# Cooperative Simultaneous Tracking and Jamming
for Disabling a Rogue Drone

Savvas Papaioannou, Panayiotis Kolios, Christos G. Panayiotou and
Marios M. Polycarpou The authors are with the KIOS Research and Innovation
Centre of Excellence (KIOS CoE) and the Department of Electrical and Computer
Engineering, University of Cyprus, Nicosia, 1678, Cyprus.
E-mail:{papaioannou.savvas, pkolios, christosp<EMAIL_ADDRESS>
###### Abstract

This work investigates the problem of simultaneous tracking and jamming of a
rogue drone in 3D space with a team of cooperative unmanned aerial vehicles
(UAVs). We propose a decentralized estimation, decision and control framework
in which a team of UAVs cooperate in order to a) optimally choose their
mobility control actions that result in accurate target tracking and b) select
the desired transmit power levels which cause uninterrupted radio jamming and
thus ultimately disrupt the operation of the rogue drone. The proposed
decision and control framework allows the UAVs to reconfigure themselves in 3D
space such that the cooperative simultaneous tracking and jamming (CSTJ)
objective is achieved; while at the same time ensures that the unwanted inter-
UAV jamming interference caused during CSTJ is kept below a specified critical
threshold. Finally, we formulate this problem under challenging conditions
i.e., uncertain dynamics, noisy measurements and false alarms. Extensive
simulation experiments illustrate the performance of the proposed approach.

##  I Introduction

In the recent years the demand for consumer drones (i.e., UAVs) has been
skyrocketed [1]. This new gadget has become extremely appealing to the
consumers and has nowadays become ubiquitous. Drones however, like every new
emerging technology, can potentially introduce new threads and risks for the
public safety. Indeed, consumer drones have created a big risk for public
safety especially around airports and restricted airspaces. In fact, numerous
times airports have been shut down [2] because of rogue drones, causing long
delays to the flights schedule and inconvenience to the passengers.

Unfortunately, no adequate solution exist for this problem as of today.
Various approaches developed in academic and industrial labs have been focused
on a) drone detection techniques [3] based on RF signal sniffing, computer
vision and sensor fusion, and b) interception techniques [4] such as net-
casting, RF denial systems, high-power laser and even trained eagles [5].
According to [6], solutions for safeguarding drones are still in their infancy
and considerable more work is needed in order for this technology to reach the
required level of maturity.

This paper deals with the deployment of a team of UAVs (i.e., agents) in 3D
space with the ultimate purpose of intercepting and downing a single rogue
drone (i.e., target). In particular, in this work we propose a multi-UAV
cooperative simultaneous tracking and jamming (CSTJ) framework in which a team
of autonomous UAVs cooperate in order to continuously track and jam a rogue
drone in the air, thus forcing it to enter failsafe mode [7] and auto-land or
return to its base. More specifically, we focus on a realistic scenario in
which the rogue drone moves in 3D space with uncertain and noisy dynamics.
Each UAV is equipped with a 3D range-finding sensor [8] which returns noisy
target measurements. In addition, we assume that the UAVs exhibit a limited
sensing range and that they can detect the presence of the rogue drone inside
their sensing range with probability less that 1. Due to imperfections of the
3D range-finding sensor we assume that, in addition to the target measurements
the UAVs receive multiple false-alarm measurements at each time-step. Finally,
we assume that the UAVs are equipped with a directional antenna which they use
to transmit power to the rogue drone at discrete power-levels.

Figure 1: The figure illustrates the problem of cooperative simultaneous
tracking and jamming tackled in this work.

In the proposed framework the UAVs cooperate in order to: a) accurately track
the rogue drone and b) choose the optimal transmit power levels from their on-
board antennas which maximize the received power at the rogue drone and thus
jamming its communications and sensing circuitry. At the same time the
cooperative UAVs make sure that the power interference between them is kept
below a specified critical interference threshold which in this work is
assumed to be fixed and known. The main contributions of this work are the
following:

  * •

We formulate the problem of cooperative simultaneous tracking and jamming of a
rogue drone in challenging conditions e.g., uncertain dynamics, noisy
measurements, false alarms, uncertain detection and with UAVs that exhibit
limited sensing range.

