MONTE CARLO LOCALIZATION IN HIGHLY SYMMETRIC

ENVIRONMENTS

Stephan Sehestedt, Frank E. Schneider

Research Establishment for Applied Sciences (FGAN)

Neuenahrer Strasse, 53343 Wachtberg, Germany

Keywords:

Monte Carlo Localization, MCL, Symmetric Environments, Two-Stage Sampling.

Abstract:

The localization problem is a central issue in mobile robotics. Monte Carlo Localization (MCL) is a popular

method to solve the localization problem for mobile robots. However, usual MCL has some shortcomings in

terms of computational complexity, robustness and the handling of highly symmetric environments. These

three issues are adressed in this work. We present three Monte Carlo localization algorithms as a solution to

these problems. The focus lies on two of these, which are especially suitable for highly symmetric environ-

ments, for which we introduce two-stage sampling as the resampling scheme.

1 INTRODUCTION

Localization is a crucial ability for a mobile robot in

order to be able to navigate. It is a key competence in

many successful robotic systems (see e.g. (S. Thrun,

2001), (S. Lenser, 2000)). Obviously, having a good

idea of ones position is also a prerequisite for decision

making and interaction with the environment. The

localization problem is often divided into three sub-

problems. The most basic problem is position track-

ing, where the robot starts with a known initial posi-

tion and just has to keep track of its pose. Whereas

global localization requires the robot to localize itself

from scratch. The most complex subproblem is the

kidnapped robot problem. Here the mobile unit has

to estimate its pose with a wrong initial believe. All

three subproblems can be solved with a particle ﬁl-

ter algorithm (L. Ronghua, 2004), (Fox, 2003) and

(S. Lenser, 2000).

Here we also consider a fourth subproblem: Highly

symmetric environments. The basic particle ﬁlter of-

ten fails to localize a robot when there are a lot of

symmetries in the environment (L. Ronghua, 2004).

Therefore, it is necessary to introduce a technique that

allows to track multiple distinct positions on the map.

Symmetric environments need to be considered for

many mobile robot applications. Examples are robots

for surveillance in ofﬁce buildings, where we fre-

quently ﬁnd symmetries. We observe the same sit-

uation for different kinds of buildings. Furthermore,

robots operating in sewers and other tunnel like envi-

ronments may have to deal with ambiguities.

In the past years several approaches to Monte Carlo

localization had been proposed. Most of these fail

in symmetric environments (Fox, 2003), (S. Lenser,

2000) and (S. Thrun, 2001) or use large sample sets

(A. Milstein, 2002). A lot of research deals with the

efﬁcient use of the samples, e.g. (S. Lenser, 2000), in

order to use small sample sets and only a few publi-

cations are concerned with variable sample set sizes

(L. Ronghua, 2004) and (Fox, 2003).

In this work we present three algorithms to solve

the localization problem based on Monte Carlo Meth-

ods. The ﬁrst is derived from Dieter Fox’s KLD-

Sampling (Fox, 2003) and includes a procedure simi-

lar to Sensor Resetting Localization (S. Lenser, 2000).

We call this method KLD-Sampling with Sensor Re-

setting (KLD-SRL). The other two algorithms use

two-stage sampling for resampling in order to master

highly symmetric environments.

The remainder of this paper is organized as fol-

lows. Section 2 summarizes Monte Carlo Localiza-

tion. Section 3 outlines related work. Section 4 intro-

duces an efﬁcient and robust localization algorithm

based on KLD-Sampling. Furthermore, in Section 5

we present an extension to MCL to handle symmet-

ric environments and show how to include it into the

schemes of existing MCL algorithms. Finally, In sec-

tion 7 we show the experimental results followed by

the conclusions.

2 MONTE CARLO

LOCALIZATION

Monte Carlo Localization (MCL) is a Bayesian ap-

proach to localization. There are numerous publica-

249

Sehestedt S. and E. Schneider F. (2006).

