Multi-sensor ATTenuation Estimation (MATTE):
Signal-strength prediction for teams of robots
Johannes Strom and Edwin Olson

Abstract— Multi-robot teams are often constrained by communications; better signal-strength models enable more efficient
coordination while still maintaining adequate communication.
This work discusses several prediction algorithms applicable to
this scenario. Whereas previous approaches typically focus on
prediction in the presence of deployed base-stations, we consider
the more general problem where all nodes in the network
can be mobile. Our new algorithm, Multi-sensor ATTenuation
Estimation (MATTE), addresses this problem by leveraging
other forms of sensor data in combination with signal-strength
measurements to infer the locations of attenuating materials in
the robots’ environment. We also extend prior tomographic and
correlation-based approaches to the multi-robot case, allowing
a competitive evaluation. All methods are evaluated on a large
corpus of real-world indoor and outdoor environments.

I. I NTRODUCTION
The inherently parallel nature of search-and-rescue missions creates an opportunity for teams of collaborating robots
to assist in emergency response. However, robot teams that
wish to collaborate must communicate to coordinate effectively. Current approaches to multi-robot coordination typically only incorporate a simple fixed-radius communication
model as a planning constraint [1], [2], [3]. In environments
with variable attenuation (e.g. urban environments), picking
a single radius may not be appropriate since communication
will be easier in open spaces, and harder in densely-built
neighborhoods.
Attempts to incorporate more complicated models of signal propagation typically focus on the case where robots
communicate with an exiting fixed base station [4]. Unfortunately, many domains lack a usable communications
infrastructure (e.g. disaster zones), forcing robots to deploy
their own. Achieving good performance from fully-mobile
networks is challenging because more complicated signalpropagation models must be incorporated into the pathplanning process to ensure connectivity. Furthermore, existing models for fixed-transmitters do not extend directly to
the case where all nodes are mobile. While prediction in the
former case is analogous to regression in a two dimensional
space, the later requires making predictions for a four dimensional space, but without a corresponding increase in data
density. The result is that achieving similar generalization
performance from observed signal-strength measurements
becomes more challenging. This paper explores how existing
methods can be modified to better cope with this reduction
This work was supported by U.S. DoD Grant FA2386-11-1-4024.
The authors are with Department of Computer Science and Engineering, University of Michigan, Ann Arbor, MI 48109 USA

{jhstrom,ebolson}@umich.edu

Fig. 1.
Attenuation estimation computed from a single traversal of a
160×100m environment by 3 robots conducting an exploration mission.
Blue indicates regions where signals are attenuated, white and orange
regions indicate where signals pass more easily. Black denotes building
structure. Our algorithm, MATTE, is designed for predicting signal strength
in the challenging case where all transmitters and receivers are mobile.

in data density, and explores methods for using additional
sensor data to inform a better prior over the locations of
significant attenuators in the environment. Specifically, the
main contributions of this paper are:
• Extension of tomographic and correlative signal prediction techniques to the case of multiple mobile robots
without a base station.
• A new signal-strength prediction method, Multi-sensor
ATTenuation Estimation (MATTE), which additionally
leverages the robots’ LIDAR data to better predict the
location of attenuating objects.
• Extensive evaluation on real world datasets covering
over 40,000 m2 of both indoor and outdoor environments
II. R ELATED W ORK
In the robotics domain, planning with guaranteed communication is a well studied topic [5], [6]. However, communication between two agents cannot be ensured in general,
so these methods are limited in their application to the real
world. Others have studied collaborative planning under a
fixed-radius communication assumption [1], but such methods result in unnecessarily conservative strategies because
they fail to exploit long-range links when possible, reducing
the effective speed of the robots. More realistic models can
be obtained by first predicting the signal strength along a
given link, which can then be used to predict packet success
rates. There are two main approaches to signal prediction
in the literature – the first is correlative in nature, that is

