SCAN MATCHING WITHOUT ODOMETRY INFORMATION

Francesco Amigoni, Simone Gasparini

Dipartimento di Elettronica e Informazione, Politecnico di Milano

Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Maria Gini

Dept of Computer Science and Engineering, University of Minnesota

Minneapolis, USA

Keywords:

Scan Matching. Robot Mapping. Laser Range Scanner.

Abstract:

We present an algorithm for merging two partial maps obtained with a laser range scanner into a single map.

The most unique original aspect of our algorithm is that it does not require any information on the position

where the scans were collected but uses only geometrical features of the scans.

1 INTRODUCTION

The increasing use of mobile robots equipped with

laser range scanners has stimulated the development

of methods for matching and aligning scan data col-

lected by these sensors. Usually these methods align

two scans starting from some information about the

relative position of the sensors obtained from odome-

try (Lu and Milios, 1997; Cox, 1991; R

ofer, 2001).

Some methods (Einsele, 1997; Martignoni III and

Smart, 2002) do not use information coming from

odometry, but they work well only in rectilinear en-

vironments and for small displacements of the robot.

In this paper we present a method for matching two

scans that does not require any odometry informa-

tion and that works for signiﬁcant displacements of

the robot. For the purposes of this paper, a scan is

a collection of segments. In our experimental setting

each scan is obtained by acquiring with a SICK LMS

200 laser range scanner (mounted on a Robuter mo-

bile platform at a height of approximatively 50 cm) a

sequence of distance measurements along directions

separated by a programmable angle (1

◦

, in our case)

sweeping 180

◦

. The result of the sensing operation

is thus a set of points expressed in polar coordinates,

with the origin of the coordinate frame in the sensor

itself. We approximate these points with a set of seg-

ments following the method described in (Gonz

ales-

nos and Latombe, 2002). The use of segments in-

stead of points reduces the computational complexity

of ﬁnding the match between scans. Since the method

does not use odometry information, it relies exclu-

sively on the geometry of the scans. In particular, we

consider the angles between pairs of segments in the

scans as a sort of “geometrical landmarks” on which

the matching process is based. We assume that the

robot moves on an indoor 2D surface and that walls

and vertical objects are at the height of the laser scan.

The method integrates two scans, S

and S

, into a

map S

1,2

. It is composed of three major steps:

1. ﬁnd the possible transformations of S

on S

;

2. evaluate the transformations to identify the best

transformation

t of S

on S

;

3. apply the best transformation to S

(obtaining S

)

and fuse the segments of S

and of S

to obtain

1,2

This paper is structured as follows. The next Sec-

tion describes in detail our method, while in Section 3

we discuss the experimental activity performed to val-

idate it. Section 4 concludes the paper.

2 THE PROPOSED METHOD

In the algorithms, two points are considered to co-

incide when they are closer than POINTDISTANC E-

TOLERANCE (in our experiments we set this param-

eter to 15 mm) and two angles are considered equal

when their values differ for less than ANGLE DI FFER-

ENCETOLERANCE (in our experiments we set this pa-

rameter to 0.2 rad).

2.1 Finding Transformations

This step, given the scans S

and S

, ﬁrst ﬁnds the an-

gles between the segments in S

and between the seg-

349

Amigoni F., Gasparini S. and Gini M. (2004).

SCAN MATCHING WITHOUT ODOMETRY INFORMATION.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 349-352

DOI: 10.5220/0001141303490352

 SciTePress

ments in S

and, second, ﬁnds the possible transfor-

mations (namely, the rotations and translations) that

superimpose at least one angle α

of S

to an equal

angle α

of S

. Angles between pairs of segments in a

scan are the geometrical landmarks we adopt. Finding

the possible transformations is a difﬁcult combinato-

rial problem since in principle, without any informa-

tion about the relative positions of the two scans, there

are O(n

) possible transformations, where n

and

are the numbers of segments in S

and S

, respec-

tively. We have therefore devised three heuristics for

reducing this complexity and ﬁnding a set of (hope-

fully) signiﬁcant transformations between two scans.

They are described in the following.

1. Considering Angles between Consecutive Seg-

ments. In each scan, we select the angles between

two consecutive segments; let A

and A

be the sets

of such angles for S

and S

, respectively. Two seg-

ments are considered consecutive when they have an

extreme point in common. Then, we ﬁnd the set of

all the transformations that make an angle in A

correspond to an equal angle in A

. The number

of possible transformations found by this method is

O(n

). We note that ﬁnding the sets A

and A

greatly facilitated when the segments in S

and in S

are ordered. This is usually the case with laser range

scanners, since the points returned by the sensor are

ordered counterclockwise and it is straightforward to

maintain the same order in the segments that approx-

imate the points.

Although this method seems to perform well in in-

door environments where the angles are usually nor-

mal, the errors introduced by the sensor and by the

algorithm that approximates points with segments al-

ter the representation of these angles.