MONTE CARLO LOCALIZATION IN HIGHLY SYMMETRIC ENVIRONMENTS.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 249-254

DOI: 10.5220/0001215802490254

Copyright

c

SciTePress

Table 1: The three steps of the MCL algorithm.

1. Prediction: Draw x

i

t

∼ p(x

i

t

| x

i

t−1

, u

t−1

).

2. Update: Compute the importance factors ω

i

t

=

ηp(y

t

| x

i

t

), with η being a normalization factor

to ensure that the weights sum up to 1. Here, y

t

is a sensor reading at time t.

3. Resample.

tions presenting Monte Carlo Methods in detail, ex-

amples are (A. Doucet, 2000) and (S. Arulampalam,

2002). The belief is represented by a set of weighted

samples X(t) = hx

i

t

, ω

i

t

| i = 1, ..., Ni, where each

x

i

t

is a possible position of the robot and ω

i

t

the ac-

cording importance factor at time step t. The calcula-

tion of the posterior probability distribution consists

of three steps. Prediction, update and resampling.

Prediction is done by moving the samples in the

prior distribution as speciﬁed by the motion model.

Then the weights are calculated, using a sensor

model. Finally, in the additional resampling step, par-

ticles with low weight are replaced by those with sig-

niﬁcant weight. Doing this, the samples get concen-

trated in areas of high probability. The downside is,

that this procedure can lead to premature convergence

in ambiguous situations (A. Bienvenue, 2002). For

more details on resampling see (A. Doucet, 2000).

MCL is summarized in table 1.

2.1 Discussion of MCL

Beside the very appealing attributes of particle ﬁlters

(S. Arulampalam, 2002), there are some issues that

need to be taken into account. Firstly, the basic parti-

cle ﬁlter uses large numbers of particles, what results

in high computational effort. Above this, MCL is

not able to track multiple hypotheses stable, but con-

verges to exactly one position. This can be harmful,

as we loose track of possibly right positions and do

not take these into account for pathplanning. Finally,

the basic particle ﬁlter recovers from localization fail-

ure very slowly.

3 RELATED WORK

3.1 Sensor Resetting Localization

Sensor Resetting Localization (SRL) is an extension

to Monte Carlo Localization (S. Lenser, 2000). In

each time step, a number of particles is replaced with

samples drawn from the sensor model. Using the

equation

Ns = (1 −

˜

P

P

t

) ∗ N (1)

we determine how many samples are to be replaced.

With

˜

P =

P

N

i=0

ω

i

t

/N and P

t

the probability thresh-

old, that can be adjusted to let the algorithm react

more or less sensitive. The logic behind this pro-

cedure is, that if the particles have a high average

weight, we can be certain that most samples are in

regions of high probability. If the average weight is

low, we can conclude that many particles are in un-

likely positions. Then we try to replace a number of

samples by sampling from the sensor model.

Therefore, we draw the new samples according to

the most recent sensor reading. Hence, these particles

are in areas of high probability. The problem with this

procedure is, that it may be very difﬁcult to sample

from p(y

i

t

| x

i

t

) (S. Thrun, 2001).

SRL is mathematically questionable as it simply re-

moves a number of particles and replaces them with

new samples, as already mentioned in (S. Thrun,

2001).

4 ADAPTIVE PARTICLE FILTERS

4.1 KLD-Sampling

Variable sample set sizes allow us to adjust the num-

ber of particles to the complexity of the problem.

Clearly, we need much more particles for global lo-

calization than for tracking (Fox, 2003). Following

this logic, KLD-Sampling adjusts the number of par-

ticles based on the quality of the approximation of the

posterior distribution. We now give a brief description

of KLD-Sampling as introduced by Dieter Fox (Fox,

2003).

It is assumed that the true posterior is given by a

discrete, piecewise constant distribution. Then it can

be guaranteed that with probability 1 − δ the distance

between the true posterior and the sample-based ap-

proximation is less than ǫ.

