Improved Techniques for Grid Mapping with RaoBlackwellized Particle Filters


 Horatio Reed
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1 1 Improved Techniques for Grid Mapping wih RaoBlackwellized Paricle Filers Giorgio Grisei Cyrill Sachniss Wolfram Burgard Universiy of Freiburg, Dep. of Compuer Science, GeorgesKöhlerAllee 79, D Freiburg, Germany Diparimeno Informaica e Sisemisica, Universiá La Sapienza, Via Salaria 113, I Rome, Ialy Eidgenössische Technische Hochschule Zurich (ETH), IRIS, ASL, 8092 Zurich, Swizerland Absrac Recenly, RaoBlackwellized paricle filers have been inroduced as an effecive means o solve he simulaneous localizaion and mapping problem. This approach uses a paricle filer in which each paricle carries an individual map of he environmen. Accordingly, a key quesion is how o reduce he number of paricles. In his paper, we presen adapive echniques for reducing his number in a RaoBlackwellized paricle filer for learning grid maps. We propose an approach o compue an accurae proposal disribuion aking ino accoun no only he movemen of he robo bu also he mos recen observaion. This drasically decreases he uncerainy abou he robo s pose in he predicion sep of he filer. Furhermore, we presen an approach o selecively carry ou resampling operaions which seriously reduces he problem of paricle depleion. Experimenal resuls carried ou wih real mobile robos in largescale indoor as well as in oudoor environmens illusrae he advanages of our mehods over previous approaches. Index Terms SLAM, RaoBlackwellized paricle filer, adapive resampling, moionmodel, improved proposal I. INTRODUCTION Building maps is one of he fundamenal asks of mobile robos. In he lieraure, he mobile robo mapping problem is ofen referred o as he simulaneous localizaion and mapping (SLAM) problem [4, 6, 9, 15, 16, 26, 29, 32, 39]. I is considered o be a complex problem, because for localizaion a robo needs a consisen map and for acquiring a map a robo requires a good esimae of is locaion. This muual dependency beween he pose and he map esimaes makes he SLAM problem hard and requires searching for a soluion in a highdimensional space. Murphy, Douce, and colleagues [6, 32] inroduced Rao Blackwellized paricle filers as an effecive means o solve he simulaneous localizaion and mapping problem. The main problem of he RaoBlackwellized approaches is heir complexiy measured in erms of he number of paricles required o build an accurae map. Therefore, reducing his quaniy is one of he major challenges for his family of algorihms. Addiionally, he resampling sep can poenially eliminae he correc paricle. This effec is also known as he paricle depleion problem or as paricle impoverishmen [44]. In his work, we presen wo approaches o subsanially increase he performance of RaoBlackwellized paricle filers applied o solve he SLAM problem wih grid maps: A proposal disribuion ha considers he accuracy of he robo s sensors and allows us o draw paricles in a highly accurae manner. An adapive resampling echnique which mainains a reasonable variey of paricles and in his way enables he algorihm o learn an accurae map while reducing he risk of paricle depleion. The proposal disribuion is compued by evaluaing he likelihood around a paricledependen mos likely pose obained by a scanmaching procedure combined wih odomery informaion. In his way, he mos recen sensor observaion is aken ino accoun for creaing he nex generaion of paricles. This allows us o esimae he sae of he robo according o a more informed (and hus more accurae) model han he one obained based only on he odomery informaion. The use of his refined model has wo effecs. The map is more accurae since he curren observaion is incorporaed ino he individual maps afer considering is effec on he pose of he robo. This significanly reduces he esimaion error so ha less paricles are required o represen he poserior. The second approach, he adapive resampling sraegy, allows us o perform a resampling sep only when needed and in his way keeping a reasonable paricle diversiy. This resuls in a significanly reduced risk of paricle depleion. The work presened in his paper is an exension of our previous work [14] as i furher opimizes he proposal disribuion o even more accuraely draw he nex generaion of paricles. Furhermore, we added a complexiy analysis, a formal descripion of he used echniques, and provide more deailed experimens in his paper. Our approach has been validaed by a se of sysemaic experimens in largescale indoor and oudoor environmens. In all experimens, our approach generaed highly accurae meric maps. Addiionally, he number of he required paricles is one order of magniude lower han wih previous approaches. This paper is organized as follows. Afer explaining how a RaoBlackwellized filer can be used in general o solve he SLAM problem, we describe our approach in Secion III. We hen provide implemenaion deails in Secion IV. Experimens carried ou on real robos are presened in Secion VI. Finally, Secion VII discusses relaed approaches. II. MAPPING WITH RAOBLACKWELLIZED PARTICLE FILTERS According o Murphy [32], he key idea of he Rao Blackwellized paricle filer for SLAM is o esimae he join poserior p(x 1:, m z 1:, u 1: 1 ) abou he map m and he
2 2 rajecory x 1: = x 1,..., x of he robo. This esimaion is performed given he observaions z 1: = z 1,...,z and he odomery measuremens u 1: 1 = u 1,..., u 1 obained by he mobile robo. The RaoBlackwellized paricle filer for SLAM makes use of he following facorizaion p(x 1:, m z 1:, u 1: 1 ) = p(m x 1:, z 1: ) p(x 1: z 1:, u 1: 1 ). (1) This facorizaion allows us o firs esimae only he rajecory of he robo and hen o compue he map given ha rajecory. Since he map srongly depends on he pose esimae of he robo, his approach offers an efficien compuaion. This echnique is ofen referred o as RaoBlackwellizaion. Typically, Eq. (1) can be calculaed efficienly since he poserior over maps p(m x 1:, z 1: ) can be compued analyically using mapping wih known poses [31] since x 1: and z 1: are known. To esimae he poserior p(x 1: z 1:, u 1: 1 ) over he poenial rajecories, one can apply a paricle filer. Each paricle represens a poenial rajecory of he robo. Furhermore, an individual map is associaed wih each sample. The maps are buil from he observaions and he rajecory represened by he corresponding paricle. One of he mos common paricle filering algorihms is he sampling imporance resampling (SIR) filer. A Rao Blackwellized SIR filer for mapping incremenally processes he sensor observaions and he odomery readings as hey are available. I updaes he se of samples ha represens he poserior abou he map and he rajecory of he vehicle. The process can be summarized by he following four seps: 1) Sampling: The nex generaion of paricles {x (i) } is obained from he generaion {x (i) 1 } by sampling from he proposal disribuion π. Ofen, a probabilisic odomery moion model is used as he proposal disribuion. 