A Probabilistic Approach to Collaborative Multi-Robot Localization

Transcription

1 Autoomous Robots 8, , 2000 c 2000 Kluwer Academic Publishers. Maufactured i The Netherlads. A Probabilistic Approach to Collaborative Multi-Robot Localizatio DIETER FOX School of Computer Sciece, Caregie Mello Uiversity, Pittsburgh, PA 15213, USA dfox@cs.cmu.edu WOLFRAM BURGARD Departmet of Computer Sciece, Uiversity of Freiburg, D Freiburg, Germay burgard@iformatik.ui-freiburg.de HANNES KRUPPA Departmet of Computer Sciece, ETH Zürich, CH-8092 Zürich, Switzerlad kruppa@if.ethz.ch SEBASTIAN THRUN School of Computer Sciece, Caregie Mello Uiversity, Pittsburgh, PA 15213, USA thru@cs.cmu.edu Abstract. This paper presets a statistical algorithm for collaborative mobile robot localizatio. Our approach uses a sample-based versio of Markov localizatio, capable of localizig mobile robots i a ay-time fashio. Whe teams of robots localize themselves i the same eviromet, probabilistic methods are employed to sychroize each robot s belief wheever oe robot detects aother. As a result, the robots localize themselves faster, maitai higher accuracy, ad high-cost sesors are amortized across multiple robot platforms. The techique has bee implemeted ad tested usig two mobile robots equipped with cameras ad laser rage-fiders for detectig other robots. The results, obtaied with the real robots ad i series of simulatio rus, illustrate drastic improvemets i localizatio speed ad accuracy whe compared to covetioal sigle-robot localizatio. A further experimet demostrates that uder certai coditios, successful localizatio is oly possible if teams of heterogeeous robots collaborate durig localizatio. Keywords: mobile robots, localizatio, multi-robot systems, ucertaity 1. Itroductio Sesor-based robot localizatio has bee recogized as oe of the fudametal problems i mobile robotics. The localizatio problem is frequetly divided ito two subproblems: Positio trackig, which seeks to compesate small dead reckoig errors uder the assumptio that the iitial positio is kow, ad global self-localizatio, which addresses the problem of localizatio with o a priori iformatio. The latter problem is geerally regarded as the more difficult oe, ad recetly several approaches have provided soud solutios to this problem. I recet years, a flurry of publicatios o localizatio which icludes a book solely dedicated to this problem (Borestei et al., 1996) documet the importace of the problem. Accordig to Cox (Cox ad Wilfog, 1990), Usig sesory iformatio to locate the robot i its eviromet is the most fudametal problem to providig a mobile robot with autoomous capabilities.

2 326 Foxetal. However, virtually all existig work addresses localizatio of a sigle robot oly. The problem of cooperative multi-robot localizatio remais virtually uexplored. At first glace, oe could solve the problem of localizig N robots by localizig each robot idepedetly, which is a valid approach that might yield reasoable results i may eviromets. However, if robots ca detect each other, there is the opportuity to do better. Whe a robot determies the locatio of aother robot relative to its ow, both robots ca refie their iteral beliefs based o the other robot s estimate, hece improve their localizatio accuracy. The ability to exchage iformatio durig localizatio is particularly attractive i the cotext of global localizatio, where each sight of aother robot ca reduce the ucertaity i the estimated locatio dramatically. The importace of exchagig iformatio durig localizatio is particularly strikig for heterogeeous robot teams. Cosider, for example, a robot team where some robots are equipped with expesive, highaccuracy sesors (such as laser rage-fiders), whereas others are oly equipped with low-cost sesors such as soar sesors. By trasferrig iformatio across multiple robots, sesor iformatio ca be leveraged. Thus, collaborative multi-robot localizatio facilitates the amortizatio of high-ed high-accuracy sesors across teams of robots. Cosequetly, phrasig the problem of localizatio as a collaborative oe offers the opportuity of improved performace from less data. This paper proposes a efficiet probabilistic approach for collaborative multi-robot localizatio. Our approach is based o Markov localizatio (Nourbakhsh et al., 1995; Simmos ad Koeig, 1995; Kaelblig et al., 1996; Burgard et al., 1996), a family of probabilistic approaches that have recetly bee applied with great practical success to sigle-robot localizatio (Burgard et al., 2000; Koolige, 1999; Fox et al., 1999b; Thru et al., 1999a). I cotrast to previous research, which relied o grid-based or coarsegraied topological represetatios of a robot s state space, our approach adopts a samplig-based represetatio (Dellaert et al., 1999; Fox et al., 1999a), which is capable of approximatig a wide rage of belief fuctios i real-time. To trasfer iformatio across differet robotic platforms, probabilistic detectio models are employed to model the robots abilities to recogize each other. Whe oe robot detects aother, these detectio models are used to sychroize the idividual robots beliefs, thereby reducig the ucertaity of both robots durig localizatio. To accommodate the oise ad ambiguity arisig i real-world domais, detectio models are probabilistic, capturig the reliability ad accuracy of robot detectio. The costrait propagatio is implemeted usig samplig, ad desity trees (Koller ad Fratkia, 1998; Moore et al., 1997; Omohudro, 1987, 1991) are employed to itegrate iformatio from other robots ito a robot s belief. While our approach is applicable to ay sesor capable of (occasioally) detectig other robots, we preset a implemetatio that uses color cameras ad laser rage-fiders for robot detectio. The parameters of the correspodig probabilistic detectio model are leared usig a maximum likelihood estimator. Extesive experimetal results, carried out with two robots i a idoor eviromet, illustrate the appropriateess of the approach. I what follows, we will first describe the ecessary statistical mechaisms for multi-robot localizatio, followed by a descriptio of our samplig-based ad Mote Carlo localizatio techique i Sectio 3. I Sectio 4 we preset our visio-based method to detect other robots. Experimetal results are reported i Sectio 5. Fially, related work is discussed i Sectio 6, followed by a discussio of the advatages ad limitatios of the curret approach. 2. Multi-Robot Localizatio Let us begi with a mathematical derivatio of our approach to multi-robot localizatio. I the remaider we assume that robots are give a model of the eviromet (e.g., a map (Thru, 1998b)), ad that they are give sesors that eable them to relate their ow positio to this model (e.g., rage fiders, cameras). We also assume that robots ca detect each other, ad that they ca perform dead-reckoig. All of these seses are typically cofouded by oise. Further below, we will make the assumptio that the eviromet is Markov (i.e., the robots positios are the oly measurable state), ad we will also make some additioal assumptios ecessary for factorial represetatios of joit probability distributios as explaied further below. Throughout this paper, we adopt a probabilistic approach to localizatio. Probabilistic methods have bee applied with remarkable success to sigle-robot localizatio (Nourbakhsh et al., 1995; Simmos ad Koeig, 1995; Kaelblig et al., 1996; Burgard et al., 1996; Fox et al., 1999b; Burgard et al., 1998; Gutma et al., 1999), where they have bee demostrated to solve

