Interval-based Robot Localization with Uncertainty Evaluation

Yuehan Jiang, Aaronkumar Ehambram and Bernardo Wagner

a

Real Time Systems Group (RTS), Institute for Systems Engineering, Leibniz Universit

¨

at Hannover,

D-30167 Hannover, Germany

Keywords:

Landmark-based Localization, Uncertainty Estimation, Interval Analysis, Factor Graph, Probabilistic

Uncertainty.

Abstract:

Being able to provide trustworthy localization for a robot in a map is essential for various tasks with safety-

related requirements. In contrast to classical probabilistic approaches that represent the uncertainty as a Gaus-

sian distribution, we use interval error bounds for the uncertainty estimation of a localization problem. To

tackle and identify the limitations of probabilistic localization uncertainty estimation, we carry out compari-

son experiments between an interval-based method and a factor graph-based probabilistic method. Different

measurement error models are propagated by the two methods to derive the robot pose uncertainty estimates.

Results show that the probabilistic approach can provide very good pose uncertainty when there is no non-

Gaussian systematic sensor error. However, if the measurements have unmodeled systematic errors, the in-

terval approach is able to robustly contain the true poses whereas the probabilistic approach gives completely

wrong results.

1 INTRODUCTION

For safe navigation and operation, a mobile robot not

only has to accurately determine its localization in

the environment but also needs to provide trustwor-

thy results. In indoor environments where no Global

Positioning System (GPS) is available, the robot usu-

ally localizes itself by using sensory information (e.g.

wheel odometry, 3D point cloud, or camera images)

and updates its belief of position with respect to a

given map. A variety of probabilistic techniques have

been proposed to reach the localization goal on a prior

map (Gutmann et al., 1998) (Wilbers et al., 2019).

Nevertheless, most approaches focus on providing ac-

curate point-wise solutions, while the quality of the

uncertainty estimation of environmental perception

and localization results are not thoroughly investi-

gated.

In probabilistic localization, the Kalman ﬁlter

and graph-based optimization approaches provide ex-

plicit estimates of robot poses efﬁciently, with the as-

sumption that the measurement and pose uncertainty

have zero-mean Gaussian distributions (Chen, 2012)

(Kummerle et al., 2011). The Gaussian uncertainty

assumption, however, could be inaccurate and unreli-

able due to the following reasons.

a

https://orcid.org/0000-0001-5900-0935

On one hand, there exist non-Gaussian sensor

errors which are hard to be identiﬁed. For exam-

ple, light detection and ranging (LiDAR) sensors are

widely used for robot localization applications. It

has been pointed out by (Voges, 2020) that stochas-

tic distributions can not always truly represent the er-

ror model of LiDARs due to the presence of unmod-

eled systematic errors. Moreover, analysis in (Mal-

let et al., 2008) shows that the LiDAR measurement

model is subject to mixed effects of geometric (e.g.

the incidence angle between the laser beam and the

surface) and radiometric object properties (e.g. dif-

ferent materials of the surface). These problems in-

duce non-Gaussian measurement uncertainty, which

can not be represented by the mean and variance. On

the other hand, due to the nonlinearity of the system, a

linearization process is required which can only give

an approximation of the uncertainty estimate. Due to

these limitations of probabilistic approaches, the re-

sulting pose estimation and its uncertainty might be

erroneous and may signiﬁcantly differ from the true

result.

In this paper, we aim to tackle and identify the lim-

itations of probabilistic localization uncertainty when

the Gaussian assumption is violated. First, we provide

a solution to a 2D landmark-based localization prob-

lem using interval analysis (Jaulin et al., 2001). In

296

Jiang, Y., Ehambram, A. and Wagner, B.

Interval-based Robot Localization with Uncertainty Evaluation.