2) Imporance Weighing: An individual imporance weigh is assigned o each paricle according o he imporance sampling principle = p(x(i) 1: z 1:, u 1: 1 ) π(x (i) 1: z 1:, u 1: 1 ). (2) The weighs accoun for he fac ha he proposal disribuion π is in general no equal o he arge disribuion of successor saes. 3) Resampling: Paricles are drawn wih replacemen proporional o heir imporance weigh. This sep is necessary since only a finie number of paricles is used o approximae a coninuous disribuion. Furhermore, resampling allows us o apply a paricle filer in siuaions in which he arge disribuion differs from he proposal. Afer resampling, all he paricles have he same weigh. 4) Map Esimaion: For each paricle, he corresponding map esimae p(m (i) x (i) 1:, z 1:) is compued based on he rajecory x (i) 1: of ha sample and he hisory of observaions z 1:. The implemenaion of his schema requires o evaluae he weighs of he rajecories from scrach whenever a new observaion is available. Since he lengh of he rajecory increases over ime, his procedure would lead o an obviously inefficien algorihm. According o Douce e al. [7], we obain a recursive formulaion o compue he imporance weighs by resricing he proposal π o fulfill he following assumpion π(x 1: z 1:, u 1: 1 ) = π(x x 1: 1, z 1:, u 1: 1 ) π(x 1: 1 z 1: 1, u 1: 2 ). (3) Based on Eq. (2) and (3), he weighs are compued as = p(x(i) π(x (i) 1: z 1:, u 1: 1 ) 1: z 1:, u 1: 1 ) = ηp(z x (i) 1:, z 1: 1)p(x (i) x (i) 1, u 1) π(x (i) x (i) 1: 1, z 1:, u 1: 1 ) p(x(i) 1: 1 z 1: 1, u 1: 2 ) π(x (i) 1: 1 z 1: 1, u 1: 2 ) } {{ } 1 (4) (5) p(z m (i) 1, x(i) )p(x (i) x (i) 1, u 1) π(x x (i) 1: 1, z 1 1:, u 1: 1 ).(6) Here η = 1/p(z z 1: 1, u 1: 1 ) is a normalizaion facor resuling from Bayes rule ha is equal for all paricles. Mos of he exising paricle filer applicaions rely on he recursive srucure of Eq. (6). Whereas he generic algorihm specifies a framework ha can be used for learning maps, i leaves open how he proposal disribuion should be compued and when he resampling sep should be carried ou. Throughou he remainder of his paper, we describe a echnique ha compues an accurae proposal disribuion and ha adapively performs resampling. III. RBPF WITH IMPROVED PROPOSALS AND ADAPTIVE RESAMPLING In he lieraure, several mehods for compuing improved proposal disribuions and for reducing he risk of paricle depleion have been proposed [7, 30, 35]. Our approach applies wo conceps ha have previously been idenified as key prerequisies for efficien paricle filer implemenaions (see Douce e al. [7]), namely he compuaion of an improved proposal disribuion and an adapive resampling echnique. A. On he Improved Proposal Disribuion As described in Secion II, one needs o draw samples from a proposal disribuion π in he predicion sep in order o obain he nex generaion of paricles. Inuiively, he beer he proposal disribuion approximaes he arge disribuion, he beer is he performance of he filer. For insance, if we were able o direcly draw samples from he arge disribuion, he imporance weighs would become equal for all paricles and he resampling sep would no longer be needed. Unforunaely, in he conex of SLAM a closed form of his poserior is no available in general. As a resul, ypical paricle filer applicaions [3, 29] use he odomery moion model as he proposal disribuion. This moion model has he advanage ha i is easy o compue for
3 3 likelihood }{{} L (i) robo posiion p(z x) p(x x,u) Fig. 1. The wo componens of he moion model. Wihin he inerval L (i) he produc of boh funcions is dominaed by he observaion likelihood in case an accurae sensor is used. mos ypes of robos. Furhermore, he imporance weighs are hen compued according o he observaion model p(z m, x ). This becomes clear by replacing π in Eq. (6) by he moion model p(x x 1, u 1 ) = ηp(z m (i) 1, x(i) )p(x (i) x (i) 1, u 1) 1 p(x (i) x (i) 1, u 1) (7) 1 p(z m (i) 1, x(i) ). (8) This proposal disribuion, however, is subopimal especially when he sensor informaion is significanly more precise han he moion esimae of he robo based on he odomery, which is ypically he case if a robo equipped wih a laser range finder (e.g., wih a SICK LMS). Figure 1 illusraes a siuaion in which he meaningful area of he observaion likelihood is subsanially smaller han he meaningful area of he moion model. When using he odomery model as he proposal disribuion in such a case, he imporance weighs of he individual samples can differ significanly from each oher since only a fracion of he drawn samples cover he regions of sae space ha have a high likelihood under he observaion model (area L (i) in Figure 1). As a resul, one needs a comparably high number of samples o sufficienly cover he regions wih high observaion likelihood. A common approach especially in localizaion is o use a smoohed likelihood funcion, which avoids ha paricles close o he meaningful area ge a oo low imporance weigh. However, his approach discards useful informaion gahered by he sensor and, a leas o our experience, ofen leads o less accurae maps in he SLAM conex. To overcome his problem, one can consider he mos recen sensor observaion z when generaing he nex generaion of samples. By inegraing z ino he proposal one can focus he sampling on he meaningful regions of he observaion likelihood. According o Douce [5], he disribuion p(x m (i) 1, x(i) 1, z, u 1 ) = p(z m (i) 1, x )p(x x (i) 1, u 1) p(z m (i) 1, x(i) 1, u 1) is he opimal proposal disribuion wih respec o he variance of he paricle weighs. Using ha proposal, he compuaion (9) of he weighs urns ino = ηp(z m (i) 1, x(i) )p(x (i) x (i) 1, u 1) 1 p(x m (i) 1, x(i) 1, z (10), u 1 ) 1 p(z m (i) 1, x(i) )p(x (i) x (i) 1, u 1) p(z m (i) 1,x)p(x x(i) 1,u 1) p(z m (i) 1,x(i) 1,u 1) (11) = 1 p(z m (i) 1, x(i) 1, u 1) (12) = 1 p(z x )p(x x (i) 1, u 1) dx. (13) When modeling a mobile robo equipped wih an accurae sensor like, e.g., a laser range finder, i is convenien o use such an improved proposal since he accuracy of he laser range finder leads o exremely peaked likelihood funcions. In he conex of landmarkbased SLAM, Monemerlo e al. [26] presened a RaoBlackwellized paricle filer ha uses a Gaussian approximaion of he improved proposal. This Gaussian is compued for each paricle using a Kalman filer ha esimaes he pose of he robo. This approach can be