# Measurement Errors in Range-Based Localization Algorithms for UAVs:
Analysis and Experimentation

Francesco Betti Sorbelli, Cristina M. Pinotti, Simone Silvestri, and Sajal K.
Das  Francesco Betti Sorbelli is with the Department of Computer Science,
Missouri University of Science and Technology, Rolla, MO 65409, USA. Part of
this work was done while F. Betti Sorbelli was working for the NALP-SAPR
Project in Perugia. Cristina M. Pinotti is with the Department of Computer
Science and Mathematics, University of Perugia, Italy. Simone Silvestri is
with the Department of Computer Science, University of Kentucky, Lexington, KY
40506, USA. Sajal K. Das is with the Department of Computer Science, Missouri
University of Science and Technology, Rolla, MO 65409, USA. Part of this work
was done while S. K. Das was Satish Dhawan Visiting Chair Professor at the
Indian Institute of Science, Bangalore - 560012, India.

###### Abstract

Localizing ground devices (GDs) is an important requirement for a wide variety
of applications, such as infrastructure monitoring, precision agriculture,
search and rescue operations, to name a few. To this end, unmanned aerial
vehicles (UAVs) or drones offer a promising technology due to their
flexibility. However, the distance measurements performed using a drone, an
integral part of a localization procedure, incur several errors that affect
the localization accuracy. In this paper, we provide analytical expressions
for the impact of different kinds of measurement errors on the ground distance
between the UAV and GDs. We review three range-based and three range-free
localization algorithms, identify their source of errors, and analytically
derive the error bounds resulting from aggregating multiple inaccurate
measurements. We then extend the range-free algorithms for improved accuracy.
We validate our theoretical analysis and compare the observed localization
error of the algorithms after collecting data from a testbed using ten GDs and
one drone, equipped with ultra wide band (UWB) antennas and operating in an
open field. Results show that our analysis closely matches with experimental
localization errors. Moreover, compared to their original counterparts, the
extended range-free algorithms significantly improve the accuracy.

###### Index Terms:

Drone, localization algorithm, range-based, ground error, trilateration error,
localization error

##  1 Introduction

In recent years, unmanned aerial vehicles (UAVs) or drones have received
increasing attention from both the research and industry community. Of
particular interest are applications enabled by combining UAVs with the
Internet of Things (IoT) [1, 2], such as environmental monitoring [3],
structural health monitoring [4], precision agriculture [5], search and rescue
operations [6], and so on.

An important requirement of such applications is the ability to accurately
localize the position of ground devices (GDs), making the collected data more
meaningful. Since it is costly to equip each GD in the network with a GPS
module, a fixed set of anchor devices, whose positions are known a-priori, is
generally used [7]. Moreover, given the anchors use wireless transmissions to
localize other GDs and their range is often limited, the number of required
anchors could dramatically increase with the size of the network, thus
increasing the cost of the localization procedure. This problem can be solved
by replacing fixed anchor devices with a single mobile anchor (MA) equipped
with a GPS unit and periodically broadcasting its position to help nearby GDs
localize.

Although there exists some work in the literature on localization based on
ground MAs, such as rovers [8], relatively less has been proposed using flying
MAs, like UAVs or drones [9], which is the focus of this paper. Compared to
the ground MAs, the flying MAs are able to reach remote locations, move at a
faster speed, and cover a wider area than terrestrial rovers [10]. Due to
these advantages, in this paper we concentrate on localization algorithms
involving flying MAs. When using UAVs as MAs, the distance can be estimated
wirelessly by measuring the time of flight (ToF) between the MA and GDs. For
distance measurements, in this paper we adopt the ultra wide band (UWB)
technology [11].

Localization algorithms can be broadly categorized as range-free and range-
based approaches [8]. In the range-free algorithms, the position is estimated
without any type of measurements, but by only discovering if the GD and MA are
in range. Among these, the radius-based approaches assume the knowledge of
transmission radius [12], while the radius-free ones do not [13]. Such
algorithms are often based on the assumption that the antenna radiation
pattern is isotropic, which is unrealistic in general. In fact, our recent
works [14, 15] have shown that the localization accuracy depends on the
quality (pattern and radius) of the antenna, and how much they differ from the
assumed isotropic pattern. On the other hand, in the range-based localization
algorithms, the position of the GD is estimated by taking several measurements
between it and the MA. These algorithms are known to be more accurate than
range-free algorithms but at the cost of additional specialized hardware. For
example, the estimation of distance exploits techniques like the received
signal strength indicator (RSSI), the time of arrival (ToA), or the time
difference of arrival (TDoA) [16].

Now, in any range-based localization procedure, measurement errors are
unavoidable, and can seriously impact the localization accuracy. This is
particularly relevant when the MA is a drone because the measurement errors
can occur while calculating the distance between the MA and the GD. The
magnitude of such errors depends on the adopted technology and on the quality
of the air-to-ground (A2G) link between the MA and the GD. For example, the
distance measurement error in a fully line of sight (LoS) link using the Wi-Fi
technology is about $7$–$10\mathrm{\,m}$; using Bluetooth it can be up to
$15\mathrm{\,m}$ while it is only $10\mathrm{\,cm}$ using the UWB technology
[17]. Additional errors may be caused by non-optimal weather conditions or the
drone’s GPS accuracy [18]. For instance, the 3DR Solo drone used in our
experiments has a GPS accuracy of $1\mathrm{\,m}$ [19]. In general, these
errors propagate when combined and projected to the ground to localize the
GDs. The error propagation depends on the specific localization technique,
such as trilateration [20], intersection of points, centroid [21, 22], and so
on.

##### Our Contributions

In this paper, we first provide bounds on various errors (e.g., instrumental
error, rolling error, altitude error) impacting the estimated ground distance
between the MA and GD. Then we focus on the commonly used trilateration based
localization, and derive bounds on the propagation of ground distance errors
on the estimated position of the GD. Finally, we perform extensive in-field
experiments to quantify the localization accuracy of several existing state-
of-the-art localization algorithms. Specifically, we consider the Drb-C [14]
range-based algorithm, the Drf [23], IoC [13], and IoA [24] range-free
algorithms extended to distance measurements, and two trilateration based
algorithms like Scan [25] and Omni [26]. Our testbed uses two UWB DecaWave
kits, namely EVK1000 kit [17] and MDEK1001 kit [27], and a 3DR Solo drone as
the MA.

To the best of our knowledge, ours is the first work that provides an
extensive in-field evaluation on the localization accuracy of the most
relevant algorithms in the literature in real experimental settings using
drones. Our novel contributions are summarized as follows.

  * •

We derive bounds on various measurement errors (instrumental, rolling, and
altitude) to estimate the impact on the estimated ground distance between the
UAV (MA) and GD.

  * •

We validate our theoretical analysis on the ground error with a simple set of
static experiments using two UWB antennas. We observe the impact of
measurement errors on the trilateration technique.

  * •

Through experiments, we comprehensively compare three range-based and three
range-free state-of-the-art localization algorithms using a UAV as MA,
extending the range-free algorithms with distance measurements to
significantly improve their localization accuracy. We also implement these
algorithms employing 3DR Solo drone and ten GDs, the first such realistic
testbed built.

The rest of the paper is organized as follows. Section 2 reviews the existing
literature on localization approaches relevant to our context. Section 3
derives expressions to approximate the measurement and ground errors, and
introduces how our results are interpreted in the light of A2G communications.
Section 4 investigates the localization error affecting the estimated position
of the GD when the trilateration procedure is applied, and describes two
localization algorithms based on trilateration compared in Section 6. Section
5 introduces four more localization algorithms not based on trilateration
which are also compared using the testbed. Three of them are transformed from
range-free to range-based algorithms. Section 6 presents a rich set of real
experiments on the field aiming to evaluate the localization error of the
different localization algorithms. Finally, Section 7 offers conclusions with
directions of future research.

##  2 Related Works

This section reviews the relevant literature on localization of GDs using MAs
and also efforts on testbed implementations considering UAVs as MAs.

###  2.1 MA-based Localization Algorithms

There exist many algorithms for ground MAs that can be classified as range-
free and range-based. In the range-free localization algorithms, such as IoC
[13] (intersection of circles) and IoA [24] (intersection of annuli), a rover
broadcasts its current position at regular time intervals while following a
path. From the heard and not-heard rover’s positions (informally, HnH
technique), the GD builds a limited area where it may reside and places itself
at the “center”. (More details about these algorithms are in Section 5.)

Usually range-free algorithms have relatively low localization accuracy,
leading to the development of range-based algorithms as like Scan and Double-
Scan [26]. In Scan and Double-Scan, the MA follows a path formed by vertical
straight lines interconnected by horizontal lines. However, such algorithms
result in a large number of collinear anchor points. Collinearity can be
reduced by increasing the changes of direction in the path, as in Hilbert [26]
and Lmat [28]. The path generated by Lmat logically tessellates the deployment
area by equilateral triangles so that each GD falls inside a triangle. The
vertices of the triangle where the GD resides are used to trilaterate the GD
position, thus completely solving the collinearity issue. The above algorithms
are designed for ground MAs.

