Robotics and Autonomous Systems 58 (2010) 133–139

Contents lists available at ScienceDirect

Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

Maximum-likelihood sample-based maps for mobile robotsDaniel Meyer-Delius ∗, Wolfram BurgardUniversity of Freiburg, Department of Computer Science, Georges-Köhler-Allee 79, 79110 Freiburg, Germany

a r t i c l e i n f o

Article history:Available online 22 September 2009

Keywords:Mobile robotsMappingLikelihood maximization

a b s t r a c t

The problem of representing the environment of a mobile robot has been studied intensively in thepast. The predominant approaches for geometric representations are grid-based or line-based maps. Inthis paper, we consider sample-based maps which use the data points obtained by range scanners torepresent the environment. The main advantage of this representation over the other techniques is thatit is able to represent arbitrary structures and at the same time provide an arbitrary accuracy. However,range measurements come in large amounts and not every measurement necessarily contributes to therepresentation in the sameway.Wepresent a novel approach for calculatingmaximum-likelihood subsetsof the data points by sub-sampling laser range data. In particular, ourmethod applies a variant of the fuzzyk-means algorithm to find a map that maximizes the likelihood of the original data. Experimental resultswith real data show that the resulting maps are better suited for robot localization than maps obtainedwith other sub-sampling techniques.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Geometric representations of the environment play an impor-tant role inmobile robotics since they support various fundamentaltasks such as path planning or localization. One of the most popu-lar approaches are occupancy grids, which provide a discrete prob-abilistic representation of the environment. Other frequently usedapproaches are representations based on geometric primitives thatare typically found in the environment. In this context, lines play amajor role, since many man-made buildings are composed of lin-ear structures like walls, for example. Although these techniqueshave been successfully applied in the past, they have certain disad-vantages coming from discretization errors or because of missingfeatures in the environment.In this paper we consider a non-parametric representation

of the environment which uses the data points gathered bythe robot for the representation. In particular, this point-basedrepresentation, which will be denoted as sample-based mapsthroughout this paper, utilizes the endpoints of the rangemeasurements projected into the global coordinate system torepresent the objects in the environment. This representation,which rests on the assumption that the data is its best ownmodel, has several advantages. First it provides a high accuracyat floating point resolution. Additionally, it does not suffer fromdiscretization errors like grid maps. Furthermore it is flexible, as itdoes not rely on any pre-defined features which can be regarded

∗ Corresponding author. Tel.: +49 7612038010; fax: +49 7612038007.E-mail address:meyerdel@informatik.uni-freiburg.de (D. Meyer-Delius).

0921-8890/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.robot.2009.09.008

as a certain a priori information about the environment. From thispoint of view, sample-based maps inherit desirable properties ofgrid maps and feature-based maps as they are able to representarbitrary structures and at the same time provide an arbitraryaccuracy. Finally, sample-based maps are consistent with the dataperceived by the robot since they do not introduce artifacts in therepresentation, and do not require any procedures for extractingfeatures from the data.One drawback of using range measurements to represent the

environment, however, is that range measurements come in largeamounts which typically introduce large memory requirements.At the same time, many of the points provided by a range sensorare redundant because nearby points already provide relevantinformation about the existence of an object in the environment.One popular approach to reduce the number of points in a sample-based map is to down-sample the data points. However, thisdirectly introduces the question of how todown-sample, especiallybecause not every sample necessarily contributes to the model inthe same way.As an example, consider the maps depicted in Fig. 1. The

topmost map shows a complete data set obtained with a PioneerII robot equipped with a laser range finder in building 79 at theUniversity of Freiburg. The complete dataset consists of 706,528points. Themap in themiddlewas obtained by down-sampling thewhole dataset using grid-based sampling, a technique that will bedescribed later in this paper. This map contains only 5749 points,which is less than 1% of the points in the original dataset. Evenwiththis enormous reduction, the structure of the environment is stillrecognizable. The map at the bottom is the result of improving themap in the middle using the approach described in this paper tomaximize the likelihood of the complete dataset. This improved

