From: AAAI-00 Proceedngs. Copyrght 0, AAAI (www.aaa.org. All rghts reserved. Monte Carlo Localzaton Wth Mxture Proposal Dstrbuton Sebastan Thrun Deter Fox School of Computer Scence Carnege Mellon Unversty http://www.cs.cmu.edu/fthrun,dfoxg Wolfram Burgard Computer Scence Department Unversty of Freburg, Germany http://www.nformatk.un-freburg.de/burgard Abstract Monte Carlo localzaton (MCL s a Bayesan algorthm for moble robot localzaton based on partcle flters, whch has enjoyed great practcal success. Ths paper ponts out a lmtaton of MCL whch s counter-ntutve, namely that better sensors can yeld worse results. An analyss of ths problem leads to the formulaton of a new proposal dstrbuton for the Monte Carlo samplng step. Extensve expermental results wth physcal robots suggest that the new algorthm s sgnfcantly more robust and accurate than plan MCL. Obvously, these results transcend beyond moble robot localzaton and apply to a range of partcle flter applcatons. Introducton Monte Carlo Localzaton (MCL s a probablstc algorthm for moble robot localzaton that uses samples (partcles for representng probablty denstes. MCL s a verson of partcle flters [4, 10, 12, 15]. In computer vson, partcle flters are known under the name condensaton algorthm [9]. They have been appled wth great practcal success to vsual trackng problems [9, 2] and moble robot localzaton [3, 6, 11]. The basc dea of MCL s to approxmate probablty dstrbutons by sets of samples. When appled to the problem of state estmaton n a partally observable dynamcal system, MCL successvely calculates weghted sets of samples that approxmate the posteror probablty over the current state. Its practcal success stems from the fact that t s nonparametrc, hence can represent a wde range of probablty dstrbutons. It s also computatonally effcent, and t s easly mplemented as an any-tme algorthm, whch adapts the computatonal load by varyng the number of samples n the estmaton process [6]. Ths paper proposes a modfed verson of MCL, whch uses a dfferent samplng mechansm. Our study begns wth the characterzaton of a key lmtaton of MCL (and partcle flters n general. Whle MCL works well wth nosy sensors, t actually fals when the sensors are too accurate. Ths effect s undesrable: Ideally, the accuracy of any sound statstcal estmator should ncrease wth the accuracy of the sensors. An analyss of ths effect leads to the formulaton of a new samplng mechansm (.e., the proposal dstrbuton, whch changes the way samples are generated n MCL. We propose three dfferent ways of computng the mportance factors for ths new proposal dstrbuton. Our approach, whch can be vewed as the natural dual to MCL, works well n cases where Copyrght c 0, Amercan Assocaton for Artfcal Intellgence (www.aaa.org. All rghts reserved. conventonal MCL fals (and vce versa. To gan the best of both worlds, the conventonal and our new proposal dstrbuton are mxed together, leadng to a new MCL algorthm wth a mxture proposal dstrbuton that s extremely robust. Emprcal results llustrate that the new mxture proposal dstrbuton does not suffer the same lmtaton as MCL, and yelds unformly superor results. For example, our new approach wth 50 samples consstently outperforms standard MCL wth 1,000 samples. Addtonal experments llustrate that our approach yelds much better solutons n challengng varants of the localzaton problem, such as the kdnapped robot problem [5]. These experments have been carred out both n smulaton and wth data collected from physcal robots, usng both laser range data and camera mages for localzaton. Our approach generalzes a range of prevous extensons of MCL that have been proposed to allevate these problems. Exstng methods nclude the addton of random samples nto the posteror [6], the generaton of samples at locatons that are consstent wth the sensor readngs [11], or the use of sensor models that assume an artfcally hgh nose level [6]. Whle these approaches have shown superor performance over strct MCL n certan settngs, they all lack mathematcal rgor. In partcular, nether of them approxmates the true posteror, and over tme they may dverge arbtrarly. Vewed dfferently, our approach can be seen as a theory that leads to an algorthm related to the ones above (wth mportant dfferences, but also establshes a mathematcal framework that s guaranteed to work n the lmt. The paper