  * •

We propose a novel decentralized estimation, decision and control framework
that allows a team of cooperative UAVs to reposition themselves in 3D space at
each time-step in order to accurately track-and-jam a rogue drone in the air,
while at the same time avoiding the critical interference between them and
thus remaining operational.

  * •

We demonstrate the performance of the proposed approach through extensive
simulation experiments.

The rest of the paper is organized as follows. Section II provides a brief
overview of the existing literature on this topic by single and multiple
agents. Section III develops the system model and Section IV formulates the
problem and illustrates the proposed system architecture. Section V discuses
the details of the proposed approach and Section VI conducts an extensive
performance evaluation. Finally, Section VII concludes the paper and discusses
future directions.

##  II Related Work

A recent report by the U.S. Homeland Security Committee regarding the security
threat posed by UAVs and possible countermeasures states that the interceptor
UAV technology is currently immature and that interceptor UAVs equipped with
reliable and accurate detection, tracking and defense capabilities are of
essence [9]. Initial works [3, 10] on this problem investigate the detection
and tracking of small UAVs using various methodologies including thermal, RF
and audio/video signals. The work in [11] provides a comprehensive overview of
the technologies utilized for drone surveillance and discusses the state-of-
the-art anti-drone systems. The works in [12, 13, 14, 15] investigate the
problem of target tracking with single and multiple autonomous UAVs but
without considering jamming capabilities. The work in [16] develops a UAV
based solution for localizing a GPS jammer whereas the authors in [17] propose
a low-cost ground jamming system that uses a 3D MIMO radar and a directional
antenna to counteract the operation of small UAVs. A more relevant work to the
proposed system is described in [18] where a consumer UAV is outfitted with
antennas and commodity radios in order to autonomously localize another drone
using its telemetry radio signature. Moreover, the work in [19], develops an
approach where a swarm of defense UAVs forms a cluster around a single rogue
UAV in order to restrict its movement and escort it outside of the restricted
airspace. The authors assume however, that a high-quality UAV monitoring
system is in place which is able to accurately detect and track the rogue UAV.
Finally, in [20] the authors investigate the problem of cooperative control
and task allocation in mission-planning and demonstrate their approach in
radar jamming.

Complementary to the aforementioned techniques, the proposed system
investigates the problem of simultaneous tracking and jamming of a rogue drone
with a team of cooperative autonomous UAVs. Compared to the existing
techniques the proposed framework proposes a novel estimation, decision and
control framework for the combined problem of tracking-and-jamming by multiple
UAVs while considering the induced interference between them. This problem
formulation, to the best of our knowledge has not been investigated before.

##  III System Model

###  III-A Rogue Drone Dynamics

Let us assume that the rogue drone evolves in 3D space with dynamics that can
be expressed by the following discrete-time dynamical model:

$x_{t}=\Phi x_{t-1}+\Gamma\nu_{t}$ (1)

where
$x_{t}=[\text{x},\text{y},\text{z},\dot{\text{x}},\dot{\text{y}},\dot{\text{z}}]_{t}^{\top}\in\mathcal{X}$
denotes the drone’s state at time $t$ which consists of the position and
velocity components in 3D cartesian coordinates and
$\nu_{t}\sim\mathcal{N}(0,\Sigma_{\text{v}})$ denotes the perturbing
acceleration noise which is drawn from a zero mean multivariate normal
distribution with covariance matrix $\Sigma_{\text{v}}$. The matrices $\Phi$
and $\Gamma$ are defined as:

$\Phi=\begin{bmatrix}\text{I}_{3}&\Delta T\cdot\text{I}_{3}\\\
\text{0}_{3}&\text{I}_{3}\end{bmatrix},\Gamma=\begin{bmatrix}0.5\Delta
T^{2}\cdot\text{I}_{3}\\\ \Delta T\cdot\text{I}_{3}\end{bmatrix}$ (2)

where $\Delta T$ is the sampling period, $\text{I}_{3}$ is the identity matrix
of dimension $3\times 3$ and $\text{0}_{3}$ is the zero matrix of dimension
$3\times 3$. In this work we assume that the drone dynamics obey the Markov
property i.e., the state of the drone at the next time step depends only upon
the state of the previous time step as shown in Eqn. (1).