134 D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139

Fig. 1. Typical sample-based maps. The top image depicts a map consisting of706,628 data points obtained in an office building by projecting the endpoints oflaser range measurements. The map depicted in the middle is obtained by down-sampling the topmost map. It consists of less than 1% of the samples. The lowerimage shows a maximum-likelihood map with the same number of samples thanthe map in the middle obtained using our approach.

map contains the samenumber of points as the one in themiddle. Itcan be observed how, as result of the maximization, the structuresin the environment are definedmore clearly,making themapmoreaccurate.In this paper, we consider the problem of finding a subset

of data points that provides an efficient representation of theenvironment and at the same time maximizes the likelihood of allthe data observed by the robot. Our approach applies a variant ofthe fuzzy k-means algorithm to find a subset that maximizes thelikelihood of the whole dataset. This algorithm can be regarded asan instance of the expectation maximization (EM) algorithm anddescribes an approximative optimization technique that takes aninitial map and iteratively improves it in a hill-climbing fashionmaximizing the likelihood of the dataset. Since the result of thefuzzy k-means algorithm highly depends on the initialization, wealso describe two alternative techniques, grid-based sampling andBIC optimization, for choosing the initial point set. As a result, weobtain highly accurate maps which consist only of a small fractionof the original data and at the same time maximize its likelihood.Additionally, the maps generated by our approach improve thecapabilities of the robot to localize itself in the environment usinga range scanner.This paper is organized as follows. After discussing relatedwork

in Section 2, we will provide a probabilistic interpretation of theoverall problem and describe how the fuzzy k-means algorithm

can be applied to find the subset of samples that maximizesthe likelihood of the original data. Then, in Section 4, we willpresent grid-based sampling and BIC optimization, as alternativetechniques for generating the initial map for the fuzzy k-meansalgorithm. Finally, we will present results obtained with ourapproach in Section 5 and conclude in Section 6.

2. Related work

In the robotic literature many different strategies have beenproposed to learn geometric representations from range data. Oneof the most popular approaches, occupancy grid maps, have beenintroduced by Moravec and Elfes [1]. Whereas this approach has anice probabilistic formulation and additionally allows a robot tomaintain a map over time, occupancy grid maps require a hugeamount of memory and suffer from discretization errors.Besides occupancy grid maps, line-based representations have

been also very popular. One of the first approaches for generatingline-based maps has been described by Crowley [2] who uses aKalman filter to fit lines on range measurements obtained fromsonar sensors. In the method proposed by Gonzalez et al. [3], pointclusters are computed from each range scan based on the distancebetween consecutive points. Linear regression is then applied tofit lines to these clusters. Arras and Siegwart [4] use a hierarchicalclustering approach to extract lines from the points obtained fromlaser data. The strategy proposed by Leonard et al. [5] uses a Houghtransform to extract linear features from a sequence of consecutivesonarmeasurements. The approach presented by Schröter et al. [6]clusters scans using the split-and-merge algorithm and combinesnearby segments using a weighted variant of linear regression.Other geometrical primitives have also been used for describingthe environment. González-Baños and Latombe [7] use polylinesas primitives for the representation. A polyline is a sequence ofline segments that are connected at their endpoints. Veeck andBurgard [8] describe an approach for learning polyline maps thatoperates on an arbitrary set of points.Whereas all these techniquesprovide highly accuratemaps, they rest on the assumption, that theenvironment actually consists of linear structures, which are notalways adequate to represent the environment.In contrast to this, our work described in this paper does not

extract geometrical structures from the data points, but uses thepoints themselves as primitives for the representation. In thecomputer graphics field, Levoy andWhitted [9] proposed points asa universal meta-primitive for geometric modeling and renderingapplications for three-dimensional geometry. Pauly et al. [10]explore the usage of points as primitives for modeling three-dimensional objects, and present several techniques for modifyingand reducing the size of the original set of points. However, thesetechniques produce set of points that are not necessarily a subset ofthe original dataset, creating models that can be inconsistent withthe data. Alexa et al. [11] work on a set of samples that is a truesubset of the original dataset, but their method relies on the ideathat the given set of points defines a surface, imposing in this way,an a priori structure on the model. In contrast to this, our workdoes not make any a priori assumptions on the dataset and uses asubset of the data as its best model. This subset is calculated usinga probabilistic formulation of the overall problem and applyingthe EM algorithm to find a maximum-likelihood subset. In thepast, Thrun [12] has utilized the EM algorithm to find grid mapsthat maximize the likelihood of the data. In this paper we borrowthis idea and calculate subsets of sample points that maximize thelikelihood of the original data.The idea of using samples to represent the environment is

