selecon for Mone Carlo localzaon s gven n able 4. As s necessary for local selecon, he world s dvded no bns. These bns are of sze 2 by 2 by 36º (he oal envronmen s º). In he frs sweep hrough he populaon, he moon and sensor model are appled o all parcles. A he same me, he bns are flled. In he second 25

26 sweep, he energy of parcles s calculaed. The accumulaed energy, E, s ncreased by he wegh of he parcle, whch s he lkelhood ha represens he robo s poson, dvded by he bn coun, E n. The accumulaed energy s furhermore decreased by a fxed amoun, E ou. Dependng on he energy of a parcle, eher reproduces (f E > θ ), des (f E < 0 ) or connues lvng (oherwse). If he parcle reproduces, an exac copy s added o he populaon. Boh parcles share he fness of he paren. The complexy of he algorhm s lnear n he number of parcles Θ ( n), bu wo sweeps hrough he parcle populaon are needed, one for fllng he world bns and one for reproducon. We changed he orgnal ELSA algorhm [21] wh respec o wo pons. Frsly, Menczer e al. propose o use a fxed amoun of energy ha eners he envronmen every me sep. Ths flls an energy reservor. If he reservor s empy, he members of he populaon canno accumulae more energy. Ths energy replenshmen conrols he maxmum sze of he populaon. However, we would lke he maxmum populaon sze emerge from he fness landscape. We herefore om he lmed energy replenshmen. Noe ha hs does no mean ha he populaon sze can grow ndefnely. Secondly, we supply he members of he nal populaon wh θ (he hreshold for mulplcaon) energy nsead of half of hs amoun, as used n [21]. The reason for hs s ha gves he nal populaon a b more me o explore he envronmen. If a parcle does no gan any energy self, wll de n 1 2θ Eou me seps n he orgnal algorhm. Wh our sengs, hs resul n a lfe expecancy of 2.5 me seps. Ths gves he parcles oo lle me o explore he envronmen, hereby ncreasng he probably ha none of he parcles near a poenal robo poson fnds hs poson before sarves. An nal lfe expecancy wce as bg yelds beer resuls. The number of parcles ha s mananed depends on he fness landscape (whch self depends on he poson of he robo n he envronmen), and on he parameers θ and E ou (see secon 3.2). Afer some nal expermenal runs, urned ou ha he bes resuls were acheved 26

27 Table 5. The carryng capacy usng he local selecon algorhm wh dfferen values of θ. θ n when E depended on θ. We had good resuls wh E = 0.2 θ. A smaller fracon resuls n ou ou oo many parcles, whch makes nfeasble o acheve real-me performance. Hgher fracons cause he parcles o dsappear quckly, resulng n a populaon ha s oo small. Alogeher, hs resuls n an algorhm wh θ as he only varable o conrol he populaon sze Expermenal Seup Dversy Manenance The goal of he expermens s o show ha soluons for premaure convergence n genec algorhms can successfully be appled o Mone Carlo localzaon. Moreover, we would lke o measure he performance of he nchng algorhms n erms of her ably o manan dversy. Ths s he reason why he envronmen has square symmery. I provdes four ambguous suaons o he robo. The ably of he nchng algorhms o manan dversy s measured by he number of meseps (wh a maxmum of 500) over whch hey manan all four possble soluons. The performance s measured as a funcon of he number of hypoheses. For algorhm 1 6, he performance s measured for 100, 200, 500, 1000, 1500, 2000 and 2500 parcles. For algorhm 7, he suaon s a b more complcaed, snce he populaon sze s no consan. I s, however, possble o conrol he populaon sze by he hreshold θ, as dscussed n secon 4.2. To es he carryng capacy for dfferen hresholds, we ran expermens wh large numbers of nal parcles. The reason o use a large number of nal parcles s ha he carryng capacy s srongly nfluenced f one of he possble soluons s no covered n he nal phase. The 27