For this, an approximation of the Kullback-Leibler

distance is used. We compute the number N of sam-

ples that is needed for the approximation based on the

number k of bins with support (Fox, 2003).

N =

k − 1

2ǫ

1 −

2

9(k − 1)

+

q

2

9(k−1)

z

1−δ

3

(2)

z

1−δ

is the upper 1−δ quantile of the standard normal

distribution. Then N relies on the inverse of the error

bound ǫ and is in the ﬁrst order linear to k. To estimate

the number k of bins with support we used a ﬁxed

ICINCO 2006 - ROBOTICS AND AUTOMATION

250

Table 2: The KLD-SRL algorithm.

1. Prediction: Draw x

i

t

∼ p(x

i

t

| x

i

t−1

, u

t−1

).

2. calculate N according to (2).

If N > N old:

draw N − Nold samples from p(y

i

t

| x

i

t

).

3. Update: Compute the importance factors ω

i

t

=

p(y

t

| x

i

t

).

4. normalize weights to sum up to 1.

5. Resample.

spatial grid, where a grid cell has support if at least

one particle lies in that cell.

It is easy to integrate KLD-Sampling into the parti-

cle ﬁlter scheme. In the prediction phase we count the

number of bins with support. If a sample falls into an

empty bin, we increase k. With each processed sam-

ple N is updated according to equation 2. Sampling

is stopped as soon as the number of samples reaches

N. For a complete description of this method refer to

(Fox, 2003).

4.2 KLD-Sampling with Sensor

Resetting

KLD-Sampling still leaves one bottleneck. While

global localization and when localization errors oc-

cur we have to use large numbers of particles. This

comes from the fact that new samples are uniformly

distributed over the state space. This is very inefﬁ-

cient in applications where it is possible to sample

from the sensor model p(y

t

| x

i

t

).

For this, we remember the number of samples we

used in the last step. We then compute N according to

equation (2) after the prediction phase. If N is larger

than the number of samples we used in the last time

step we add new samples to the sample set. The newly

generated samples are not just uniformly distributed

over the map but they are generated according to the

most recent sensor reading.

Sampling from the sensor model can be a very dif-

ﬁcult task depending on the sensors one wants to use.

In this work we focused on laser range ﬁnders. For a

deﬁned number of positions on the map, we syntheti-

cally produce the joint probability distribution p(y, x)

that contains the information about the occurrence of

a sensor reading y on a certain position x on the map.

This distribution is represented by a piecewise con-

stant distribution using a kd-tree.

In this way we place these particles in areas of high

probability and ignore areas of low probability. This

allows us to reduce the maximum size of the sample

set to a few hundred particles.

5 CLUSTERED PARTICLE

FILTERS

Usual MCL and KLD-Sampling often fail to correctly

localize a robot in ambiguous situations. Both meth-

ods prematurely converge to possibly wrong positions

on the map. As soon as the robot leaves the symmet-

ric parts of the environment, KLD-Sampling quickly

recovers from the localization failure.

Nevertheless, the pathplanner had possibly wrong

data about the pose before recovery. This may have

fatal consequences for the mobile unit. In the follow-

ing we describe the proposed algorithms for localiz-

ing a mobile robot in symmetric environments.

5.1 Background

In this section we present the mathematical back-

ground of our approach to Monte Carlo Localization

with clustering.

Using Sequential Importance Sampling (SIS), we

take account of the mismatch between the proposal

distribution and the target distribution (S. Thrun,

2001). In usual SIS the update equation for the im-

portance factors can be shown to be (S. Arulampalam,

2002)

ω

i

t

=

p(y

t

| x

i

t

)p(x

i

t

| x

i

t−1

)p(x

i

0:t−1

| y

0:t−1

)

q(x

i

t

| x

i

0:t−1

, y

1:t

)q(x

i

0:t−1

| y

1:t−1

)