not new. Lu and Milios [13], for example, use raw data from alaser range scanner to build a globally consistent representationof the environment. Biber and Duckett [14] propose a dynamic

D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139 135

map that adapts continuously over time using a sample-basedrepresentation of the environment. Their representation consistson multiple local maps related to each other by their globalcoordinates. In contrast to these works our representation consistsof a single global map that uses only a subset of the raw data.

3. Sample-based maps

A sample-based map is a non-parametric representation of theenvironment which uses points to represent the objects in theenvironment. By using points as primitive for the representation,sample-based maps are able to represent arbitrary structures andat the same time provide an arbitrary accuracy. The points ina sample-based map correspond to the endpoints of the rangemeasurements gathered by the robot projected into the globalcoordinate system. In this paper we assume that the pose of therobot is known, or an accurate pose estimation is given for all rangemeasurements. Our system applies the scan-matching techniquedeveloped by Hähnel et al. [15] to estimate the pose of the robot.Projecting all range measurements, a set D = {x1, . . . , xN} ofpoints is obtained. Formally, a sample-basedmap S = {x1, . . . , xK }is a subset S ⊆ D of the datasetD .

3.1. Probabilistic interpretation

A sample-based map S can be interpreted as a probabilisticmixture model Θ with K components, where each point xi ∈ Sin the map corresponds to a component ωi ∈ Θ in the model.The following assumptions are made about the probabilistic

structure of the model:(i) The number K of componentsωj, that is, the size of themodel,is known.

(ii) Each component ωj has the same prior probability P(ωj) =1/K .

(iii) Each sample xi ∈ D is independent from the other samplesand distributed according to p(xi|ωj, θj) ∼ N (µj,Σj), whereθj represents the mean µj and covariance matrix Σj of anormal distribution.

The samples x in the data setD are assumed to be generated by firstselecting a component ωj in the model with probability P(ωj) andthen selecting x according to p(x|ωj, θj). The probability densityfunction for a sample x is then given by the following mixturedensity

p(x|Θ) =K∑j=1

p(x|ωj, θj)P(ωj), (1)

where Θ = {θ1, . . . , θK } represents the set of model parameters.We further assume that the covariance matrices Σj of theprobabilities p(x|ωj, θj) are isotropic, known, and identical foreach component ωj. Therefore, the unknown parameter vector Θconsists of only the means µj of the normal distributions fromwhich the samples are selected, that isΘ = {µ1, . . . , µK }.

3.2. Likelihood maximization

IfD contains N samples, the likelihood of the dataD given themodelΘ is computed as

p(D|Θ) =N∏i=1

p(xi|Θ). (2)

The likelihood is a measure of the goodness of the model. Itindicates, for a fixed datasetD , that the model Θ with the higherlikelihood p(D|Θ) is more likely to be the true model.We are interested in finding the subset S ⊆ D of points that

maximizes the likelihood of the data D . Since evaluating every

possible subset S is computationally infeasible but for the mosttrivial datasets, our approach to find a maximum-likelihood mapis to start with a given subset S and then use an approximativeiterative optimization technique to modify it so as to maximizethe likelihood of the data. Starting with some given subset S, werepresent it as a probabilistic modelΘ (as described in Section 3.1)and use a variant of the fuzzy k-means algorithm for finding thevalue of the parameter of Θ that maximize the likelihood of D .The fuzzy k-means algorithm is basically a gradient ascent orhill-climbing procedure which seeks a minimum of the followingheuristic cost function [16]

J =K∑j=1

N∑i=1

P(ωj|xi, µj)‖xi − µj‖2, (3)

where P(ωj|xi, µj) is an estimate for the posterior probability formodel component ωj defined as