###  III-B UAV Dynamics

Suppose that we have in our disposal a set of controllable UAV agents
$S=\\{1,2,...,|S|\\}$, where $|S|$ denotes the cardinality of the set, i.e.,
the number of available UAVs. Each UAV $j\in S$ is subject to the following
discrete time dynamics:

$s^{j}_{t}=s^{j}_{t-1}+\begin{bmatrix}\Delta_{R}[l_{1}]\sin(l_{2}\Delta_{\phi})\cos(l_{3}\Delta_{\theta})\\\
\Delta_{R}[l_{1}]\sin(l_{2}\Delta_{\phi})\sin(l_{3}\Delta_{\theta})\\\
\Delta_{R}[l_{1}]\cos(l_{2}\Delta_{\phi})\end{bmatrix},\begin{array}[]{l}l_{1}=1,...,|\Delta_{R}|\\\
l_{2}=0,...,N_{\phi}\\\ l_{3}=1,...,N_{\theta}\end{array}$ (3)

where
$s^{j}_{t-1}=[s^{j}_{\text{x}},s^{j}_{\text{y}},s^{j}_{\text{z}}]^{\top}_{t-1}\in\mathbb{R}^{3}$
denotes the state (i.e., position) of UAV $j$ (i.e., $(x,y,z)$ coordinates) at
time $t-1$, $\Delta_{R}$ is a vector of possible radial step sizes
($\Delta_{R}[l_{1}]$ returns the value at index $l_{1}$),
$\Delta_{\phi}=\pi/N_{\phi}$, $\Delta_{\theta}=2\pi/N_{\theta}$ and the
parameters $(|\Delta_{R}|,N_{\phi},N_{\theta})$ determine the number of
possible control actions. We denote the set of all admissible control actions
of UAV $j$ at time $t$ as
$\mathbb{U}^{j}_{t}=\\{s^{j,1}_{t},s^{j,2}_{t},...,s^{j,|\mathbb{U}_{t}|}_{t}\\}$
as computed by Eqn. (3).

###  III-C UAV Sensing Model

The UAVs exhibit a limited sensing range for detecting a target, which is
modeled by the function $p_{D}(x_{t},s_{t})$ that gives the probability that a
target with state
$x_{t}=[\text{x},\text{y},\text{z},\dot{\text{x}},\dot{\text{y}},\dot{\text{z}}]_{t}^{\top}$
at time $t$ is detected by the UAV with state
$s_{t}=[s_{\text{x}},s_{\text{y}},s_{\text{z}}]^{\top}_{t}$. More specifically
a target with state $x_{t}$ and position coordinates $Hx_{t}$ (where $H$ is a
matrix that extracts the $(x,y,z)$ coordinates of a target from its state
vector) is detected by a UAV with state $s_{t}$ with probability which is
given by:

$p_{D}(x_{t},s_{t})=\begin{cases}p^{\max}_{D}&\text{if }d_{t}<R_{0}\\\
\text{max}\\{0,p^{\max}_{D}-\eta(d_{t}-R_{0})\\}&\text{if }d_{t}\geq
R_{0}\end{cases}$ (4)

where $d_{t}=\left\lVert Hx_{t}-s_{t}\right\rVert_{2}$ denotes the Euclidean
distance between the UAV and the target in 3D space, $p^{\max}_{D}$ is the
detection probability for a target which resides within $R_{0}$ distance from
the UAV’s position and finally $\eta$ captures the reduced effectiveness of
the UAV to detect a distant target.