broadcasts from nearby positions are assumed to have similar
signal strengths, allowing prediction at nearby points using
what is essentially a locally-weighted average. The second
technique uses principles from tomography, where changes
in signal-strength are used to infer the presence of attenuating
objects, which can in turn be used to predict signal-strength
at unknown locations.
Malmirchegini and Mostofi have explored correlative
methods of predicting the link quality between a fixed
base station and a mobile robot, taking into account the
spatial correlations of signal strength measurements [7].
Fink and Kumar use a similar approach to allow a robot
to automatically localize a base-station [8]. However, these
approaches, also demonstrated in a simulated robotic domain
in [4], assume signals are uniformly correlated in all directions without regard to the location of attenuating objects.
Nonetheless, this approach has successfully been applied to
prediction of signal strength in real-world environments, and
forms the basis for one of the methods we benchmark in this
paper. Our extension of their approach addresses the more
general problem when all nodes are potentially mobile. This
makes the problem significantly more challenging because
of a substantial reduction in data density.
An alternative approach to predicting signal strength, using
principles from the field of tomography, is to explicitly
estimate the location and properties of attenuating objects
affecting signal propagation. Knowing the location of attenuating objects enables future signal strength to be predicted,
even when they are not spatially adjacent to previous measurements. The central challenge with this technique is that
directly estimating the positions of attenuating objects from
signal-strength measurements results in an ill-posed estimation problem. This is because for any given set of signal
measurements, there are many possible world configurations
which explain the data. Prior work by Wilson and Patwari
has explored the capabilities of RF tomography to recover the
motion of moving people inside a region bounded by a large
number of regularly placed radios [9]. Their work estimated
the derivative of the attenuation over a grid of pixels inside
the perimeters of radios to extract the positions of moving
people. By only estimating the derivative, their signal model
was simplified, allowing a single calibration step to be used
to later recover the target’s position. Despite the large number
of radios (28), their estimation problem was still ill-posed,
requiring application of regularization techniques to make
computing the attenuation-derivative possible [10]. However,
their approach does not directly apply to the case where
multiple robots traverse arbitrary paths, since in such cases
no prior calibration is possible, and the number of links
constraining the attenuation computed at each pixel can
vary considerably. Our signal-strength prediction algorithm
MATTE addresses these problems by employing a novel
fusion of laser range-finder data with signal-strength measurements to better constrain the attenuations estimated at
each pixel. Furthermore, we show that our approach scales
to environments hundreds of meters in diameter.

III. BACKGROUND
Standard macroscopic models of RF propagation describe
the expected signal strength, which depends on the location
of the receiver r and transmitter t, as having three main
components [2]:
ydBm = L0 − a log10 (||r − t||) − g(r, t) −
path-loss

shadowing

(1)
multipath

In simplified, ideal environments, signal-strength can be
determined purely by path-loss which has two degrees of
freedom: L0 corresponds to the power of the transmitter and
a is the path-loss exponent that determines how quickly the
signal attenuates with distance. In real environments both
shadowing and multipath can additionally affect the signal
strength: shadowing corresponds to the attentuation a signal
experiences as it passes through dense objects, and multipath
corresponds to amplification or cancellation which occurs
when waves travel multiple paths of different lengths from
source to destination. In general, multipath is very difficult
to predict because it results from complex reflection and
diffraction interactions. Shadowing, on the other hand, is
easier to predict, since the scale of its effects are larger and
more spatially coherent. For the remainder of this paper, we
will focus on models which predict path-loss and shadowing
but ignore multipath.
Robots are able to measure the signal-strength from each
other robot during the robot’s mission. These noisy values
form a vector y , with corresponding vectors of positions
ˆ
R and T containing all pairs of receiver and transmitter
positions, respectively. We treat predicting y as a linear regression problem. For the case of the simple path-loss model,
estimating the two variables (L0 , a) from the data is sufficient
to predict signal measurements for arbitrary positions. This
can be done using standard least-squares approaches: if
x = [L0 a] and the ith row of A is [1 log10 (||ri − ti ||)] then
the least-squared error estimate for x is x = (AT A)−1 AT y .
¯
ˆ
Path-loss-only models are useful due to their simplicity
and correspondence with theoretical propagation equations:
the low degree of freedom reduces the chance of overfitting. However, as this model ignores shadowing effects,
its application in environments with varying attenuation is
limited. In our evaluation, we will use this simple log-fit as
a baseline.
Correlative and tomographic methods include the same
path-loss model, but also explicitly incorporate shadowing,
allowing for improved predictive performance. The way in
which shadowing is captured varies between the two types
of models. In the tomographic case, the shadowing function,
g(·) is computed by integrating the effect of all attenuators
between transmitter and receiver. In practice, we model the
individual attenuation of a grid of infinitely tall columns,
represented by a regular 2D grid of pixels. If the signal
passes through a series of p discretized pixels, the shadowing
is computed as [9]:
p