P(ωj|xi, µj) =p(xi|ωj, µj)P(ωj)

p(xi|Θ)

=p(xi|ωj, µj)K∑l=1p(xi|ωl, µl)

. (4)

The algorithm uses an initial estimate of the model parametersΘ [0] = {µ

[0]1 , . . . , µ

[0]K }, to compute the posteriors P(ωj|xi, µ

[0]j )

using Eq. (4) and then updates the estimates µ[0]j according to

µ[1]j =

N∑i=1P(ωj|xi, µ

[0]j )xi

N∑i=1P(ωj|xi, µ

[0]j )

. (5)

The improved estimates are then used as starting point for the nextiteration until the algorithm converges.After convergence, the values of the resulting estimatesΘ [T ] do

not necessarily correspond to points in the dataset. If this is thecase, they are replaced with the closest point in D . This ensuresthat the resulting set of points is a true subset of the originaldataset. Since duplicates are eliminated, this final step causes, ingeneral, the final estimate Θ [T ] to contain less points than in theoriginal estimateΘ [0].The time complexity of the fuzzy k-means algorithm is O(TKN),

where T is the number of iterations required for convergence, Nthe number of samples in D , and K the number of componentsin the model. The likelihood of the final model and the numberof iterations T needed for convergence depend strongly on thedistribution of the samples in the initial model Θ [0]. If theinitial estimates are good, convergence can be quite rapid. Afterconvergence, like all hill-climbing procedures, our algorithm findsonly local maxima and there is no guarantee that the solutionfound constitutes a global optimum.

4. Generating the initial map

The maps produced by the algorithm described in the previoussection depend strongly on the initial model Θ [0]. A commontechnique for generating an initial model is to randomly selectpoints out of the datasetD . However, this strategy produces mapsthat, in general, do not represent the structure of the environmentthoroughly and are not well suited for robotic applications. Inthis section we present two alternative techniques, grid-basedsampling and BIC optimization, for generating an initial model.

136 D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139

Fig. 2. The map on the left was generated by randomly selecting samples from adataset. Observe how the samples are unevenly distributed, concentrated in areaswhere the original distribution has a high sample density. The map on the rightcontains the same number of points and was generated using grid-based sampling.

4.1. Grid-based sampling

Fig. 2 depicts twomaps of 389 points each (0.25% of the 155,644points in the complete dataset). Themap on the left was generatedby randomly selecting points from the dataset. Observe how thepoints are unevenly distributed, concentrated in areas where theoriginal distribution has a high sample density. The map on theright was generated using grid-based sampling. It can be seen howthe points are distributed more evenly, representing the structureof the environment more thoroughly.Grid-based sampling takes as input a set D of points and the

size of the grid cells, and returns a subset S ⊆ D . The algorithmdivides the environment into cells of equal size and replaces thepoints that fall within the same cell with a common representativepoint. The resulting set S of samples is constituted by the meancenters xc of each non-empty cell C . The mean center xc of a cell Cis a point xc ∈ C such that∑x∈C

‖xc − x‖2 ≤∑x∈C

‖y− x‖2 ∀y ∈ C . (6)

The time complexity for building the grid isO(N), whereN is thenumber of samplesD . The mean center of a cell can be computedin O(n), where n is the number of points in the cell, by keepingtrack of themean of the points in the cell as the grid is build. At theend, we just need to look within the cell for the closest sample tothe mean, what can be done in O(n) using a naive search. The totalcomplexity of the algorithm is given by O(N + Kn), where K is thenumber of non-empty cells in the grid and n is the average numberof points per non-empty cell.Grid-based sampling does not attempt to find an optimal

model according to a global evaluation criterion, but utilizes anapproximative heuristic. It tries to partition the points according toregions, assuming that each region describes an equally importantpart of the environment. The main disadvantage of grid-basedsampling is that the exact number of resulting samples cannotbe specified directly. The number of samples in the resulting setdepends on the resolution of the grid. A finer resolution willproduce a larger set than a coarser one.