(3)

with the numerator being the target distribution and

the denominator the proposal distribution. Seeing that

q(x

i

t

| x

i

0:t−1

, y

1:t

) = q(x

i

t

| x

i

t−1

, y

t

) we can calcu-

late the weights according to

ω

i

t

= η

p(y

t

| x

i

t

)p(x

i

t

| x

i

t−1

)

q(x

i

t

| x

i

t−1

, y

t

)

(4)

η is a normalizing constant, which ensures that the

weights sum up to 1. In that way, ω

i

t

is only de-

pendent on x

i

t

, y

t

, and y

t−1

. For our clustering ap-

proach we have to take account of the existence of the

clusters. Using a modiﬁed proposal distribution, we

have a function η(x

i

t

) which calculates the normaliz-

ing factor for the sample x

i

t

depending on the asso-

ciated cluster in which x

i

t

lies, in order to consider

multiple hypotheses.

ω

i

t

= η(x

i

t

)

p(y

t

| x

i

t

)p(x

i

t

| x

i

t−1

)

q(x

i

t

| x

i

t−1

, y

t

)

(5)

Consequently, we use the following equation to

compute the weights of the samples.

MONTE CARLO LOCALIZATION IN HIGHLY SYMMETRIC ENVIRONMENTS

251

ω

i

t

= η(x

i

t

)p(y

t

| x

i

t

) (6)

For C

j

t

being the sum of weights of the jth cluster

η(x

i

t

) = 1/C

j

t

(7)

This method allows us to use a procedure simi-

lar to Two-Stage Cluster Sampling as our resampling

scheme. In Two-Stage Cluster Sampling one divides

the whole sample set into clusters. In the ﬁrst stage, a

cluster is randomly drawn. Then, in the second stage,

the actual sample is drawn (J. Hartung, 1989). Here

we ﬁrst draw a cluster and then apply systematic re-

sampling on the individual particles in that cluster.

Hence, we draw samples from all clusters and there-

fore are able to keep track of multiple hypotheses.

Note, that we actually draw each of the clusters. If we

only detect one cluster, the resampling procedure be-

haves exactly like usual (systematic) resampling. Fur-

thermore, the association of a sample with a cluster is

determined in every time step. Thus, a sample can

be associated with different clusters in different time

steps.

5.2 Clustering

In the following, we present an extension to Monte

Carlo localization that allows for adequate localiza-

tion in highly symmetric environments as frequently

occurring in ofﬁce like environments. Furthermore, it

is shown how to incorporate this into the functionality

of Sensor Resetting Localization and KLD-Sampling

with Sensor Resetting.

In order to use two-stage sampling, we need to de-

termine clusters in the sample set. Clustering is a well

known technique to classify data and build groups of

similar objects (W. Lioa, 2004). Here we use cluster-

ing to ﬁnd groups of particles that occupy the same

area in the state space. Doing this, we try to extract

signiﬁcant clusters that represent a possible position

of our robot.

When choosing a clustering method, we mainly

need to care about two things. The speed of this

routine and the accuracy. Regarding the speed, grid-

based algorithms are a good choice (W. Lioa, 2004).

For us, a clustering algorithm is accurate, if it ﬁnds

all signiﬁcant particle clusters. That are those clusters

with a high average weight. Clusters with low weight

are not likely to represent the correct pose and can

therefore be discarded. That implies, that the output

of the clustering procedure only includes the signiﬁ-

cant clusters.

We used a ﬁxed spatial grid for a ﬁrst estimate

of the clusters. Here a coarse grid of approximately

2 m ∗ 2 m ∗ 45 degree is sufﬁcient and since we have

to do the clustering for just a few hundred samples it

is fast as well. If we detect more than one cluster, it

is checked if clusters can be fused. This is the case,

if clusters are in close proximity of each other in the

state space.