used when he map is represened by a se of feaures and if he error affecing he feaure deecion is assumed o be Gaussian. In his work, we ransfer he idea of compuing an improved proposal o he siuaion in which dense grid maps are used insead of landmarkbased represenaions. B. Efficien Compuaion of he Improved Proposal When modeling he environmen wih grid maps, a closed form approximaion of an informed proposal is no direcly available due o he unpredicable shape of he observaion likelihood funcion. In heory, an approximaed form of he informed proposal can be obained using he adaped paricle filer [35]. In his framework, he proposal for each paricle is consruced by compuing a sampled esimae of he opimal proposal given in Eq. (9). In he SLAM conex, one would firs have o sample a se of poenial poses x j of he robo from he moion model p(x x (i) 1, u 1). In a second sep, hese samples need o be weighed by he observaion likelihood o obain an approximaion of he opimal proposal. However, if he observaion likelihood is peaked he number of pose samples x j ha has o be sampled from he moion model is high, since a dense sampling is needed for sufficienly capuring he ypically small areas of high likelihood. This resuls in a similar problem han using he moion model as he proposal: a high number of samples is needed o sufficienly cover he meaningful region of he disribuion. One of our observaions is ha in he majoriy of cases he arge disribuion has only a limied number of maxima and i mosly has only a single one. This allows us o sample posiions x j covering only he area surrounding hese maxima. Ignoring he less meaningful regions of he disribuion saves a significan amoun of compuaional resources since i requires less samples. In he previous version of his work [14], we approximaed p(x x (i) 1, u 1) by a consan k wihin he inerval L (i) (see also Figure 1) given by L (i) = { x p(z m (i) 1 }., x) > ǫ (14)
4 4 = 1 η(i). (19) Noe ha η (i) is he same normalizaion facor ha is used in he compuaion of he Gaussian approximaion of he proposal in Eq. (17). (a) (b) (c) Fig. 2. Paricle disribuions ypically observed during mapping. In an open corridor, he paricles disribue along he corridor (a). In a dead end corridor, he uncerainy is small in all dimensions (b). Such poseriors are obained because we expliciely ake ino accoun he mos recen observaion when sampling he nex generaion of paricles. In conras o ha, he raw odomery moion model leads less peaked poseriors (c). In our curren approach, we consider boh componens, he observaion likelihood and he moion model wihin ha inerval L (i). We locally approximae he poserior p(x m (i) 1, x(i) 1, z, u 1 ) around he maximum of he likelihood funcion repored by a scan regisraion procedure. To efficienly draw he nex generaion of samples, we compue a Gaussian approximaion N based on ha daa. The main differences o previous approaches is ha we firs use a scanmacher o deermine he meaningful area of he observaion likelihood funcion. We hen sample in ha meaningful area and evaluae he sampled poins based on he arge disribuion. For each paricle i, he parameers µ (i) and Σ (i) are deermined individually for K sampled poins {x j } in he inerval L (i). We furhermore ake ino accoun he odomery informaion when compuing he mean µ (i) and he variance Σ (i). We esimae he Gaussian parameers as µ (i) = Σ (i) = 1 η (i) K j=1 x j p(z m (i) 1, x j) p(x j x (i) 1, u 1) (15) 1 K η p(z (i) m (i) 1, x j) j=1 p(x j x (i) 1, u 1) (x j µ (i) )(x j µ (i) ) T (16) wih he normalizaion facor K η (i) = p(z m (i) 1, x j) p(x j x (i) 1, u 1). (17) j=1 In his way, we obain a closed form approximaion of he opimal proposal which enables us o efficienly obain he nex generaion of paricles. Using his proposal disribuion, he weighs can be compued as = 1 p(z m (i) = 1 1 K j=1 1, x(i) 1, u 1) (18) p(z m (i) 1, x ) p(x x (i) 1, u 1) dx p(z m (i) 1, x j) p(x j x (i) 1, u 1) C. Discussion abou he Improved Proposal The compuaions presened in his secion enable us o deermine he parameers of a Gaussian proposal disribuion for each paricle individually. The proposal akes ino accoun he mos recen odomery reading and laser observaion while a he same ime allowing us efficien sampling. The resuling densiies have a much lower uncerainy compared o siuaions in which he odomery moion model is used. To illusrae his fac, Figure 2 depics ypical paricle disribuions obained wih our approach. In case of a sraigh feaureless corridor, he samples are ypically spread along he main direcion of he corridor as depiced in Figure 2 (a). Figure 2 (b) illusraes he robo reaching he end of such a corridor. As can be seen, he uncerainy in he direcion of he corridor decreases and all samples are cenered around a single poin. In conras o ha, Figure 2 (c) shows he resuling disribuion when sampling from he raw moion model. As explained above, we use a scanmacher o deermine he mode of he meaningful area of he observaion likelihood funcion. In his way, we focus he sampling on he imporan regions. Mos exising scanmaching algorihms maximize he observaion likelihood given a map and an iniial guess of he robo s pose. When he likelihood funcion is mulimodal, which can occur when, e.g., closing a loop, he scanmacher reurns for each paricle he maximum which is closes o he iniial guess. In general, i can happen ha addiional maxima in he likelihood funcion are missed since only a single mode is repored. However, since we perform frequen filer updaes (afer each movemen of 0.5 m or a roaion of 25 ) and limi he search area of he scanmacher, we consider ha he disribuion has only a single mode when sampling daa poins o compue he Gaussian proposal. Noe ha in siuaions like a loop closure, he filer is sill able o keep muliple hypoheses because he iniial guess for he saring posiion of he scanmacher when reenering a loop is differen for each paricle. Neverheless, here are siuaions in which he filer can a leas in heory become overly confiden. This migh happen in exremely cluered environmens and when he odomery is highly affeced by noise. A soluion o his problem is o rack he muliple modes of he scanmacher and repea he sampling process separaely for each node. However, in our experimens carried ou using real robos we never encounered such a siuaion. During filering, i can happen ha he scanmaching process fails because of poor observaions or a oo small overlapping area beween he curren scan and he previously compued map. In his case, he raw moion model of he robo which is illusraed in Figure 2 (c) is used as a proposal. Noe ha such siuaions occur rarely in real daases (see also Secion VIE).