3 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 327 problems like global localizatio ad localizatio i dese crowds Data Let N be the umber of robots, ad let d deote the data gathered by the -th robot, with 1 N. Obviously, each d is a sequece of three differet types of iformatio: 1. Odometry measuremets. Each robot cotiuously moitors its wheel ecoders (dead-reckoig) ad geerates, i regular itervals, odometric measuremets. These measuremets, which will be deoted a, specify the relative chage of positio accordig to the wheel ecoders. 2. Eviromet measuremets. The robots also query their sesors (e.g., rage fiders, cameras) i regular time itervals, which geerates measuremets deoted by o. The measuremets o establish the ecessary referece betwee the robot s local coordiate frame ad the eviromet s frame of referece. I our experimets below, o will be laser rage scas or ultrasoud measuremets. 3. Detectios. Additioally, each robot queries its sesors for the presece or absece of other robots. The resultig measuremets will be deoted r. Robot detectio might be accomplished through differet sesors tha eviromet measuremets. Below, i our experimets, we will use a combiatio of visual sesors (color camera) ad rage fiders for robot detectio. The data of all robots is deoted d with 2.2. Markov Localizatio d = d 1 d 2 d N. (1) Before turig to the topic of this paper collaborative multi-robot localizatio let us first review a commo approach to sigle-robot localizatio, which our approach is built upo: Markov localizatio. Markov localizatio uses oly dead reckoig measuremets a ad eviromet measuremets o; it igores detectios r. I the absece of detectios (or similar iformatio that ties the positio of oe robot to aother), iformatio gathered at differet platforms caot be itegrated. Hece, the best oe ca do is to localize each robot idividually, idepedetly of all others. The key idea of Markov localizatio is that each robot maitais a belief over its positio. The belief of the -th robot at time t will be deoted Bel (L). Here L is a three-dimesioal radom variable composed of a robot s x-y positio ad its headig directio θ (we will use the terms positio, pose ad locatio iterchageably). Accordigly, Bel (L = l) deotes the belief of the -th robot of beig at a specific locatio l. Iitially, at time t = 0, Bel (0) (L) reflects the iitial kowledge of the robot. I the most geeral case, which is beig cosidered i the experimets below, the iitial positio of all robots is ukow, hece Bel (0) (L) is iitialized by a uiform distributio. At time t, the belief Bel (L) is the posterior with respect to all data collected up to time t: Bel (L) = P( L ) d (2) where d deotes the data collected by the -th robot up to time t. By assumptio, the most recet sesor measuremet i d is either a eviromet or a odometry measuremet. Both cases are treated differetly, so let s cosider the former first: 1. Sesig the eviromet: Suppose the last item i d is a eviromet measuremet, deoted o. Usig the Markov assumptio (ad exploitig that the robot positio does ot chage whe the eviromet is sesed), we obtai for ay locatio l Bel (L = l) = P ( L = P( o = l d L ) ) ( = l, d (t 1) P L P ( o d (t 1) ) = l ) d (t 1) = P( o L = l ) P ( L = l ) d (t 1) P ( o d (t 1) ) = α P ( o L = l ) P ( L = l d (t 1) ) = α P ( o = α P ( o L L = l ) P ( L (t 1) = l d (t 1) = l ) Bel (t 1) (L = l) (3) where α is a ormalizer that does ot deped o the robot positio l. Notice that the posterior belief Bel (L = l) of beig at locatio l after icorporatig o is obtaied by multiplyig the )

4 328 Foxetal. perceptual model P(o belief Bel (t 1) (L = l). L = l) with the prior This observatio suggests the followig icremetal update equatio (we omit the time idex t ad the state variable L for brevity): Bel (l) α P(o l) Bel (l) (4) The coditioal probability P(o l) is called the eviromet perceptio model of robot ad describes the likelihood of perceivig o give that the robot is at positio l. I Markov localizatio, it is assumed to be give ad costat over time. For proximity sesors such as ultrasoud sesors or laser rage-fiders, the probability P(o l) ca be approximated by P(o o l ), which is the probability of observig o coditioed o the expected measuremet o l at locatio l. The expected measuremet, a distace i this case, is easily computed from the map usig ray tracig. Figure 1 shows this perceptio model for laser rage-fiders. Here the x-axis is the distace o l expected give the world model, ad the y-axis is the distace o measured by the sesor. The fuctio is a mixture of a Gaussia (cetered aroud the correct distace o l ), a Geometric distributio (modelig overly short readigs) ad a Dirac distributio (modelig max-rage readigs). It itegrates the accuracy of the sesor with the likelihood of receivig a radom measuremet (e.g., due to obstacles ot modeled i the map (Fox et al., 1999b)). is. Usig the 2. Odometry: Now suppose the last item i d a odometry measuremet, deoted a Theorem of Total Probability ad exploitig the Markov property, we obtai Bel (L = l) = P ( L = = = P ( L P ( L (t 1) P ( L P ( L (t 1) P ( L = l d ) = l d, L(t 1) = l ) = l ) d dl = l a, L(t 1) = l ) = l ) d (t 1) dl = l a, L(t 1) = l ) Bel (t 1) (L = l ) dl (5) which suggests the icremetal update equatio: Bel (l) P(l a, l ) Bel (l ) dl (6) Here P(l a, l ) is called the motio model of robot. Figure 2 illustrates the resultig desities for two example paths. As the figure suggests, a motio model is basically a model of robot kiematics aotated with ucertaity. These equatios together form the basis of Markov localizatio, a icremetal probabilistic algorithm for estimatig robot positios. Markov localizatio relies o kowledge of P(o l) ad P(l a, l ). The former coditioal typically requires a model (map) of the eviromet. As oticed above, Markov localizatio has bee applied with great practical success to mobile robot localizatio. However, it is oly applicable to sigle-robot localizatio, ad caot take advatage of robot detectio measuremets. Thus, i its curret form it caot exploit relative iformatio betwee differet robots positios i ay sesible way. Figure 1. Perceptio model for laser rage fiders. The x-axis depicts the expected measuremet, the y-axis the measured distace, ad the vertical axis depicts the likelihood. The peak marks the most likely measuremet. The robots are also give a map of the eviromet, to which this model is applied Multi-Robot Markov Localizatio The key idea of multi-robot localizatio is to itegrate measuremets take at differet platforms, so that each