DOI: 10.5220/0011143700003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 296-303

ISBN: 978-989-758-585-2; ISSN: 2184-2809

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

comparison to probabilistic methods that use Gaus-

sian distributions for the measurement model (i.e.

range and bearing) as well as the robot pose (position

and heading), the interval method models them with

interval boxes with lower and upper bounds, where

no probability distribution is assumed. The interval-

based solution can provide guaranteed uncertainty es-

timation for the robot poses that enclose the true val-

ues. We focus on the uncertainty estimation of the lo-

calization problem and compare the results from the

interval-based method and a factor graph-based prob-

abilistic method, to ﬁnd advantages and limitations

of both. In particular, we investigate the uncertainty

propagation from the measurement model to the re-

sulting robot pose.

The contributions of the paper are:

• We evaluate the uncertainty estimation of an

interval-based solution and a factor graph-based

probabilistic solution for a localization problem

by designing four simulated experiments.

• We ﬁnd failure cases where the probabilistic ap-

proach leads to incorrect uncertainty estimation

with the presence of non-Gaussian measurement

error, whereas the results of the interval-based

method still hold true.

In the remainder of this paper, we ﬁrst introduce

the related work in Section 2. In Section 3 we in-

troduce the basics of interval analysis. Subsequently,

we formulate our localization problem and establish

our interval-based localization solution, as well as the

state-of-the-art probabilistic solution in Section 4. In

Section 5, we evaluate the uncertainty estimation of

the two solutions by four small simulated experiments

and analyze the results.

2 RELATED WORK

In the existing map-based localization literature,

landmark-based methods are widely used by differ-

ent sensor modalities beneﬁting from the identiﬁable

landmarks in the environments like poles (Brenner,

2010) (Schaefer et al., 2019), visual features (Wilbers

et al., 2019) and natural landmarks (Rohou et al.,

2020). In this work, we also use a set of landmarks

as the prior map. Typically, the landmarks observed

by the sensor are associated with the map to track the

robot in the map. The data association problem is

challenging and false associations introduce outliers

that need to be dealt with. Since data association is

not the focus of this paper, we assume that the asso-

ciation between locally-observed landmarks and the

map landmarks is known.

In recent years, interval methods have shown their

potential in robot localization and simultaneous local-

ization and mapping problems. The work in (Rohou

et al., 2020) solves the localization problem for an un-

derwater vehicle in a known map but with unreliable

data association. By formulating a constraint network

with a constellation contractor, the data association

and state estimation problems are solved together. In

an unmanned aerial vehicle (UAV) localization appli-

cation (Kenmogne et al., 2017) interval has been used

to compute the 6 DOF pose of the UAV by tracking

known image features with bounded-error. The work

in (Voges and Wagner, 2021) ﬁrstly proposes a guar-

anteed visual-LiDAR odometry for autonomous ve-

hicles that uses LiDAR and a monocular camera to

compute the 6 DOF pose estimate with bounded un-

certainty. Evaluation of real data from large-scale ex-

periments shows that the vehicle’s real poses are en-

closed in a guaranteed way. Authors in (Ehambram

et al., 2021) extend this work (Voges and Wagner,

2021) by fusing information from LiDAR and stereo

cameras to solve the dead-reckoning of a vehicle and

use the interval contraction method for consecutive

robot pose estimation. The above methods show that

interval analysis can provide robust uncertainty es-

timates of the robot pose which always enclose the

ground truth data. However, few investigations have

been done to compare the interval results with those

of the probabilistic methods. Interval analysis has

the beneﬁt of providing a guaranteed estimation of

the robot pose, however, it can also suffer from pes-

simism due to large initial intervals, wrapping effect,

and dependencies of parameters (Jaulin et al., 2001).

Thus, we think it is important to validate the use of in-

terval analysis by showing under what cases interval

has advantages over probability approaches and also

disadvantages in terms of uncertainty estimation.