5 5 D. Adapive Resampling A furher aspec ha has a major influence on he performance of a paricle filer is he resampling sep. During resampling, paricles wih a low imporance weigh are ypically replaced by samples wih a high weigh. On he one hand, resampling is necessary since only a finie number of paricles are used o approximae he arge disribuion. On he oher hand, he resampling sep can remove good samples from he filer which can lead o paricle impoverishmen. Accordingly, i is imporan o find a crierion for deciding when o perform he resampling sep. Liu [23] inroduced he socalled effecive sample size o esimae how well he curren paricle se represens he arge poserior. In his work, we compue his quaniy according o he formulaion of Douce e al. [7] as 1 N eff = N ( ) i=1 w (i) 2, (20) where refers o he normalized weigh of paricle i. The inuiion behind N eff is as follows. If he samples were drawn from he arge disribuion, heir imporance weighs would be equal o each oher due o he imporance sampling principle. The worse he approximaion of he arge disribuion, he higher is he variance of he imporance weighs. Since N eff can be regarded as a measure of he dispersion of he imporance weighs, i is a useful measure o evaluae how well he paricle se approximaes he arge poserior. Our algorihm follows he approach proposed by Douce e al. [7] o deermine wheher or no he resampling sep should be carried ou. We resample each ime N eff drops below he hreshold of N/2 where N is he number of paricles. In exensive experimens, we found ha his approach drasically reduces he risk of replacing good paricles, because he number of resampling operaions is reduced and hey are only performed when needed. E. Algorihm The overall process is summarized in Algorihm 1. Each ime a new measuremen uple (u 1, z ) is available, he proposal is compued for each paricle individually and is hen used o updae ha paricle. This resuls in he following seps: 1) An iniial guess x (i) pose represened by he paricle i is obained from he = x (i) 1 u 1 for he robo s previous pose x (i) 1 of ha paricle and he odomery measuremens u 1 colleced since he las filer updae. Here, he operaor corresponds o he sandard pose compounding operaor [24]. 2) A scanmaching algorihm is execued based on he map saring from he iniial guess x (i). The search performed by he scanmacher is bounded o a limied m (i) 1 region around x (i). If he scanmaching repors a failure, he pose and he weighs are compued according o he moion model (and he seps 3 and 4 are ignored). 3) A se of sampling poins is seleced in an inerval repored scanmacher. Based on his poins, he mean and he covariance marix of he proposal are compued by poinwise evaluaing he around he pose ˆx (i) arge disribuion p(z m (i) 1, x j)p(x j x (i) 1, u 1) in he sampled posiions x j. During his phase, also he weighing facor η (i) is compued according o Eq. (17). 4) The new pose x (i) of he paricle i is drawn from he Gaussian approximaion N(µ (i), Σ (i) ) of he improved proposal disribuion. 5) Updae of he imporance weighs. 6) The map m (i) of paricle i is updaed according o he drawn pose x (i) and he observaion z. Afer compuing he nex generaion of samples, a resampling sep is carried ou depending on he value of N eff. IV. IMPLEMENTATION ISSUES This secion provides addiional informaion abou implemenaion deails used in our curren sysem. These issues are no required for he undersanding of he general approach bu complee he precise descripion of our mapping sysem. In he following, we briefly explain he used scanmaching approach, he observaion model, and how o poinwise evaluae he moion model. Our approach applies a scanmaching echnique on a per paricle basis. In general, an arbirary scanmaching echnique can be used. In our implemenaion, we use he scanmacher vasco which is par of he Carnegie Mellon Robo Navigaion Toolki (CARMEN) [27, 36]. This scanmacher aims o find he mos likely pose by maching he curren observaion agains he map consruced so far where x (i) ˆx (i) = argmaxp(x m (i) 1, z, x (i) ), (21) x is he iniial guess. The scanmaching echnique performs a gradien descen search on he likelihood funcion of he curren observaion given he grid map. Noe ha in our mapping approach, he scanmacher is only used for finding he local maximum of he observaion likelihood funcion. In pracice, any scanmaching echnique which is able o compue he bes alignmen beween a reference map m (i) 1 and he curren scan z given an iniial guess x (i) can be used. In order o solve Eq. (21), one applies Bayes rule and seeks for he pose wih he highes observaion likelihood p(z m, x). To compue he likelihood of an observaion, we use he so called beam endpoin model [40]. In his model, he individual beams wihin a scan are considered o be independen. Furhermore, he likelihood of a beam is compued based on he disance beween he endpoin of he beam and he closes obsacle from ha poin. To achieve a fas compuaion, one ypically uses a convolved local grid map. Addiionally, he consrucion of our proposal requires o evaluae p(z m (i) 1, x j)p(x j x (i) 1, u 1) a he sampled poins x j. We compue he firs componen according o he previously menioned beam endpoin model. To evaluae he second erm, several closed form soluions for he moion esimae are available. The differen approaches mainly differ in he way he kinemaics of he robo are modeled. In our curren implemenaion, we compue p(x j x 1, u 1 ) according o
6 6 Algorihm 1 Improved RBPF for Map Learning Require: S 1, he sample se of he previous ime sep z, he mos recen laser scan u 1, he mos recen odomery measuremen Ensure: S, he new sample se S = {} for all s (i) 1 S 1 do < x (i) 1, w(i) 1, m(i) 1 // scanmaching x (i) ˆx (i) if ˆx (i) x (i) = x (i) 1 u 1 >= s(i) 1 = argmax x p(x m (i) 1, z, x (i) = failure hen p(x x (i) 1, u 1) = 1 p(z m (i) 1, x(i) ) ) else // sample around he mode for k = 1,...,K do x k {x j x j ˆx (i) < } end for // compue Gaussian proposal µ (i) = (0, 0, 0) T η (i) = 0 for all x j {x 1,..., x K } do µ (i) = µ (i) +x j p(z m (i) 1, x j) p(x x (i) 1, u 1) η (i) = η (i) + p(z m (i) 1, x j) p(x x (i) 1, u 1) end for µ (i) Σ (i) = µ (i) /η (i) = 0 for all x j {x 1,..., x K } do Σ (i) = Σ (i) + (x j µ (i) )(x j µ (i) ) T p(z m (i) 1, x j) p(x j x (i) 1, u 1) end for = Σ (i) /η (i) // sample new pose Σ (i) x (i) N(µ (i), Σ (i) ) // updae imporance weighs = 1 η(i) end if // updae map m (i) = inegraescan(m (i) 1, x(i), z ) // updae sample se S = S {< x (i),, m (i) >} end for 1 N eff = N ( i=1 w(i) ) 2 if N eff < T hen S = resample(s ) end if he Gaussian approximaion of he odomery moion model described in [41]. We obain his approximaion hrough Taylor expansion in an EKFsyle procedure. In general, here are more sophisicaed echniques esimaing he moion of he robo. However, we use ha model o esimae a movemen beween wo filer updaes which is performed afer he robo raveled around 0.5 m. In his case, his approximaion works well and we did no observed a significan difference beween he EKFlike model and he in general more accurae samplebased velociy moion model [41]. V. COMPLEXITY This secion discusses he complexiy of he presened approach o learn grid maps wih a RaoBlackwellized paricle filer. Since our approach uses a sample se o represen he poserior abou maps and poses, he number N of samples is he cenral quaniy. To compue he proposal disribuion, our approach samples around he mos likely posiion repored by he scanmacher. This sampling is done for each paricle a consan number of imes (K) and here is no dependency beween he paricles when compuing he proposal. Furhermore, he mos recen observaion z used o compue µ (i) and Σ (i) covers only an area of he map m (bounded by he odomery error and he maximum range of he sensor), so he complexiy depends only on he number N of paricles. The same holds for he updae of he individual maps associaed o each of he paricles. During he resampling sep, he informaion associaed o a paricle needs o be copied. In he wors case, N 1 samples are replaced by a single paricle. In our curren sysem, each paricle sores and mainains is own grid map. To duplicae a paricle, we herefore have o copy he whole map. As a resul, a resampling acion inroduces a wors case complexiy of O(NM), where M is he size of he corresponding grid map. However, using he adapive resampling echnique, only very few resampling seps are required during mapping. To decide wheher or no a resampling is needed, he effecive sample size (see Eq. (20)) needs o be aken ino accoun. Again, he compuaion of he quaniy inroduces a complexiy of O(N). As a resul, if no resampling operaion is required, he overall complexiy for inegraing a single observaion depends only linearly on he number of paricles. If a resampling is required, he addiional facor M which represens he size of he map is inroduced and leads o a complexiy of O(NM). The complexiy of each individual operaion is depiced in Table I. Noe ha he complexiy of he resampling sep can be reduced by using a more inelligen map represenaion as done in DPSLAM [9]. I can be shown, ha in his case he complexiy of a resampling sep is reduced o O(AN 2 log N), where A is he area covered by he sensor. However, building an improved map represenaion is no he aim of his paper. We acually see our approach as orhogonal o DPSLAM because boh echniques can be combined. Furhermore, in our experimens using real world daa ses, we figured ou he resampling seps are no he dominan par and hey occur rarely due o he adapive resampling sraegy.