frst revews Bayes flters, the basc mathematcal framework, followed by a dervaton of MCL. Based on experments characterzng the problems wth plan MCL, we then derve dual MCL. Fnally, the mxture proposal dstrbuton s obtaned by combnng MCL and ts dual. Emprcal results are provded that llustrate the superor performance of our new extenson of MCL. Bayes Flterng Bayes flters address the problem of estmatng the state x of a dynamcal system (partally observable Markov chan from sensor measurements. For example, n moble robot localzaton, the dynamcal system s a moble robot and ts envronment, the state s the robot s pose theren (often specfed by a poston n a Cartesan x-y space and the robot s headng drecton. Measurements may nclude range measurements, camera mages, and odometry readngs. Bayes flters assume that the envronment s Markov, that s, past and future data are (condtonally ndependent f one knows
(a (b (c Robot poston Robot poston Robot poston Fgure 1: Global localzaton of a moble robot usng MCL (10,000 samples. the current state. The key dea of Bayes flterng s to estmate a probablty densty over the state space condtoned on the data. Ths posteror s typcally called the belef and s denoted Bel(x(t recursve equaton Bel(x(t p(o(t jx(t (2 p(o(tja(t,1; : : :; o(0 p(x(tjx(t,1; a(t,1 Bel(x(t,1 dx(t,1 p(x(tjd(0:::t Here x denotes the state, x(t s the state at tme t, and d(0:::t denotes the data startng at tme 0 up to tme t. For moble robots, we dstngush two types of data: perceptual data such as laser range measurements, and odometry data or controls, whch carres nformaton about robot moton. Denotng the former by o (for observaton and the latter by a (for acton, we have Bel(x(t p(x(tjo(t ; a(t,1; o(t,1; a(t,2 : : :; o(0 (1 Wthout loss of generalty, we assume that observatons and actons arrve n an alternatng sequence. Bayes flters estmate the belef recursvely. The ntal belef characterzes the ntal knowledge about the system state. In the absence of such, t s typcally ntalzed by a unform dstrbuton over the state space. In moble robotcs, the state estmaton wthout ntal knowledge s called the global localzaton problem whch wll be the focus throughout much of ths paper. To derve a recursve update equaton, we observe that Expresson (1 can be transformed by Bayes rule to p(o(tjx(t; a(t,1; : : :; o(0 p(x(t ja(t,1; : : :; o(0 p(o(t ja(t,1; : : :; o(0 Under our Markov assumpton, p(o(tjx(t; a(t,1; : : :; o(0 can be smplfed to p(o(tjx(t, hence we have p(o(t jx(t p(x(tja(t,1; : : :; o(0 p(o(t ja(t,1; : : :; o(0 We wll now expand the rghtmost term n the denomnator by ntegratng over the state at tme t, 1 p(o(t jx(t (t (t,1 (t,1 (0 p(o(t ja(t,1; : : :; o(0 p(x jx ; a ; : : :; o p(x(t,1ja(t,1; : : :; o(0 dx(t,1 Agan, we can explot the Markov assumpton to smplfy p(x(tjx(t,1; a(t,1; : : :; o(0 to p(x(tjx(t,1; a(t, Usng the defnton of the belef Bel, we obtan the mportant p(o(t jx(t p(x(tjx(t,1; a(t,1bel(x(t,1 dx(t,1 where s a normalzaton constant. Ths equaton s of central mportance, as t s the bass for varous MCL algorthms studed here. We notce that to mplement (2, one needs to know three dstrbutons: the ntal belef Bel(x(0 (e.g., unform, the next state probabltes p(x(t jx(t,1; a(t,1, and the perceptual lkelhood p(o(t jx(t. MCL employs specfc next state probabltes p(x(tjx(t,1; a(t,1 and perceptual lkelhood models p(o(tjx(t that descrbe robot moton and percepton probablstcally. Such models are descrbed n detal elsewhere [7]. Monte Carlo Localzaton The dea of MCL (and other partcle flter algorthms s to represent the belef Bel(x by a set of m weghted samples dstrbuted accordng to Bel(x: Bel(x fx; wg1;:::;m Here each x s a sample (a state, and w s a non-negatve numercal factor (weght called mportance factors, whch sums up to one over all. In global moble robot localzaton, the ntal belef s a set of poses drawn accordng to a unform dstrbuton over the robot s unverse, and annotated by the unform mportance 1 factor m. The recursve update s realzed n three steps. t,1 Sample x Bel(x(t,1 usng mportance samplng from the (weghted sample set representng Bel(x(t,1. ( (t p(x(t jx(t,1; a(t, Obvously, the par t (t,1 hx ; x s dstrbuted accordng to the product ds- Sample x ( trbuton q(t : p(x(tjx(t,1; a(t,1 Bel(x(t,1 whch s commonly called proposal dstrbuton. (3