Additionally, each UAV is equipped with a 3D range finding sensor which
provides noisy measurements
$y_{t}=[\rho,\theta,\phi]^{\top}_{t}\in\mathcal{Y}$ (i.e., radial distance
$\rho$, azimuth angle $\theta$ and inclination angle $\phi$) from the detected
target $x_{t}$ according to the following measurement model:

$y_{t}=h(x_{t},s_{t})+w_{t}$ (5)

where $h(x_{t},s_{t})$ is given by:

$\left[\left\lVert
Hx_{t}-s_{t}\right\rVert_{2},~{}\tan^{-1}(\frac{\delta_{y}}{\delta_{x}}),~{}\tan^{-1}(\frac{\sqrt{\delta_{x}^{2}+\delta_{y}^{2}}}{\delta_{z}})\right]^{\top}$
(6)

where $\delta_{y}=\text{y}-s_{y}$, $\delta_{x}=\text{x}-s_{x}$,
$\delta_{z}=\text{z}-s_{z}$ and $w_{t}\sim\mathcal{N}(0,\Sigma_{w})$ is zero
mean Gaussian measurement noise with covariance matrix
$\Sigma_{w}=\text{diag}[\sigma^{2}_{\rho},~{}\sigma^{2}_{\theta},~{}\sigma^{2}_{\phi}]$
and $\sigma_{\rho}$ is range dependent and given by
$\sigma_{\rho}=\sigma_{\rho_{0}}+\beta_{\rho}\left\lVert
Hx_{t}-s_{t}\right\rVert_{2}$.

Finally, due to the imperfection of the 3D range-finding sensor, at time $t$,
a UAV agent receives (in addition to the target measurement) with an average
Poisson rate of $\lambda_{c}$ a number of false alarm measurements
$c^{1}_{t},\ldots,c^{n}_{t}\in\mathcal{Y},~{}\mathbb{E}(n)=\lambda_{c}$ which
are uniformly distributed inside the measurement space according to the
density function $p_{c}(c_{t})$. To summarize, at each time-step each UAV
receives a collection of measurements:

$\Upsilon_{t}=\bigcup\Big{\\{}a\subset\\{\emptyset,y_{t}\\},b\subseteq\\{c^{1}_{t},\ldots,c^{n}_{t}\\}\Big{\\}}$
(7)

###  III-D UAV Wireless Propagation Model

Each UAV carries a directional antenna which is used as the main mechanism for
jamming the rogue drone. The antenna profile in cartesian 3D coordinates is
described by a circular right angle cone which is given by:
$[x=u\cos(v),~{}y=u\sin(v),~{}z=\frac{h_{a}}{r}u]$ for
$u\in[0,r],~{}v\in[0,2\pi]$ where $h_{a}$ characterizes the effective antenna
range, $r=\tan(\frac{\theta_{a}}{2})h_{a}$ is the circular base radius and
$\theta_{a}$ is the opening angle of the cone which determines the antenna’s
conic lobe $\mathcal{A}$. At any point in time $t$, the antenna is auto-
rotated to point in the direction of the vector
$\vec{e}_{t}=\tilde{x}_{t}-s_{t}$, where $\tilde{x}_{t}$ and $s_{t}$ are the
predicted position of the rogue drone and the state of the UAV, respectively.
This is illustrated in Fig. (2).

Figure 2: The figure illustrates the agent antenna model used in this work and
described in Sec. III-D. In the illustrative example above 3 UAV agents aiming
their antennas (with $\theta_{a}=60\deg$ and $h=15$m) at the position of the
target (i.e., rogue drone) to be jammed.

The UAV emitted transmit power varies over a discrete set of power levels,
$P^{j}_{t,\omega},~{}\omega\in\\{1,\dots,\Omega\\},j\in S$ while the received
power at the rogue drone $x_{t}$ from a UAV with state $s_{t}$ is governed by
the following path-loss model:

$\Lambda(s_{t},x_{t})=EF+10n_{e}\text{log}_{10}\left(\left\lVert
s_{t}-x_{t}\right\rVert_{2}\right)+AF$ (8)

where the term $EF$ models the near field loss effects, $n_{e}$ is the path-
loss exponent and $AF$ models the attenuation effects [17]. We should point
out here that due to the directional antenna of the UAVs, Eqn. (8) is valid
only when $x_{t}$ resides inside the conic lobe $\mathcal{A}$ of the antenna
of the UAV. When the rogue drone $x_{t}$ resides outside of $\mathcal{A}$, it
receives no power from the UAV.