7 7 TABLE I COMPLEXITY OF THE DIFFERENT OPERATIONS FOR INTEGRATING ONE OBSERVATION. Operaion Complexiy Compuaion of he proposal disribuion O(N) Updae of he grid map O(N) Compuaion of he weighs O(N) Tes if resampling is required O(N) Resampling O(NM) Fig. 3. Differen ypes of robo used o acquire real robo daa used for mapping (AcivMedia Pioneer 2 AT, Pioneer 2 DX8, and an irobo B21r). VI. EXPERIMENTS The approach described above has been implemened and esed using real robos and daases gahered wih real robos. Our mapping approach runs online on several plaforms like AcivMedia Pioneer2 AT, Pioneer 2 DX8, and irobo B21r robos equipped wih a SICK LMS and PLS laser range finders (see Figure 3). The experimens have been carried ou in a variey of environmens and showed he effeciveness of our approach in indoor and oudoor seings. Mos of he maps generaed by our approach can be magnified up o a resoluion of 1 cm, wihou observing considerable inconsisencies. Even in big real world daases covering an area of approximaely 250 m by 250 m, our approach never required more han 80 paricles o build accurae maps. In he reminder of his secion, we discuss he behavior of he filer in differen daases. Furhermore, we give a quaniaive analysis of he performance of he presened approach. Highly accurae grid maps have been generaed wih our approach from several daases. These maps, raw daa files, and an efficien implemenaion of our mapping sysem are available on he web [38]. Fig. 4. The Inel Research Lab. The robo sars in he upper par of he circular corridor, and runs several imes around he loop, before enering he rooms. The lef image depics he resuling map generaed wih 15 paricles. The righ image shows a cuou wih 1cm grid resoluion o illusrae he accuracy of he map in he loop closure poin. Fig. 5. The Freiburg Campus. The robo firs runs around he exernal perimeer in order o close he ouer loop. Aferwards, he inernal pars of he campus are visied. The overall rajecory has a lengh of 1.75 km and covers an area of approximaely 250 m by 250 m. The depiced map was generaed using 30 paricles. A. Mapping Resuls The daases discussed here have been recorded a he Inel Research Lab in Seale, a he campus of he Universiy of Freiburg, and a he Killian Cour a MIT. The maps of hese environmens are depiced in Figures 4, 5, and 6. a) Inel Research Lab: The Inel Research Lab is depiced in he lef image of Figure 4 and has a size of 28 m by 28 m. The daase has been recorded wih a Pioneer II robo equipped wih a SICK sensor. To successfully correc his daase, our algorihm needed only 15 paricles. As can be seen in he righ image of Figure 4, he qualiy of he final map is so high ha he map can be magnified up o 1cm of resoluion wihou showing any significan errors. b) Freiburg Campus: The second daase has been recorded oudoors a he Freiburg campus. Our sysem needed only 30 paricles o produce a good qualiy map such as he one shown in Figure 5. Noe ha his environmen parly violaes he assumpions ha he environmen is planar. Addiionally, here were objecs like bushes and grass as well as moving objecs like cars and people. Despie he resuling spurious measuremens, our algorihm was able o generae an accurae map. c) MIT Killian Cour: The hird experimen was performed wih a daase acquired a he MIT Killian cour 1 and he resuling map is depiced in Figure 6. This daase is exremely challenging since i conains several nesed loops, which can cause a RaoBlackwellized paricle filer o fail due o paricle depleion. Using his daase, he selecive resampling procedure urned ou o be imporan. A consisen and opologically correc map can be generaed wih 60 paricles. However, he resuling maps someimes show arificial double walls. By employing 80 paricles i is possible o achieve high qualiy maps. 1 Noe ha here exis wo differen versions of ha daase on he web. One has a precorreced odomery and he oher one has no. We used he raw version wihou precorreced odomery informaion.
8 8 g f 100 d e c a b Neff/N [%] A B ime C D Fig. 8. The graph plos he evoluion of he N eff funcion over ime during an experimen in he environmen shown in he op image. A ime B he robo closes he small loop. A ime C and D resampling acions are carried afer he robos closes he big loop. Fig. 6. The MIT Killian Cour. The robo sars from he poin labeled a and hen raverses he firs loop labeled b. I hen moves hrough he loops labeled c, d and moves back o he place labeled a and he loop labeled b. I he visis he wo big loops labeled f and g. The environmen has a size of 250 m by 215 m and he robo raveled 1.9km. The depiced map has been generaed wih 80 paricles. The recangles show magnificaions of several pars of he map. TABLE II THE NUMBER OF PARTICLES NEEDED BY OUR ALGORITHM COMPARED TO THE APPROACH OF HÄHNEL e al. [16]. Proposal Disribuion Inel MIT Freiburg our approach approach of [16] B. Quaniaive Resuls In order o measure he improvemen in erms of he number of paricles, we compared he performance of our sysem using he informed proposal disribuion o he approach done by Hähnel e al. [16]. Table II summarizes he number of paricles needed by a RBPF for providing a opologically correc map in a leas 60% of all applicaions of our algorihm. I urns ou ha in all of he cases, he number of paricles required by our approach was approximaely one order of magniude smaller han he one required by he oher approach. Moreover, he resuling maps are beer due o our improved sampling process ha akes he las reading ino accoun. Figure 7 summarizes resuls abou he success raio of our success rae [%] number of paricles Inel Lab Freiburg Campus MIT MIT2 Fig. 7. Success rae of our algorihm in differen environmens depending on he number of paricles. Each success rae was deermined using 20 runs. For he experimen MIT2 we disabled he adapive resampling. algorihm in he environmens considered here. The plos show he percenage of correcly generaed maps, depending on he number of paricles used. The quesion if a map is consisen or no has been evaluaed by visual inspecion in a blind fashion (he inspecors were no he auhors). As a measure of success, we used he opological correcness. C. Effecs of Improved Proposals and Adapive Resampling The increased performance of our approach is due o he inerplay of wo facors, namely he improved proposal disribuion, which allows us o generae samples wih an high likelihood, and he adapive resampling conrolled by monioring N eff. For proposals ha do no consider he whole inpu hisory, i has been proven ha N eff can only decrease (sochasically) over ime [7]. Only afer a resampling operaion, N eff recovers is maximum value. I is imporan o noice ha he behavior of N eff depends on he proposal: he worse he proposal, he faser N eff drops. We found ha he evoluion of N eff using our proposal disribuion shows hree differen behaviors depending on he informaion obained from he robo s sensors. Figure 8 illusraes he evoluion of N eff during an experimen. Whenever he robo moves hrough unknown errain, N eff ypically drops slowly. This is because he proposal disribuion becomes less peaked and he likelihoods of observaions ofen differ slighly. The second behavior can be observed when he robo moves hrough a known area. In his case, each paricle keeps localized wihin is own map due o he improved proposal disribuion and he weighs are more or less equal. This resuls in a more or less consan behavior of N eff. Finally, when closing a loop, some paricles are correcly aligned wih heir map while ohers are no. The correc paricles have a high weigh, while he wrong ones have a low weigh. Thus he variance of he imporance weighs increases and N eff subsanially drops. Accordingly, he hreshold crierion applied on N eff ypically forces a resampling acion when he robo is closing a loop. In all oher cases, he resampling is avoided and in his way he filer keeps a variey of samples in he paricle se. As a resul, he risk of paricle depleion problem is seriously reduced. To analyze his, we performed an experimen in