4.2. BIC optimization

The likelihood of the data given a model is directly affected bythe number of components in the model. In general, the likelihoodincreases with the number of components. Clearly, the data setitself is its best model. However, we want to find a tradeoffbetween the number of components in the models and its qualitywith respect to the data.A commonly used measure that balances the accuracy of the

model with its complexity is the Bayesian Information Criterion(BIC) [17]

-1e+06

-1e+070 10000 20000 30000 40000 50000 60000 70000

BIC

k

max BIC

Fig. 3. Bayesian Information Criterion (BIC) value as a function of the size k of themodel for a dataset with more than 73,000 points. For better visualization the BICvalues are displayed in logarithmic scale. Themarked point indicates themaximumBIC value for k = 25,893.

BIC(D,Θ) = log p(D|Θ)−12k log n, (7)

where p(D|Θ) is the likelihood of the data D given model Θ (seeEq. (2)), k is the number of components in Θ and n the size of D .When using the BIC criterion to determine the size of a model, wechoose the size k forwhich the BIC ismaximal. According to the BIC,increasing the size of the model beyond this point would increaseits complexity without proportionately increasing its accuracy.Fig. 3 plots the BIC value as a function of the size k of the modelfor a dataset with more than 73,000 points.To find the best model we first compute the BIC value for the

model corresponding to the complete dataset (i.e. k = n). Acomponent is then removed and the BIC for the new model isrecomputed. Components are removed until k = 1. Finally, themodel with the maximum BIC value is selected.To determine the component that is to be removed in each

iteration, we want to choose the one that least fits the data.According to Eq. (4) a small posterior P(ωj|xi, µj) indicates a poorfit of ωj relative to the other components. We use the sum of acomponent’s posterior over all data points xi as an heuristic ofhow bad the component fits the data, and select the one with thesmallest sum as the component to be deleted for the next iteration.The main disadvantage of this approach is its time complexity

which isO(N3) forN the size of the data set. A substantial speed-upcan be achieved by considering only points within a given radiusof the components and recomputing the posteriors only in thevicinity on the component that was removed. Nevertheless thecomputational complexity of the approach can be restrictive forlarge data sets. Although the approach is heuristic since there is noactual correct model, using the BIC allows us to generate an initialmodel without specifying any parameters.

5. Experimental results

The algorithms presented previously were implemented andevaluated using various datasets gathered with real robots.The goal of the experiments presented in this section is todemonstrate howour approach canbeused to generatemaximum-likelihood sample-based maps of the environment from laserrange measurements. We also demonstrate how the improvedsample-based maps increase the accuracy of the pose estimationduring robot localization when compared to the maps beforeimprovement.The datasets used consisted of different structured and

unstructured environments that differ considerably in the numberand distribution of the samples. In particular, the results presentedin this section correspond to the Intel Research Lab dataset freelyavailable online from The Robotics Data Set Repository (Radish) [18].When projecting the corrected rangemeasurements, only readingsshorter than 10 m were considered. The resulting dataset isconformed by 155,644 samples (see Fig. 4). Similar results as theones presented in this section were obtained when using otherdatasets.

D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139 137

Fig. 4. The dataset of the Intel Research Lab in Seattle (29 m×29 m) with 155,644samples.

5.1. Likelihood maximization

The first experiment is designed to illustrate how our approachcan be used to obtain maximum-likelihood sample-based mapsof range data gathered by the robot. To produce the initial maps,grid-based sampling (Section 4) was used since we wanted toevaluate the results with different model sizes. We also comparedour results using random sampling to generate the initial map.We tested our approach using initial maps of different sizes. Fig. 5plots the likelihood for the initial and resulting maps for thedifferent sizes (expressed as fraction of the number of samples inthe dataset). The maps obtained after applying the fuzzy k-meansalgorithm are labeled with +f-km. The different initializations arelabeled with gbs and rnd for grid-based sampling and randomsampling respectively.As can be seen in the figure, our approach does increase the