2. Considering Angles between Randomly Se-

lected Segments. In each scan, we examine a num-

ber of angles between pairs of segments selected ran-

domly. We assign a higher probability to be selected

to longer segments, since they provide more precise

information about the environment. Let A

and A

be the sets of the selected angles for S

and S

, re-

spectively. We ﬁnd the set of all the transformations

that brings an angle in A

to correspond to an equal

angle in A

. The number of transformations gener-

ated by this method is O(a

), where a

= |A

| and

= |A

| are the number of selected angles in A

and A

, respectively.

Instead of assigning directly to each segment the

probability of being selected (according to its length)

and of selecting the a

(respectively a

) pairs, the

following approximate and easy-to-implement tech-

nique is employed. Initially only segments longer

than SEGMENTDIVISIONFAC TOR times the length of

the longest segment in S

(resp. S

) are considered

for selection. All the segments considered have equal

probability of being selected. Then, we proceed to it-

erate with k = 1, . . . , K. In the k-th iteration, we use

a threshold equal to SEGMENTDIVISIONFACTOR

times the length of the longest segment in S

(resp.

). Out of the segments longer than this threshold

we select one with equal probability. Thus, the pa-

rameter SEGMENTDIV IS IO NFAC TOR determines the

length of the segments that are considered for selec-

tion and, implicitly, the probability of selection. This

technique tries ﬁrst to ﬁnd transformations based on

angles between long segments; then it progressively

considers transformations based on angles between

shorter and shorter segments.

3. Considering Angles between Perpendicular Seg-

ments. In each scan, we select only angles between

perpendicular segments. This heuristic is particularly

convenient for indoor environments, where the pres-

ence of regular walls usually involves perpendicular

segments. The heuristic is based on histograms. The

histogram of S

(and, in similar way, that of S

)

is an array of nslots elements, where nslots is the

number of buckets of the histogram. Each bucket

(i = 0, 1, . . . , nslots − 1) contains the segments

with orientation comprised between π × i/nslots and

π × (i + 1)/nslots, measured with respect to a given

reference axis. To each element L

of the histogram

of S

is associated a value calculated as the sum of the

lengths of the segments in L

. The principal direction

of an histogram is the element with maximum value.

The normal direction of an histogram is the element

that is π/2 rad away from the principal direction. Let

and A

be the sets of angles formed by a seg-

ment in the principal direction and by a segment in

the normal direction of the histograms of S

and S

respectively. The set of possible transformations is

then found comparing the angles in A

and A

. The

number of possible transformations generated by the

above heuristic is O(p

), where p

and n

are

the number of segments in the principal and normal

directions of the histogram of scan S

2.2 Evaluating Transformations

Every transformation found in the previous step

is evaluated in order to identify the best one. To

determine the goodness of a transformation t we

transform S

on S

(in the reference frame of S

)

according to t (obtaining S

), then we calculate

the approximate length of the segments of S

that

correspond to (namely, match with) segments of

. The measure of a transformation is the length

of the corresponding segments that the transforma-

tion produces. More precisely, the measure of a

transformation is the sum of all the matching values

calculated for every pair of segments s

∈ S

and

∈ S

. The matching value between two segments

and s

is calculated as follows. We project s

on the line supporting s

thus getting a projected

ICINCO 2004 - ROBOTICS AND AUTOMATION

350

segment s

and then we compute the length l

the common part of s1 and s

; we do the same but

projecting s

on s

, obtaining l

. The matching value

of s

and s

is calculated as the average of l

and l

When s

and s

do not intersect, the matching value

is multiplied by 0.95

d(s

)/PO I N T DI S TANCETO L E R A N C E

to penalize the match between segments that are

far away. Note that 0.95 is an empirical constant

whose value has been determined during exper-

imental activities and d(s

, s

) is the distance

between two segments, calculated as d(s

, s

) =

min(max(dist(s

, start(s

)), dist(s

, end(s

))),

max(dist(s

, start(s

)), dist(s

, end(s

)))) where

start(s) and end(s) are the extremes of seg-

ment s. Finally, two special cases can appear

during the evaluation of the matching values

of s

and s

. The matching value is set to 0

when the two segments are too far away, namely

when d(s

, s

)/POI NT DI STANCETOLERANCE >

SEGMENTDISTANCE TH RE SH OL D. SEGMENTDIS-

TANCE TH RE SH OL D is usually set to 5 to obtain good

experimental results. The matching value is set to

−1 when the two segments intersect and are longer

than SEGMENTLENGTHREFUSE; in this case the

transformation is discarded.

The above algorithm evaluates a single transforma-

tion by considering all the pairs of segments of the

two scans that are O(n

2.3 Transforming and Fusing Scans

Once the best transformation

t has been found, the

third and last step of our method transforms the sec-

ond scan S

in the reference frame of S

according to

t obtaining S

The map that constitutes the output of our scan

matching method is obtained by fusing the seg-

ments of S

with the segments of S

. To this

end, we use the idea of matching chains. A

matching chain of the pair of scans S

and S

is a set C = {hs

, s

i|s

∈ S

and s

∈

have a positive matching value for

t} alge-

braically closed under segment belong-to relation.