5 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 329 Figure 2. Motio model represetig the ucertaity i robot motio: The robot s belief starts with a Dirac distributio ad the lies represet the trajectories of the robot. Both distributios are three-dimesioal (i x, y,θ -space) ad show are their 2D projectios ito x, y -space. robot ca beefit from data gathered by robots other tha itself. At first glace, oe might be tempted to maitai a sigle belief over all robots locatios, i.e., L = L 1 L 2 L N (7) Ufortuately, the dimesioality of this vector grows with the umber of robots. Distributios over L are, hece, expoetial i the umber of robots. Moreover, sice each robot positio is described by three values (its x-y positio ad its headig directio θ), L is of dimesio 3N. Thus, modelig the joit distributio of the positios of all robots is ifeasible already for small values of N. Our approach maitais factorial represetatios; i.e., each robot maitais its ow belief fuctio that models oly its ow ucertaity, ad occasioally, e.g., whe a robot sees aother oe, iformatio from oe belief fuctio is trasfered from oe robot to aother. The factorial represetatio assumes that the distributio of L is the product of its N margial distributios: P ( L 1,...,L N d ) = P ( L 1 d )... P ( L N d ) (8) Strictly speakig, the factorial represetatio is oly approximate, as oe ca easily costruct situatios where the idepedece assumptio does ot hold true. However, the factorial represetatio has the advatage that the estimatio of the posteriors is coveietly carried out locally o each robot. I the absece of detectios, this amouts to performig Markov localizatio idepedetly for each robot. Detectios are used to provide additioal costraits betwee the estimated pairs of robots, which will lead to refied local estimates. To derive how to itegrate detectios ito the robots beliefs, let us assume that robot is detected by robot m ad the last item i d m is a detectio variable, deoted r m. For the momet, let us assume this is the oly such detectio variable i d, ad that it provides iformatio about the locatio of the -th robot relative to robot m (with m ). The Bel (L = l) = P ( L = l d ) = P ( L = P ( L = d = l d ) P ( L P ( L m = l d m (t 1) ) P ( L ) dl = l d m ) = l L m = l, r m which suggests the icremetal update equatio: Bel (l) Bel (l) P ( L = l L m = l ), r m ) (9) Bel m (l ) dl (10) Here P(L = l L m = l, r m ) Bel m (l ) dl describes robot m s belief about the detected robot s positio. The reader may otice that, by symmetry, the same detectio ca be used to costrai the m-th robot s positio based o the belief of the -the robot. The derivatio is omitted sice it is fully symmetrical. Table 1 summarizes the multi-robot Markov localizatio algorithm. The time idex t ad the state variable L is omitted wheever possible. Of course, this

6 330 Foxetal. Table 1. Multi-robot Markov localizatio algorithm for robot umber. for each locatio l do /* iitialize the belief */ Bel (l) P(L (0) = l) ed for forever do if the robot receives ew sesory iput o do for each locatio l do /* apply the perceptio model */ Bel (l) α P(o l) Bel (l) ed for ed if if the robot receives a ew odometry readig a do for each locatio l do /* apply the motio model */ Bel (l) P(l a,l )Bel (l ) dl ed for ed if if the robot is detected by the m-th robot do for each locatio l do /* apply the detectio model */ ed for ed if ed forever Bel (l) Bel (l) P(L = l L M = l, r m ) Bel m (l ) dl algorithm is oly a approximatio, sice it makes certai idepedece assumptios (e.g. it excludes that a sesor reports I saw a robot, but I caot say which oe ), ad strictly speakig it is oly correct if there is oly a sigle r i the etire ru. Furthermore, repeated itegratio of aother robot s belief accordig to (9) results i usig the same evidece twice. Hece, robots ca get overly cofidet i their positio. To reduce the dager arisig from the factorial distributio, our approach uses the followig two rules. 1. Our approach igores egative sights, i.e., evets where a robot does ot see aother robot. 2. It icludes a couter that, oce a robot has bee sighted, blocks the ability to see the same robot agai util the detectig robot has traveled a prespecified distace (2.5 m i our experimets). I our curret approach, this distace is based purely o experiece ad i future work we will test the applicability of formal iformatio-theoretic measures for the errors itroduced by our factorized represetatio (see e.g., Boye ad Koller (1999)). I our practical experimets described below we did ot realize ay evidece that these two rules are ot sufficiet. Istead, our approach to collaborative localizatio based o the factorial represetatio still yields superior performace over robot teams with idividual localizatio ad without ay robot detectio capabilities. 3. Samplig ad Mote Carlo Localizatio The previous sectio left ope how the belief about the robot positio is represeted. I geeral, the space of all robot positios is cotiuous-valued ad o parametric model is kow that would accurately model arbitrary beliefs i such robotic domais. However, practical cosideratios make it impossible to model arbitrary beliefs usig digital computers Mote Carlo Localizatio The key idea here is to approximate belief fuctios usig a Mote Carlo method. More specifically, our

7 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 331 approach is a extesio of Mote Carlo localizatio (MCL), which was recetly proposed i Dellaert et al. (1999) ad Fox et al. (1999a). MCL is a versio of Markov localizatio that relies o sample-based represetatios ad the samplig/importace re-samplig algorithm for belief propagatio (Rubi, 1988). MCL represets the posterior beliefs Bel (L) by a set of K weighted radom samples, or particles, deoted S ={s i i = 1..K }. A sample set costitutes a discrete distributio ad samples i MCL are of the type s i = l i,p i (11) where l i = x i,y i,θ i deotes a robot positio, ad p i 0 is a umerical weightig factor, aalogous to a discrete probability. For cosistecy, we assume K i=1 p i = 1. I the remaider we will omit the subscript i wheever possible. I aalogy with the geeral Markov localizatio approach outlied i Sectio 2, MCL proceeds i two phases: 1. Robot motio. Whe a robot moves, MCL geerates K ew samples that approximate the robot s positio after the motio commad. Each sample is geerated by radomly drawig a sample from the previously computed sample set, with likelihood determied by their p-values. Let l deote the positio of this sample. The ew sample s l is the geerated by geeratig a sigle, radom sample from P(l l, a), usig the odometry measuremet a. The p-value of the ew sample is K 1. Figure 3 Figure 3. Samplig-based approximatio of the positio belief for a o-sesig robot. The solid lie displays the trajectory, ad the samples represet the robot s belief at differet poits i time. shows the effect of this samplig techique for a sigle robot, startig at a iitial kow positio (bottom ceter) ad executig actios as idicated by the solid lie. As ca be see there, the sample sets approximate distributios with icreasig ucertaity, represetig the gradual loss of positio iformatio due to slippage ad drift. 2. Eviromet measuremets are icorporated by reweightig the sample set, which is aalogous to Bayes rule i Markov localizatio. More specifically, let l, p be a sample. The p α P(o l) (12) where o is a sesor measuremet, ad α is a ormalizatio costat that eforces K i=1 p i = 1. The icorporatio of sesor readigs is typically performed i two phases, oe i which p is multiplied by P(o l), ad oe i which the various p-values are ormalized. A algorithm to perform this resamplig process efficietly i O(K ) time is give i Carpeter et al. (1997). I practice, we have foud it useful to add a small umber of uiformly distributed, radom samples after each estimatio step (Fox et al., 1999a). Formally, these samples ca be uderstood as a modified motio model that allows, with very small likelihood, arbitrary jumps i the eviromet. The radom samples are eeded to overcome local miima: Sice MCL uses fiite sample sets, it may happe that o sample is geerated close to the correct robot positio. This may be the case whe the robot loses track of its positio. I such cases, MCL would be uable to re-localize the robot. By addig a small umber of radom samples, however, MCL ca effectively re-localize the robot, as documeted i our experimets described i Fox et al. (1999a) (see also the discussio o loss of diversity i Doucet (1998)). Aother modificatio to the basic approach is based o the observatio that the best sample set sizes ca vary drastically (Koller ad Fratkia, 1998). Durig global localizatio, a robot may be completely igorat as to where it is; hece, it s belief uiformly covers its full three-dimesioal state space. Durig positio trackig, o the other had, the ucertaity is typically small. MCL determies the sample set size o-the-fly: It typically uses may samples durig global localizatio or if the positio of the robot is lost, ad oly a small umber of samples is used durig positio trackig (see Fox et al. (1999a) for details).