A few localization algorithms have been proposed for flying MAs. Since the
drone flies at a certain altitude, there are new constraints on the anchor
points. In [29], one can find simulation comparisons of the above algorithms
extended to flying MAs, with particular attention to the path length. The Omni
[25] algorithm is the first localization algorithm for drones that selects the
anchor points in such a way that a certain accuracy is guaranteed. A simple
and lightweight range-based algorithm called Drb-C [14] localizes the GDs by
determining the correct intersection point of two circles exploiting a third
reference point for the disambiguation. Finally, the Drf [23] algorithm, the
first range-free algorithm for UAVs as MAs, exploits lightweight geometrical
rules for position estimation. (See Section 5 for more details on Omni and
Drf.)

Very recently, in [30], a novel framework based on reinforcement learning (RL)
has been proposed to enable a UAV to autonomously find a suitable trajectory.
This improves the localization accuracy of multiple GDs minimizing the flight
time, path length, and UAV’s energy consumption. As usual for RL techniques,
an initial step is required for allowing the UAV to be trained for an online
real scenario. However, this work does not mention how long this training
phase lasts.

###  2.2 UAV based Testbeds for Localization Experiments

Recently, we performed preliminary experiments evaluating the range-free,
radius-free Drf [23] algorithm using a UAV as MA and relatively inexpensive
antennas [14]. Our conclusion was that the performance of Drf heavily depends
on the shape of the antenna radiation pattern; precisely, the more the
omnidirectionality of the antenna, the higher is the localization accuracy.
Employing the DecaWave MDEK1001 kit, in [15], it has been experimentally shown
that other range-free radius-based algorithms like IoC [13] and IoA [24], also
exhibit higher accuracy with the omnidirectionality of antennas.

A real outdoor implementation of a range-based localization algorithm is
presented in [31], which aims to localize a radio-frequency identification
(RFID) ground tag with the help of a UAV. Initially, the RFID tag is detected
by the UAV using its RFID reader. Then, the UAV takes hundreds of RSSI
measurements and estimates the tag’s position using a multilateration
procedure. Experimental results show an average localization error of
$6\mathrm{\,m}$. Note that this algorithm considers only random paths and
random measurements, and does not investigate the relationship between the
localization error and the UAV’s altitude.

A range-based algorithm is experimentally tested in [32], in which a UAV
regularly broadcasts its current GPS position while flying. The GD aims to
detect a set of at least three equidistant RSSI distance measurements from its
center in order to apply trilateration. Experimental results show an average
error of $4\mathrm{\,m}$ under normal conditions that reduces to
$1\mathrm{\,m}$ using GPS corrections provided by a Real Time Kinematic (RTK)
algorithm. However, an RTK positioning system is not always available.

In [33] a testbed deploying five UWB ground anchors is implemented for
evaluating and tracking the 3D position of a UAV equipped with UWB antenna.
The anchors and the UAV are also equipped with GPS receivers with RTK
capabilities. The experimental error between UWB and RTK distance measurements
ranges from $2$–$24\mathrm{\,cm}$, while the GPS positioning error alone is
$2\mathrm{\,m}$ on an average. Another similar set-up employing UWB technology
is proposed in [34] to localize a moving UAV. In this scenario, four fixed UWB
anchor devices are placed on the ground. Experimental results show an average
localization error of $1\mathrm{\,m}$. Notice that, differently from the
setting considered in this paper, the last two approaches [33, 34] aim at
localizing or tracking the UAV (MA) instead of the GDs. Nonetheless, they
employ UWB and UAV technologies.

##  3 Measurements and Ground Errors

In this section, we provide analytical bounds on the impact of different
measurement errors that may occur on the estimated ground distance between the
MA and a GD.

###  3.1 Terminology and Notations

Let the slant distance $s$ denote the 3D distance between the drone as MA and
the GD. We define the accuracy as the maximum error in absolute value, and we
let $\epsilon_{s}$ denote the instrumental accuracy, i.e., the maximum error
in estimating the slant distance. Let the point $P$ be the GD’s position on
the ground, the point $\widetilde{W}$ be the actual drone’s position, and the
point $W$ be the scheduled drone’s position (see Figure 1). The measured slant
distance $s^{\prime}=\overline{\widetilde{W}P}$ can be different from the
exact slant distance $s=\overline{WP}$ due to the instrumental error and due
to the accuracy of the drone’s position (i.e., the drone resides at
$\widetilde{W}$ and not at the scheduled position $W$). Let the slant error
$E_{s}$ be the 3D measurement error that affects the measured slant distance
$s$. The slant and the instrumental errors are depicted along with $s$ in
Figure 1.

$d$$\widetilde{W}$$h$$W$$\gamma_{h}$$\alpha$$s^{\prime}$$s$$P^{\prime}$$\epsilon_{s}$$s^{\prime}-s$$P$$W^{\prime}$$\gamma_{d}$$E_{d}(\gamma_{d},\gamma_{h},\epsilon_{s})$$E_{s}$$Q$
Figure 1: The ground error: the point $P$ is the GD.

In reality, the accuracy of the drone’s position $\widetilde{W}$ depends on
its drift with respect to its trajectory, and on the changes of its altitude.
Indeed, a drone is more unstable than a rover, even if it hovers in a given
position. We say that the drone rolls when it drifts to some directions on a
fixed plane, and that the drone uplifts or downfalls when it elevates or
decreases its altitude, respectively. We denote with $\gamma_{d}$ and
$\gamma_{h}$, respectively, the rolling and the altitude accuracy that depend
on the GPS, and the barometer. Figure 1 depicts the cylinder where the drones
may reside due to the rolling and altitude errors. Interestingly, the drone
resides inside a cylinder instead of a sphere since we consider $\gamma_{d}$
and $\gamma_{h}$ independently to each other. Let $\alpha$ in Figure 1 be the
elevation angle as when there are no errors. That is, $\alpha$ is given
assuming the scheduled position of the drone.

To localize the GD, we must convert the 3D slant distance $s$ into the ground
distance $d$, which is a distance derived on the 2D plane. The exact ground
distance $d$ is the distance $\overline{W^{\prime}P}$ between $P$ and the
projection $W^{\prime}$ on the ground of the drone’s position $W$. That is,
$d$ assumes the drone to be in the scheduled position $W$. However, we do not
know $s$, but we know $s^{\prime}$. Then, let the ground error $E_{d}$ be the
measurement error $\overline{PP^{\prime}}$, where $P^{\prime}$ is the position
of $P$ estimated on the ground by using the measured slant distance
$s^{\prime}$ and the scheduled elevation angle $\alpha$ which assumes the
exact ground distance $d$ and the scheduled altitude $h$. Finally, let the
ground accuracy $\epsilon_{d}$ be the maximum $E_{d}$.

TABLE I: Summary of Notations for errors and accuracies. symbol | description
---|---
$\alpha$ | elevation angle
$\epsilon_{s}$ | instrumental accuracy
$\gamma_{d}$ | rolling accuracy
$\gamma_{h}$ | altitude accuracy
$\epsilon_{d}$ | ground accuracy (max error)
$E_{d}$ | ground error
$E_{s}$ | slant error
$E_{L}$ | localization error
$E^{T}_{L}$ | localization trilateration error
$\epsilon^{T}_{L}$ | localization trilateration accuracy (max error)

The ground error $E_{d}$ is the 3D slant error $E_{s}$ as it is perceived on
the ground. With a single measurement, we only know the relative distance
between the MA and the GD, thus the GD is not yet localized. Beyond the ground
error, there is the localization error $E_{L}$, which is instead the distance
from the GD’s estimated position (by any localization algorithm) and the GD’s
actual position. This error also depends on the invoked algorithm and its
implicit rules to find the GD’s position, and will be investigated farther.

Table I summarizes the notations used in this paper.

###  3.2 The Ground Error

In this section, we analytically study the ground error $E_{d}$ by breaking it
up into three independent components $E_{d}(\epsilon_{s})$,
$E_{d}(\gamma_{d})$, and $E_{d}(\gamma_{h})$ that, respectively, depend on:
1. the instrumental accuracy,  2. the rolling accuracy, and  3. the altitude
accuracy.  We recall that we define accuracy as the maximum error in absolute
value. $E_{d}(\gamma_{d})$ and $E_{d}(\gamma_{h})$ model the error in the
drone’s position. Note that each component depends on an independent hardware
part, namely, UWB, GPS, and barometer, and thus it makes sense to study them
separately. Whenever we study one component, we assume the other errors to be
null.