Thereby, the importance factors are computed ac-

cording to equation (6). In that way, we do not loose

possible positions while resampling. A cluster is con-

sidered signiﬁcant if it’s weight is above a manually

chosen threshold. This method showed to be stable

and accurate. Moreover, it allows us to simply dis-

tribute the free samples from clusters with low weight

over the signiﬁcant clusters. While resampling these

samples are replaced with the high weighted particles

of the according cluster. Alternatively, one can re-

place these samples by newly generated ones.

5.3 Clustered Sensor Resetting

Localization

To incorporate two-stage sampling we need to regard

the fact that SRL simply replaces particles in the sam-

ple set if needed. Thus, we cannot guarantee that all

clusters will still exist after this replacement. In order

to exclude this case, we disable the replacement of

particles whenever we ﬁnd more than one signiﬁcant

cluster. The algorithm is summarized in table 3.

The intuition behind CSRL is as follows. When the

robot gets kidnapped while our ﬁlter is tracking mul-

tiple distinct hypotheses, we observe the following.

The weights of the clusters become negligible and are

therefore not considered signiﬁcant any more. Thus,

we do regard all samples as though they were in one

cluster. We then can calculate a number of samples

to be replaced according to Sensor Resetting Local-

ization. The newly generated particles are distributed

according to the most recent sensor reading. Then, for

the new sample set, clustering is done and it is decided

if we have to track multiple or a single position.

In this way, we are able to combine the ability to

track multiple hypotheses stable with the advantage

of using small sample sets.

5.4 Clustered KLD-Sampling with

Sensor Resetting

Clustered Sensor Resetting Localization still suffers

from using a sample set of constant size. Thus, we

are not taking account of the different complexities of

the localization problem. To overcome this issue, we

developed a method to combine two-stage sampling

with KLD-SRL.

Analogous to KLD-SRL the number of particles is

adjusted according to equation (2). Only, if we detect

more than one signiﬁcant cluster, we do not allow the

number of samples to be reduced. So we can guaran-

tee that we won’t loose any of the detected clusters.

As soon as there is only one cluster left, the number

ICINCO 2006 - ROBOTICS AND AUTOMATION

252

Table 3: The CSRL algorithm.

1. Prediction: Draw x

i

t

∼ p(x

i

t

| x

i

t−1

, u

t−1

).

2. Update: Compute the importance factors ω

i

t

=

p(y

t

| x

i

t

).

3. Normalize weights to sum up to 1.

4. Determine the number of signiﬁcant clusters.

5. Resample using two-stage sampling.

6. If there is one cluster:

Compute Ns according to equation (1).

If Ns > 0: Replace Ns samples with sam-

ples drawn from p(y

i

t

| x

i

t

).

Table 4: The Clustered KLD-SRL algorithm.

1. Prediction: Draw x

i

t

∼ p(x

i

t

| x

i

t−1

, u

t−1

).

2. calculate N according to (2).

3. If N > Nold:

draw N − Nold samples from p(y

i

t

| x

i

t

).

4. If N < Nold and number of clusters is one:

Decrease number of samples to N.

5. Update: Compute the importance factors ω

i

t

=

p(y

t

| x

i

t

).

6. normalize weights to sum up to 1.

7. Resample using two-stage clustering.

of samples can be adjusted in the usual way. Note,

that the number samples can still be increased while

tracking multiple clusters and thus, allowing to ﬁnd

more clusters.

In ambiguous situations we usually have a number

of signiﬁcant clusters that we do not want to disappear

prematurely. As soon as we detect distinct features in

the environment, all but one cluster will have negli-

gible weights and therefore be merged into the one

cluster with high weight. Then, the number of sam-

ples can be decreased again.

Clustered KLD-SRL joins the advantages of using

a small and variable sample set and the ability to track

multiple positions over extended periods of time. Ex-

periments show this algorithm to be accurate and ro-

bust. The algorithm is summarized in table 4.

6 EXPERIMENTAL RESULTS

In this section we present the results of our exper-

iments. Due to the fact that it is very difﬁcult to

obtain quantitative results in real world experiments,

we focus on simulations. Different maps have been

used to test accuracy and tracking multiple hypothe-

ses. Simulations were conducted on a 1.7 GHz Pen-

tium 4 computer with 512 MB of memory.