8 332 Foxetal Properties of MCL. MCL is based o a family of techiques geerically kow as particle filters, or samplig/importace re-samplig (Rubi, 1988). A overview ad discussio of the properties of these filters ca be foud i Doucet (1998). Particle filters are kow alteratively as the bootstrap filter (Gordo et al., 1993), the Mote-Carlo filter (Kitagawa, 1996), the Codesatio algorithm (Isard ad Blake, 1996, 1998), or the survival of the fittest algorithm (Kaazawa et al., 1995). A ice property of particle filters is that they ca uiversally approximate arbitrary probability distributios. As show i Taer (1993), the sample-based distributios smoothly approximate the correct oe at a rate of 1/ K as K goes to ifiity ad uder coditios that are true for MCL. The sample set size aturally trades off accuracy ad computatio. The true advatage, however, lies i the way MCL places computatioal resources. By samplig i proportio to the likelihood, MCL focuses its computatioal resources o regios with high likelihood, where thigs really matter. MCL also leds itself icely to a ay-time implemetatio (Dea ad Boddy, 1988; Zilberstei ad Russell, 1995). Ay-time algorithms ca geerate a aswer at ay time; however, the quality of the solutio icreases over time. The samplig step i MCL ca be termiated at ay time. Thus, whe a sesor readig arrives, or a actio is executed, samplig is termiated ad the resultig sample set is used for the ext operatio A Global Localizatio Example. Figure 4(a) (c) illustrates MCL whe applied to localizatio of a sigle mobile robot. Show there is a series of sample sets (projected ito 2D) geerated durig global localizatio of the mobile robot Rhio operatig i a office buildig. I Fig. 4(a), the robot is globally ucer- tai; hece the samples are spread uiformly over the free-space. Figure 4(b) shows the sample set after approximately 1.5 meters of robot motio, at which poit MCL has disambiguated the robot s positio maily up to a sigle symmetry. Fially, after aother 4 meters of robot motio, the ambiguity is resolved, the robot kows where it is. The majority of samples is ow cetered tightly aroud the correct positio, as show i Fig. 4(c). All ecessary computatio is carried out i real-time o a low-ed PC Multi-Robot MCL The extesio of MCL to collaborative multi-robot localizatio is ot straightforward. This is because uder our factorial represetatio, each robot maitais its ow, local sample set. Whe oe robot detects aother, both sample sets are sychroized usig the detectio model, accordig to the update equatio Bel (L = l) Bel (L = l) P(L = l L m = l, r m ) Bel m (L = l ) dl (13) Notice that this equatio requires the multiplicatio of two desities. Sice samples i Bel (L) ad Bel m (L) are draw radomly, it is ot straightforward to establish correspodece betwee idividual samples i Bel (L) ad P(L = l L m = l, r m ) Bel m (L = l ) dl. To remedy this problem, our approach trasforms sample sets ito desity fuctios usig desity trees (Koller ad Fratkia, 1998; Moore et al., 1997; Omohudro, 1987, 1991). These methods approximate sample sets usig piecewise costat desity fuctios represeted by a tree. Each ode i a desity tree is aotated with a hyper-rectagular subspace of the three-dimesioal state space of the robot. Iitially, all Figure 4. Global localizatio: (a) Iitializatio, (b) ambiguity due to symmetry, ad (c) achieved localizatio.

9 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 333 Figure 5. (a) Map of the eviromet alog with a sample set represetig the robot s belief durig global localizatio, ad (b) its approximatio usig a desity tree. The tree trasforms the discrete sample set ito a cotiuous distributio, which is ecessary to geerate ew importace factors for the idividual sample poits represetig the belief of aother robot. samples are assiged to the root ode, which covers the etire state space. The tree is grow by recursively splittig each ode util a certai stoppig coditio is fulfilled (see Thru et al. (1999b) for details). If a ode is split, its iterval is divided ito two equally sized itervals alog its logest dimesio. Figure 5 shows a example sample set alog with the tree extracted from this set. The resolutio of the tree is a fuctio of the desities of the samples: the more samples exist i a regio of space, the fier-graied the tree represetatio. After the tree is grow, each leaf s desity is give by the quotiet of the sum of all weights p of all samples that fall ito this leaf, divided by the volume of the regio covered by the leaf. The latter amouts to maximum likelihood estimatio of (piecewise) costat desity fuctios. To implemet the update equatio, our approach approximates the desity i Eq. (13) usig samples, just as described above. The resultig sample set is the trasformed ito a desity tree. These desity values are the multiplied ito each idividual sample l, p of the detected robot accordig to Eq. (14). p α P(l L = l, r ) Bel(L =l )dl (14) The resultig sample set is a refied desity for the -th robot, reflectig the detectio ad the belief of the m-th robot. Please ote that the same update rule ca be applied i the other directio, from robot to robot m. Sice the equatios are completely symmetric, they are omitted here. 4. Probabilistic Detectio Model To implemet the multi-robot Mote-Carlo localizatio techique, robots must possess the ability to sese each other. The crucial compoet is the detectio model P(L = l L m = l, r m ) which describes the coditioal probability that robot is at locatio l, give that robot m is at locatio l ad perceives robot with measuremet r m. From a mathematical poit of view, our approach is sufficietly geeral to accommodate a wide rage of sesors for robot detectio, assumig that the coditioal desity P(L L m, r m ) is adjusted accordigly. We will ow describe a specific detectio method that itegrates iformatio from multiple sesor modalities. This method, which itegrates camera ad rage iformatio, will be employed throughout our experimets (see Kruppa (1999) for more details) Detectio To determie the relative locatio of other robots, our approach combies visual iformatio obtaied from a o-board camera, with proximity iformatio comig from a laser rage-fider. Camera images are used to detect other robots, ad laser rage-fider scas are used to determie the relative positio of the detected robot ad its distace. The top row i Fig. 6 shows examples of camera images recorded i a corridor. Each image shows a robot, marked by a uique, colored marker to facilitate its recogitio. Eve though the robot is oly show with a fixed orietatio i this figure, the marker ca be detected regardless of the robot s orietatio. To fid robots i a camera image, our approach first filters the image by employig local color histograms ad decisio trees tued to the colors of the marker. Thresholdig is the employed to search for the marker s characteristic color trasitio. If foud, this implies that a robot is preset i the image. The small