####  3.2.1 Instrumental error

Let us investigate $E_{d}(\epsilon_{s})$, i.e., the impact of the instrumental
error $e_{s}$ on $E_{d}$. Note that $e_{s}$ is defined as the difference,
positive (overestimation) or negative (underestimation), between the measured
distance and the actual distance. Moreover, $\epsilon_{s}$ is the absolute
value of the maximum instrumental error. Accordingly,
$|e_{s}|\leq\epsilon_{s}$. Here we assume $\gamma_{d}=\gamma_{h}=0$. Let $s$
be the exact 3D distance between the drone and the object $P$ ($P$ denotes the
GD’s position). Then, let $s^{\prime}=s+e_{s}$ be the measure of the segment
$\overline{WP}$, where $-\epsilon_{s}\leq e_{s}\leq\epsilon_{s}$. In the
following, we geometrically show how the measured slant distance $s^{\prime}$
is converted into the ground distance. Figure 2 illustrates the reasoning
behind the choice of $e_{s}=\pm\epsilon_{s}$. We draw a circumference of
radius $s^{\prime}$ centered at the waypoint $W$. Such circumference will
intersect the line that passes between $W$ and $P$ in $Q$ (see Figure 2).
Since the measured slant distance is different from the exact one, i.e.,
$s^{\prime}\not=s$, $Q$ does not coincide with $P$, and $Q$ is not at the
ground level. Specifically, the segment $\overline{PQ}$ of length
$e_{s}=s^{\prime}-s$ is on the extension of $\overline{WP}$ if $e_{s}>0$;
whereas $\overline{PQ}$ is on the radius $\overline{WP}$ if $e_{s}<0$.

$h$$s$$d$$\epsilon_{s}$$\alpha$$E_{d}(\epsilon_{s})$$P$$P^{\prime}$$W$$s^{\prime}$$Q$$W^{\prime}$
Figure 2: Overestimation in the instrumental accuracy $\epsilon_{s}$.

Since in general $\epsilon_{s}\ll s$, we can approximate the circumference of
radius $s^{\prime}$ with its tangent in $Q$. The point $P^{\prime}$, where the
tangent intersects the ground111Figure 2 shows the intersection $P^{\prime}$
between the tangent and the ground, which approximates the intersection (white
dot) between the circumference and the ground. However, the two intersections
become closer and closer when $s$ increases., is the estimated position for
$P$ according to the measurement $s^{\prime}$. Thus, recalling that
$W^{\prime}$ is the projection of $W$ on the ground, $\overline{PP^{\prime}}$
is the error on the ground derived from the slant error $e_{s}$. By elementary
geometric rules applied to the right-angled triangle $PQP^{\prime}$, we obtain
$\overline{PP^{\prime}}=e_{s}\cdot\frac{1}{\cos(\alpha)}=e_{s}\cdot\sqrt{1+\frac{h^{2}}{d^{2}}}$,
where $h$ is the drone’s altitude. because $\angle{QPP^{\prime}}$ is equal to
the elevation angle $\alpha$. The error $E_{d}(\epsilon_{s})$, when the
instrumental error is maximum and the object is at ground distance $d$ from
the drone, is given by:

$E_{d}(\epsilon_{s})=\epsilon_{s}\cdot\frac{1}{\cos(\alpha)}=\epsilon_{s}\cdot\sqrt{1+\frac{h^{2}}{d^{2}}}.$
(1)

The ground error $E_{d}(\epsilon_{s})$ varies with the distance $d$ on the
ground. When $h\not=0$, the error increases when $d$ decreases (whereas, when
$h=0$ the error does not depend on $d$). When $h\not=0$, the worst case occurs
when the drone is perpendicular to the point to be measured (i.e.,
$W^{\prime}=P$, $d=0$, $E_{d}\rightarrow\infty$). From this observation, we
can assert that, when the measurements are taken by a UAV, rather than a
rover, in order to bound $E_{d}(\epsilon_{s})$, it is convenient to add the
constraint that all the measurements have to respect a given minimum ground
distance $d\textsubscript{min}$.

####  3.2.2 Rolling error

In this section, we only consider the rolling error (i.e.,
$\epsilon_{s}=\gamma_{h}=0$). When the drone hovers in position $W=(x,y,z)$,
it may not be in $W$, but rather in position $\widetilde{W}$ due to the GPS
accuracy or the bad weather conditions (see Figure 3).

$s$$d$$\alpha$$E_{d}(\gamma_{d})$$\widetilde{W}$$h$$W$$\gamma_{d}$$P$$s^{\prime}$$P^{\prime}$$s^{\prime}-s$$W^{\prime}$$d$$Q$
Figure 3: The rolling accuracy $\gamma_{d}>0$ and ground error.

To better define the rolling error, we set a 3D-Cartesian coordinate system
whose origin is the projection $W^{\prime}=(0,0,0)$ on the ground of the exact
drone’s position $W$, whose $x$-axis passes through the object to measure $P$,
and $z$-axis passes through $W$. Thus, $W=(0,0,h)$ and $P=(d,0,0)$. Then, let
the actual drone’s position be $\widetilde{W}=(e_{x},e_{y},h)$, with
$-\gamma_{d}\leq e_{x},e_{y}\leq\gamma_{d}$, where $\gamma_{d}$ is the rolling
accuracy. Obviously, $\widetilde{W}^{\prime}=(e_{x},e_{y},0)$ is the
projection of $\widetilde{W}$ on the ground, which is inside a circle of
radius $\gamma_{d}$ centered at the origin $W^{\prime}$. For each point of the
circle, it holds $e_{x}=\gamma_{d}\cos(\psi)$ and
$e_{y}=\gamma_{d}\sin(\psi)$, where
$\psi=\angle{\widetilde{W}^{\prime}W^{\prime}P}$ and $0\leq\psi\leq 2\pi$. The
measured slant distance $s^{\prime}$ between $\widetilde{W}$ and $P$ given by:

$\displaystyle s^{\prime}$
$\displaystyle=\sqrt{h^{2}+(d-e_{x})^{2}+e_{y}^{2}}$
$\displaystyle=\sqrt{h^{2}+(d-\gamma_{d}\cos(\psi))^{2}+(\gamma_{d}\sin(\psi))^{2}}$
$\displaystyle=\sqrt{h^{2}+d^{2}-2d\gamma_{d}\cos(\psi)+\gamma_{d}^{2}\cos^{2}(\psi)+\gamma_{d}^{2}\sin^{2}(\psi)}$
$\displaystyle=\sqrt{h^{2}+d^{2}-2d\gamma_{d}\cos(\psi)+\gamma_{d}^{2}}$

Recalling that $h>0$, $d>0$, $\gamma_{d}\geq 0$ and $0\leq\psi\leq 2\pi$, we
note that $s^{\prime}$ is maximum when $f=-2d\gamma_{d}\cos(\psi)$ is minimum.

Including the instrumental error, the slant error is:

$E_{s}=s^{\prime}-s=\sqrt{h^{2}+d^{2}-2d\gamma_{d}\cos(\psi)+\gamma_{d}^{2}}-\sqrt{h^{2}+d^{2}}$

which is maximum when $\cos(\psi)=-1$ and $\epsilon_{s}>0$.

In order to project $E_{s}$ on the ground, we repeat the same construction as
in Section 3.2.1. We draw a circumference of radius $s^{\prime}$ centered in
the waypoint $W$, which intersects the line that passes for $W$ and $P$ in
$Q$. The tangent in $Q$ intersects the ground in the estimated position
$P^{\prime}$. Applying elementary trigonometry to the right-angled triangle
$PQP^{\prime}$ whose $\angle{QPP^{\prime}}$ is equal to the elevation angle
$\alpha$,

$\displaystyle E_{d}(\gamma_{d})$
$\displaystyle=\frac{E_{s}(\gamma_{d})}{\cos(\alpha)}=\frac{\left|s^{\prime}-s\right|}{\cos(\alpha)}=\frac{|(s^{\prime})^{2}-(s)^{2}|}{\cos(\alpha)\left(s^{\prime}+s\right)}$
$\displaystyle=\frac{|\gamma_{d}^{2}-2d\gamma_{d}\cos(\psi)|\sqrt{h^{2}+d^{2}}}{\left(s^{\prime}+s\right)d}=\frac{|\gamma_{d}^{2}-2d\gamma_{d}\cos(\psi)|s}{\left(s^{\prime}+s\right)d}$

When $s^{\prime}>s$ (i.e., $\frac{\pi}{2}<\psi<\frac{3\pi}{2}$), that is, when
the drone rolls away from the object, it holds:

$\displaystyle E_{d}(\gamma_{d})$
$\displaystyle\leq\frac{|\gamma_{d}^{2}-2d\gamma_{d}|}{2d}$ and assuming
$\gamma_{d}\ll d$ $\displaystyle E_{d}(\gamma_{d})$
$\displaystyle\leq\gamma_{d}$ (2)

When $s^{\prime}<s$ (i.e., $0\leq\psi\leq\frac{\pi}{2}$ or
$\frac{3\pi}{2}\leq\psi\leq 2\pi$), that is, when the drone rolls close to the
object, since $s+s^{\prime}>s$, we obtain a weaker bound:

$\displaystyle E_{d}(\gamma_{d})$
$\displaystyle<\frac{|\gamma_{d}^{2}-2d\gamma_{d}|}{d}\frac{s}{s}$
$\displaystyle<2\gamma_{d}$

Now, if $\gamma_{d}\ll d$ holds, $\frac{s}{s^{\prime}}\rightarrow 1$. Since
$s+s^{\prime}\geq 2s^{\prime}$, we have:

$\displaystyle E_{d}(\gamma_{d})$
$\displaystyle<\frac{|\gamma_{d}^{2}-2d\gamma_{d}|}{2d}\frac{s}{s^{\prime}}$
$\displaystyle<\gamma_{d}\frac{s}{s^{\prime}}\rightarrow\gamma_{d}$ (3)

We will see in our experiments that indeed the stricter bound in Eq. (3.2.2)
well approximates the rolling error even when the drone rolls close to the GD.