Recently, graph-based optimization has become

the standard solution for SLAM and widely applied

for localization applications (Kummerle et al., 2011)

(Kaess et al., 2011) (Wu et al., 2017) (Wilbers et al.,

2019). The graph representation allows adding robot

poses and landmark positions as nodes in the graph,

while the constraints between the nodes are added as

edges (or factors). A non-linear least-squares prob-

lem is formulated and the error function is minimized

in a bundle-adjustment manner to determine the opti-

mal robot pose and landmark positions. In this paper,

we use the factor graph (Kaess et al., 2011) to formu-

late a 2D localization problem. We establish a factor

graph-based localization solution as the state-of-the-

art probabilistic approach. The uncertainty estimation

of this method will be compared with the interval-

based uncertainty and evaluated.

Interval-based Robot Localization with Uncertainty Evaluation

297

3 INTERVAL ANALYSIS

Interval analysis is a numerical tool based on the idea

of enclosing real numbers in intervals (Jaulin et al.,

2001). Instead of an exact value, variables are de-

ﬁned by an interval that is a subset of R , denoted

by [x] = [x, x]. The variable is bounded by a lower

bound x and an upper bound x , without any assump-

tions about its probability distribution. By using the

simple error bounds, interval provides a guaranteed

way to represent the uncertainty of a variable. For

example, if we know that the accuracy of a distance

measurement is ± 0.3 m, then the true value x

∗

is en-

closed by an interval x

∗

∈ [x] , with an uncertainty of

the radius r([x]) = (x−x)/2 = 0.3 m. An interval box

[x] is deﬁne as a vector of intervals.

Interval computation enables classical real arith-

metic operations, +,−,×, and ÷, as well as elemen-

tary functions like sin,cos, exp ... on intervals based

on set theory. Here is an example of addition opera-

tion on intervals:

[−2, 4] + [5, 6] = [−2 + 5, 4 + 6] = [3, 10] (1)

For a 2D robot localization problem, where we

consider the set of possible robot poses described in

the interval domain X , the problem can be character-

ized as:

X = {x ∈ R

3

| f (x) ∈ Y} = f

−1

(Y), (2)

where f : R

3

→ R

2

is a non-linear function, which

can be the measurement constraint, and Y the land-

mark positions. The robot pose X can be obtained

by using Set Inverter Via Interval Analysis (SIVIA)

(Jaulin et al., 2001) to calculate a subpaving solution

set from an initial search domain.

Alternatively, this can also be solved by formulat-

ing a Constraint Satisfaction Problem (CSP) as a set

of constraints and variables. We give an exemplary

CSP:

H :

Variable : x

n

Constraint : g(x

n

) = 0

Domain : [x

n

]

, (3)

where x

n

∈ [x

n

] is the unknown variable vector, and

g(x

n

) is the constraints to fulﬁll in the CSP H .

Contractors (Jaulin et al., 2001) can be built to

contract the variable’s initial domain, to get a result-

ing interval box that satisﬁes all the constraints. Here

the contraction means that the initial interval domain

is reduced by removing all values that do not sat-

isfy the constraints. The forward-backward contrac-

tor is the most convenient contractor to contract the

domain of CSP H where all constraints are decom-

posed into primitive constraints (i.e. a sequence of

functions containing only a single operator) and each

of the primitive constraints is contracted until the in-

terval boxes converge towards the smallest possible

sizes. The beneﬁt of contraction is that all possible

solutions are guaranteed to be contained in the con-

tracted domain.

In this work, interval analysis is used for solving a

2D robot localization problem. A CSP is formulated

to compute the uncertainty of the robot pose. Contrac-

tors are built to propagate the uncertainty of the map

landmarks and measurement constraints, resulting in

a guaranteed robot pose uncertainty that satisﬁes the

constraints in CSP H .

4 PROBLEM FORMULATION

The goal is to track a moving robot in a known map

of landmarks M . As shown in Figure 1 the robot

moves in a square-shaped indoor corridor from a start-

ing point.

Figure 1: The 2D view of a robot moving in a 10m × 10 m

square-shape corridor (bounded by two black boxes). The

robot starts from the red point (0, 0) and follows the clock-

wise green trajectory (ground truth). Orange dots are land-

marks to be measured.