10 334 Foxetal. Figure 6. Traiig data of successful detectios for the robot perceptio model. Each image i the top row shows a robot, marked by a uique, colored marker to facilitate recogitio. The bottom row shows the correspodig laser scas ad the dark lie i each diagram depicts the extracted locatio of the robot i polar coordiates, relative to the positio of the detectig robot (the laser scas are scaled for illustratio purposes). black rectagles, superimposed o each marker i the images i Fig. 6, illustrate the ceter of the marker as idetified by this visual routie. Curretly, images are aalyzed at a rate of 1 Hz, with the mai delay beig caused by the camera s parallel port iterface. 1 This slow rate is sufficiet for the applicatio at had. Oce a robot has bee detected, the curret laser sca is aalyzed for the relative locatio of the robot i polar coordiates (distace ad agle). This is doe by searchig for a covex local miimum i the distaces of the sca, usig the agle obtaied from the camera image as a startig poit. Here, tight sychroizatio of photometric ad rage data is very importat, especially because the detectig robot might sese ad rotate simultaeously. I our framework, sesor sychroizatio is fully cotrollable because all data is tagged with timestamps. We foud that the described multi-sesor method is robust ad gives accurate results eve i cluttered eviromets. The bottom row i Fig. 6 shows laser scas ad the result of our aalysis for the example situatios depicted i the top row of the same figure. Each sca cosists of 180 distace measuremets with approximately 5 cm accuracy, spaced at 1 degree agular distace. The dark lie i each diagram depicts the extracted locatio of the robot i polar coordiates, relative to the positio of the detectig robot. All scas are scaled for illustratio purposes. detectio evet by the m-th robot, which comprises the idetity of the detected robot (if ay), ad its relative locatio i polar coordiates. The variable L describes the locatio of the detected robot (here with m refers to a arbitrary other robot), ad L m rages over locatios of the m-th robot. As described above, we will restrict our cosideratios to positive detectios, i.e., cases where a robot m did detect a robot. Negative detectio evets (a robot m does ot see a robot ) are beyod the scope of this paper ad will be igored. The detectio model is traied usig data. More specifically, durig traiig we assume that the exact locatio of each robot is kow. Wheever a robot takes a camera image, its locatio is aalyzed as to whether other robots are i its visual field. This is doe by a geometric aalysis of the eviromet, exploitig the fact that the locatios of all robots are kow durig traiig. The, the image is aalyzed, ad for each detected robot the idetity ad relative locatio is recorded. This data is sufficiet to trai the detectio model P(L L m, r m ). I our implemetatio, we employ a parametric mixture model to represet P(L L m, r ). Our approach models false-positive ad false-egative detectios usig a biary radom variable. Table 2 shows the ratios Table 2. Rates of false-positives ad false-egatives for our detectio routie Learig the Detectio Model Robot detected No robot detected Next, we have to devise a probabilistic detectio model of the type P(L L m, r m ). To recap, r m deotes a Robot i field of view 93.3% 6.7% No robot i field of view 3.5% 96.5%

11 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 335 Figure 7. Gaussia desity represetig the robot perceptio model. The x-axis represets the deviatio of relative agle ad the y-axis the error i the distace betwee the two robots. of these errors estimated from a traiig set of 112 images, i half of which aother robot is withi the field of view. As ca be see, our curret visual routies have a 6.7% chace of ot detectig a robot i their visual field, ad oly a 3.5% chace of erroeously detectig a robot whe there is oe. The Gaussia distributio show i Fig. 7 models the error i the estimatio of a robot s locatio. Here the x-axis represets the agular error, ad the y-axis the distace error. This Gaussia has bee obtaied through maximum likelihood estimatio based o the traiig data. As is easy to be see, the Gaussia is zero-cetered alog both dimesios, ad it assigs low likelihood to large errors. The correlatio betwee both compoets of the error, agle ad distace, are approximately zero, suggestig that both errors might be idepedet. Assumig idepedece betwee the two errors, we foud the mea error of the distace estimatio to be 48.3 cm, ad the mea agular error to be 2.2 degree. To obtai the traiig data, the true locatio was ot determied maually; istead, MCL was applied for positio estimatio (with a kow startig positio ad very large sample sets). Empirical results i Dellaert et al. (1999) suggest that MCL is sufficietly accurate for trackig a robot with oly a few cetimeters error. The robots positios, while movig at speeds like 30 cm/sec through our eviromet, were sychroized ad the further aalyzed geometrically to determie whether (ad where) robots are i the visual fields of other robots. As a result, data collectio is extremely easy as it does ot require ay maual labelig; however, the error i MCL leads to a slightly less cofied detectio model tha oe would obtai with maually labeled data (assumig that the accuracy of maual positio estimatio exceeds that of MCL). 5. Experimetal Results I this sectio we preset experimets coducted with real ad simulated robots. The cetral questio drivig our experimets was: To what extet ca cooperative multi-robot localizatio improve the localizatio quality, whe compared to covetioal sigle-robot localizatio? I the first set of experimets, our approach was tested usig two Pioeer robots (Robi ad Maria) marked optically by a colored marker, as show i Fig. 6. I order to evaluate the beefits of multirobot localizatio i more complex scearios, we additioally performed experimets i simulated eviromets. These experimets are described i Sectio Experimets Usig Real Robots Figure 8 shows the setup of our experimets alog with a part of the occupacy grid map (Thru, 1998b) Figure 8. Map of the eviromet alog with a typical path take by Robi durig a experimet. Mario is operatig i the lab facig towards the opeig of the hallway.