likelihood of the initial maps. A two-sample t test revealed thatthe improvement in the likelihood is significant on the α =0.01 level for all the evaluated sizes and initialization techniques.The amount of improvement depends on the distribution andnumber of points in the initial map. As the number of pointsin the map increases, the amount of improvement that can beobtained decreases. In proportionally large maps, the contributionof each point to the likelihood of the data is relatively small. Thus,as the number of points in the map increases, the strategy forselecting the samples becomes less important, since all pointsare almost equally relevant. The top plot in Fig. 5 reveals thatfor maps containing more than 10% of the points of the dataset,the sub-sampling strategy becomes essentially unimportant. Forproportionally large models, the application of our algorithm isno longer justified. However, we are only interested in mapsthat are considerably smaller than the original dataset, where theapplication of our algorithm provides an important increment inthe likelihood as can be seen in the figure. The bottom plot in Fig. 5represents the likelihood for the initial and resulting maps withless than 1% of the samples of the complete dataset. The size ofthemap atwhich the sub-sampling strategy becomes unimportantdepends on the specific dataset. In particular, it depends on thesample density of the complete dataset.

5.2. Generating the initial map

In Fig. 5, the influence of the initial map on the result ofour algorithm can also be seen. Observe how when randomlyinitializing the algorithm, the likelihood of the resulting map issometimes smaller than the likelihood of a map of the same sizeobtained using grid-based sampling even without applying ouralgorithm. In general, different initialization techniques producedifferent results. A standard approach to alleviate this problem is torun the algorithm several times using different initializations andto keep the best result.

-2.8e+06

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likel

ihoo

d

size of the map (%) - log scale

rn

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log

likel

ihoo

d

size of the map (%)

gbs + f-kmrnd

rnd + f-km

gbs

gbsgbs + f-km

drnd + f-km

Fig. 5. Likelihood for initial and improved maps of different sizes. The mapsobtained after applying the fuzzy k-means algorithm are labeled with +f-km. Thedifferent initializations are labeled with gbs and rnd for grid-based sampling andrandom sampling respectively.

-155000

-150000

-145000

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-135000

-130000

-125000

gbs rnd bic

log

likel

ihoo

d

gbsgbs + f-km

rndrnd + f-km

bicbic + f-km

Fig. 6. Likelihood for initial and improved maps using different initializationtechniques: grid-based sampling gbs, random sampling rnd, and BIC optimizationbic. Themaps obtained after applying the fuzzy k-means algorithm are labeled with+f-km.

We also evaluated the initial maps obtained using BICoptimization (Section 4). Fig. 6 plots the likelihood for the initialand improved maps for the different initialization techniques. Weapplied BIC optimization on a relatively small dataset with 20,814points. The map obtained had 8925 points (approx. 4% of thedataset). Then grid-based sampling and random sampling wereused to generate maps of approximately the same size. Finally, weapplied the fuzzy k-means algorithm to improve the initial maps.It can be seen that the map obtained using the BIC optimizationstrategy had the lowest likelihood. Examining the resulting map,

138 D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139

it can be observed that the sampled points were concentrated inareas where the original distribution has a high sample density.Although the same occurs when random sampling the points, theheuristic used to remove the points during the BIC optimizationsearch, concentrates the points in the high density areas evenmore. Still, the main advantage of the BIC optimization approachis that the number of points of the resulting map does not have tobe specified in any way.

5.3. Localization using sample-based maps

This second experiment is designed to evaluate the sample-based maps generated using our approach in the context ofposition tracking. The estimated pose of the robot using MonteCarlo localization [19] is compared using different maps. We usethe maps obtained in the previous experiment and compare theaccuracy of the localization algorithm in both the initial andimproved maps. As error between the true and the estimated posewe use the distance d(x, x′) described in [20] as

d(x, x′) =[ξδ(x3 − x′3)

2+ (1− ξ)((x1 − x′1)

2+ (x2 − x′2)