g(r, t) =

wi vi
i

(2)

where vi is the attenuation in the ith pixel and wi is the
importance weight of that pixel for that signal’s path. (e.g.
in our implementation, the weights wi correspond to the
length of the line between transmitter and receiver which
is contained in the pixel.) In other words, we compute
attenuation per pixel by assuming that signal strength is
reduced linearly according to the sum of the pixels along
the path between receiver and transmitter. Given a set of
signal strength measurements, we can attempt to find the
attenuation values xi of each pixel using a similar leastsquares approach as described above. However, there are
many possible attenuation assignments to the pixels which
can adequately explain the signal strength measurements,
resulting in an under-determined system of equations. In the
next section we will discuss applying regularization techniques to encode a prior that prefers real-world environments,
thereby over-constraining the system of equations.
In the correlative case, the shadowing component is considered to be uniformly correlated in all directions, allowing
predictions to be made by making inferences from spatiallyproximate training points. In particular, prior approaches
have had success modeling shadowing using a Gaussianprocess (GP) [2], [7].
In the case of a fixed base-station (located at b), we
first use the log-fit as a mean function, and then use a
standard squared-exponential kernel to specify the expected
covariance between signals at two locations, x and x :
2
k(x, x ) = σf exp{− ||x−x || }. Using this covariance (kernel)
l2
function, we can apply standard GP regression techniques to
make predictions for a set of sample points [11]:
1) Compute Log-fit: Fit L0 and a using least-squared
approach described above.
2) Fit Hyperparameters: Choose correlation distance l
and function variance σf to maximize likelihood of
training data.
3) Compute Covariances: Compute covariance matrix
Ky of training data, and covariance vector k∗ of sample
2
points with respect to training data. Ky = Kf + σd I,
where each entry ki,j ∈ Ky = k(xi , xj ).
T
−1
4) Evaluate prediction: ysample = k∗ Ky (yobserved −ylog )
These correlative methods have good predictive performance, especially in the case when many training points
are available. The main shortcomings of this method are
that prediction assumes signals are uniformly correlated in
all directions – an assumption which breaks down in the
presence of discrete attenuating objects. Direct extension of
this method to the case of mobile nodes is also problematic,
since training points are in R4 , requiring significantly more
data to achieve the same performance. Finally, this method
is computationally expensive, requiring the inversion of a
matrix whose dimension is determined by the number of
training points.
IV. M ETHODS
In this section, we extend both the tomographic and correlative methods to the case of multiple mobile nodes. We will
describe our modifications to these existing approaches, and

introduce our new approach, MATTE, which also leverages
other sensors to infer the location of attenuating objects.
A. Correlative Methods for Mobile Nodes
Extending previous approaches of correlative prediction to
the case of moving nodes exacerbates the data-sparsity problem. Instead of producing signal predictions for R2 (all points
in the plane), we now must produce predictions in R4 (all
possible pairs of points in the plane). Since we can’t increase
the number of signal-strength observations that robots make
without slowing down the speed of exploration, this means
we have significantly reduced data density. However, we
were able to mitigate this problem somewhat by recognizing
that our signal-strength models are symmetric with regard to
where the transmitter and receiver are – that is, we assume
the signal strength is the same from robot A to B as it is
from robot B to A. This observation allows us to construct a
symmetric distance function,ds4 , which effectively doubles
the data density:
ds4 (ra , ta , rb , tb ) = min

||ra − rb || + ||ta − tb ||
||ra − tb || + ||ta − rb ||

(3)