Firstly, we ran simulations with KLD-SRL. Since

this algorithm is not able to cope with symmetric en-

vironments properly, accuracy, robustness and speed

are the measures of interest. In situations like the one

illustrated in ﬁgures 1(a) and 1(b) KLD-SRL usually

fails. The average localization failure was approxi-

mately 3 cm, with a mean update time of 0.03 sec-

onds. Kidnapping experiments revealed the time for

recovery from localization failure to be approximately

2 seconds in the average. Further experiments showed

that the according values are in the same magnitude

for CSRL and Clustered KLD-SRL.

To test the ability of tracking multiple hypotheses

stable over extended periods of time, we generated

a completely symmetric map consisting of a cross

shaped area of approximately 15 m ∗ 15 m (ﬁgures 1

and 2). The green arrows indicate the estimated head-

ing of the robot according the position estimate. This

looks the same for Clustered Sensor Resetting Lo-

calization and Clustered KLD-Sampling with Sensor

Resetting. Methods that do not facilitate techniques

to handle symmetries converge to one single position

and therefore this estimate is wrong in most cases.

Figure 1(a) shows the initialization of the ﬁlter,

where all particles are distributed according to the

most recent sensor reading. Thus, the samples cover

four possible positions on the map. Since there is no

way to ﬁgure out which of these poses is the right

one, the ﬁlter keeps track of all of them (ﬁgures 1(a)

to 2(b)). Even as the estimates cross in the middle, we

do not loose any of them.

Furthermore, we conducted experiments with maps

of ofﬁce buildings, where we observe a lot of symme-

tries and asymmetries. Global localization was per-

formed at random starting points and was success-

fully done in all cases. Above this, we frequently kid-

napped the robot and let it relocalize itself. Most of

the time we placed it on symmetric parts of the map

and kidnapping was also done while the ﬁlter tracked

multiple hypotheses.

In these experiments, relocalization was usually

performed within 2 seconds and as soon as the robot

left the symmetric part of the environment, the ﬁl-

ter converged to the correct position. Thus, both,

CSRL and Clustered KLD-SRL, are able to localize a

robot in highly symmetric environments and thereby

showed to be very robust and accurate.

Regarding computational complexity, CSRL is in-

MONTE CARLO LOCALIZATION IN HIGHLY SYMMETRIC ENVIRONMENTS

253

(a) (b)

Figure 1: CSRL: a) After initialization on a cross shaped

symmetric map. The green arrows denote the actual head-

ing of the robot, according to the four possible positions. b)

The robot moves. Simulated data.

(a) (b)

Figure 2: CSRL: a) The four distinct position estimates

come to close proximity in the middle of the cross. b) The

robot moves on. Simulated data.

ferior, since it uses a sample set of constant size.

The drawback of using variable sample set sizes is,

that sometimes we need signiﬁcantly more computa-

tion time. Thus, we always have to regard the worst

case or design the robotic system in a way, that some

processes can sleep for a while without affecting the

usual operation too much.

7 CONCLUSIONS

The focus of this work is on handling highly symmet-

ric environments with a particle ﬁlter algorithm. Nev-

ertheless, computational complexity and robustness

are still important issues, that need to be regarded.

We introduced a resampling scheme that is easy to

integrate into a particle ﬁlter.

Furthermore, we presented two algorithms that are

able to localize a mobile unit correctly in symmetric

environments at all times. CSRL uses small but con-

stant sample sets. Therefore, CKLD-SRL is superior

to this method, for its use of variable sample sets. Ex-

perimental results show the good quality of the posi-

tion estimate and high robustness produced by these

methods.

Further research is concerned with the viability of

such methods for 3D/6D localization. Furthermore,

we are interested in using such ﬁlters for multi target

tracking (A. Kraeussling, 2005).

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