####  3.2.3 Altitude error

In this section, we only consider the altitude error (i.e.,
$\gamma_{d}=\epsilon_{s}=0$). When the drone is subject to an uplift (resp.,
downfall), the measured slant distance $s^{\prime}$ is overestimated (resp.,
underestimated). The overestimate case is illustrated in Figure 4.

$d$$\widetilde{W}$$h$$W$$\gamma_{h}$$\alpha$$s^{\prime}$$s$$P^{\prime}$$s^{\prime}-s$$E_{d}(\gamma_{h})$$P$$W^{\prime}$$Q$
Figure 4: The altitude accuracy $\gamma_{h}>0$ and ground error.

The measured slant distance $s^{\prime}$ between $\widetilde{W}$ and $P$ is:
$s^{\prime}=\sqrt{h+{\gamma_{h}}^{2}+d^{2}}$. Recalling that $h>0$, $d>0$, and
$\gamma_{h}\geq 0$, including the instrumental error, the slant error is:

$E_{s}=s^{\prime}-s=\sqrt{h+{\gamma_{h}}^{2}+d^{2}}-\sqrt{h^{2}+d^{2}}$

Moreover,

$\displaystyle E_{d}(\gamma_{h})$
$\displaystyle=\frac{E_{s}(\gamma_{h})}{\cos(\alpha)}=\frac{\left|s^{\prime}-s\right|}{\cos(\alpha)}=\frac{|\gamma_{h}^{2}-2h\gamma_{h}|s}{\left(s^{\prime}+s\right)d}$

Repeating calculations similar to those above, and assuming that the altitude
accuracy $\gamma_{h}$ is very small with respect to $h$, and that
$\frac{s}{s^{\prime}}\rightarrow 1$, we find that the ground error can be
approximated as:

$E_{d}(\gamma_{h})\approx\gamma_{h}\frac{h}{d}$ (4)

####  3.2.4 Overall ground error

From the previous discussions we can estimate the overall ground error as
stated by the following:

######  Fact 1.

Let $\epsilon_{s}$, $\gamma_{d}$, and $\gamma_{h}$ be respectively the
instrumental accuracy, rolling accuracy, and altitude accuracy that may affect
the slant measurement. By projecting the slant distance on the ground, the
largest error $E_{d}$ given the ground distance $d$, is:

$E_{d}(\gamma_{d},\gamma_{h},\epsilon_{s})\approx\gamma_{d}+\frac{h}{d}\gamma_{h}+\epsilon_{s}\sqrt{1+\frac{h^{2}}{d^{2}}}$
(5)

Analyzing Eq. (5), it is clear that when $d$ is very small, the ground error
is very large. Increasing $d$, the impact of both instrumental and altitude
accuracies decreases, but $E_{d}$ cannot be smaller than the rolling accuracy
$\gamma_{d}$.

In conclusion, the ground error can be bounded by adding a constraint on the
minimum ground distance ($d\textsubscript{min}$) between the drone and the GD.
If it is ensured that $d\geq d\textsubscript{min}$ using the drone, then the
ground accuracy $\epsilon_{d}$, i.e., the maximum error on the ground
distance, is bounded by:

$\epsilon_{d}=\epsilon_{d}(\gamma_{d},\gamma_{h},\epsilon_{s})\approx\gamma_{d}+\frac{h}{d\textsubscript{min}}\gamma_{h}+\epsilon_{s}\sqrt{1+\frac{h^{2}}{d\textsubscript{min}^{2}}}$
(6)

Our first takeaway is that the ground accuracy $\epsilon_{d}$ can be monitored
by monitoring the ratio $h/d$ between altitude and ground distance.

##### A2G links and ground error

In this paragraph, we explain how the A2G communication link quality between
GD and MA impacts on our results. According to the model in [35], each A2G
link has a certain probability $P(\text{LoS})$ to be in LoS and
$P(\text{NLoS})$ to be in NLoS. $P(\text{LoS})$ depends on the elevation angle
$\alpha$ between drone and GD and on the environment type, i.e., sub-urban,
urban, dense, and highrise. Clearly, in crowded environments, links have a
higher probability to be mixed LoS and NLoS scenarios.

TABLE II: The line of sight probabilities $P(\text{LoS})$ in different environments [35]. $h/d\textsubscript{min}$ | $\alpha$ | sub-urban | urban | dense | highrise
---|---|---|---|---|---
$5.67$ | $80^{\circ}$ | $100\%$ | $100\%$ | $100\%$ | $100\%$
$\sqrt{3}$ | $60^{\circ}$ | $100\%$ | $100\%$ | $100\%$ | $60\%$
$1$ | $45^{\circ}$ | $100\%$ | $97\%$ | $85\%$ | $30\%$
${1}/{2}$ | $26.5^{\circ}$ | $100\%$ | $75\%$ | $30\%$ | $5\%$
${1}/{3}$ | $20^{\circ}$ | $100\%$ | $40\%$ | $20\%$ | $\to 0\%$

The UWB distance measurements are possible as long as the antennas keep an A2G
link. Up to $35\mathrm{\,m}$, even if the elevation angle is small,
communications can be established since UWB works in both LoS and NLoS [36].
Beyond $35\mathrm{\,m}$ UWB only works in LoS, and hence only LoS links can be
guaranteed. Suitable values for $h/d\textsubscript{min}$ such that the
elevation angle $\alpha=\arctan(h/d\textsubscript{min})$ gives LoS links with
high probability, have to be selected. For example, as reported in Table II,
in sub-urban environment $h/d\textsubscript{min}=1/3$ because A2G link has
$100\%$ probability to be LoS whenever $\alpha\geq 20^{\circ}$. In urban
environment, links are $100\%$ in LoS when $h/d\textsubscript{min}\geq 1/2$
($\alpha\geq 26.5^{\circ}$). Similarly, in highrise the minimum ratio for LoS
links is $h/d\textsubscript{min}=5.67$. Note that for
$h/d\textsubscript{min}<1/3$ ($\alpha<20^{\circ}$) links could be mixed and
hence UWB might work or not. Recall that the ground accuracy in Eq. (6) can be
bounded selecting a small $h/d\textsubscript{min}$. Keeping in mind the
maximum UWB NLoS range of $35\mathrm{\,m}$, $h/d\textsubscript{min}$ cannot be
freely chosen. However, in our experiments since we work in a sub-urban,
obstacle-free, and flat environment, any ratio $h/d\textsubscript{min}\geq
1/3$ is sufficient to be in LoS.

##  4 Localization Error for Trilateration Based Algorithms

Once a GD has collected a suitable number of distance measurements from the
MA, it can be localized by invoking any localization algorithm. A very common
approach for localization is trilateration. In Section 4.1 we analytically
derive the localization trilateration error $E^{T}_{L}$ and the localization
trilateration accuracy $\epsilon^{T}_{L}$, which are incurred by any algorithm
based on this method. Subsequently, in Section 4.2, we discuss the
trilateration based algorithms that are considered in our experiments.

$W^{\prime}_{1}$$W^{\prime}_{3}$$W^{\prime}_{2}$$P$ (a) Linearization of each
measurement.

$W^{\prime}_{1}$$W^{\prime}_{3}$$W^{\prime}_{2}$$P$$\beta$$E_{d}$$\beta/2$$E_{L}$$P^{\prime}$$d_{3}(-)$$d_{2}(-)$
(b) Same signs estimation.

$W^{\prime}_{1}$$W^{\prime}_{3}$$W^{\prime}_{2}$$P$$\overline{\beta}$$E_{d}$$\overline{\beta}/2$$E_{L}$$P^{\prime}$$d_{3}(+)$$d_{1}(-)$
(c) Different signs estimation.

Figure 5: Trilateration error with different conditions.

###  4.1 Trilateration Error

This section discusses the localization error $E^{T}_{L}$ that may affect the
estimated GD’s position when the trilateration procedure is applied.

Let us briefly recall that the trilateration procedure for estimating the
position of the object $P$, takes as input three ground distances $d_{1}$,
$d_{2}$, and $d_{3}$ of $P$ from three waypoints $W_{1}$, $W_{2}$, and $W_{3}$
respectively. The procedure returns, as the GD’s estimated position $P$, the
intersection of the three circumferences corresponding to the radii $d_{1}$,
$d_{2}$, and $d_{3}$ centered at the projections $W^{\prime}_{1}$,
$W^{\prime}_{2}$, and $W^{\prime}_{3}$ of the waypoints. Due to the ground
errors, however, the three circumferences do not intersect at a single point,
but they delimit a small star area, as depicted in Figure 5a. In fact, a pair
of extreme circumferences, one obtained by considering the radius affected by
the maximum positive $E_{d}$ error ($d_{i}+E_{d}$, measurement overestimation)
and one whose radius is affected by the maximum negative $E_{d}$ error
($d_{i}-E_{d}$, measurement underestimation) is drawn in place of each
circumference of radius $d_{i}$.