600 3. 4. 5. 7. 10. error (n centmeter samples 1,000 samples 10,000 samples error (n centmeter MCL (dashed: hgh error model 0 20 40 60 80 step Fgure 2: Average error of MCL as a functon of the number of robot steps/measurements. 3. To offset the dfference between the proposal dstrbuton and the desred dstrbuton (c.f., Equaton (2 p(o (t jx (t p(x (t jx (t,1 ;a (t,1 Bel(x (t,1 (4 the sample s weghted by the quotent p(o (t jx (t p(x (t jx (t,1 ;a (t,1 Bel(x (t,1 Bel(x (t,1 p(x (t jx (t,1 ;a (t,1 / p(o (t jx (t w (5 Ths s exactly the new (non-normalzed mportance factor w. After the generaton of m samples, the new mportance factors are normalzed so that they sum up to 1 (hence defne a probablty dstrbuton. It s known [17] that under mld assumptons (whch hold n our work, the sample set converges to the true posteror Bel(x (t as m goes to nfnty, wth a convergence speed n O( p 1 m. The speed may vary by a constant factor, whch depends on the proposal dstrbuton and can be sgnfcant. Examples Fgure 1 shows an example of MCL n the context of localzng a moble robot globally n an offce envronment. Ths robot s equpped wth sonar range fnders, and t s also gven a map of the envronment. In Fgure 1a, the robot s globally uncertan; hence the samples are spread unformly trough the free-space (projected nto 2D. Fgure 1b shows the sample set after approxmately 1 meter of robot moton, at whch pont MCL has dsambguated the robot s poston up to a sngle symmetry. Fnally, after another 2 meters of robot moton the ambguty s resolved, and the robot knows where t s. The majorty of samples s now centered tghtly around the correct poston, as shown n Fgure 1c. Unfortunately, data collected from a physcal robot makes t mpossble to freely vary the level of nose n sensng. Fgure 2 shows results obtaned from a robot smulaton, modelng a B21 robot localzng an object n 3D wth a mono camera whle movng around. The nose smulaton ncludes a smulaton of measurement nose, false postves (phantoms and false negatves (falures to detect the target object. MCL s drectly applcable; wth the added advantage that we can vary the level of nose arbtrarly. Fgure 2 shows systematc error curves for MCL n global localzaton for dfferent 3. 4. 5. 7. 10. sensor nose level (n % Fgure 3: Sold curve: error of MCL after steps, as a functon of the sensor nose. 95% confdence ntervals are ndcated by the bars. Notce that ths functon s not monotonc, as one mght expect. Dashed curve: Same experment wth hgh-error model. sample set szes m, averaged over 1,000 ndvdual experments. The bars n ths fgure are confdence ntervals at the 95% level. Wth 10,000 samples, the computaton load on a Pentum III ( Mh s only 14%, ndcatng that MCL s well-suted for real-tme applcatons. The results also ndcate good performance as the number of samples s large. The reader should notce that these results have been obtaned for perceptual nose level of 20% (for both false-negatve and false-postve and an addtonal poston nose that s Gaussan-dstrbuted wth a varance of 10 degrees. For our exstng robot system, the errors are n fact much lower. A Problem wth MCL As notced by several authors [4, 11, 12, 15], the basc partcle flter performs poorly f the proposal dstrbuton, whch s used to generate samples, places too lttle samples n regons where the desred posteror Bel(x t s large. Ths problem has ndeed great practcal mportance n the context of MCL, as the followng example llustrates. The sold curve n Fgure 3 shows the accuracy MCL acheves after steps, usng 1,000 samples. These results were obtaned n smulaton, enablng us to vary the amount of perceptual nose from 50% (on the rght to 1% (on the left; n partcular, we smulated a moble robot localzng an object n 3D space from mono-camera magery. It appears that MCL works best for 10% to 20% perceptual nose. The degradaton of performance towards the rght, when there s a lot of nose, barely surprses. The less accurate a sensor, the larger an error one should expect. However, MCL also performs poorly when the nose level s too small. In other words, MCL wth accurate sensors may perform worse than MCL wth naccurate sensors. Ths fndng s a bt counter-ntutve n that t suggests that MCL only works well n specfc stuatons, namely those where the sensors possess the rght amount of nose. At frst glance, one mght attempt to fx the problem by usng a perceptual lkelhood p(o (t jx (t that overestmates the sensor nose. In fact, such a strategy partally allevates the problem: The dashed curve n Fgure 3b shows the accuracy f the error model assumes a fxed 10% nose (shown there only for smaller true error rates. Whle the performance s better, ths s barely a fx. The overly pessmstc sensor model s naccurate, throwng away precous nformaton n