28 carryng capaces for dfferen hresholds are gven n able 5. In he man expermens, we used hese hresholds, and an nal number of parcles equal o he assocaed carryng capacy. We consder ha a soluon s mananed by he parcle fler f a leas one parcle s n s vcny, where vcny s defned as a sphere wh radus 10 cenered on he locaon and orenaon of ha soluon, as calculaed usng he real poson of he smulaed robo. To be able o compare locaon and orenaon, a dfference of 180 n orenaon s consdered commensurae o a dsance n locaon of 50. Two performance measures are used: he percenage of successful runs, and he me o premaure convergence. A successful run s a run where he parcle populaon manans all four ambguous soluons hroughou he whole perod of a run (500 cycles). The me o premaure convergence s he me, measured n cycles, when he parcle fler loses one or more of he four poenal soluons. Each run s ermnaed afer 500 cycles. If no premaure convergence has occurred, he me o convergence s se o 500. A oal of 100 runs s used o es all algorhms for each populaon sze. To esmae varaon n he percenage of successful runs, he 100 runs are spl up n 10 chunks of 10 runs, over whch he varance s calculaed Compacness of he Parcle Subpopulaons Besdes he performance measures for dversy preservaon, we measured he compacness of he parcle subpopulaons. The compacness s an neresng measure, snce ells us somehng abou he precson of he nchng mehods. More compac subpopulaons wll resul n more precse esmaons of he locaon of he robo. Consder for nsance a nchng mehod ha would smply unformly place parcles on he map. Tha mehod would manan all possble soluons, bu s oal lack of compacness would make mpossble o correcly esmae he robos poson. More compac subpopulaons are preferred. We used wo measures for compacness. Frs, he proporon of all parcles ha are whn he vcny, ρ = 10, of he four opma. The dsance from each parcle o he neares opmum s calculaed by he Eucldean 28

29 Fgure 4. The non-ambguous map used n expermens C. dsance n hree dmensons, he x- and y- values of he locaon as well as he orenaon. The proporon of parcles whn he vcny gves a measure of he compacness of he subpopulaons. The second measure s he mean sum of squared errors (MSSE), where he error s aken as he dsance owards he neares opmum. Ths gves he varance of he parcles n he subpopulaons, whch s nversely relaed o he compacness of he subpopulaons Esmaon Accuracy Fnally, we esed he dfferen nchng mehods on her accuracy n esmang he robo s poson. Alhough hs research focuses on he ably of he dfferen algorhms o manan dversy, hs should no conflc wh he power o correcly esmae he poson of he robo. We esed he esmaon performance on he symmercal map ha s used n he prevous expermens, as well as on a non-symmercal map (see fgure 4). The laer does no provde any long-lasng ambguous suaons for he robo. To esmae he poson of he robo, we used he kernel densy esmaon [7]. Ths mehod esmaes he poson of he robo by fndng he hghes densy of parcles. 29

30 successful runs (%) A me o convergence (cycles) B nr of hypoheses nr of hypoheses Fgure 5. (A) The percenage of runs whou convergence ploed agans he number of hypoheses. The error bars gve he 95% confdence nervals. Means and confdences are calculaed from 10 ses of 10 runs. The number of hypoheses for he local selecon algorhm s varable and depends on θ. The correspondng horzonal error bars show he 95% confdence nerval for he average number of hypoheses per run. (B) The me o convergence as a funcon of he populaon sze. A run s consdered converged f one or more of he four opma are los. The run has a maxmum lengh of 500 cycles. Nonconverged runs receve a me o convergence of 500. The error bars gve he 95% confdence nervals. The daa comes from a oal of 100 runs. 5. Resuls Ths secon gves he resuls of he expermens. In he frs hree subsecons, he resuls of he ndvdual expermens are gven. The secon ends wh a summary of all he resuls Dversy Manenance The resuls of he expermens are shown n fgure 5. (A) shows he percenage of successful runs ploed agans he number of parcles. (B) gves he me o convergence ploed agans he number of parcles. The error bars show he 95% confdence nervals. As dscussed n he 30

31 prevous secon, he graph of he local selecon algorhm dffers a b from he ohers, caused by he nably o drecly conrol he number of hypoheses. Each pon n he local selecon graph corresponds o a value for θ, as gven n able 5. The x-values of he pons are he means for he average number of hypoheses per run. Whn a run, he populaon sze flucuaes, dependng on he curren suaon of he robo.. In our expermens, he populaon sze ncreases when he robo s n he mddle of a hallway, because locaons n fron and behnd he robo gve smlar sensor readngs, whch flaens he fness landscape. On he oher hand, he number of parcles decreases when he robo s near a corner. In our expermens, hs flucuaon per run was approxmaely ±40%. We are, however, only neresed n he average number of parcles, whch concdes wh he average compuaonal load. The horzonal error bars of he local selecon graph n fgures 5A and 5B show ha he average populaon sze over he runs s que sable. All graphs n boh plos show an ncrease n performance wh he number of hypoheses. Ths s expeced, snce he me o premaure convergence s exended wh larger populaon sze. For small numbers of hypoheses, all algorhms perform badly. Ths reflecs he fac ha here needs o be a mnmal number of parcles o fnd and manan he four opma. Ths s especally acue n he nal phase, when all parcles are randomly placed on he map and enough parcles mus be presen such ha a leas one parcle s near each opmum. As can be seen n boh plos, he dversy preservng ably of he sandard parcle fler s very lmed. Even for 2500 parcles, he algorhm s successful n mananng he dversy n only 6% of he runs, wh an average me o convergence of 183 cycles. Even he smples and compuaonally cheapes nchng mehod, frequency dependen selecon wh a sample sze of 1, sgnfcanly ouperforms he sandard algorhm. However, hs algorhm does no acheve more han 37% success on average, wh a me o convergence of 303 cycles for he maxmum populaon sze. The wo sharng algorhms wh equal complexy, sandard sharng and frequency dependen selecon, boh wh a sample sze of 20%, perform equally well. Ther performance n mananng dversy s smlar o ha of local selecon. Among hese hree 31