12 336 Foxetal. Figure 9. Detectio evet: (a) Sample set of Maria as it detects Robi i the corridor. (b) Sample set reflectig Maria s belief about Robi s positio. (c) Tree-represetatio of this sample set ad (d) correspodig desity. Figure 10. Sample set represetig Robi s belief (a) as it passes Maria ad (b) after icorporatig Maria s measuremet. used for positio estimatio. Maria operates i our lab, which is the cluttered room adjacet to the corridor. Because of the o-symmetric ature of the lab, the robot kows fairly well where it is (the samples represetig Maria s belief are plotted i Fig. 9(a)). Figure 8 also shows the path take by Robi, which was i the process of global localizatio. Figure 10(a) represets the typical belief of Robi whe it passes the lab i which Maria is operatig. Sice Robi already moved several meters i the corridor, it developed a belief which is cetered alog the mai axis of the corridor. However, the robot is still highly ucertai about its exact locatio withi the corridor ad eve does ot kow its global headig directio. Please ote that due to the lack of features i the corridor the robots geerally have to travel a log distace util they ca resolve ambiguities i the belief about their positio. The key evet, illustratig the utility of cooperatio i localizatio, is a detectio evet. More specifically, Maria, the robot i the lab, detects Robi, as it moves through the corridor (see Fig. 6 for the camera image ad laser rage sca of a characteristic measuremet of this type). Usig the detectio model described i Sectio 4, Maria geerates a ew sample set as show i Fig. 9(b). This sample set is coverted ito a desity usig desity trees (see Fig. 9(c) ad (d)). Maria the trasmits this desity to Robi which itegrates it ito its curret belief. The effect of this itegratio o Robi s belief is show i Fig. 10(b). It shows Robi s belief after itegratig the desity represetig Maria s detectio. As this figure illustrates, this sigle icidet almost completely resolves the ucertaity i Robi s belief. We coducted te experimets of this kid ad compared the performace to covetioal MCL for sigle robots which igores robot detectios. To measure the performace of localizatio we determied the true locatios of the robot by measurig the startig positio of each ru ad performig positio trackig off-lie usig MCL. For each ru, we the computed the estimatio error at the referece positios. The estimatio error is measured by the average distace of all samples from the referece positio. The results are summarized i Fig. 11. The graph plots the estimatio error as a fuctio of time, averaged over the te experimets, alog with their 95% cofidece itervals (bars). As ca be see i the figure, the quality of positio estimatio icreases much faster whe usig multi-robot localizatio. Please ote that the detectio evet typically took place secods after the start of a experimet. Obviously, this experimet is specifically wellsuited to demostrate the advatage of detectios i multi-robot localizatio, sice the robots ucertaities

13 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 337 Figure 11. Compariso betwee sigle-robot localizatio ad localizatio makig use of robot detectios. The x-axis represets the time ad the y-axis represets the estimatio error obtaied by averagig over te experimets. are somewhat orthogoal, makig the detectio highly effective. I order to test the performace of our approach i more complex situatios, we additioally performed experimets i two simulatio eviromets Simulatio Experimets I the followig experimets we used a simulatio tool which simulates robots o the sesor level, providig raw odometry ad proximity measuremets (see Schulz et al. (1999) for details). Sice the simulatio icludes sesor oise, the results are directly trasferable to real robots. Robot detectios were simulated by usig the positios of the robots ad visibility costraits extracted from the map. Noise was added to these detectios accordig to the errors extracted from the traiig data usig our real robots. It should be oted that falsepositive detectios were ot cosidered i these experimets (see Sectio 7.2 for a discussio of false-positive detectios) Homogeeous Robots. I the first simulatio experimet we use eight robots, which are all equipped with ultrasoud sesors. The task of the robots is to perform global localizatio i the hallway eviromet show i Fig. 12(a). This eviromet is particularly Figure 12. (a) Symmetric hallway eviromet. (b) Localizatio error for eight robots performig global localizatio simultaeously. The dashed lie shows the error over time whe performig sigle-robot MCL ad the solid lie plots the error usig our multi-robot method.

14 338 Foxetal. Figure 13. Hexagoal eviromet with edges of legth 8 meters. Distiguishig obstacles ca oly be detected either with (a) soar sesors or (b) laser rage-fiders. Typical sample sets represetig the positio ucertaity of robots equipped with (a) soar sesors or (b) laser rage-fiders. challegig for sigle robot systems sice a robot has to either pass the ope space o the left corridor marked A, or it has to move through all other hallways marked B, C, ad D to uiquely determie its positio. However, the localizatio task remais hard eve if there are multiple robots which ca detect each other ad ca exchage their beliefs. Sice all robots have to perform global localizatio at the same time, several robot detectios ad belief trasfers are ecessary to sigificatly reduce the distace to be traveled by each robot. As i the previous experimet, we compare the performace of our multi-robot localizatio approach to the performace of sigle-robot localizatio igorig robot detectios. Figure 12(b) shows the localizatio errors for both methods averaged over eight rus of global localizatio usig eight robots simultaeously i each ru. The plot shows that the exploitatio of detectios i robot teams results i a highly superior localizatio performace. The surprisigly high error values for teams ot performig collaborative localizatio are due to the fact that eve after 600 secods, some of the robots are still ucertai about their positio. Aother measure of performace is the average time it takes for a robot to fid out where it is. We assume that a robot has successfully localized itself, if the localizatio error falls below 1.5 meters. As metioed above, this error is give by averagig over the distace of all samples from a referece positio. Without makig use of robot detectios, a robot eeds 379 ±37 secods to uiquely determie its positio. Our approach to multi-robot localizatio reduces this time by 60% to 153 ±17 secods Heterogeeous Robots. The goal of this experimet is to demostrate the potetial beefits for heterogeeous teams of robots. Here, the heterogeeity is due to differet types of sesors: Oe group of robots uses soar sesors for localizatio ad the other robots are equipped with laser rage-fiders. The tests are carried out i the eviromet show i Fig. 13. This eviromet is highly symmetric ad oly certai objects allow the robots to reduce their positio ucertaity. These objects ca be detected either by soar sesors or by laser rage-fiders (see Fig. 13(a) ad (b)). The positio of these obstacles is chose so that ay robot equipped with oly oe of the sesor types is ot able to determie uiquely where it is. Whereas robots usig soar sesors for localizatio caot distiguish betwee three possible robot locatios (see Fig. 13(c)), robots equipped with laser rage-fiders remai ucertai about two possible locatios (see Fig. 13(d)). As i the previous experimet, eight robots are placed i the eviromet ad their task is to fid out where they are. Four of the robots are equipped with ultraasoud sesors ad the other four robots use laser rage-fiders. The localizatio error for the differet settigs is plotted i Fig. 14. Not surprisigly, the error for sigle-robot localizatio decreases i the begiig of the experimets, but remais at a sigificatly high level. The correspodig curves are depicted by the dashed lies (soar black, laser grey) i Fig. 14. The results obtaied whe the robots are able to make use of detectios are preseted as solid lies (soar black, laser grey). As ca be see, both teams of robots beefit from the additioal iformatio provided by the sesors of the other robots. As a result, each robot is able to uiquely determie its positio. 6. Related Work Mobile robot localizatio has frequetly bee recogized as a key problem i robotics with sigificat