3. 4. 5. 7. 10. 3. 4. 5. 7. 10. error (n centmeter error (n centmeter 3. 4. 5. 7. 10. sensor nose level (n % Fgure 4: Error of MCL wth the dual (dashed lne and the mxture (sold lne proposal dstrbuton the latter s the dstrbuton advocated here. Compare the sold graph wth dashed one, and the curves n Fgure 3! the sensor readngs. In fact, the resultng belef s not any longer a posteror, even f nfntely many samples were used. As we wll see below, a mathematcally sound method exsts that produces much better results. To analyze the problem more thoroughly, we frst notce that the true goal of Bayes flterng s to calculate the product dstrbuton specfed n Equaton (4. Thus, the optmal proposal dstrbuton would be ths product dstrbuton. However, samplng from ths dstrbuton drectly s too dffcult. As notced above, MCL samples nstead from the proposal dstrbuton q (t defned n Equaton (3, and uses the mportance factors (5 to account for the dfference. It s wellknown from the statstcal lterature [4, 12, 15, 17] that the dvergence between (3 and (4 determnes the convergence speed. Ths dfference s accounted by the perceptual densty p(o (t jx (t : If the sensors are entrely unnformatve, ths dstrbuton s flat and (3 s equvalent to (4. For low-nose sensors, however, p(o (t jx (t s typcally qute narrow, hence MCL converges slowly. Thus, the error n Fgure 3 s n fact caused by two dfferent types of errors: one arsng from the lmtatonof the sensor data (nose, and one that arses from the msmatch of (3 and (4 n MCL. As we wll show n ths paper, an alternatve verson of MCL exsts that practcally elmnates the second error source. Alternatve Proposal Dstrbutons An alternatve proposal dstrbuton, whch allevates ths problem, can be obtaned by samplng drectly from q (t p(o(t jx (t (o (t wth (o (t p(o (t jx (t dx (t (6 Ths proposal dstrbuton leads to the dual of MCL. It can be vewed as the logcal nverse of the samplng n regular MCL: Rather than forward-guessng and then usng the mportance factors to adjust the lkelhood of a guess based on an observaton, dual MCL guesses backwards from the observaton and adjusts the mportance factor based on the belef Bel(x (t,1. Consequently, the dual proposal dstrbuton possesses complmentary strengths and weaknesses: whle t s deal for hghly accurate sensors, ts performance s negatvely affected by measurement nose. The key advantage of dual MCL s that when the dstrbuton of p(ojx s 3. 4. 5. 7. 10. sensor nose level (n % Fgure 5: Error of plan MCL (top curve and MCL wth the mxture proposal dstrbuton (bottom curve wth 50 samples (nstead of 1,000 for each belef state. narrow whch s the case for low-nose sensors dual samplng can be much more effectve than conventonal MCL. Importance Factors We wll now provde three alternatve ways to calculate the mportance factors for q (t. Approach 1 (proposed by Arnaud Doucet, personal communcaton: Draw x (t,1 Bel(x (t,1. Hence, the par hx (t ;x (t,1 s dstrbuted accordng to p(o (t jx (t (o (t Bel(x (t,1 (7 and the mportance factor s obtaned as follows: " # p(o (t jx (t,1 w (o (t Bel(x (t,1 p(o (t jx (t p(x (t jx (t,1 ;a (t,1 Bel(x (t,1 p(o (t ja (t,1 ;:::;o (0 p(x(t jx (t,1 ;a (t,1 (o (t p(o (t ja (t,1 ;:::;o (0 / p(x (t jx (t,1 ;a (t,1 (8 Ths approach s mathematcally more elegant than the two alternatves descrbed below, n that t avods the need to transform sample sets nto denstes (whch wll be the case below. We have not yet mplemented ths approach. However, n the context of global moble robot localzaton, we suspect the mportance factor p(x (t ja (t,1 ;x (t,1 wll be ;x (t,1. zero (or very close to zero for many pose pars hx (t Approach 2 Alternatvely, one may n an explct forward phase sample x (t,1 j Bel(x (t,1 and then x (t j p(x(t jx (t,1 j ;a (t,1, whch represents the robot s belef before ncorporatng the sensor measurement. The trck s then to transform the samples x (t j nto a kdtree [1, 14] that represents the densty p(x (t ja (t,1 ;d (0:::t, whch s agan the pose belef just before ncorporatng the most recent observaton o (t.