2)]1/2

,

(8)

where x1 and x′1 are the x-coordinates, x2 and x′

2 are the y-coordinates, and δ(x3 − x′3) is the difference in the orientationof x and x′. The constant ξ is a weighing factor that was set to0.8 in all our experiments. As ground truth we use the correctedpose obtained using the scan-matching algorithm of the CarnegieMellon Robot Navigation Toolkit (CARMEN) [21]. In order to makeour results general, we add some Gaussian noise to the odometryof the robot. As observation model for the localization we use thelikelihood fields model as described in [22].Fig. 7 plots the average localization error obtained when using

the different maps from the previous experiment. For the particlefilter implementation of theMonte Carlo localization, 500 particleswere used and the average of the particles was used as estimationof the current pose of the robot. The localization algorithm wasrun 10 times for each map and the errors were averaged. Forthe selected map sizes there were no serious localization errorsfrom which the particle filter could not recover from. When usingless than 0.25% of the samples for the maps, the particle filterdid diverge in some runs. It can be observed in the topmostplot of Fig. 7 that using more than 1% of the samples did notalways improve the accuracy of the localization, and in some caseseven made it slightly worst. The reason for this lies in the usedobservation model. The likelihood fields model neglects the dataassociation problem. Adding more samples to the map increasesthe likelihood associated to slightly delocalized particles, reducingthe overall localization accuracy. The size of the map for whichaccurate localization could no longer be guaranteed, or the sizefor which additional samples cease to improve the localizationaccuracy depend on the specific environment and dataset.As Fig. 7 suggests, the improved maps obtained using our ap-

proach are better suited for localization tasks. The localization er-ror is in general smaller when using the improved representations.This is specially true for small maps, where the choice of the sam-ples is more relevant. It can also be observed that small maps thattry to distribute the samples uniformly throughout the structureof the environment, like the ones obtained using grid-based sam-pling (gbs) are more appropriate for localization when comparedwith maps of the same size that only concentrate the samples insome parts of the environment, like random sampling (rnd).

6. Conclusions

In this paper, we present a novel approach for learningpoint-based maps that maximize the likelihood of the sensormeasurements. The key idea of our approach is to obtain a non-

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1.000.750.500.25

aver

age

erro

r

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gbsgbs + f-km

rndrnd + f-km

Fig. 7. Average localization error during position tracking for the different mapsobtained in the previous experiment.

parametric representation which uses a subset of the data as itsbest model. We consider this representation as a probabilisticmodel and apply a variant of the fuzzy k-means algorithm to finda map that maximizes the likelihood of the whole data. We alsodescribe two alternative techniques for choosing the initial pointset.Our approach was evaluated using different datasets gathered

with real robots. Experimental results illustrate how our approachcan be used to generate maximum-likelihood sample-basedmaps of the environment from laser range measurements.Results also indicate that the improved sample-based mapsincrease the accuracy during robot localization when comparedto the maps before improvement. We believe that maximum-likelihood sample-based maps constitute an accurate and generalrepresentation of the environment and are therefore suitable forrobotic applications.Despite these encouraging results, the presented approach

suffers from several limitations that are the aim for future research.First, our approach is strongly dependent on the map used forinitialization, and gets easily stuck in local maxima. In the futurewe therefore will investigate strategies to escape local maxima.Another interesting area for future work would be to try ourapproach beyond two dimensions. The framework can be easilyextended to three dimensions, and evaluated, for example in three-dimensional scene interpretation and maps.

References

[1] H.P. Moravec, A. Elfes, High resolution maps from wide angle sonar, in: Proc.of the IEEE Int. Conf. on Robotics & Automation, ICRA, 1985.

[2] J.L. Crowley, World modeling and position estimation for a mobile robot usingultrasonic ranging, in: Proc. of the IEEE Int. Conf. on Robotics & Automation,ICRA, 1989.

D. Meyer-Delius, W. Burgard / Robotics and Autonomous Systems 58 (2010) 133–139 139

[3] J. González, A. Ollero, A. Reina, Map building for a mobile robot equipped witha 2d laser rangefinder, in: Proc. of the IEEE Int. Conf. on Robotics & Automation,ICRA, 1994, pp. 1904–1909.

[4] Kai O. Arras, Roland Siegwart, Feature extraction and scene interpretation formap-based navigation and map building, in: Mobile Robotics XII, Pittsburgh,USA, Proceedings of the SPIE 3210 (1997) 42–53.