The robot carries a LiDAR sensor and measures

the radial distance r and horizontal angle θ of its

surrounding landmarks to the robot itself. For each

LiDAR scan a subset of the map landmarks m

l

can

be measured, where l ∈ {1,..., L} . L is the num-

ber of landmarks the robot observes from its current

scan. Note that this scenario is built in simulation,

with the positions of map landmarks M known. The

landmarks are assumed to be randomly distributed in

the environment and can be perceived by the LiDAR

accurately. Furthermore, the locally observed land-

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

298

marks m

l

are acquired according to the robot mo-

tion model and the geometry of the corridor which

we establish by simulation. We assume that a feature

extraction algorithm provides measurements of land-

marks in the current scan and association between the

locally observed landmarks m

l

and the map M is al-

ready solved.

This localization problem can be formulated by

the state estimation equations:

(

Measurement : G(x, z , m

l

) = 0

Association : m

l

∈ M

, (4)

where we denote x = (x y ψ)

T

as the unknown robot

pose, z = (r θ)

T

as the measurement for each m

l

in current scan, and m

l

= (m

l

x

m

l

y

)

T

the position

of the landmark. More speciﬁcally, the measurement

constraint in Equation 4 can be broken down as fol-

lows:

G(x, z ) =

x + r · cos(ψ + θ) − m

l

x

y + r · sin(ψ + θ) − m

l

y

. (5)

We illustrate the above measurement model in Figure

2 where the robot measures its distance and azimuthal

angle with respect to a landmark.

Figure 2: The robot measurement model.

Next, we will introduce our interval-based so-

lution and factor graph-based probabilistic solution

for the aforementioned localization problem, respec-

tively.

4.1 Interval-based Localization

All variables in Equation 4 are described in the inter-

val domain. This allows us to formulate a CSP L:

L :

Variable : x, z, m

l

, M

Constraint :

G(x, z , m

l

) = 0

m

l

∈ M

Domain : [x], [z], [m

l

], [M]

, (6)

where [x] is the set of feasible robot poses that we

want to determine and [z] = ([r] [θ])

T

is our land-

mark measurement. Interval boxes are suitable to

represent the uncertainty of LiDAR measurements

because we usually only know the accuracy of the

distance and beam divergence (Voges, 2020). Be-

sides the measurement uncertainty, the uncertainty of

known map landmarks M is also speciﬁed. The un-

certainty of map landmarks can be provided by the

accuracy of the given map or feature extraction algo-

rithm. In this CSP we assume no information about

the prior robot position, thus its initial domain is set

to [x] = ([−∞, ∞] [−∞, ∞] [−π, π])

T

.

Then, we build forward-backward contractors for

solving each constraint in CSP L and get the inter-

section to contract the interval pose. Before using the

contractors, we ﬁrst break down our constraints into

simple equations according to Equation 5 and 6 (Ro-

hou et al., 2020):

a = ψ + θ

d

x

= r · cos (a)

d

y

= r · sin (a)

m

l

x

= x + d

x

m

l

y

= y + d

y

m

l

∈ M

, (7)

where a and d = (d

x

d

y

)

T

are intermediate variables

for decomposing complex constraints into elementary

equations. Subsequently, we apply addition contrac-

tor C

+

(e.g. C

+

([a], [ψ], [θ]) ) for addition equations

and polar contractor C

polar

([d

x

], [d

y

], [r], [a]) for the

polar equations in Equation 7 (Desrochers and Jaulin,

2016). As a result, related interval domains are re-

duced until smallest interval boxes are reached.

In our localization problem, the CSP L will be

solved for each robot pose, which means that when-

ever the robot measures landmarks from one LiDAR

scan, the contractors will be applied to contract the

initial box [x] for the contracted robot pose. The re-

sulted interval box represents the uncertainty of the

robot pose.

4.2 Factor Graph-based Localization

A factor graph includes variables and factors which

represent constraints between variables in a graph and

solve a maximum a posteriori (MAP) estimate prob-

lem (Kaess et al., 2011). Here we illustrate the factor

graph of our localization problem in Figure 3.