(a map wth sample poses (b scan Fgure 6: Robot poses sampled accordng to q for the scan shown on the rght, usng a pre-compled verson of the jont dstrbuton p(o; x represented by kd-trees. Approach 3 The thrd approach combnes the best of both worlds, n that t avods the explct forward-samplng phase of the second approach, but also tends to generate large mportance factors. In partcular, t transforms the ntal belef Bel(x (t,1 nto a kd-tree. For each sample x (t q (t,we now draw a sample x (t,1 from the dstrbuton p(x (t ja (t,1 ;x (t,1 (x (t ja (t,1 (10 where (x (t ja (t,1 p(x (t ja (t,1 ;x (t,1 dx (t,1 (11 In other words, our approach projects x (t back to a possble successor pose x (t,1. Consequently, the par of poses hx (t ;x (t,1 s dstrbuted accordng to p(o (t jx (t p(x(t (o (t ja (t,1 ;x (t,1 (x (t ja (t,1 (12 whch gves rse to the followng mportance factor: " # p(o (t jx (t w p(x(t (o (t ja (t,1 ;x (t,1,1 (x (t ja (t,1 p(o (t jx (t p(x (t jx (t,1 ;a (t,1 Bel(x (t,1 p(o (t jd (0:::t,1 Fgure 7: Left: The nteractve tourgude robot Mnerva. Rght: Celng Map of the Smthsonan Museum. (o(t (x (t ja (t,1 Bel(x (t,1 p(o (t jd (0:::t,1 After ths frst phase, the mportance weghts of our samples x (t q (t are then calculated as follows: " # p(o (t jx (t,1 p(o (t (t jx w (o (t p(x (t ja (t,1 ;d (0:::t,1 p(o (t jd (0:::t,1 ;a (t,1 sentng ths belef densty. The only complcaton arses from the need to calculate (x (t ja (t,1, whch depends on / p(x (t ja (t,1 ;d (0:::t,1 both x (t and a (t,1. Luckly, n moble robot localzaton, (9 (x (t ja (t,1 can safely be assumed to be a constant, although ths assumpton may not be vald n general. Ths approach avods the danger of generatng pars of poses hx (t ;x (t,1 for whch w 0, but t nvolves an explct The reader should notce that all three approaches requre forward samplng phase. a method for samplng poses from observatons accordng to q (t whch can be non-trval n moble robot applcatons. The frst approach s the easest to mplement and mathematcally most straghtforward. However, as noted above, we suspect that t wll be neffcent for moble robot localzaton. The two other approaches rely on a densty estmaton method (such as kd-trees. The thrd also requres a method for samplng poses backwards n tme, whch further complcates ts mplementaton. However, the superor results gven below may well make ths addtonal work worthwhle. / (x (t ja (t,1 Bel(x (t,1 (13 where Bel(x (t,1 s calculated usng the kd-tree repre- The Mxture Proposal Dstrbuton Obvously, nether proposal dstrbuton s suffcent, as they both fal n certan cases. To llustrate ths, the dashed lne n Fgure 4 shows the performance for the dual. As n the prevous fgure, the horzontal axs depcts the amount of nose n percepton, and the vertcal axs depcts the error n centmeters, averaged over 1,000 ndependent runs. Two thngs are remarkable n these expermental results: Frst, the accuracy f now monotonc n perceptual nose: More accurate sensors gve better results. Second, however, the overall performance s much poorer than that of conventonal MCL. The poor performance of the dual s due to the fact that erroneous sensor measurements have a devastatng effect on the estmated belef, snce almost all samples are generated at the wrong place. Ths consderaton leads us to the central algorthm proposed n ths paper, whch uses the followng mxture (1, q (t + q (t (14 wth 0 1 as the proposal dstrbuton. In our experments, the mxng rate s set to throughout. Experments