15 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 339 Figure 14. Localizatio error for robots equipped with soar sesors (black lies) or laser rage-fiders (grey lies). The solid lies summarize results obtaied by multi-robot localizatio ad the dashed lies are obtaied whe igorig robot detectios. practical importace. A recet book by Borestei et al. (1996) provides a excellet overview of the stateof-the-art i localizatio. Localizatio plays a key role i various successful mobile robot architectures preseted i Cox (1991), Fukuda et al. (1993), Hikel ad Kierieme (1988), Leoard ad Durrat-Whyte (1992), Leoard et al. (1992), Neve ad Schöer (1996), Peters et al. (1994), Recke (1993) ad Weiß et al. (1994) ad various chapters i Kortekamp et al. (1998). While some localizatio approaches, such as those described i Horswill (1994), Kortekamp ad Weymouth (1994), Simmos ad Koeig (1995) ad Kaelblig et al. (1996) localize the robot relative to ladmarks i a topological map, our approach localizes the robot i a metric space, just like those methods proposed i Betke ad Gurvits (1993), Thru (1998a) ad Thru et al. (1998). Almost all existig approaches address sigle-robot localizatio oly. Moreover, the vast majority of approaches is icapable of localizig a robot globally; istead, they are desiged to track the robot s positio by compesatig small odometric errors. Thus, they differ from the approach described here i that they require kowledge of the robot s iitial positio; ad they are ot able to recover from global localizig failures. Probably the most popular method for trackig a robot s positio is Kalma filterig (Gutma ad Schlegel, 1996; Gutma et al., 1999; Lu ad Milios, 1997; Maybeck, 1990; Schiele ad Crowley, 1994; Smith et al., 1990), which represets ucertaity by the first ad secod momets of the desity. These approaches are uable to localize robots uder global ucertaity a problem which Egelso called the kidapped robot problem (Egelso, 1994). Recetly, several researchers proposed Markov localizatio, which eables robots to localize themselves uder global ucertaity (Burgard et al., 1996; Kaelblig et al., 1996; Nourbakhsh et al., 1995; Simmos ad Koeig, 1995; Koolige, 1999). Global approaches have two importat advatages over local oes: First, the iitial locatio of the robot does ot have to be specified ad, secod, they provide a additioal level of robustess, due to their ability to recover from localizatio failures. Amog the global approaches those usig metric represetatios of the space such as MCL ad (Burgard et al., 1996, 1998; Koolige, 1999) ca deal with a wider variety of eviromets tha those methods relyig o topological maps. For example, they are ot restricted to orthogoal eviromets cotaiig pre-defied features such as corridors, itersectios ad doors. I additio, most existig approaches are restricted i the type of features that they cosider. May approaches reviewed i Borestei et al. (1996) are limited i that they require modificatios of the eviromet. Some require artificial ladmarks such as bar-code reflectors (Everett et al., 1994), reflectig tape, ultrasoic beacos, or visual patters that are easy to recogize, such as black rectagles with white dots (Borestei, 1987). Of course, modifyig the eviromet is ot a optio i may applicatio domais. Some of the more advaced approaches use more atural ladmarks that do ot require modificatios of the eviromet. For example, the approaches of

16 340 Foxetal. Kortekamp ad Weymouth (1994) ad Matarić (1990) use gateways, doors, walls, ad other vertical objects to determie the robot s positio. The Helpmate robot uses ceilig lights to positio itself (Kig ad Weima, 1990). Dark/bright regios ad vertical edges are used i Collet ad Cartwright (1985), Wolfart et al. (1995) ad hallways, opeigs ad doors are used by the approaches described i Kaelblig et al. (1996), Shatkey ad Kaelblig (1997) ad Simmos ad Koeig (1995). Others have proposed methods for learig what feature to extract, through a traiig phase i which the robot is told its locatio (Greier ad Isukapalli, 1994; Oore et al., 1997; Thru, 1998a). These are just a few represetative examples of may differet features used for localizatio. Our approach differs from all these approaches i that it does ot extract predefied features from the sesor values. Istead, it directly processes raw sesor data. Such a approach has two key advatages: First, it is more uiversally applicable sice fewer assumptios are made o the ature of the eviromet; ad secod, it ca utilize all sesor iformatio, typically yieldig more accurate results. Other approaches that process raw sesor data ca be foud i Koolige (1999), Gutma ad Schlegel (1996) ad Lu ad Milios (1997). The issue of cooperatio betwee multiple mobile robots has gaied icreased iterest i the past (see Cao et al. (1997) ad Arki ad Balch (1998) for overviews). I this cotext most work o localizatio has focused o the questio of how to reduce the odometry error usig a cooperative team of robots. Kurazume ad Shigemi (1994), for example, divide the robots ito two groups. At every poit i time oly oe of the groups is allowed to move, while the other group remais at its positio. Whe a motio commad has bee executed, all robots stop, perceive their relative positio, ad use this to reduce errors i odometry. While this method reduces the odometry error of the whole team of robots it is ot able to perform global localizatio; either ca it recover from sigificat sesor errors. Rekleitis ad colleagues (1997) preset a cooperative exploratio method for multiple robots, which also addresses localizatio. To reduce the odometry error, they use a approach closely related to the oe described i Kurazume ad Shigemi (1994). Here, too, oly oe robot is allowed to move at ay poit i time, while the other robots observe the movig oe. The statioary robots track the positio of the movig robot, thus providig more accurate positio estimates tha could be obtaied with pure dead-reckoig. Fially, i Borestei (1995), a method is preseted that relies o a compliat likage of two mobile robots. Special ecoders o the likage estimate the relative positios of the robots while they are i motio. The author demostrates that the dead-reckoig accuracy of the compliat likage vehicle is substatially improved. However, all these approaches oly seek to reduce the odometry error. Noe of them icorporates evirometal feedback ito the estimatio, ad cosequetly they are uable to localize robots relative to each other, or relative to their eviromets, from scratch. Eve if the iitial locatio of all robots are kow, they ultimately will get lost but at a slower pace tha a comparable sigle robot. The problem addressed i this paper differs i that we are iterested i collaborative localizatio i a global frame of referece, ot just reducig odometry error. I particular, our approach addresses cooperative global localizatio i a kow eviromet. 7. Coclusio 7.1. Summary We have preseted a statistical method for collaborative mobile robot localizatio. At its core, our approach uses probability desity fuctios to represet the robots estimates as to where they are. To avoid expoetial complexity i the umber of robots, a factorial represetatio is advocated where each robot maitais its ow, local belief fuctio. A fast, uiversal sampligbased scheme is employed to approximate beliefs. The probabilistic ature of our approach makes it possible that teams of robots perform global localizatio, i.e., they ca localize themselves from scratch without iitial kowledge as to where they are. Durig localizatio, robots ca detect each other. Here we use a combiatio of camera images ad laser rage scas to determie aother robot s relative locatio. The reliability of the detectio routie is modeled by learig a parametric detectio model from data, usig the maximum likelihood estimator. Durig localizatio, detectios are used to itroduce additioal probabilistic costraits, that tie oe robot s belief to aother robot s belief fuctio. To combie sample sets geerated at differet robots (each robot s belief is represeted by a separate sample set), our approach trasforms detectios ito desity trees, which approximate discrete sample sets by piecewise costat desity fuctios. These trees are the used to refie the