32 algorhms, no sgnfcan dfferences are found. Wh he maxmum populaon sze, he algorhms have an average success of 96%, 97% and 92% and a me o convergence of 480, 485 and 461 cycles respecvely. I mus be kep n mnd ha he maxmum populaon sze of local selecon s on average 2258 n conras wh 2500 for all oher nchng mehods. The bes performng algorhms are he wo crowdng algorhms, sandard crowdng and closes-of-hewors. For populaon szes of 1000 parcles and more, boh algorhms ouperform he oher algorhms. Boh algorhms show a sgnfcan dfference, usng he -es wh p < 0.05, wh all oher algorhms for 1500 parcles. Wh more parcles, he performance s no sgnfcanly beer hen he oher algorhms, bu hs s caused by he fac ha he performance s close o wha s maxmally possble, whch nherenly flaens he curves. The dfference would reman f a larger maxmal number of runs was chosen. In he case of 2500 parcles, boh algorhms have a perfec score. In 100% of he runs, he dversy s mananed over all 500 cycles. 32

33 proporon of nearby parcle A Sandard PF Freq.dep 20% Freq.dep 1 Crowdng Cl. o/ wors Sharng Local sel. msse B Crowdng Cl. o/ wors Sandard PF Sharng Freq.dep 20% Freq.dep 1 Local sel cycle cycle Fgure 6. Compacness of he parcle subpopulaons. Expermens are performed wh 2500 parcles, and θ = 0.35 for local selecon. (A) Proporon of parcles ha s whn he radus ρ = 10 of he four possble soluons. Ths s a measure of he compacness of a subpopulaon. Afer an nal phase, he graphs show perodc behavor. Ths s a resul of he shape of he robo s envronmen. When he robo approaches a corner, he subpopulaons ge more compac, whle hey expand n he corrdors. (B) The mean sum of squared errors (MSSE). Ths shows he varance whn a subpopulaon and s nversely proporonal o he compacness of he populaon. The wo crowdng algorhms show a consan ncrease n he MSSE. The oher algorhms reman sable afer he nal phase Compacness of he Parcle Subpopulaons The compuaonal complexy of he wo sharng algorhms, sandard sharng and frequency dependen selecon, s equal when a sample sze of 20% s used. Boh algorhms also show smlar performance n mananng he dversy. However, fgure 6 reveals ha boh algorhms have a dfferen compacness of he subpopulaons. Boh measures of compacness show ha he subpopulaons are more compac wh frequency dependen selecon han wh sandard sharng. The compacness wh frequency dependen selecon s smlar o he sandard parcle fler, whereas sharng resuls n sgnfcanly less compac subpopulaons. The dfference n 33

34 compacness can be explaned by he dfference beween equaon (4.4), used by frequency dependen selecon, and (4.3), used by sharng. The laer demoes parcles ha are close o oher parcles more han he former. Ths resuls n less compac parcle populaons. 34

35 error (relave o random) error (relave o random) 1.0 A sandard PF freq.dep. 20% freq.dep. 1 cl. o/ wors crowdng local selecon sharng sandard PF cl. o/ wors local selecon freq.dep. 1 sharng crowdng freq.dep. 20% Fgure 7. The esmaon error, usng kernel densy esmaon, of he dfferen algorhms. The error s relave o he expeced error wh a random esmaon. Expermens are performed wh 2500 parcles. For local selecon, θ = (A) shows he esmaon error n he ambguous envronmen, where he error s measured by he dsance from he esmaon o he neares of he four possble robo posons. (B) shows he esmaon error n a non-ambguous envronmen. Here, he error s he dsance from he esmaon o he robo s poson. The numbers above he boxes show he number of successful runs. An unsuccessful run s a run where here are less han 10 parcles whn he vcny of he robo ( ρ = 10 ) over an exended perod of me. All unsuccessful runs are caused by he fac ha, by chance, he poson of he robo s no covered by he random nalzaon of parcles. Snce hs s an nalzaon problem, and no an esmaon problem, only successful runs are consdered n he error esmaon. Fgure 5 shows ha local selecon has he bes performance n mananng dversy of all algorhms wh lnear complexy. Is performance s comparable o ha of sandard sharng and frequency dependen selecon wh 20% sample sze, whch boh have quadrac complexy. The 35