In other words, ds4 is a distance metric for pairs of lines that
is invariant to rotations of 180 degrees.
In practice, many thousands of observations may be
available for use in training the Gaussian process. Due to
the O(n3 ) computation cost of matrix inversion, it quickly
becomes impractical to include all training data. However,
prediction performance is improved as more training points
are used. In the case where K is very sparse, the inversion
can be done more quickly, enabling use of more training
data. However, the squared exponential kernel is not sparse;
two measurements will never have a covariance of exactly
zero, even if they are very far apart. By modifying the kernel
to have compact support – that is, the kernel is exactly zero
once some distance Θ is reached – the matrix K becomes
sparse [12]:
ks (x, x ) = max(0, 1 −

||x − x || γ
) ∗ k(x, x )
Θ

(4)

The resulting sparsity allows computing K −1 much faster,
enabling the use of more training points. Although sparse
kernels generally have worse performance, we found in practice the that the speed improvement enabled the incorporation
of enough extra training points to achieve a net improvement
in performance. This sparsification also increases the number
of hyper-parameters that must be estimated to a total of 5.
In principle, we can learn these additional parameters the
same way we learn the parameters for the original covariance
function. However, increasing Θ will always result in a lower
training error and an increased computation due to a lesssparse covariance matrix. To limit worst-case computation
time, we do an offline parameter sweep to estimate the best
Θ and γ subject to a fixed CPU budget.
Together, the symmetric distance metric and the forced
sparsification of the correlation function enabled us to adapt
existing correlative prediction techniques to the case of

multiple mobile robots, and achieve results competitive with
our proposed method. These additions to existing methods
form the Multi-robot Gaussian Process “MRGP” method
which we include in our evaluation.
B. Multi-robot Tomography (MRT)
Conceptually, extension of the tomographic methods to
the case of multiple robots is relatively straight-forward. The
attenuation of each pixel through which a signal passed is
estimated in least-squares fashion. The number of pixels, p,
is determined by the grid size such that the pixels completely
fill the workspace of the robots. For a grid of width w and
height h, p = w ×h and pixels are labeled v0,0 · · · vw−1,h−1 .
In addition, the two path-loss parameters, L0 and a are also
estimated simultaneously, resulting in n = p + 2 unknowns:
x = [L0 a v0,0 · · · vw−1,h−1 ]. Each of the m signal strength
observations yi provide one equation partially constraining a
subset of the pixels in addition to the path loss parameters
(See Eqns 2 and 1). Together these equations are stacked
to form the rows of an n × m matrix A. If A is full rank,
x = (AT A)−1 AT y. However, even though m is generally
greater than n, A remains rank deficient, resulting in many
possible solutions for x. Related approaches have solved
this by using Tikhanov regularization that enforces smooth
changes in attenuation between neighboring pixels [9]. This
corresponds to constructing a Tikhanov matrix Γ whose rows
correspond to equations of the form vi,j − vi+1,j = 0 and
vi,j − vi,j+1 = 0 for each i, j < w, h. The over-constrained
solution for x now takes the form
x = (AT A + λ2 ΓT Γ)−1 AT y

