![](bg3.png)
and upper bounds are used, defined respectively by x
and x. We assume the true measurement is enclosed
in the interval [x] = [x,x]. However, no assumption is
made as to which value is most likely. By conducting
the intervals of two distance measurements A = [9,10]
and B = [2, 3] from the same reference point, we cal-
culate their interval distance as follows:
[9,10]− [2,3] = [9 − 3,10 − 2] = [6,8]. (1)
Hence, it is guaranteed that the distance is between
6 and 8. More dimensional intervals are represented
as an interval vector [x], resulting in an interval box.
Furthermore, given a measurement Y ∈ R
m
(e.g. the
TDoA) and a non-linear measurement function f :
R
n
→ R
m
the relationship to an unknown set X (e.g.
the position of the sound sources) is characterized as
follows:
X = {x ∈ R
n
| f(x) ∈ Y} = f
−1
(Y). (2)
The unknown set X can be calculated with a branch
and bound algorithm Set Inversion Via Interval Anal-
ysis (SIVIA) (Jaulin et al., 2001). Another approach
is to formulate f as a Constraint Satisfaction Prob-
lem (CSP) and use so-called contractors (Chabert and
Jaulin, 2009). The main idea is to start with an initial
search space containing a set of boxes. On each box
a calculation is performed and inconsistent parts are
removed, resulting in a smaller box.
3.2 Application to SSM
As we will show in Section 5, the main advantage of
using interval analysis in SSM is the simple method-
ology of solving the CSP resulting from the relation-
ship between the microphone signals (represented by
the TDoA) and their position. The solution (possible
position of sound sources) is represented by a set of
interval boxes. After conducting a subsequent mea-
surement from a different position, a new restriction is
added to the CSP. Fortunately, by using interval anal-
ysis, this process can be performed by the intersection
of the interval boxes of the previous solution. Impor-
tantly, both the microphone position and the TDoA
can also be modeled as interval boxes. By doing this,
the solution of the CSP will result in larger interval
boxes, compared to fixed values for both quantities.
However, uncertainties can be modeled in a simple
manner, making it applicable for real scenarios.
First, the accuracy of the transformation between
the microphones and the reference coordinate sys-
tem at the robot is affected by the used calibration
method. This knowledge needs to be integrated to
the transformation by specifying an interval box [x].
Next, the localization accuracy of the robot within the
map depends on the used sensor and the resolution
of the map. (Langerwisch and Wagner, 2012) pro-
pose an interval-based approach for guaranteed robot
localization. In (Sliwka et al., 2011) interval meth-
ods are used in the context of robust localization of
underwater robots. Furthermore, extracting multiple
TDoA’s from the microphone signals in a noisy envi-
ronments is a challenging task. In many cases an es-
timation is given by extracting peaks from the cross-
correlation function. However, signals are sampled
at a discrete timestamp. Therefore, the uncertainty of
the TDoA highly depends on the sampling frequency.
A interval-based method to estimate the timestamps
between two sensors are proposed in (Voges and Wag-
ner, 2018).
4 PROBLEM DEFINITION AND
NOTATION
Let us assume, various sound sources s ∈ {1,...,n
s
}
are emitting acoustical signals with the velocity of
sound c in the current environment. n
s
is the total
number of sources, which is unknown in advance.
Their positions are characterized by x
s
∈ R
3
. Fur-
ther, a mobile robot perceives these acoustic signals
using a microphone array, equipped with n
m
micro-
phones. We model the relationship between micro-
phone pairs. Therefore, we denote the first and second
position of a microphone pair i ∈ {1,..., n
p
} as x
(n)
m
P
i,1
and x
(n)
m
P
i,2
∈ R
3
. n
p
is the total number of used micro-
phone pairs. Due to the movement of the robot, the
positions of the microphones are changing. There-
fore, the superscript n ∈ {1,.., n
l
} denotes the index
of location and n
l
is the total number of locations.
Moreover, the time a signal arrives at the first and
the second microphone results in a time difference -
so-called Time Difference of Arrival (TDoA) - which
we denote as
(s)
∆t
(n)
i
. The superscript s indicates the
TDoA resulting from source s. Further, the TDoA
depends on the position of microphone pair i at loca-
tion index n. Finally, we form a relationship between
the position of a microphone pair and the TDoA mea-
sured for a single source s as followed:
||x
(n)
m
P
i,1
− x
s
||
2
− ||x
(n)
m
P
i,2
− x
s
||
2
=
(s)
∆t
(n)
i
· c. (3)
For an easier understanding we show the relation-
ship in Fig. 2a dropping the superscript (n). It can
be noted that, the left-hand side of Equation (3) con-
tains the geometrical properties of the microphone
configuration, whereas the right-hand side includes
the measurements of the microphone pair in form of
Interval-based Sound Source Mapping for Mobile Robots
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