1 error rate (n percentage of lost postons 0.8 0.6 0.4 0.2 Hybrd MCL MCL wthout random samples MCL wth random samples Fgure 8: Part of the map of the Smthsonan s Museum of Natonal Hstory and path of the robot. wth an adaptve mxng rate (usng pursut methods dd not mprove the performance n a notceable way. Samplng Form q The remanng prncple dffculty n applyng our new approach to robotcs s that t may not be easy to sample poses x based on sensor measurements o. In partcular, the prevous MCL algorthm only requres an algorthm for calculatng p(ojx; whle there s obvous ways to extend ths nto a samplng algorthm, samplng effcently from q may not a straghtforward matter ths s specfcally the case for the experments wth laser range fnders descrbed below. Unfortunately, space lmtatons prohbt a detaled descrpton of our soluton. In our mplementaton, a kd-tree representng the jont dstrbuton p(o; x s learned n a preprocessng phase, usng real robot data (a log-fle as a sample of o, andp(ojx wth randomly generated poses x to generate a weghted sample that represents p(o; x. The nce aspect of the tree s that t permts effcent samplng of the desred condtonal. Fgure 6 shows a set of poses generated for a specfc laser range measurement o. Expermental Results Our experments were carred out both n smulaton and for data collected wth our tour-gude robot Mnerva (shown n Fgure 7, collected durng a two-week perod n whch t gave tours to thousands of people n the Smthsonan s Museum of Natonal Hstory [18]. The smulaton experments were carred out usng the thrd method for calculatng mportance factors outlned above. A comparatve study showed no notceable dfference between ths and the second method. All real-world results were carred out usng the second approach, n part because t avods backwards samplng of poses. As noted above, we dd not yet mplement the frst method. Smulaton Fgure 4 shows the performance of MCL wth the mxture proposal dstrbuton, under condtons that are otherwse dentcal to those n Fgures 3. As these results suggest, our new MCL algorthm outperforms both MCL and ts dual by a large margn. At every sngle nose level, our new algorthm outperforms ts alternatves by a factor that ranges from 07 (hgh nose level to 9.7 (low nose level. 0 250 0 0 0 number of samples Fgure 9: Performance of conventonal (top curve, conventonal wth random samples (mddle curve and our new mxture (bottom curve MCL for the kdnapped robot problem n the Smthsonan museum. The error rate s measured n percentage of tme durng whch the robot lost track of ts poston. For example, at a nose level of 1%, our new MCL algorthm exhbts an average error of 24:6cm, whereas MCL s error s 238cm and that of dual MCL s 293cm. In comparson, the average error wth nose-free sensors and the optmal estmator s approxmately 19:5cm (t s not zero snce the robot has to face the object to see t. Our approach also degrades ncely to very small sample sets. Fgure 5 plots the error of conventonal MCL (top curve and MCL wth mxture proposal dstrbuton (bottom curve for dfferent error levels, usng m 50samples only. Wth 50 samples, the computatonal load s 0.126% on a MHz Pentum Computer meanng that the algorthm s approxmately 800 faster than real-tme. Whle plan MCL bascally fals under ths crcumstances to track the robot s poston, our new verson of MCL performs excellently, and s only slghtly nferor to m 1; 000 samples. Real-World Experments wth Lasers Our approach was tested usng data recorded durng a two-week deployment of the moble robot Mnerva as a museum tour-gude n the Smthsonan s Museum of Natonal Hstory [18]. The data contans logs of odometry measurements and sensor scans taken by Mnerva s two laser range-fnders. Fgure 8 shows part of the map of the museum and the path of the robot used for ths experment. As reported n [2, 3, 6], conventonal MCL relably succeeds n localzng the robot. To test our new approach under even harder condtons, we repeatedly ntroduced errors nto the odometry nformaton. These errors made the robot lose track of ts poston wth probablty of 0.01 when advancng one meter. The resultng localzaton problem s known as the kdnapped robot problem [5], whch s generally acknowledged as the most challengng localzaton problem. As argued n [7], ths problem tests the ablty to recover from extreme falures of the localzaton algorthm. Fgure 9 shows comparatve results for three dfferent approaches. The error s measured by the percentage of tme, durng whch the estmated poston devates by more than 2 meters from the reference poston. Obvously, the mxture proposal dstrbuton yelds sgnfcantly better results, even f the basc proposal dstrbuton s mxed wth 5% random samples (as suggested n [6] as a soluton to the kdnapped