Assuming that all the ground distances are sufficiently large compared to the
ground error, these extreme circumferences can be linearized (i.e., replaced
by the tangent to the radius) without significantly changing the area. Each
different non-parallel pair of linearized circumferences intersects at a
single point forming overall $12$ different points, that correspond to the
vertices of the star shape. Note that $P$ is at the center of the star. The
trilateration procedure returns as the estimated position, instead of the
exact intersection $P$, a point $P^{\prime}$ in the star. The point
$P^{\prime}$ is selected by means of the least-squares-error method. In fact,
given three ground measurements, the estimated position of $P$ is the point
$(x_{P},y_{P})$ that minimizes the sum of the least squares, i.e.:

$\begin{array}[]{r@{}r<EMAIL_ADDRESS>\text{s.t.}\qquad\sqrt{(x_{W^{\prime}_{i}}-x_{P})^{2}+(y_{W^{\prime}_{i}}-y_{P})^{2}}&{}+\delta_{i}=\overline{W^{\prime}_{i}P}\\\
\lx@intercol\hfil\hskip 139.41832pt\textrm{for}\quad
<EMAIL_ADDRESS>(7)

The largest value of the positioning error, i.e., $\overline{PP^{\prime}}$,
called localization trilateration error $E^{T}_{L}$, or simply trilateration
error, occurs when the estimated position $P^{\prime}$ is at the furthest
vertex of the star shape. In other words, the positioning error is bounded by
the distance between the center of the star $P$ (i.e., the actual position of
the GD) and its farthest vertex.

As an example, in Figure 5b, the distance between the actual point $P$ and the
estimated point $P^{\prime}$ at the intersection of two measurement
underestimations $d_{2}(-)$ and $d_{3}(-)$ is $\frac{E_{d}}{\cos(\beta/2)}$,
where $\beta$ is one of the three different angles in which the turn angle in
$P$ is divided by the lines $\overline{W^{\prime}_{1}P}$,
$\overline{W^{\prime}_{2}P}$, and $\overline{W^{\prime}_{3}P}$ (see Figure
6a). In Figure 5c, the distance between $P$ and $P^{\prime}$ that results from
the measurement underestimation, i.e., $d_{1}(+)$, and the measurement
overestimation, i.e., $d_{3}(-)$, is depicted. In this case, the distance
$\overline{PP^{\prime}}=\frac{E_{d}}{\sin(\overline{\beta}/2)}$.

For each vertex of the star, depending on the signs of the estimations ($+$
overestimation, $-$ underestimation) of each pair of circumferences, we have:
$\frac{E_{d}}{\sin(\beta_{i}/2)}$ if the signs are the same; and
$\frac{E_{d}}{\cos(\beta_{i}/2)}$ if the signs are different, where
$\beta_{1}\leq\beta_{2}\leq\beta_{3}$ are the three different angles formed in
$P$ such that $\sum_{i}\beta_{i}=\pi$. In the following, we prove that the
farthest vertex occurs when the measurement estimations have the same signs
and the angle is minimum.

######  Lemma 1 ([37]).

Let $\beta\textsubscript{min}=\min_{i}\\{\beta_{i}\\}$,
$\beta\textsubscript{max}=\max_{i}\\{\beta_{i}\\}$ and
$\sum_{i}\beta_{i}=\pi$. Then
$\sin(\frac{\beta\textsubscript{min}}{2})\leq\cos(\frac{\beta\textsubscript{max}}{2})$.

###### Proof:

Let $\beta\textsubscript{min}=\min_{i}\\{\beta_{i}\\}$ and
$\beta\textsubscript{max}=\max_{i}\\{\beta_{i}\\}$. Then, we have:
$\beta\textsubscript{max}\leq\pi-2\beta\textsubscript{min}\Rightarrow\frac{\beta\textsubscript{max}}{2}\leq\frac{\pi}{2}-\beta\textsubscript{min}$
from which
$\cos(\frac{\pi}{2}-\beta\textsubscript{min})\leq\cos(\frac{\beta\textsubscript{max}}{2})$,
and thus
$\sin(\beta\textsubscript{min})\leq\cos(\frac{\beta\textsubscript{max}}{2})$.
Since $0\leq\beta\textsubscript{min}\leq\pi/3$, it yields:

$\displaystyle\sin\left(\frac{\beta\textsubscript{min}}{2}\right)\leq\sin(\beta\textsubscript{min})\leq\cos\left(\frac{\beta\textsubscript{max}}{2}\right)$

Thus, the furthest vertex is at distance
$\frac{E_{d}}{\sin(\frac{\beta\textsubscript{min}}{2})}$ from $P$. ∎

######  Theorem 1 ([37]).

Given the accuracies $\epsilon_{s}$, $\gamma_{d}$, and $\gamma_{h}$, given
$d\textsubscript{min}$, and recalling that $\epsilon_{d}(\gamma_{d},$
$\gamma_{h},\epsilon_{s})\approx|\gamma_{d}|+\frac{h}{d\textsubscript{min}}|\gamma_{h}|+|\epsilon_{s}|\sqrt{1+\frac{h^{2}}{d\textsubscript{min}^{2}}}$,
the localization trilateration accuracy defined as the maximum trilateration
error is obtained as:

$\epsilon^{T}_{L}(\gamma_{d},\gamma_{h},\epsilon_{s})=\frac{\epsilon_{d}(\gamma_{d},\gamma_{h},\epsilon_{s})}{\sin\left(\frac{\beta\textsubscript{min}}{2}\right)}$
(8)

Therefore, from Eq. (8), we learn that, given a certain ground error, the
localization error is minimized when
$\beta\textsubscript{min}\rightarrow\frac{\pi}{3}=60^{\circ}$. Figure 6b
reports an example of the trilateration error ${E^{T}_{L}}$ computed by
varying both the values $d$ and $\beta\textsubscript{min}$, and assuming only
the instrumental error, i.e., $\gamma_{d}=\gamma_{h}=0\mathrm{\,m}$ and
$\epsilon_{s}=0.10\mathrm{\,m}$. As expected, when both $d$ and
$\beta\textsubscript{min}$ tend to $0$, ${E^{T}_{L}}$ grows quickly.

$W^{\prime}_{3}$$W^{\prime}_{1}$$W^{\prime}_{2}$$\beta_{1}=\beta\textsubscript{min}$$P$$\beta_{2}$$\beta_{3}$
(a) The three angles $\beta_{i}$.

(b) $E^{T}_{L}$.

Figure 6: The trilateration error $E^{T}_{L}$: when $d$ and
$\beta\textsubscript{min}$ are very small, the error is extremely high.

Analyzing Eq. (8) it is clear that the localization trilateration accuracy
$\epsilon^{T}_{L}$ can be bounded keeping the minimum angle
$\beta\textsubscript{min}$ as large as possible, i.e., closer to $60^{\circ}$.

Our second takeaway is that a good localization accuracy in trilateration
methods can be obtained keeping as low as possible the ratio
$h/d\textsubscript{min}$, the elevation angle $\alpha$ as small as possible
respecting also the communication conditions in LoS, and making the minimum
angle $\beta\textsubscript{min}$ as large as $60^{\circ}$.

###  4.2 Omni and Scan Localization Algorithms

In this section, we review trilateration based algorithms Omni and Scan
considered in our experiments. Based on the above discussion, the localization
error for these algorithms is bounded by $\epsilon^{T}_{L}$, which is a
function of the accuracies $\epsilon_{s}$, $\gamma_{d}$, and $\gamma_{h}$, the
minimum angle $\beta\textsubscript{min}$, altitude $h$, and minimum distance
$d\textsubscript{min}$. Both algorithms are based on a static path $\Pi$
formed by a series of vertical lines (each called as vertical scan) connected
by horizontal lines.

$I_{w}$$O$$P$$A_{0}$$A_{1}$$A_{2}$$A_{3}$$B_{1}$$d$ (a) Drf.

$O$$I_{w}$$A_{0}$$A_{1}$$A_{2}$$A_{3}$$P_{1}$$P_{3}$$P_{4}$$P_{2}$$d$$d$ (b)
IoC.

$d-I_{w}$$d$$O$$A_{1}$$A_{2}$$d$$P_{1}$$P_{2}$$P_{3}$$P_{4}$ (c) IoA.

Figure 7: The Drf, IoC, and IoA localization algorithms. In IoC e IoA there
are two symmetric intersection areas: a third point is required to find and
disambiguate the correct intersection area.

##### The Scan Algorithm

Scan [26] is one of the first range-based localization algorithms designed for
rovers. Each GD is localized employing trilateration using three waypoints.
The main drawback is the collinearity between points in the estimation phase.
Since we wish to avoid such undesirable conditions, in our experiments we
perform single trilateration selecting three non-collinear waypoints from at
least two distinct vertical scans. In this slightly improved version of Scan,
the $\beta\textsubscript{min}$ and $d\textsubscript{min}$ constraints may not
be satisfied, resulting in large localization errors.