(5)

where λ is a parameter to determine the weight of the
smoothness constraints in Γ relative to the observation equations in A. In contrast to the correlative methods, which
scale O(m3 ) with respect to the number of observations,
the tomographic methods scale O(p3 ) with respect to the
number of pixels. However, unlike the correlative methods,
AT A is naturally sparse since each pixel is only jointly
constrained with a small number of other pixels. This means
that sparse matrix-inversion methods, such as a Cholesky
decomposition, can perform significantly better than O(p3 ).
In practice, AT A is still dense enough (95% zeros) that we
are limited to solving grids on the order of 100×100 (10000
pixels). For the datasets in our evaluation, this translates to
a grid size between 2 and 4 meters. Such large grid sizes
poorly approximate the sharp spatial changes in attentuation
found in real environments, such as at the border between a
building and neighboring free space. Furthermore, in order
to avoid over-fitting, λ must be set large enough to allow
very little spatial gradient in the attenuation of each pixel
(See Fig. 1). In areas where attenuation changes quickly,
the predictive power is limited. Our implementation of this
approach, Mutli-robot Tomography (MRT), is included in our
evaluation.
C. Multi-sensor ATTenuation Estimation (MATTE)
The approaches we’ve discussed so far focus solely on
using signal-strength readings for prediction of future mea-

surements. Many robots already carry additional sensors
for mapping and navigation. Intuitively, since the physical
structure of an environment (the position of walls and other
solid surfaces) influences signal propagation, a map of the
environment should help predict signal propagation. This is
especially true for sensors like laser range-finders, which
tend to have a range of at least 10-30 meters. Incorporating
a map does not solve the problem completely though, since
the attenuation of a wall depends on the material and LIDAR
generally can’t distinguish between a reinforced concrete
wall and drywall.
Specifically, we propose the use of occupancy grids derived from laser range-finders to provide a more informed
regularization constraint to tomographic methods. The occupancy grids collected by our robots label the world with three
classes: known free space, known structure and unknown.
This information can provide a much better prior about the
attenuating properties of the environment – for example, we
generally expect areas which are marked as free space to
pass signals with little interference. Similarly, knowing the
location of structures can provide a prior about where attenuation should increase dramatically. Besides providing a better
prior about the magnitude of attenuation, the occupancy grids
also provide a much finer view of the environment. Using the
MRT method, we are typically limited to coarse grid sizes
(e.g. 3 m for our datasets) due to computational constraints.
On the other hand, LIDAR-based occupancy grids can be
computed cheaply even for fine-grained grid sizes (we used
10 cm grid in our experiments).
Of the many ways of incorporating this data, we explored
explicitly estimating the attenuation of each of the three
classes separately. A simple approach to this problem is to
fit a single attenuation value to all pixel of the same class.
For example, known free space might have an attenuation of
−0.01dBm per meter, whereas structure (e.g. walls) could
have an attenuation of −0.3dBm per meter, and unknown
space could be approximated as somewhere in between. This
approach is notable in its simplicity – it has no parameters to
tune and only 5 degrees of freedom to fit the observed data:
two for path loss parameters and three more for the attenuation assigned to each class in the occupancy grid. In general,
however, it is a poor assumption that all objects detected by
the robots as structure will have the same attenuation. For
example, a wooden fence and a brick building have very
different effects on a signal, but can appear very similar
to a robot’s laser range finder. As our initial experiments
confirmed, this model performed poorly in explaining realworld data.
Another way to use the occupancy grids is to better
inform regularization methods, for example by enforcing
smoothness constraints only between neighboring pixels of
the same class. Using a more flexible model is difficult,
however, since individually estimating the attenuation for
each pixel in the fine-grained occupancy grid results in
a poorly constrained, computationally prohibitive problem.
The largest map we present has nearly 5 million pixels!
Downsampling to a coarser resolution can make the problem

more tractable, but reduces the ability to incorporate features
such as doors or walls which may be lost by resampling.
Algorithm 1 MATTE UPDATE(trainPoints, map)
init A, b from bounds(map)
for all p ∈ trainPoints do
r ← [1, log10 (||pr − pt ||), 0 · · · , 0]
for all locations vi ∈ losPath(p) do
c ← map.class(vi )
r(vi , c) ← 1
end for
A ← appendRow(r)
b ← appendElement(pz )
end for
{A, b} ← appendRegularization(A, b)
x ← cholesky.solve(AT A, Atb)
store x
store map
return