[5] J. Leonard, P Newman, R. Rikoski, J. Neira, J. Tardós, Towards robust dataassociation and feature modeling for concurrent mapping and localization, in:10th Int. Symposium on Robotics Research, ISRR, 2001.

[6] D. Schröter, M. Beetz, J.-S. Gutmann, Rg mapping: Learning compact andstructured 2d line maps of indoor environments, in: Proceedings of the IEEEInternational Workshop on Robot and Human Interactive Communication,ROMAN’02, 2002.

[7] H. González-Baños, J.-C. Latombe, Robot navigation for automatic modelconstruction using safe regions, in: Int. Symposium on Experimental Robotics,ISER, 2000, pp. 405–415.

[8] M. Veeck, W. Burgard, Learning polyline maps from range scan data acquiredwith mobile robots, in: Proc. of the IEEE/RSJ International Conference onIntelligent Robots and Systems, IROS, 2004.

[9] M. Levoy, T. Whitted, The use of points as a display primitive, ComputerScience Technical Report 85-022, UNC-Chapel Hill, 1985.

[10] M. Pauly, M. Gross, L.P. Kobbelt, Efficient simplification of point-sampledsurfaces, Theoret. Comput. Sci. 284 (1) (2002) 67–108.

[11] M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. David Levin, C.T. Silva, Point setsurfaces, in: Proc. of the Conference on Visualization, VIS, 2001.

[12] S. Thrun, Learning occupancy grids with forward sensor models, in: Au-tonomous Robots, vol. 15, 2003, pp. 111–127.

[13] F. Lu, E. Milios, Globally consistent range scan alignment for environmentmapping, Autonomous Robots 4 (4) (1997) 333–349.

[14] Peter Biber, Tom Duckett, Dynamic maps for long-term operation of mobileservice robots, in: Robotics: Science and Systems, 2005, pp. 17–24.

[15] D. Hähnel, R. Triebel, W. Burgard, S. Thrun, Map building with mobile robotsin dynamic environments, in: Proc. of the IEEE Int. Conf. on Robotics andAutomation, ICRA, 2003.

[16] R.O. Duda, P.E. Hart, D.G. Stork, Pattern Classification,Wiley Interscience, 2000.[17] Gideon Schwarz, Estimating the dimension of amodel, The Annals of Statistics

6 (2) (1978) 461–464.[18] A. Howard, N. Roy, The robotics data set repository (radish), http://radish.

sourceforge.net/, 2003.

[19] D. Fox, W. Burgard, F. Dellaert, S. Thrun, Monte carlo localization: Efficientposition estimation for mobile robots, in: Proc. of the National Conference onArtificial Intelligence, 1999.

[20] P. Pfaff, W. Burgard, D. Fox, Robust monte-carlo localization using adaptivelikelihood models, in: H.I. Christiensen (Ed.), European Robotics Symposium2006, vol. 22, Springer-Verlag, BerlinHeidelberg, Germany, 2006, pp. 181–194.

[21] N. Roy, M. Montemerlo, S. Thrun, Perspectives on standardization in mobilerobot programming, in: Proc. of the IEEE/RSJ Int. Conf. on Intelligent Robotsand Systems, IROS, 2003.

[22] S. Thrun, D. Fox, W. Burgard, F. Dellaert, Robust monte carlo localization formobile robots, Artificial Intelligence 128 (1–2) (2001).

Daniel Meyer-Delius is a Ph.D. student at the Universityof Freiburg. He got a degree in computer engineering fromthe Universidad Simón Bolívar in Caracas, Venezuela, in2002, and a M.Sc. degree in Applied Computer Sciencesfrom the University of Freiburg in 2006. His researchinterests lie in mobile robot localization, mapping, andmachine learning.

Wolfram Burgard is a professor for computer science atthe University of Freiburg where he heads the Laboratoryfor Autonomous Intelligent Systems. In the past, he and hisgroup have developed numerous novel algorithms formo-bile robot state estimation and control including localiza-tion,mapping, SLAM, path planning, and exploration.Wol-fram Burgard’s group is also known for numerous installa-tions of mobile robots in real-world environments. Wol-fram Burgard is a member of the IEEE and the AAAI.