As a probabilistic approach, all quantities and fac-

tors involved are labeled with a probability distribu-

tion. Given that all measurements are affected by

Gaussian noise, the goal of our localization problem

is to estimate a Gaussian approximation of the poste-

rior of the robot poses. In the following, we formulate

our localization problem as an optimization problem

Interval-based Robot Localization with Uncertainty Evaluation

299

Figure 3: Factor graph representation of the localization

problem. We have robot poses x (grey circles), observed

landmarks m

l

(orange circles) as variables. Blue boxes are

the measurement factors between poses and observed land-

marks m

l

. Green boxes represent the map factors between

the observed landmarks and map M .

that determines the robot poses and landmark posi-

tions based on a set of measurements.

Let v = (x m

l

)

T

be the state vector that consists

of robot poses x = (x

1

... x

T

) and landmarks m

l

=

(m

l

1

... m

l

K

) where T, K are the numbers of all poses

and landmarks, respectively. Here we deﬁne the robot

poses as x

t

∈ SE(2) and landmarks as m

l

k

∈ R

2

. The

optimization problem can be represented as a MAP

estimate according to Bayes’ law:

v

∗

= argmax

v

p(v|z) = argmax

v

p(z|v) p(v) (8)

where z is the measurements gathered by the robot.

The term p(v) is the prior over all states which re-

quire to be given. With the assumption of indepen-

dent Gaussian noise on the measurements, Equation

8 can be written as:

v

∗

= argmin

v

∑

i

e

i

(v, z

i

)

T

Ω

i

e

i

(v, z

i

) + F

map

(m

l

)

(9)

where e

i

(v, z

i

) is the error function that measures

how well the state vector v satisfy the constraint z

i

,

Ω

i

is the information matrix of measurement z

i

and

F

map

is the prior knowledge of the map landmarks.

In our localization problem, apart from deﬁning mea-

surement factors which can be described by the left

part on the right side of Equation 9, we also de-

ﬁne the map factors, which is a Gaussian prior about

the landmark positions in the known map (similarly,

in the interval approach we deﬁne the prior of map

landmarks as interval boxes). To solve Equation 9

and compute the Gaussian approximation of the robot

poses as well as landmark positions, we use the stan-

dard least-squares optimization method Levenberg-

Marquardt using the GTSAM solver for factor graphs

(Dellaert, 2012).

5 EXPERIMENTAL EVALUATION

Our simulated experiments are designed to 1) support

the claim that the interval-based localization method

can ensure a guaranteed uncertainty estimation with-

out assuming any probability distribution over the

measurement; 2) compare the uncertainty estimation

of the interval-based method and the factor graph-

based probabilistic method; 3) to ﬁnd out the advan-

tages and disadvantages of using interval method and

probabilistic method when there are unidentiﬁed non-

Gaussian measurement errors.

We use simulated data (as explained in Section 4)

for our experiments to leave out the inﬂuence of in-

accurate data association and outliers so that we can

investigate how the measurement model and uncer-

tainty propagation strategy can affect the ﬁnal uncer-

tainty estimation. Firstly, we introduce important pa-

rameters that we use for the experiments.

We assume the robot moving in a 10 m × 10 m

square-shaped indoor corridor with an initial position

at (0, 0) as illustrated in Figure 1. To calculate com-

parable uncertainty estimations for the interval-based

and factor graph-based methods, we carefully choose

the uncertainty of map landmarks and measurements.

For the factor graph based method, we deﬁne the

standard deviations for the prior map landmarks as

σ

M

x

= σ

M

y

= 0.01 m , and for the angle and dis-

tance measurements as σ

θ

= 0.01° , σ

r

= 0.1 m . For

the interval-based method, we intend to deﬁne the in-

terval box region of the map and measurements as

such, that it will always include the 99% conﬁdence

region of Gaussian error ellipse. According to the χ

2

distribution, we choose 3 σ as the radius of the in-

tervals. Speciﬁcally, for the map landmarks the un-

certainty is [∆

M

x

] = [∆

M

y

] = [−0.03, 0.03]m , and

for the measurements the uncertainties are [∆

θ

] =

[−0.03, 0.03]°, [∆

r

] = [−0.3, 0.3]m.