Algorithm 2 MATTE PREDICT(testPoints)
retrieve x
z ← vector(testP oints.len)
for all p ∈ testPoints do
zp ← x(0) + log10 (||p||)
zs ← attenuationIntegral(x, p, map)
p ← append(p, zp + zs )
end for
return p
Instead, we propose a method that benefits from the detailed resolution of the map, but has computational complexity comparable to the MRT method. Our technique, called
MATTE, separately estimates spatially-varying attenuation
for each of the three classes in the occupancy grid. These
estimates allow querying any point in the environment and
determining, for example, if there was structure there, what
attenuation it would have. When predicting signal-strength,
we first examine pixels in the occupancy grid to determine
which class they are, and then perform a look-up on the
appropriate attenuation estimate. The benefit of this approach
is that we can estimate the spatially-varying attenuation
at a resolution independent of the resolution required to
adequately map the environment structure. Typically grid
sizes of 10 to 20 cm are required to preserve the presence of
doors or thin walls – but attenuation rates of a particular
material, (e.g. building) tend to vary at a much slower
rate. The computational implications of this approach are
significant – whereas the resolution of the occupancy grid is
typically 10 cm, we have achieved good results with a grid
size of 4 m for the spatially varying per-class attenuations,
resulting in a reduction by a factor of 1600 in the number
of unknowns.
Similar to the previous tomographic approach, we simultaneously estimate the path-loss parameters and the spatially

(a) nw1
(b) nw2

(c) bbb3

(d) eecs4

(e) dc1

Fig. 2. Maps collected in each of the five datasets we used in our evaluation.
From left to right they are: two outdoor sections of the Northwood
residential housing complex, the 3rd floor of the Bob and Betty Beyster
Building, the 4th floor of the Electrical Engineering and Computer Science
Building and the first floor of the Duderstadt Library. White is unexplored,
gray is known free-space and black is structure. Each evaluation set consists
of back-to-back traversals of three of these environments; meta-parameters
are tuned using the remaining two.

varying attenuation. For a workspace w × h pixels in dimensions, we estimate a total of 2 + 3 ∗ w ∗ h variables that
comprise x. Although this represents a threefold increase
in the number of degrees of freedom over our previous
approach, AT A is generally more sparse (e.g. 99% sparse
in our datasets), ultimately requiring less time to compute
a result. An overview of the update and prediction steps
are shown in Algorithms 1 and 2. The MATTE algorithm
has several important parameters which can either be tuned
by hand or estimated automatically from training data. The
most important parameter is the relative weight λ given to the
smoothing regularizer we described in Eqn. 5. In addition we
introduce three parameters to govern the expected attenuation
in areas where we have not collected any signal data: a
weighting factor φ determines the relative weight of this
prior, and prior attentuation values in units of dBm per meter
for each class are ρf , ρs , ρu , for free-space, structure, and
unknown space respectively. In practice, we set ρs = ρu ,
allowing free space to have a distinct prior from other areas.
An additional potential advantage of MATTE is that
new occupancy grids can be incorporated cheaply, as the
underlying spatially varying attenuation does not necessarily
need to be updated when the map changes. For example, our
approach might encode the knowledge that a structure-class
pixels in a particular area tend to have an attenuation of X
dBm per meter. If the map of that area is later expanded,
that information will be able to provide an estimate for the
attenuation expected there, without needing to recompute x.
This allows our approach to potentially be adapted to run
online.

Eval.