##### The Omni Algorithm

Omni [25] is the first range-based localization algorithm that takes into
account the impact of the drone’s altitude on the measurement accuracy and on
the geometry of the waypoints from which trilateration is performed. It
logically tessellates the deployment area into diamonds. Then, each GD, once
it has acquired a sufficient number of waypoints/distances from the drone,
performs two trilaterations. The first trilateration is invoked using any
three non-collinear waypoints in order to compute the logical diamond in which
the GD resides. Since each diamond is associated with an optimal triple of
waypoints which satisfy the minimum angle/distance constraints [25], any GD
belonging to such diamond can be finally trilaterated for a second time using
that triple. In conclusion, Omni has been proved to be highly accurate but
requires two trilaterations, as opposed to the single trilateration performed
by Scan.

##  5 Other Localization Algorithms

In this section, we describe four more localization algorithms, namely Drf,
IoA, IoC, and Drb-C, not based on trilateration. The first three of such
algorithms are range-free, therefore the localization accuracy depends on the
antenna radiation pattern quality. Recently, Betti et al. [14] experimentally
showed the poor accuracy of Drf using relatively inexpensive hardware. In this
paper, motivated by these results, we extend these algorithms by considering
distance measurements to improve on the localization accuracy. Specifically,
as also detailed in the experiments in Section 6, the GD stores, for each
waypoint that it hears, the relative distance between itself and that
waypoint. Exploiting this information, we reformulate all the range-free
techniques making them actually range-based. This way, we mean to overcome the
poor localization accuracy resulting from the low-quality of the radio
antenna, while still keeping the original procedures for the localization.

In the following, for each of these extended algorithms, as well as for Drb-C,
we identify the sources of the localization error. However, we do not derive
any analytical expression of $E_{L}$, since the analysis would involve too
many variables to be expressed in a closed formula. Nevertheless, we study
their error through real experiments in Section 6.

##### The Drf Algorithm

Drf [23] is a lightweight range-free radius-free algorithm designed for
drones. This algorithm is based on the notion of chord. In general, the
perpendicular bisector of any chord passes through the center $O$ of the
circle itself. So, the bisector of another non-parallel chord and the previous
one intersect at $O$ point. In Figure 7a, two chords are identified by the
pairs $A_{1}A_{2}$ and $A_{2}B_{1}$. Theoretically, the circle is identified
by the receiving disk of GD which is centered at $O$. Accordingly, the GD
starts to estimate its position when it detects two chords. The two chords are
detected using the HnH technique [15] on each scan. The detection of chords
incurs several problems that eventually affects the localization accuracy.
First, recalling that the MA regularly broadcasts its current position
(waypoint) at discrete intervals of time and that two consecutive waypoints
are at distance inter-waypoint $I_{w}$, the endpoints of the chords may not
exactly fall on the circumference of the receiving disk, even if the receiving
disk is a perfect circle (e.g., $A_{2}$ and $A_{3}$ in Figure 7a). However,
the chords can be improperly defined if the antenna pattern has “holes” and
“bubbles”, as experienced in the field as reported in [15].

Range-based extension: Exploiting the fact that our tested kit allows us to
take distance measurements, the choice of the chords can be performed
selecting three waypoints at a certain fixed distance $d$ from the GD,
relaxing the range-free constraint. In this way, with three waypoints on the
same circumference of radius $d$, two chords can be derived. Accordingly, we
can obtain a localization error which depends only on the length of $I_{w}$
and on the error $E_{d}$. A more detailed explanation of the original version
of Drf can be found in [14].

##### The IoC Algorithm

IoC [13] is a range-free radius-based localization algorithm initially
developed for ground MAs. Like Drf, the IoC algorithm exploits the HnH method
in order to detect special points used for building a constraint area that
bounds the GD’s position. However, diversely from Drf, IoC relies also on the
value of the communication radius $d$. In fact, initially the GD detects the
pair of endpoints ($A_{1}$ and $A_{2}$, in Figure 7b) using the HnH method.
Successively, two more points called pre-arrival and post-departure ($A_{0}$
and $A_{3}$), respectively, are determined using the value of $I_{w}$, since
MA sends its current position at discrete intervals. Note that such four
points belong on the same straight line. Then, four circles of radius $d$
centered at each of these four points are drawn. Those circles create two
symmetrical intersection areas where the GD may reside. In order to select the
correct intersection area, the GD needs to detect a third point. Finally, the
GD is localized at the “center” of the correct intersection area. This
definition of center slightly varies depending on the shape of the
intersection area, which may vary from four to five vertices.

Range-based extension: As for Drf, also in IoC we exploit the distance
measurements for computing all the required points. That is, we select the two
waypoints on the same line at distance $d$ from GD as $A_{1}$ and $A_{2}$, and
the preceding and subsequent waypoints as $A_{0}$ and $A_{3}$.

##### The IoA Algorithm

IoA [24], is a range-free radius-based algorithm very similar to IoC. Indeed,
it builds a similar constrained area using the HnH method and the knowledge of
both $d$ and $I_{w}$. Once the GD has detected the two extreme endpoints
($A_{1}$ and $A_{2}$, in Figure 7c), it traces two circles of radius $d$ and
$d-I_{w}$ on both the points. These circles, which create two annuli,
intersect in two distinct and symmetrical intersection areas, so also in this
case a third point is required. Finally, the GD estimates its position at the
center of such an area, using easy geometrical rules.

Range-based extension: As for the previous algorithms, we select the two
extreme endpoints $A_{1}$ and $A_{2}$ as two waypoints at distance $d$ from
the GD.

##### The Drb-C Algorithm

Drb-C [14] is a lightweight range-based technique designed for UAVs. The goal
of GD is to detect two waypoints at distance $d_{1}$ and $d_{2}$, and drawing
two circumferences centered at these waypoints, of radius $d_{1}$ and $d_{2}$,
respectively. Then, the GD knows to reside simultaneously on the two
intersections of two circumferences, and a third point is required to
disambiguate the correct intersection point. In conclusion, we can note that,
differently from the trilateration based algorithms Omni and Scan in which the
least-squares-error method is employed (see Eq. (7)), Drb-C only demands a few
algebraic calculations.

Finally, Table III summarizes the six algorithms that will be compared in our
testbed.

TABLE III: Summary of the compared algorithms. name | method | points | error source
---|---|---|---
Omni [25] | trilaterations | $3+3$ | geometry
Scan [26] | trilateration | $3$ | geometry
Drb-C [14] | circles intersection | $2+1$ | center
Drf [23] | bisector intersection | $3$ | chords
IoC [13] | points “center” | $2+1$ | center
IoA [24] | points “center” | $2+1$ | center

##  6 Evaluation on a Real Testbed

In this section we propose our experimental evaluation. Initially, in Section
6.1 we describe the adopted hardware for our testbed. In Section 6.2, we study
the ground error $E_{d}$. In Section 6.3, we study the localization error
${E^{T}_{L}}$ of the trilateration method. Finally, in Section 6.4, we run a
campaign of experiments with the goal of comparing the localization error of
different algorithms.

###  6.1 Performance of UWB Antennas

The experiments in Sections 6.2 and 6.3 are done using the DecaWave EVK1000
kit (see Figure 8a), formed by two UWB antennas which are based on the DW1000
UWB chip [17]. For the experiments done in Section 6.4, we rely on a more
consistent set of twelve antennas, using the MDEK1001 kit (see Figure 8b),
based on the same DW1000 UWB chip [27].

According to DecaWave, those chips have a $6.5\mathrm{\,GHz}$ center
frequency, and have a declared and reliable point-to-point range up to
$60\mathrm{\,m}$ LoS and $35\mathrm{\,m}$ NLoS [27] on a typical use-case.
Although the DW1000 chip transmitting power is set to
$-41.3\mathrm{\,dBm/MHz}$, and the typical receiver sensitivity is
$-93\mathrm{\,dBm/500MHz}$ [17], the received power is influenced by the
antenna polarity. In both the DecaWave kits the antennas are vertically
polarized, meaning that the module is intended to be vertically positioned to
let another vertically polarized antenna observe an omnidirectional radiation
pattern in the azimuth plane [27]. For this reason, following the
recommendations provided by DecaWave in their datasheet, in our experiments we
always vertically placed our antennas. The antenna placed on the drone is
positioned vertically, but reversely, keeping the transceiver on the bottom,
for avoiding the MA’s body to become an obstacle between the GDs and the MA
itself.

(a) EVK1000 kit.

(b) MDEK1001 kit.

(c) An antenna.

(d) The drone.

Figure 8: The used DecaWave kits and the 3DR Solo drone.