9 9 Fig. 10. The effec of considering he odomery in he compuaion of he proposal on he paricle cloud. The lef image depics he paricle disribuion if only he laser range finder daa is used. By aking ino accoun he odomery when compuing he proposal disribuion, he paricles can be drawn in a more accurae manner. As can be seen in he righ image, he paricle cloud is more focused, because i addiionally incorporaes he odomery informaion. Fig. 9. Maps from he ACES building a Universiy of Texas, he 4h floor of he Sieg Hall a he Universiy of Washingon, he Belgioioso building, and building 101 a he Universiy of Freiburg. which we compared he success rae of our algorihm o ha of a paricle filer which resamples a every sep. As Figure 7 illusraes, our approach more ofen converged o he correc soluion (MIT curve) for he MIT daase compared o he paricle filer wih he same number of paricles and a fixed resampling sraegy (MIT2 curve). To give a more deailed impression abou he accuracy of our new mapping echnique, Figure 9 depics maps learned from well known and freely available [18] real robo daases recorded a he Universiy of Texas, a he Universiy of Washingon, a Belgioioso, and a he Universiy of Freiburg. Each map was buil using 30 paricles o represen he poserior abou maps and poses. D. The Influence of he Odomery on he Proposal This experimen is designed o show he advanage of he proposal disribuion, which akes ino accoun he odomery informaion o draw paricles. In mos cases, he purely laserbased proposal like he one presened in our previous approach [14] is wellsuied o predic he moion of he paricles. However, in a few siuaions he knowledge abou he odomery informaion can be imporan o focus he proposal disribuion. This is he case if only very poor feaures are available in he laser daa ha was used o compue he parameers of he Gaussian proposal approximaion. For example, an open free space wihou any obsacles or a long feaureless corridor can lead o high variances in he compued proposal ha is only based on laser range finder daa. Figure 10 illusraes his effec based on simulaed laser daa. In a furher experimen, we simulaed a shorrange laser scanner (like, e.g., he Hokuyo URG scanner). Due o he maximum range of 4 m, he robo was unable o see he end of he corridor in mos cases. This resuls in an high pose uncerainy in he direcion of he corridor. We recorded several rajecories in his environmen and used hem o learn maps wih and wihou considering he odomery when compuing he proposal disribuion. In his experimen, he approach considering he odomery succeeded in 100% of all cases o learn a opologically correc map. In conras o ha, our previous approach which does no ake ino accoun he odomery succeeded only in 50% of all cases. This experimen indicaes he imporance of he improved proposal disribuion. Figure 11 depics ypical maps obained wih he differen proposal disribuions during his experimen. The lef map conains alignmen errors caused by he high pose uncerainy in he direcion of he corridor. In conras o ha, a robo ha also akes ino accoun he odomery was able o mainain he correc pose hypoheses. A ypical example is depiced in he righ image. Noe ha by increasing he number of paricles, boh approaches are able o map he environmen correcly in 100% of all cases, bu since each paricle carries is own map, i is of umos imporance o keep he number of paricles as low as possible. Therefore, his improved proposal is a means o limi he number of paricles during mapping wih Rao Blackwellized paricle filers. E. Siuaions in Which he ScanMacher Fails As repored in Secion III, i can happen ha he scanmacher is unable o find a good pose esimae based on he laser range daa. In his case, we sample from he raw
10 10 alignmen errors TABLE III AVERAGE EXECUTION TIME USING A STANDARD PC. Operaion Average Execuion Time Compuaion of he proposal disribuion, he weighs, and he map updae 1910 ms Tes if resampling is required 41 ms Resampling 244 ms Fig. 11. Differen mapping resuls for he same daa se obained using he proposal disribuion which ignores he odomery (lef image) and which considers he odomery when drawing he nex generaion of paricles (righ image). odomery model o obain he nex generaion of paricles. In mos esed indoor daase, however, such a siuaion never occurred a all. In he MIT daase, his effec was observed once due o a person walking direcly in fron of he robo while he robo was moving hough a corridor ha mainly consiss of glass panes. In oudoor daases, such a siuaion can occur if he robo moves hrough large open spaces because in his case he laser range finder mainly repors maximum range readings. While mapping he Freiburg campus, he scanmacher also repored such an error a one poin. In his paricular siuaion, he robo enered he parking area (in he upper par of he map, compare Figure 5). On ha day, all cars were removed from he parking area due o consrucion work. As a resul, no cars or oher objecs caused reflecions of he laser beams and mos pars of he scan consised of maximum range readings. In such a siuaion, he odomery informaion provides he bes pose esimae and his informaion is used by our mapping sysem o predic he moion of he vehicle. F. Runime Analysis In his las experimen, we analyze he memory and compuaional resources needed by our mapping sysem. We used a sandard PC wih a 2.8 GHz processor. We recorded he average memory usage and execuion ime using he defaul parameers ha allows our algorihm o learn correc maps for nearly all real world daases provided o us. In his seing, 30 paricles are used o represen he poserior abou maps and poses and a new observaion, consising of a full laser range scan, is inegraed whenever he robo moved more han 0.5 m or roaed more han 25. The Inel Research Lab daase (see Figure 4) conains odomery and laser range readings which have been recorded over 45min. Our implemenaion required 150MB of memory o sore all he daa using a maps wih a size of approx. 40 m by 40 m and a grid resoluion of 5cm. The overall ime o correc he log file using our sofware was less han 30 min. This means ha he ime o record a log file is around 1.5 imes longer han he ime o correc he log file. Table III depics he average execuion ime for he individual operaions. VII. RELATED WORK Mapping echniques for mobile robos can be roughly classified according o he map represenaion and he underlying esimaion echnique. One popular map represenaion is he occupancy grid [31]. Whereas such gridbased approaches are compuaionally expensive and ypically require a huge amoun of memory, hey are able o represen arbirary objecs. Feaurebased represenaions are aracive because of heir compacness. However, hey rely on predefined feaure exracors, which assumes ha some srucures in he environmens are known in advance. The esimaion algorihms can be roughly classified according o heir underlying basic principle. The mos popular approaches are exended Kalman filers (EKFs), maximum likelihood echniques, sparse exended informaion filers (SEIFs), smoohing echniques, and RaoBlackwellized paricle filers. The effeciveness of he EKF approaches comes from he fac ha hey esimae a