In the following sections, we design four small ex-

periments to compare the uncertainty estimation re-

sults of the two methods. We assign non-Gaussian

measurement noises in the measurement model to ﬁnd

out how the two methods handle it. For the interval

method, the measurement model always follows the

interval uncertainty. For the probabilistic methods, no

matter there is or not non-Gaussian noise in the mea-

surements, it always assume it to be Gaussian. First,

we compare the interval uncertainty with the proba-

bilistic uncertainty when the measurements of given

map landmarks follow Gaussian distributions. The

non-zero Gaussian measurement is just a perfect as-

sumption, while in real cases, the LiDAR measure-

ments could contain systematic errors that are hard to

identify. Thus in the second and third experiment, we

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300

simulate systematic errors for both methods and show

how the two approaches handle the uncertainty prop-

agation. In the last experiment, we suppose that the

actually measured map landmarks are uniformly dis-

tributed in the interval uncertainty box of map land-

marks, then apply the two methods with interval and

probabilistic measurement models as in the ﬁrst ex-

periment.

5.1 Comparison of the Uncertainty

Estimation

Our ﬁrst experiment is designed to show the compari-

son of the robot pose uncertainty estimation using the

interval-based localization algorithm and the factor

graph-based optimization algorithm. Here the mea-

surements are acquired according to the given map

landmark positions. The measurements in the factor

graph are inserted using Gaussian distribution while

the measurements in the interval method follow inter-

val uncertainty. We show in Figure 4 the result of both

methods.

Figure 4: Uncertainty of robot poses given by interval

method and factor graph method. The green boxes repre-

sent the interval pose uncertainty. Red ellipses represent the

probabilistic pose uncertainty which is the 99% conﬁdence

region. The black dots inside the red ellipses are the ground

truth value.

As can be seen, the green boxes represent the

interval pose uncertainty and the red error ellipses

which are the 99% conﬁdence region represent the

probabilistic uncertainty. It is clear that throughout

the trajectory the probabilistic uncertainty is much

smaller than the interval uncertainty. Moreover, 100%

ground-truth poses are enclosed by both the interval

and probabilistic pose uncertainties. The results show

that both methods can give guaranteed results while

the probabilistic method can provide more accurate

pose estimation (i.e. the mean value) than the interval

method, as the interval results can only give a region

of all possible robot positions, but not explicit poses

like the probabilistic method do.

5.2 Impact of 1-σ Systematic Error

In the second experiment, we add a positive 1-σ

r

sys-

tematic error to the distance measurements of both

methods. This is to resemble unidentiﬁed systematic

errors in the range measurements from a LiDAR. The

results are shown in Figure 5.

Figure 5: Green boxes represent the interval pose uncer-

tainty. Red ellipses represent the 99% conﬁdent region of

probabilistic uncertainty. Black dots are the ground truth

value.

Note that interval uncertainty boxes still enclose

the ground truth. Also, the interval boxes shrink

slightly because solutions that are not consistent with

the new measurement constraints are removed. The

probabilistic uncertainty (red ellipse) shows an ob-

vious shift to the outer bound of the interval box,

and no longer includes the ground-truth value. The

results imply that the probabilistic method underes-

timates the uncertainty and gives wrong estimates,

while the interval method is still able to enclose the

ground truth.

5.3 Impact of 2-σ Systematic Error

In the third experiment, we decide to further investi-

gate the inﬂuence of systematic errors of the measure-

Interval-based Robot Localization with Uncertainty Evaluation

301

ments. We add a positive 2-σ

r

systematic error to the

distance measurements of both methods. The results

are shown in Figure 6.

Figure 6: Green boxes represent the interval pose uncer-

tainty. Red ellipses represent the 99% conﬁdent region of

probabilistic uncertainty. Black dots are the ground truth

value.