Method

Meta-parameters

MATTE
MRT

Train: {nw2,bbb3} Test: {eecs4,dc1,nw1}
λ=2.0 φ=.2 ρf =-.1 ρu,s =-.3
λ=0.1375

MATTE
MRT

Train: {eecs4,dc1} Test: {bbb3,nw2,nw2}
λ=1.225 φ=0.05 ρf =0.2125 ρu,s =-0.2875
λ=0.1125

MATTE
MRT

Training: {bbb3,eecs4} Test: {nw2,nw1,dc1}
λ=2.0 φ=0.275 ρf =0.2875 ρu,s =-0.4875
λ=0.1250

MATTE
MRT

Train: {nw1,eecs4} Test: {nw2,dc1,bb3}
λ=1.38 φ=0.3 ρf =-.1 ρu,s =-.3
λ=0.1125

MATTE
MRT

Train: {dc1,nw1} Test: {eecs4,bbb3,nw2}
λ=2.0 φ=0.1 ρf =-0.15 ρu,s =-.3
λ=0.5875

1

2

3

4

5

TABLE I
AUTOMATICALLY DETERMINED PARAMETER SETTINGS FOR EACH
EVALUATION USING COORDINATE DESCENT ON THE TRAINING SET.

Some parameters were fixed for performance reasons, including the grid
sizes for MATTE and MRT methods, at 4.0 meters and 3.0 meters
respectively. For MRGP, the number of training points was set to
min(600, .05 ∗ m), Θ was fixed to 20.0, and γ to 3. Also σd and σf
were fixed for numerical stability reasons at 1.5 and 1.0 respectively. For
MRGP, coordinate descent fixed l to 28.75 on all training data.

V. E VALUATION
We evaluated the three primary methods we have discussed
to determine which methods were most successful at predicting real-world signal-strength measurements. The main
purposes of our evaluation is to show that previous signalstrength predictions methods using a fixed base station can
be extended to the more general case where all nodes are
mobile. We also show that in many cases sensor data can
be effectively leveraged to further improve signal-strength
predictions. Several of our models have many degrees of
freedom, so our evaluation also seeks to show that the
methods we present can generalize well from previous observations to the prediction of future signal measurements.
A. Test Apparatus
We used a robot platform our lab custom designed for
urban reconnaissance [13]. We outfitted three of our 14
robots with additional 2.4 GHz TP-Link WiFi radios which
were programmed to report signal-strength measurements at
20 Hz. Our robots are equipped with 3D laser range finders,
in addition to IMUs and odometers, enabling them to produce
high-quality globally consistent maps using Simultaneous
Localization and Mapping (SLAM) algorithms [14], [15],
[16]. This capability allows us to quickly collect large
amounts of signal-strength data that is co-registered with a
global grid.
We used these three robots to collect a series of 5 datasets
in various indoor and outdoor environments, spanning a total
area over 40,000 m2 . The datasets consist of three indoor
environments, bbb3, eecs4 and dc1 for short, as well as two