fully correlaed poserior over landmark maps and robo poses [21, 37]. Their weakness lies in he srong assumpions ha have o be made on boh he robo moion model and he sensor noise. Moreover, he landmarks are assumed o be uniquely idenifiable. There exis echniques [33] o deal wih unknown daa associaion in he SLAM conex, however, if hese assumpions are violaed, he filer is likely o diverge [12]. Similar observaions have been repored by Julier e al. [20] as well as by Uhlmann [43]. The unscened Kalman filer described in [20] is one way of beer dealing wih he nonlineariies in he moion model of he vehicle. A popular maximum likelihood algorihm compues he mos likely map given he hisory of sensor readings by consrucing a nework of relaions ha represens he spaial consrains beween he poses of he robo [8, 13, 15, 24]. Gumann e al. [15] proposed an effecive way for consrucing such a nework and for deecing loop closures, while running an incremenal maximum likelihood algorihm. When a loop closure is deeced, a global opimizaion on he nework of relaion is performed. Recenly, Hähnel e al. [17], proposed an approach which is able o rack several map hypoheses using an associaion ree. However, he necessary expansions of his ree can preven he approach from being feasible for realime operaion. Thrun e al. [42] proposed a mehod o correc he poses of robos based on he inverse of he covariance marix. The advanage of he sparse exended informaion filers (SEIFs) is ha hey make use of he approximaive sparsiy of he informaion marix and in his way can perform predicions and updaes in consan ime. Eusice e al. [10] presened a echnique o make use of exacly sparse informaion marices in a delayedsae framework. Paskin [34] presened a soluion
11 11 o he SLAM problem using hin juncion rees. In his way, he is able o reduce he complexiy compared o he EKF approaches since hinned juncion rees provide a linearime filering operaion. Folkessen e al. [11] proposed an effecive approach for dealing wih symmeries and invarians ha can be found in landmark based represenaion. This is achieved by represening each feaure in a low dimensional space (measuremen subspace) and in he meric space. The measuremen subspace capures an invarian of he landmark, while he meric space represens he dense informaion abou he feaure. A mapping beween he measuremen subspace and he meric space is dynamically evaluaed and refined as new observaions are acquired. Such a mapping can ake ino accoun spaial consrains beween differen feaures. This allows he auhors o consider hese relaions for updaing he map esimae. Very recenly, Dellaer proposed a smoohing mehod called square roo smoohing and mapping [2]. I has several advanages compared o EKF since i beer covers he nonlineariies and is faser o compue. In conras o SEIFs, i furhermore provides an exacly sparse facorizaion of he informaion marix. In a work by Murphy, Douce, and colleagues [6, 32], Rao Blackwellized paricle filers (RBPF) have been inroduced as an effecive means o solve he SLAM problem. Each paricle in a RBPF represens a possible robo rajecory and a map. The framework has been subsequenly exended by Monemerlo e al. [28, 29] for approaching he SLAM problem wih landmark maps. To learn accurae grid maps, RBPFs have been used by Eliazar and Parr [9] and Hähnel e al. [16]. Whereas he firs work describes an efficien map represenaion, he second presens an improved moion model ha reduces he number of required paricles. Based on he approach of Hähnel e al., Howard presened an approach o learn grid maps wih muliple robos [19]. The focus of his work lies in how o merge he informaion obained by he individual robos and no in how o compue beer proposal disribuions. Bosse e al. [1] describe a generic framework for SLAM in largescale environmens. They use a graph srucure of local maps wih relaive coordinae frames and always represen he uncerainy wih respec o a local frame. In his way, hey are able o reduce he complexiy of he overall problem. In his conex, Modayil e al. [25] presened a echnique which combines merical SLAM wih opological SLAM. The opology is uilized o solve he loopclosing problem, whereas meric informaion is used o build up local srucures. Similar ideas have been realized by Lisien e al. [22], which inroduce a hierarchical map in he conex of SLAM. The work described in his paper is an improvemen of he algorihm proposed by Hähnel e al. [16]. Insead of using a fixed proposal disribuion, our algorihm compues an improved proposal disribuion on a perparicle basis on he fly. This allows us o direcly use he informaion obained from he sensors while evolving he paricles. The work presened here is also an exension of our previous approach [14], which lacks he abiliy o incorporae he odomery informaion ino he proposal. Especially, in criical siuaions in which only poor laser feaures for localizaion are available, our approach performs beer han our previous one. The compuaion of he proposal disribuion is done in a similar way as in FasSLAM2 presened by Monemerlo e al. [26]. In conras o FasSLAM2, our approach does no rely on predefined landmarks and uses raw laser range finder daa o acquire accurae grid maps. Paricle filers using proposal disribuions ha ake ino accoun he mos recen observaion are also called lookahead paricle filers. Moralez Menéndez e al. [30] proposed such a mehod o more reliably esimae he sae of a dynamic sysem where accurae sensors are available. The advanage of our approach is wofold. Firsly, our algorihm draws he paricles in a more effecive way. Secondly, he highly accurae proposal disribuion allows us o uilize he effecive sample size as a robus indicaor o decide wheher or no a resampling has o be carried ou. This furher reduces he risk of paricle depleion. VIII. CONCLUSIONS In his paper, we presened an improved approach o learning grid maps wih RaoBlackwellized paricle filers. Our approach compues a highly accurae proposal disribuion based on he observaion likelihood of he mos recen sensor informaion, he odomery, and a scanmaching process. This allows us o draw paricles in a more accurae manner which seriously reduces he number of required samples. Addiionally, we apply a selecive resampling sraegy based on he effecive sample size. This approach reduces he number of unnecessary resampling acions in he paricle filer and hus subsanially reduces he risk of paricle depleion. Our approach has been implemened and evaluaed on daa acquired wih differen mobile robos equipped wih laser range scanners. Tess performed wih our algorihm in differen largescale environmens have demonsraed is robusness and he abiliy of generaing high qualiy maps. In hese experimens, he number of paricles needed by our approach ofen was one order of magniude smaller compared o previous approaches. ACKNOWLEDGMENT This work has parly been suppored by he Marie Curie program under conrac number HPMTCT , by he German Research Foundaion (DFG) under conrac number SFB/TR8 (A3), and by he EC under conrac number FP CoSy, FP6IST BACS, and FP IST5 mufly. The auhors would like o acknowledge Mike Bosse and John Leonard for providing us he daase of he MIT Killian Cour, Parick Beeson for he ACES daase, and Dirk Hähnel for he Inel Research Lab, he Belgioioso, and he SiegHall daase. REFERENCES [1] M. Bosse, P.M. Newman, J.J. Leonard, and S. Teller. An ALTAS framework for scalable mapping. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , Taipei, Taiwan, [2] F. Dellaer. Square Roo SAM. In Proc. of Roboics: Science and Sysems (RSS), pages , Cambridge, MA, USA, 2005.