###  6.2 Experiments with Ground Error

In this section we analyze the ground error employing the DecaWave EVK1000
kit. We start with pre-arranged antenna experiments in which two antennas (one
reproduces the GD to localize and another one the MA) are used. The antenna
that acts as GD is fixed on the ground, while the other one (MA), fixed on a
pole, moves accordingly to the specific experiment emulating the rolling and
altitude error. We fixed the drone’s position on the ground and that of the GD
measuring the distance with a Bosch digital laser [38]. The ground GPS drone’s
position is then converted as the origin $W^{\prime}=(0,0,0)$ of the local
Cartesian coordinate system used during the experiments. Then, in the
subsequent experiments, we replace the pole with a drone hovering at a certain
altitude. The goal is to understand how the drone impacts the measurement
error. For each experiment, we record at least $30$ slant distances and we
finally determine the final computed value at the $95\%$ confidence level.

(a) Instrumental error.

(b) Rolling error.

(c) Altitude error.

(d) Combined error.

(e) Experimental $\overline{E_{d}}$.

(f) Experimental $\overline{E^{T}_{L}}$: $d$ varies.

(g) Experimental $\overline{E^{T}_{L}}$: $\beta\textsubscript{min}$ varies.

Figure 9: The experimental error and the theoretical error in different cases.

In the first experiment, we measure the slant distance and comparing its
projection on the ground with the exact ground distance $d$. Consequently, we
compute the experimental ground error $\overline{E_{d}}$ and compare it with
theoretical error $E_{d}$. To verify Eq. (1), Eq. (3.2.2), and Eq. (4), we
have measured and reported in Figure 9a, Figure 9b, and Figure 9c the
experimental error $\overline{E_{d}}$ when the instrumental error, rolling
error, and altitude error, separately affect the GD on the ground,
respectively. We also report the theoretical ground error bound $E_{d}$. It is
interesting to see that, in each plot, the measurement error $E_{d}$ (solid
line) almost always upper-bounds the experimental error $\overline{E_{d}}$
(dashed line). We also measured and reported in Figure 9d the combined error
where all the three components affect the error along with the bound in Eq.
(6). The curves almost coincide.

In the second experiment, we repeat the previous setting employing this time a
drone. In Figure 9e we report the experimental $\overline{E_{d}}$ for
different altitudes. Since the drone’s position is affected by the wind, air
density, humidity, the strength of the propellers, and GPS error, we know that
even the slant distance is affected at the same time. Moreover, we know from
Eq. (6) that the error $\overline{E_{d}}$ increases when $h$ increases and
when $d$ tends to $0\mathrm{\,m}$. In Figure 9e, we also plot the theoretical
error $E_{d}$ in solid lines, fixing $\epsilon_{s}=0.10\mathrm{\,m}$ and using
$\gamma_{d}=\\{0.6,0.8,1.2\\}\mathrm{\,m}$ and
$\gamma_{h}=\\{0.1,0.15,0.2\\}\mathrm{\,m}$ for each
$h=\\{10,20,30\\}\mathrm{\,m}$, respectively, that empirically fit the
experimental curves. Differently from the previous ones, this is the first
experiment that somehow simulates a real scenario. It is interesting to note
that, we can model the curve of the combined error even in a non-optimal
scenario, just tuning in advance the parameters on Eq. (6), which provides a
good approximation of the error.

In conclusion, upon this first campaign of experiments, we can confirm that
the measurement error is small when either the ground distance between the
drone and the GD is large or the altitude of the drone is low.

###  6.3 Experiments on the Trilateration Error

In this section, we describe two more comparative experiments to better
understand how the localization error can be affected when the trilateration
method is applied. From Eq. (8), it is clear that the localization error
$E_{L}$ can be bounded if the three waypoints are sufficiently apart from the
GD. In other words, the three points must respect good geometry and minimum
distance constraints. In both the experiments we use our 3DR Solo drone as a
MA and placed a single GD in $P=(0,0,0)$. Moreover, the drone’s initial
position $W^{\prime}$ was initially set at the same GD’s position, i.e., in
$W^{\prime}=P$.

In the first experiment depicted in Figure 9f, we plot the experimental
localization trilateration error $\overline{E^{T}_{L}}$ between the estimated
and the actual position of the GD. Here, we fix the best possible minimum
angle $\beta\textsubscript{min}=60^{\circ}$ and we decrease the value of the
ground distance $d$ to smaller values. For each value of $d$, we perform
trilateration using three points which satisfy the optimal geometry. As
expected and according to Eq. (8), $\overline{E_{L}^{T}}$ is high when $d$ is
short, even though the minimum angle is fixed at the best possible value
$60^{\circ}$.

In the second experiment shown in Figure 9g we do the opposite by keeping a
large and good enough ground distance $d=40\mathrm{\,m}$, and decreasing the
value of $\beta\textsubscript{min}$ to narrow values. Even here we perform
trilateration and according to Eq. (8), the error decreases when
$\beta\textsubscript{min}$ increases.

###  6.4 Comparison of Localization Algorithms

In this section, we describe the hardware and software architecture of the
comparative testbed. The goal is to evaluate the performance of different
localization algorithms in-field.

In this testbed, we cannot use the previous EVK1000 kit since it is formed by
only two antennas, hence it is definitively not sufficient for evaluating a
real scenario in which we have to localize multiple GDs at once. Instead, we
move towards the larger set of antennas relying on the new MDEK1001 kit from
DecaWave, since it comprises of a set of twelve antennas. In addition, the
testbed consists also of a Raspberry Pi which is the main component that auto-
pilots the drone via Wi-Fi and sends UWB commands via a single UWB antenna
that is physically connected to it by the serial peripheral interface (SPI).

####  6.4.1 Testbed setup

We set a rectangular deployment area of sizes $100\times 100\mathrm{\,m^{2}}$,
and fix a Cartesian coordinate system with origin at the special position Home
$(0,0,h_{0}=1\mathrm{\,m})$. Then, we deploy on the ground $n=10$ antennas
placed at the top of a tripod of height $h_{0}$. Each antenna (Figure 8c)
identified by own ID is not aware of its relative position with respect to the
Home, even though we already know its position. In fact, as illustrated in
Figure 10b, the deployed 10 antennas respect a predefined pattern, i.e., form
a series of equilateral triangles (shown in green) with the same side of
length $30\mathrm{\,m}$ in which each vertex is a GD. Thus, we are able to
measure, with reasonable accuracy before our experiments, the relative
distance between the GDs, with the help of a digital laser. Finally, the Home
position is set between antennas ID 4 and ID 5 accurately measured with the
same digital laser.

By a drone’s mission at a certain altitude $h$, we actually refer to a drone
(see Figure 8d) that flies at a fixed altitude $h_{0}+h$ (see Figure 10a)
following a certain static path $\Pi$. For each algorithm, the trajectory
$\Pi$ starts and finishes at Home and consists of vertical scans connected by
horizontal scans (see Figure 10b). Once all the GDs are deployed, the drone
starts its mission flying over the deployment area. When both GD and MA are
within the communication range of the other, the devices start a ToA based
distance measurement protocol. Then, the GD stores the computed distance along
with the current MA’s position. In other words, the GD memorizes the position
of the waypoint and the associated distance on it. At the end of the mission,
each GD estimates its position by invoking a localization algorithm.

$h$$s$$d$$P$$r$$W^{\prime}$$h_{0}$ (a) The MA and GD $P$.

$3$$0$$1$$4$$5$Home$7$$8$$9$$6$$2$ (b) The deployment area.

Figure 10: The experimental testbed on the field.

All the compared algorithms require at least three points (see Table III) to
compute and estimate the position of a GD. However, each GD has several stored
distance measurements, so it can potentially exploit all of them. In order to
better understand how either the altitude of the drone or the geometry of the
waypoints impact the quality of the localization accuracy, as already
investigated in Section 6.3, we fix two constraints during the selection of
the three points:  (i) the ground distance $d$ between the GD and the MA, and
(ii) the geometry angle $\beta$ to keep between the three waypoints.
Accordingly, we fix $d=\\{20,30,\ldots,60\\}\mathrm{\,m}$ and
$\beta=\\{0,15,30\\}^{\circ}$, where $\beta=0^{\circ}$ means an unconstrained
geometry. Moreover, we vary the altitude $h=\\{10,20,30\\}\mathrm{\,m}$.
Clearly, it is not easy to find three points at an exact distance $d$. Thus,
we relax the constraint and we search for three points at distance $d\pm\tau$,
where $\tau$ indicates a tolerance in our measurements (we fix
$\tau=1\mathrm{\,m}$) due to the fact that the drone sends its position at
discrete intervals of time, i.e., the inter-waypoint distance $I_{w}$. The
$I_{w}$ value is affected by the drone’s speed. In our experiments, we have
seen that $I_{w}=1\mathrm{\,m}$ with a drone’s speed of $10\mathrm{\,m/s}$.

####  6.4.2 Results

We compare all algorithms varying the drone’s altitude $h$, the minimum
distance $d$ among the GDs and the waypoints, and the waypoint geometry. In
Omni, by construction, we select always the furthest three waypoints that
guarantee good geometry (see Figure 6a). The Omni error is reported as a
reference for the other algorithms.

(a) $h=10\mathrm{\,m},\beta=0^{\circ}$.

(b) $h=10\mathrm{\,m},\beta=30^{\circ}$.