Speciﬁcally, a matching chain C is such that if

, s

i ∈ C, then also hs

, si ∈ C and hs, s

i ∈ C

for all the segments s that have a positive match

value (namely, have matched with) s

or with s

We explicitly note that, given an element hs

, s

the matching chain C that contains (that is generated

by) hs

, s

i is uniquely identiﬁed. A transformation

t generates a set of (disjoint) matching chains. The

main idea behind the fusion of segments is that each

matching chain (i.e., each set of matching segments)

is substituted in the ﬁnal map by a single polyline.

Therefore, the ﬁnal map is obtained by adding the

polylines that represent the matched segments to the

unmatched segments of S

and S

. The problem is

thus reduced to build a polyline that approximates the

segments in a matching chain C. With this polyline,

it is easy to smoothly connect the different segments

inserted in the ﬁnal map.

The solution to the above problem consists in itera-

tively building a sequence of approximating polylines

, P

, . . . that converges to the polyline P that ade-

quately approximates (and substitutes in the resulting

map) the matching segments in C. The polyline P

composed of a single segment connecting the pair of

farthest points in C. Given the polyline P

n−1

, call s

the segment in (a pair belonging to) C that is at max-

imum distance from its (closest) corresponding seg-

ment ¯s in P

n−1

. If the distance d(s, ¯s) is less than the

acceptable error, then P

n−1

is the ﬁnal approximation

P . Otherwise, s substitutes ¯s in P

n−1

and s is con-

nected to the two closest segments in P

n−1

to obtain

the new polyline P

3 EXPERIMENTAL RESULTS

The method presented in this paper has been coded

in ANSI C++ employing LEDA libraries 4.2 (LEDA

Library, 2004) for two-dimensional geometry and has

been run on a 1GHz Pentium III processor with Linux

SuSe 8.0. We considered 31 pairs of scans (from

− S

to S

− S

) that have been acquired by

driving the robot manually and without recording any

odometric information. The scans have been collected

in a laboratory, a very scattered environment, in a nar-

row hallway with rectilinear walls, and in a depart-

ment hall, a large open space with long perpendicular

walls. The correctness of the scan matches has been

determined by visually evaluating the initial scans and

the ﬁnal map with respect to the real environment. For

every scan match, we tested the basic method and the

three heuristics, sometimes modifying the values of

the parameters.

In general, our experimental results demonstrate

that our method performs very well: 28 pairs of scans

out of 31 have been correctly matched. Unsurpris-

ingly, the histogram-based heuristic worked well with

scans containing long and perpendicular segments, as

those taken in the hallway and in the hall. The heuris-

tic based on consecutive segments seems to work well

in all three kinds of environment, even if sometimes

it needs some parameter adjustments.

Table 1 shows the results obtained for three inter-

esting scan matches (see also Fig. 1). S

and S

were

taken inside the laboratory: they contain a large num-

ber of short segments since the environment is highly

scattered. S

and S

were taken along the hallway:

they contain fewer segments than the previous scans

SCAN MATCHING WITHOUT ODOMETRY INFORMATION

351

Figure 1: Top, left to right: scans S

, S

, and S

; bottom, left to right: ﬁnal maps S

4,5

, S

18,19

, and S

25,26

and are characterized by long rectilinear segments.

and S

were taken in the hall: they contain only

few segments since the environment is characterized

by long rectilinear and perpendicular walls.

Table 1: Some experimental results

# of segments 47 36 24 24 10 12

All 936 s [41260]

32 s [3096] 0.38 s [231]

Consecutive 1.25 s [2] 0.73 s [27] 0.13 s [4]

Random

7.69 s 2.51 s 0.78 s

Histogram 3.29 s [73] 1.97 s [192] 0.15 s [32]

[Number of possible transformations that have been evaluated]

Obtained by generating about 20000 angles

4 CONCLUSIONS

We have described a method for scan matching that

works without any information about the relative po-

sitions of the two scans but relies exclusively on the

geometrical features of the scans. This is the major

feature which distinguishes our method from most of

the scan alignment and matching methods reported

in the literature. Extensive experimental results vali-

dated the effectiveness of the approach. We are work-

ing to apply this method to the integration of n scans.

ACKNOWLEDGEMENTS

The authors would like to thank Jean-Claude Latombe

for his generous hospitality at Stanford University

where this research was started, H

ector Gonz

ales-

nos for sharing his programs and expertise with

collecting laser range scan data, Paolo Mazzoni and

Emanuele Ziglioli for the initial implementation of

the fusing algorithm. Francesco Amigoni was par-

tially supported by a Fulbright fellowship and by

“Progetto Giovani Ricercatori” 1999 grant, Maria

Gini by NSF under grants EIA-0224363 and EIA-

0324864.

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