12 12 [3] F. Dellaer, D. Fox, W. Burgard, and S. Thrun. Mone carlo localizaion for mobile robos. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), Leuven, Belgium, [4] G. Dissanayake, H. DurranWhye, and T. Bailey. A compuaionally efficien soluion o he simulaneous localisaion and map building (SLAM) problem. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , San Francisco, CA, USA, [5] A. Douce. On sequenial simulaionbased mehods for bayesian filering. Technical repor, Signal Processing Group, Dep. of Engeneering, Universiy of Cambridge, [6] A. Douce, J.F.G. de Freias, K. Murphy, and S. Russel. RaoBlackwellized parcile filering for dynamic bayesian neworks. In Proc. of he Conf. on Uncerainy in Arificial Inelligence (UAI), pages , Sanford, CA, USA, [7] A. Douce, N. de Freias, and N. Gordan, ediors. Sequenial Mone Carlo Mehods in Pracice. Springer Verlag, [8] T. Ducke, S. Marsland, and J. Shapiro. Fas, online learning of globally consisen maps. Journal of Auonomous Robos, 12(3): , [9] A. Eliazar and R. Parr. DPSLAM: Fas, robus simulainous localizaion and mapping wihou predeermined landmarks. In Proc. of he In. Conf. on Arificial Inelligence (IJCAI), pages , Acapulco, Mexico, [10] R. Eusice, H. Singh, and J.J. Leonard. Exacly sparse delayedsae filers. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , Barcelona, Spain, [11] J. Folkesson, P. Jensfel, and H. Chrisensen. Vision SLAM in he measuremen subspace. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , April [12] U. Frese and G. Hirzinger. Simulaneous localizaion and mapping  a discussion. In Proc. of he IJCAI Workshop on Reasoning wih Uncerainy in Roboics, pages 17 26, Seale, WA, USA, [13] U. Frese, P. Larsson, and T. Ducke. A mulilevel relaxaion algorihm for simulaneous localisaion and mapping. IEEE Transacions on Roboics, 21(2):1 12, [14] G. Grisei, C. Sachniss, and W. Burgard. Improving gridbased slam wih RaoBlackwellized paricle filers by adapive proposals and selecive resampling. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , Barcelona, Spain, [15] J.S. Gumann and K. Konolige. Incremenal mapping of large cyclic environmens. In Proc. of he IEEE In. Symposium on Compuaional Inelligence in Roboics and Auomaion (CIRA), pages , Monerey, CA, USA, [16] D. Hähnel, W. Burgard, D. Fox, and S. Thrun. An efficien FasSLAM algorihm for generaing maps of largescale cyclic environmens from raw laser range measuremens. In Proc. of he IEEE/RSJ In. Conf. on Inelligen Robos and Sysems (IROS), pages , Las Vegas, NV, USA, [17] D. Hähnel, W. Burgard, B. Wegbrei, and S. Thrun. Towards lazy daa associaion in slam. In Proc. of he In. Symposium of Roboics Research (ISRR), pages , Siena, Ialy, [18] A. Howard and N. Roy. The roboics daa se reposiory (Radish), hp://radish.sourceforge.ne/. [19] Andrew Howard. Mulirobo simulaneous localizaion and mapping using paricle filers. In Roboics: Science and Sysems, pages , Cambridge, MA, USA, [20] S. Julier, J. Uhlmann, and H. DurranWhye. A new approach for filering nonlinear sysems. In Proc. of he American Conrol Conference, pages , Seale, WA, USA, [21] J.J. Leonard and H.F. DurranWhye. Mobile robo localizaion by racking geomeric beacons. IEEE Transacions on Roboics and Auomaion, 7(4): , [22] B. Lisien, D. Silver D. Morales, G. Kanor, I.M. Rekleiis, and H. Chose. Hierarchical simulaneous localizaion and mapping. In Proc. of he IEEE/RSJ In. Conf. on Inelligen Robos and Sysems (IROS), pages , Las Vegas, NV, USA, [23] J.S. Liu. Meropolized independen sampling wih comparisons o rejecion sampling and imporance sampling. Sais. Compu., 6: , [24] F. Lu and E. Milios. Globally consisen range scan alignmen for environmen mapping. Journal of Auonomous Robos, 4: , [25] J. Modayil, P. Beeson, and B. Kuipers. Using he opological skeleon for scalable global merical mapbuilding. In Proc. of he IEEE/RSJ In. Conf. on Inelligen Robos and Sysems (IROS), pages , Sendai, Japan, [26] M. Monemerlo, S. Thrun D. Koller, and B. Wegbrei. FasSLAM 2.0: An improved paricle filering algorihm for simulaneous localizaion and mapping ha provably converges. In Proc. of he In. Conf. on Arificial Inelligence (IJCAI), pages , Acapulco, Mexico, [27] M. Monemerlo, N. Roy, S. Thrun, D. Hähnel, C. Sachniss, and J. Glover. CARMEN he carnegie mellon robo navigaion oolki. hp://carmen.sourceforge.ne, [28] M. Monemerlo and S. Thrun. Simulaneous localizaion and mapping wih unknown daa associaion using FasSLAM. In Proc. of he IEEE In. Conf. on Roboics & Auomaion (ICRA), pages , Taipei, Taiwan, [29] M. Monemerlo, S. Thrun, D. Koller, and B. Wegbrei. FasSLAM: A facored soluion o simulaneous localizaion and mapping. In Proc. of he Naional Conference on Arificial Inelligence (AAAI), pages , Edmonon, Canada, [30] R. MoralesMenéndez, N. de Freias, and D. Poole. Realime monioring of complex indusrial processes wih paricle filers. In Proc. of he Conf. on Neural Informaion Processing Sysems (NIPS), pages , Vancover, Canada, [31] H.P. Moravec. Sensor fusion in cerainy grids for mobile robos. AI Magazine, pages 61 74, Summer [32] K. Murphy. Bayesian map learning in dynamic environmens. In Proc. of he Conf. on Neural Informaion Processing Sysems (NIPS), pages , Denver, CO, USA, [33] J. Neira and J.D. Tardós. Daa associaion in sochasic mapping using he join compaibiliy es. IEEE Transacions on Roboics and Auomaion, 17(6): , [34] M.A. Paskin. Thin juncion ree filers for simulaneous localizaion and mapping. In Proc. of he In. Conf. on Arificial Inelligence (IJCAI), pages , Acapulco, Mexico, [35] M.K. Pi and N. Shephard. Filering via simulaion: auxilary paricle filers. Technical repor, Deparmen of Mahemaics, Imperial College, London, [36] N. Roy, M. Monemerlo, and S. Thrun. Perspecives on sandardizaion in mobile robo programming. In Proc. of he IEEE/RSJ In. Conf. on Inelligen Robos and Sysems (IROS), pages , Las Vegas, NV, USA, [37] R. Smih, M. Self, and P. Cheeseman. Esimaing uncerain spaial realionships in roboics. In I. Cox and G. Wilfong, ediors, Auonomous Robo Vehicles, pages Springer Verlag, [38] C. Sachniss and G. Grisei. Mapping resuls obained wih RaoBlackwellized paricle filers. hp://www.informaik.unifreiburg.de/ sachnis/research/rbpfmapper/, [39] S. Thrun. An online mapping algorihm for eams of mobile robos. In. Journal of Roboics Research, 20(5): , [40] S. Thrun, W. Burgard, and D. Fox. Probabilisic Roboics, chaper Robo Percepion, pages MIT Press, [41] S. Thrun, W. Burgard, and D. Fox. Probabilisic Roboics, chaper Robo Moion, pages MIT Press, [42] S. Thrun, Y. Liu, D. Koller, A.Y. Ng, Z. Ghahramani, and H. Durran Whye. Simulaneous localizaion and mapping wih sparse exended informaion filers. In. Journal of Roboics Research, 23(7/8): , [43] J. Uhlmann. Dynamic Map Building and Localizaion: New Theoreical Foundaions. PhD hesis, Universiy of Oxford, [44] R. van der Merwe, N. de Freias, A. Douce, and E. Wan. The unscened paricle filer. Technical Repor CUED/FINFENG/TR380, Cambridge Universiy Engineering Deparmen, Augus 2000.
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