larger outdoor environments, abbreviated nw1 and nw2 (See
Fig. 2 for details). Real-world rescue robots must operate
in mixed indoor-outdoor urban environments. In order to
mimic these conditions, we randomly selected a sequence
of three of the environments for testing, guaranteeing a mix
of indoor and outdoor datasets. Each of the datasets consist
of exploration missions so very few sensor measurements can
be considered duplicates, since robots do not tend to retrace
their steps. This enables us to explicitly test the predictive
performance of each method, rather than their recall abilities.
For each randomly-selected set of three areas, we replayed
the signal-strength observations and corresponding maps in
two second increments. After each step, the methods were
tested on their prediction performance of the next 10 intervals
of future signal strength data (e.g from 0-2 seconds, 24 seconds, 4-6 seconds, etc.). As the traversal played out,
methods had access to an increasing amount of training
data, but the amount of testing data was relatively similar
at each step. Several of the methods also have meta- or
hyper- parameters which need to be tuned before the start
of a traversal. We used the two remaining areas not selected
for the test set to train meta parameters using a compass
search. We include results for 5 such randomly generated
traversals. The parameters selected by the compass search
for each method on each evaluation are shown in Table I. All
of the methods we presented were tuned to use comparable
amounts of computational resources. While each method
exhibits significantly different asymptotic growth, we set
parameters which resulted in roughly equal CPU use over the
course of an evaluation set. Run-times for MATTE ranged
between 2 seconds to process the bbb3 dataset up to 30
seconds for the significantly larger nw2 dataset. For the GP
method, times varied between 15 seconds for bbb3 datasets
and 20 seconds on nw2.
B. Results
The testing results for the five evaluation sets are displayed
in Fig. 3. As expected, all methods are better at making
short-term rather than long-term predictions. This is a result
of the fact that measurements nearer in the future are more
likely to be similar to existing measurements, or the fact that
attenuating objects impacting observations in the near future
are likely to be correlated with sensor data the robots have
just now collected.
All of the methods we have implemented show competitive
performance, especially for predictions between 0 and 10
seconds in the future. However, MATTE significantly outperforms the other methods in evaluations 1 and 5. In evaluations 2 and 4, it performs comparable to the tomographic
method. However, in evaluation 3, our method exhibits worse
performance than the other methods. An examination of
the meta-parameters determined via compass search show
evidence of over-fiting the training set, which by nature
of our randomly selected test sets, happened to both be
exclusively indoors. This is manifest as a positive attenuation
prior for free-space (ρf ), consistent with the wave-guide
effect sometimes seen in hallways. In the other evaluation

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(a) Eval 1 {eecs4,dc1,nw1} (b) Eval 2 {bbb3,nw2,nw1}

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(c) Eval 3 {nw2,nw1,dc1}

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(d) Eval 4 {nw2,dc1,bbb3} (e) Eval 5 {eecs4,bbb3,nw2}

Fig. 3. MSE of prediction accuracy for each method evaluated on 5 randomly chosen sequences over the testing datasets. Error bars are bounded by 0.8
dBm2 and are omitted for clarity. Our proposed method, MATTE, is compared to extensions of previous methods MRGP and MRT. Our method exploits
generally performs better for near-term predictions where the robots’ sensor data provides an informative prior. Poor performance of MATTE in evaluation
3 is attributable to the randomly generated training set which contains only indoor datasets, while the testing set contains both indoor and outdoor sets.

sets, there was at least one outdoor dataset used for training,
which helped to mitigate this type of over fitting. In most
of the evaluation sets, we see that MATTE can out-perform
the correlative method for short-term predictions up to ten
seconds in the future. For longer-term predictions, where
training data is mostly useless, it is difficult to beat the log
baseline.

VI. C ONCLUSION

In this work, we have explored the difficult problem
of predicting signal strength when all nodes are actively
exploring a variety of real world environments. This problem is challenging because robots typically only sample
the possible signal propagation paths very sparsely, making
generalization beyond a naive log-fit difficult. We extended
several existing signal strength prediction methods to the case
of multiple robots conducting exploration-style missions in
mixed indoor-outdoor urban environments. We additionally
suggested a new method for signal prediction which uses
additional sensors already in use by many autonomous
robots to better estimate the regions of attenuation in an
environment. Our methods performs competitively with the
other approaches, and in some cases performs much better.
Key to our approach is the ability to estimate the attenuation
properties of the environment at a coarse level, but still use
a fine-grained spatial model of an environment derived from
additional sensors. It is interesting that both correlative and
tomographic methods exhibit such similar performance, as
they employ very different approaches to the same problem.
This suggests that both methods are exploiting similar aspects of signal strength propagation.
Our work has important applications to autonomous robot
teams which are collaborating to achieve a joint goal. By
introducing more advanced signal-strength modeling techniques, such teams can better predict when robots can expect
to communicate, allowing them to plan their future actions

by explicitly including communication as a constraint.
In the future we look to continue exploring methods for
effectively predicting signal strength. In particular, it may be
possible to leverage additional sensing modalities to improve
prediction. Incorporating this predictor in an online planning
system would also serve to further validate our approach.
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