Dstance [cm] 4 0 3 0 2 0 1 0 0 0 0 1 0 2 0 3 0 Tme [sec] Standard MCL Mxture MCL Fgure 10: MCL wth the standard proposal dstrbuton (dashed curve compared to MCL wth the new mxture dstrbuton (sold lne. Shown here s the error for a 4,000-second epsode of camerabased localzaton n the Smthsonan museum. robot problem. The mxture proposal dstrbuton reduces the error rate of localzaton by as much as 70% more than MCL f the standard proposal dstrbuton s employed; and 32% when compared to the case where the standard proposal dstrbuton s mxed wth a unform dstrbuton. These results are sgnfcant at the 95% confdence level, evaluated over actual robot data. Real-World Experments wth Vson We also compared MCL wth dfferent proposal dstrbutons n the context of vsual localzaton, usng only camera magery obtaned wth the robot Mnerva durng publc museum hours [2]. Fgure 7 shows on the rght a texture mosac of the museum s celng. Snce the celng heght s unknown, only the center regon n the camera mage s used for localzaton. The mage sequence used for evaluaton s of extremely poor qualty, as people often ntentonally covered the camera wth ther hand and placed drt on the lens. Fgure 10 shows the localzaton error obtaned when usng vson only (calculated usng the localzaton results from the laser as ground truth. The data covers a perod of approxmately 4,000 seconds, durng whch MCL processes a total of 20,740 mages. After approxmately 630 seconds, a drastc error n the robot s odometry leads to a loss of the poston (whch s an nstance of the kdnapped robot problem. As the two curves n Fgure 10 llustrate, the regular MCL sampler (dashed curve s unable to recover from ths event, whereas MCL wth mxture proposal dstrbuton (sold curve recovers quckly. These result are not statstcally sgnfcant n that only a sngle run s consdered, but they confrm our fndngs wth laser range fnders. Together, our result suggest that the mxture dstrbuton drastcally ncreases the robustness of the statstcal estmator for moble robot localzaton. Concluson Ths paper ntroduced a new proposal dstrbuton for Monte Carlo localzaton, a randomzed Bayesan algorthm for moble robot localzaton. Our approach combnes two proposal dstrbuton whch sample from dfferent factors of the desred posteror. By dong so, our approach overcomes a range of lmtatons that currently exst for dfferent versons of MCL, such as the nablty to estmate posterors for hghly accurate sensors, poor degradaton to small sample sets, and the ablty to recover from unexpected large state changes (robot kdnappng. Extensve expermental results suggest that our new approach consstently outperforms MCL by a large margn. The resultng algorthm s hghly practcal, and mght mprove the performance of partcle flters n a range of applcatons. Acknowledgments The authors are ndebted to Nando de Fretas and Arnaud Doucet for whose nsghtful comments on an earler draft of a related paper. We also thank Frank Dellaert and the members of CMU s Robot Learnng Lab for nvaluable suggestons and comments. References [1] J.L. Bentley. Multdmensonal dvde and conquer. Communcatons of the ACM, 23(4, 1980. [2] F. Dellaert, W. Burgard, D. Fox, and S. Thrun. Usng the condensaton algorthm for robust, vson-based moble robot localzaton. CVPR-99. [3] J. Denzler, B. Hegl, and H. Nemann. Combnng computer graphcs and computer vson for probablstc selflocalzaton. Rochester Unversty, Internal Report, 1999. [4] A Doucet. On sequental smulaton-based methods for Bayesan flterng. TR CUED/F-INFENG/TR 310, Cambrdge Unv., 1998. [5] S. Engelson and D. McDermott. Error correcton n moble robot map learnng. ICRA-9 [6] D. Fox, W. Burgard, F. Dellaert, and S. Thrun. Monte carlo localzaton: Effcent poston estmaton for moble robots. AAAI-99. [7] D. Fox, W. 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