17 A Probabilistic Approach to Collaborative Multi-Robot Localizatio 341 weightig factors (importace factors) of other robots beliefs, thereby reducig their ucertaity i respose to the detectio. As a result, our approach makes it possible to amortize data collected at multiple platforms. This is particularly attractive for heterogeeous robot teams, where oly a small umber of robots may be equipped with high-precisio sesors. Experimetal results, carried out i real ad simulated eviromets, demostrate that our approach ca reduce the ucertaity i localizatio sigificatly, whe compared to covetioal sigle-robot localizatio. I oe of the experimets we showed that uder certai coditios, successful localizatio is oly possible if teams of heterogeeous robots collaborate durig localizatio. This experimet additioally demostrates that it is ot ecessary to equip each robot with a sesor suit eeded for global localizatio. I cotrast, oe ca sigificatly decrease costs by spreadig the differet kids of sesors amog multiple platforms, thereby geeratig a team of heterogeeous robots. Thus, whe teams of robots are placed i a kow eviromet with ukow startig locatios, our approach ca yield sigificatly better localizatio results the covetioal, sigle-robot localizatio at lower sesor costs, approximate equal computatio costs, ad relatively small commuicatio overhead Limitatios ad Discussio The curret approach possesses several limitatios that warrat future research. Not seeig each other: I our curret system, oly positive detectios are processed. Not seeig aother robot is also iformative, eve though ot as iformative as positive detectios. Icorporatig such egative detectios is geerally possible i the cotext of our statistical framework (usig the iverse weighig scheme). However, such a extesio would drastically icrease the computatioal overhead, ad it is uclear as to whether the effects o the localizatio accuracy justify the additioal computatio ad commuicatio. Idetificatio of robots: Aother limitatio of the curret approach arises from the fact that it must be able to idetify idividual robots hece they must be marked appropriately. Of course, simple meas such as bar-codes ca provide the ecessary, uique labels. However, because of the iheret ucertaity of their sesors, mobile robots must be able to deal with situatios i which they ca detect but ot idetify other robots. The factorial represetatio, however, caot deal with measuremets such as either robot A or robot B is straight i frot of me. I the worst case, this would require to cosider all possible combiatios of robots ad thus would scale expoetially i the umber of robots which is equivalet to computig distributios over the joit space of all robots. Active localizatio: The collaboratio described here is purely passive. The robots combie iformatio collected locally, but they do ot chage their course of actio so as to aid localizatio. I Burgard et al. (1997) ad Fox et al. (1998a), we proposed a algorithm for active localizatio based o iformatio-theoretic priciples, where a sigle robot actively explores its eviromet so as to best localize itself. A desirable objective for future research is the applicatio of the same priciple to coordiated multi-robot localizatio. False-positive detectios: As discussed i Sectio 4, our approach to robot detectio has a false-positive rate of 3.5%. This rate describes the chace of erroeously detectig a robot whe there is oe. While a rate of 3.5% seems to be reasoably low, it turs out to cause major problems if the robots see each other very rarely, which might happe i large eviromets. I this case, the ratio betwee truepositive ad false-positive detectios ca fall below oe, which meas that more tha 50% of all detectios are false-positive. Our sample-based implemetatio of multi-robot localizatio is ot robust to such high failure-rates ad we did ot model false-positive detectios i our experimets. Oe way to hadle such failures is to filter them out. First experimets based o the filter techiques itroduced i Fox et al. (1998b, 1999b) have show very promisig results ad will be pursued i future work. Delayed itegratio: Fially, the robots update their positio istatly wheever they perceive aother robot. I situatios i which both robots are highly ucertai at the time of the detectio it might be more appropriate to delay the update. For example, if oe of the robots afterwards becomes more certai by gatherig further iformatio about the eviromet or by beig detected by aother, certai robot, the the sychroizatio result ca be much better if it is doe retrospectively. This, however, requires that the robots keep track of their actios ad measuremets after detectig other robots.

18 342 Foxetal. Despite these ope research areas, our approach does provide a soud statistical basis for iformatio exchage durig collaborative localizatio, ad empirical results illustrate its appropriateess i practice. These results suggest that robots actig as a team are superior to robots actig idividually. Ackowledgmets Special thaks go to Dirk Schulz at the Uiversity of Bo. Without his help, the simulatio rus would ot have bee possible. This research is sposored i part by the Natioal Sciece Foudatio, DARPA via TACOM (cotract umber DAAE07-98-C-L032) ad Rome Labs (cotract umber F ), ad also by the EC (cotract umber ERBFMRX-CT ) uder the TMR programme. The views ad coclusios cotaied i this documet are those of the authors ad should ot be iterpreted as ecessarily represetig official policies or edorsemets, either expressed or implied, of NSF, DARPA, TACOM, Rome Labs, the Uited States Govermet, or the EC. Note 1. 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20 344 Foxetal. Thru, S., Beewitz, M., Burgard, W., Cremers, A.B. Dellaert, F., Fox, D., Hähel, D., Roseberg, C., Schulte, J., ad Schulz, D. 1999a. MINERVA: A secod-geeratio museum tour-guide robot. I Proc. of the Iteratioal Coferece o Robotics ad Automatio (ICRA 99). Thru, S., Fox, D., ad Burgard, W A probabilistic approach to cocurret mappig ad localizatio for mobile robots. Machie Learig, 31: Also appeard i Autoomous Robots 5, pp , joit issue. Thru, S., Lagford, J., ad Fox, D. 1999b. Mote Carlo hidde Markov models: Learig o-parametric models of partially observable stochastic processes. I Proc. of the Iteratioal Coferece o Machie Learig (ICML). Weiß, G., Wetzler, C., ad vo Puttkamer, E Keepig track of positio ad orietatio of movig idoor systems by correlatio of rage-fider scas. I Proc. of the IEEE/RSJ Iteratioal Coferece o Itelliget Robots ad Systems (IROS). Wolfart, E., Fisher, R.B., ad Walker, A Positio refiemet for a avigatig robot usig motio iformatio based o hoey bee strategies. I Proc. of the Iteratioal Symposium o Robotic Systems (SIR 95), Pisa, Italy. Zilberstei, S. ad Russell, S Approximate reasoig usig aytime algorithms. I Imprecise ad Approximate Computatio, S. Nataraja (Ed.), Kluwer Academic Publishers: Dordrecht. Haes Kruppa is a PhD studet i the Perceptioal Computig ad Computer Visio group of ETH Zurich, Switzerlad. His mai research iterests are i the field of perceptual computig ad cotext modellig. I 1999 he obtaied his M.S. i Computer Sciece from ETH. Haes Kruppa is a fellow of the Germa Merit Foudatio ad of Rotary Iteratioal. Sebastia Thru is curretly a assistat professor at the Departmet of Computer Sciece at the Caregie Mello Uiversity, Pittsburgh, PA. His research iterests iclude robotics, machie learig, ad various other aspects of artificial itelligece. He received is Ph.D. from the Uiversity of Bo i Dieter Fox has a post-doctoral positio at the Departmet of Computer Sciece at the Caregie Mello Uiversity, Pittsburgh, PA. His research iterests cover robotics, probabilistic reasoig ad artificial itelligece. Fox eared his Ph.D. degree from the Uiversity of Bo i Wolfram Burgard is a associate professor at the Departmet of Computer Sciece at the Uiversity of Freiburg. He received his Ph.D. degree i Computer Sciece from the Uiversity of Bo i His research focuses o mobile robotics ad artificial itelligece.