We can clearly see that interval boxes are still en-

closing the ground truth, however, most of the prob-

abilistic error ellipses go even further away from the

true value. We believe that in this case, the probabilis-

tic method gives very wrong uncertainty estimation

that some of the error ellipses are even partly outside

of the guaranteed interval uncertainty.

To conclude, we ﬁnd that the interval method is

robust to systematic errors of measurements, while

the probabilistic method becomes signiﬁcantly unre-

liable. The results of the second and third experiment

show that the probabilistic method tends to underesti-

mate the uncertainty of pose estimation and can even

give totally wrong uncertainty estimation in the pres-

ence of large systematic errors. The interval method,

on the contrary, can guarantee to provide trustworthy

uncertainty estimates and not underestimate the pose

uncertainty.

5.4 Impact of a Uniformly Distributed

Measurement

In this part, we assume our measured map landmarks

are no longer on the given positions, but uniformly

distributed in the interval uncertainty box we deﬁned

for the map landmarks. This is a more realistic as-

sumption since the actually measured landmarks by

a LiDAR sensor could reside in the close vicinity of

the real landmark positions. In other words, we cre-

ate our map landmark positions that follow a uni-

form distribution inside the interval region [M

i

] =

M

i

+ [−0.03, 0.03] m × [−0.03, 0.03]m and create

the measurements which follow interval uncertainty

and Gaussian uncertainty according to this new map.

In this experiment, the non-Gaussian measurement

error comes from the uncertainty of measured map

landmarks, but not the uncertainty in the measure-

ment models, as in the previous two experiments. The

purpose of this design is to see how the probabilistic

method copes with a known map with interval uncer-

tainty. Results are displayed in Figure 7.

Figure 7: Green boxes represent the interval pose uncer-

tainty. Red ellipses represent the 99% conﬁdent region of

probabilistic uncertainty. Black dots are the ground truth

value.

As seen from the result, the interval uncertainty

boxes encloses the ground truth poses throughout the

whole trajectory. However, the probabilistic error el-

lipses are not always able to include the ground truth

poses with some black dots completely outside the red

ellipse region, which shows that the probabilistic un-

certainty is overconﬁdent. This experiment implies

that, when the uncertainty of map landmarks is an in-

terval, then modeling the measurements as Gaussian

can lead to underestimated pose uncertainty. Unlike

the interval-based method that gives a worst-case un-

certainty but ensures the inclusion of real result, the

probabilistic method tends to underestimate the un-

certainty.

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302

6 CONCLUSIONS

We compare an interval-based solution and a factor

graph-based probabilistic solution for a 2D localiza-

tion problem for robot pose uncertainty estimation.

For this purpose, we design four comparison exper-

iments with different measurement models. From the

results, we ﬁnd out that the probabilistic method can

provide more accurate pose estimation and smaller

uncertainty estimation than the interval method given

Gaussian measurement noise. However, the uniden-

tiﬁed non-Gaussian errors in measurements can sig-

niﬁcantly impede the performance of the probabilis-

tic approach, causing wrong pose estimates and un-

derestimated uncertainty. In comparison, the inter-

val method is not sensitive to systematic measure-

ment errors or uncertainty of measured map positions,

which is able to robustly provide guaranteed pose un-

certainty estimation. It also shows that the interval

method usually gives pose estimation with larger un-

certainty compared to the probabilistic method. How-

ever, the beneﬁts of using intervals for solving robot

localization problems is that no prior information of

the robot pose is required, and the ability to provide

all possible solutions with guarantee.

In the further work, we aim to apply the interval-

based method to real data. We plan to tackle the land-

mark extraction and association problem which can

introduce more sources of error that will inﬂuence

the uncertainty estimation. Besides, we would like

to investigate again the uncertainty estimation perfor-

mance of interval methods and probabilistic methods

with noisy real measurement data for which the real

distribution is unknown.

ACKNOWLEDGEMENTS

This work was supported by the German Academic

Exchange Service (DAAD) as part of the Research

Training Group i.c.sens [RTG 2159].

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