(c) $h=30\mathrm{\,m},\beta=0^{\circ}$.

(d) $h=30\mathrm{\,m},\beta=30^{\circ}$.

Figure 11: Errors for Scan and Omni.

In Figure 11, we show the observed errors of Scan and Omni along with their
theoretical bounds $\epsilon^{T}_{L}$ given in Eq. (8) obtained by
substituting $\epsilon_{s}=0.10\mathrm{\,m}$, and the values of $\gamma_{d}$
and $\gamma_{h}$ taken from Figure 9e. Obviously, Omni is better than Scan
because the geometry of the waypoints is enforced. The difference between the
theoretical and the observed error is smaller for Omni than Scan when
$\beta=0^{\circ}$, and almost the same when $\beta=30^{\circ}$.

(a) $h=30\mathrm{\,m},\beta=0^{\circ}$.

(b) $h=30\mathrm{\,m},\beta=30^{\circ}$.

(c) $h=10\mathrm{\,m},\beta=0^{\circ}$.

(d) $h=10\mathrm{\,m},\beta=30^{\circ}$.

Figure 12: For fixed $h$ and $\beta$, the algorithms error when $d$ varies.

Figure 12 compares the localization errors $E_{L}$ of the algorithms when $d$
varies. The localization error of Drf, IoC, and IoA is greater than that of
Scan and Omni. As said, the trilateration based algorithms, Omni and Scan, get
a good localization but pose many constraints (angle, distance) in the
selection of waypoints; they also compute the estimated position by performing
a least-squares-error optimization technique (which is complex). The Drb-C is
not as accurate as Omni or Scan because it omits the least-squares-error
optimization technique, but it is quite good. The chords based method in Drf
is the least accurate. IoC and IoA improve over Drf because their localization
technique use the radius information. The errors are large when
$\beta=0^{\circ}$, while they significantly decrease for all the algorithms
when $\beta=30^{\circ}$. This shows that all the algorithms, and not only
those based on trilateration, benefit from a good waypoint geometry. The
experiments with $h=10\mathrm{\,m}$ in Figures 12c and 12d have a smaller
elevation angle and thus a smaller error than those with $h=30\mathrm{\,m}$
reported in Figures 12a and 12b. When $h=30\mathrm{\,m}$, all the ratios yield
$h/d\geq 1/3$, and since our experiments are in sub-urban area all the
measurements are possibly in LoS; whereas when $h=10\mathrm{\,m}$, the
measurements at $d\geq 30\mathrm{\,m}$ are mixed LoS and NLoS. Nonetheless, we
cannot notice any special behavior, probably, thanks to the UWB multipath
immunity.

(a) $h=10\mathrm{\,m},d=30\mathrm{\,m}$.

(b) $h=10\mathrm{\,m},d=50\mathrm{\,m}$.

Figure 13: For fixed $h$ and $d$, the algorithms error when $\beta$ varies.

Figure 13 compares the localization error $E_{L}$ when $h=10\mathrm{\,m}$ and
$d=30\mathrm{\,m}$ or $d=50\mathrm{\,m}$, varying the geometry angle $\beta$.
From the observed errors, any localization that satisfies $\beta\geq
30^{\circ}$ has a small error, and cannot significantly improve decreasing the
elevation angle (i.e., ratio $h/d$). The decrease of the error when $h$
decreases from $50\mathrm{\,m}$ to $30\mathrm{\,m}$ is large when
$0^{\circ}\leq\beta\leq 30^{\circ}$. Finally, note that in Figure 13b, it
holds $h/d=0.2$, which is below the ratio that guarantees $100\%$ LoS in sub-
urban area in Table II, but we do not notice a meaningful worsening of the
error.

Figure 14 plots the error for different pairs of $h$ and $d$ with the same
ratio $h/d$. Precisely, we compare two ratios $h/d$: $0.5$ and $1.0$. Each
ratio can be extracted from three different combinations altitude/distance.
For example, for $h/d=0.5$ we consider the combinations
$h=\\{10,20,30\\}\mathrm{\,m}$ and $d=\\{20,40,60\\}\mathrm{\,m}$. The
improvement in the accuracy is high when the elevation angle decreases from
$45^{\circ}$ to $26^{\circ}$. Drf, the least accurate algorithm in all our
experiments, is very sensitive to the change of the elevation angle.

(a) $h/d=1.0$.

(b) $h/d=0.5$.

Figure 14: For fixed $h/d$ ratio, the algorithms error when $\beta$ varies. TABLE IV: Error between range-free and range-based algorithms, in meters (m). | Drf | IoC | IoA
---|---|---|---
| RF | RB | RF | RB | RF | RB
$h$ (m) | $10$ | $52.3$ | $16.6$ | $48.8$ | $10.8$ | $47.2$ | $10.8$
$20$ | $55.1$ | $19.8$ | $49.1$ | $11.2$ | $48.8$ | $11.0$
$30$ | $57.7$ | $20.4$ | $48.2$ | $13.8$ | $51.2$ | $13.6$

In conclusion, in Table IV compares the localization error $E_{L}$ among
range-free (RF) and range-based (RB) versions of the three original range-free
algorithms Drf, IoC, and IoA, for different altitudes $h$. In particular, we
report the localization error obtained from our previous testbed [15] on which
the three algorithms were implemented as pure range-free techniques based on
the HnH technique (RF columns), along with the average results shown in Figure
12 (RB columns). As reported in [15], on average, the experimental error of
those algorithms is very large (almost $60\mathrm{\,m}$) and variable. These
experiments show that the error of the original range-free versions is 3-4
times larger than the corresponding extended range-based version that exploit
distance measurements. Moreover, in [15], about one-third of the GDs were left
unlocalized by IoC and IoA, while Drf localized all of them. These results and
the fact that our antennas are able to take distance measurements via ToA
fully justify our transformation of Drf, IoC, and IoA in range-based
algorithms, implying that measurements help.

##  7 Conclusions

In this paper, we analytically study and experimentally evaluate thorough real
experiments on the field the errors that can affect the localization of GDs
using a drone as MA. We decompose the error in measurement error, ground
error, and localization error, and provide analytical expressions of these
errors. We also link the ground error with the theory of the A2G communication
link via the elevation angle. Our experiments confirm that our analytical
analysis is accurate. Furthermore, results also show that extending range-free
algorithms with range-based measurements, significantly increases the
localization accuracy.

In the future, we plan to extend the analysis to NLoS scenarios. We will
investigate DecaWave antenna capabilities, and also modulating and tuning the
transmitting power in different scenarios. We finally plan to extend our work
to propose a more realistic antenna radiation pattern.

##### Acknowledgments

The authors are grateful to the editor and reviewers for valuable comments
that helped us improve the quality of the manuscript. This work was partially
supported by Project NALP-SAPR granted by FSE, Project NALP-SAPR2 granted by
University of Perugia, by NATO grant G4936, Intelligent Systems Center (ISC)
at Missouri S&T, and by NSF grants CNS-1545050, CNS-1725755, CNS-1818942, and
SCC-1952045.

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|  Francesco Betti Sorbelli received the Bachelor and Master degrees cum laude
in Computer Science from the University of Perugia, Italy, in 2007 and 2010,
respectively, and his Ph.D. in Computer Science from the University of
Florence, Italy, in 2018. After his Ph.D., he was a Postdoc Researcher at
University of Perugia. Currently, he is a Postdoc at the Missouri University
of Science and Technology University, USA. His research interests include
wireless sensor networks, algorithms on drones and robots.
---|---
|  Cristina M. Pinotti received the Master degree cum laude in Computer
Science from the University of Pisa, Italy, in 1986. In 1987-1999, she was
Researcher with the National Council of Research in Pisa. In 2000-2003, she
was Associate Professor at the University of Trento. Since 2004, she is a Full
Professor at the University of Perugia. Her current research interests include
the design and analysis of algorithms for wireless sensor networks and
communication networks. She has published more than 100 refereed papers on
international journals and conferences, workshops, and book chapters.
---|---
|  Simone Silvestri graduated with honors and received his PhD in computer
science at Sapienza University of Rome, Italy. He is an Assistant Professor in
the Department of Computer Science of the University of Kentucky. His research
interests include cyber-physical-human systems, Internet of Things, hybrid
sensor networks, and network management. He servers as Associate Editor of
Elsevier’s Pervasive and Mobile Computing journal, and has served on the
Technical Program Committees of numerous international conferences. His
research is funded by several agencies such as NIFA, NSF, and NATO. He
received the NSF CAREER award in 2020.
---|---
|  Sajal K. Das is a professor of computer science and Daniel St. Clair
Endowed Chair at Missouri University of Science and Technology. His research
interests include wireless sensor networks, mobile and pervasive computing,
cyber-physical systems and IoT, smart environments, cloud computing, cyber
security, and social networks. He serves as the founding Editor-in-Chief of
Elsevier’s Pervasive and Mobile Computing journal, and as Associate Editor of
several journals including the IEEE Transactions of Mobile Computing, IEEE
Transactions on Dependable and Secure Computing, and ACM Transactions on
Sensor Networks. He is an IEEE Fellow.
---|---