##  IV Problem Statement and System Overview

In this paper we are interested in making optimized decisions on the UAV
mobility controls ($u_{t}^{j}\in\mathbb{U}_{t}^{j},\forall j\in S$) and UAV
transmit power levels ($P^{j}_{t,\omega},\forall j\in S$) of the cooperative
UAVs in order to counteract the operation of a rogue drone. The problem of
cooperative simultaneous tracking and jamming (CSTJ) tackled in this work can
now be stated as follows:  At each time step $t$ we would like to find the
optimal joint UAV mobility control actions
$u_{t}^{j}\in\mathbb{U}_{t}^{j},\forall j\in S$ and at the same time choose
the transmit power level $P^{j}_{t,\omega},\forall j\in S$ for each UAV agent
so that a) the UAVs maintain tracking of the rogue drone, b) the received
power at the rogue drone is maximized and c) the in-between UAV interference
is limited below a specified critical threshold.

A closer look at this problem however easily reveals its combinatorial
complexity. Suppose we have in our disposal $|S|$ autonomous UAVs, each of
which exhibits $|\mathbb{U}_{t}|$ mobility control actions and transmits in
$\Omega$ different power levels. Then the number of joint mobility and power
control actions that need to be evaluated at each time-step is given by
$(|\mathbb{U}_{t}|\Omega)^{|S|}$, which quickly becomes computationally
prohibitive to compute.

For this reason, instead of tackling the above joint problem we propose an
alternative suboptimal decentralized approach, as depicted in Fig. 3, which
allows us to solve instances of the problem in real-time. In essence we
decouple the joint problem and we propose a cascaded control architecture
which is composed of a Tracking control module which gives input to a Jamming
control module. This cascaded controller is used by each UAV to determine its
own optimal mobility and power control actions at each time step. Then we
process the UAVs in a sequential fashion i.e., as it is shown in Fig. 3, the
UAV $j+1$ will take into consideration the actions taken by the already
decided UAVs $1,\ldots,j$ when optimizing its CSTJ objective.

Each UAV uses stochastic filtering to compute and propagate in time the
probability distribution of the state of the rogue drone. Briefly, in
stochastic filtering [21, 22] we are interested in the posterior filtering
density $p(x_{t}|Z_{1:t})$ of some hidden state $x_{t}\in\mathcal{X}$ at time
$t$ given all measurements up to time $t$ i.e., $Z_{1:t}=z_{1},...,z_{t}$,
with $z_{t}\in\mathcal{Z}$. Assuming an initial density on the state
$p(x_{0})$, the posterior density at time $t$ can be computed using the Bayes
recursion as:

$\displaystyle p(x_{t}|Z_{1:t-1})$ $\displaystyle=\int
p(x_{t}|x_{t-1})p(x_{t-1}|Z_{1:t-1})dx_{t-1}$ (9) $\displaystyle
p(x_{t}|Z_{1:t})$ $\displaystyle=\frac{p(z_{t}|x_{t})p(x_{t}|Z_{1:t-1})}{\int
p(z_{t}|x_{t})p(x_{t}|Z_{1:t-1})dx_{t}}$ (10)

where Eqn. (9) and (10) are referred to as the prediction and update steps
respectively and the functions $p(x_{t}|x_{t-1})$ and $p(z_{t}|x_{t})$ are the
known transitional density and measurement likelihood function respectively.
At each time step the hidden state $x_{t}$ is usually extracted from the
posterior distribution using the expected a posteriori (EAP) or the maximum a
posteriori (MAP) estimators. Finally, at each time-step the UAVs fuse their
local estimation results to produce the final estimate for the state of the
rogue drone.

Figure 3: The figure illustrates the proposed cooperative simultaneous
tracking and jamming (CSTJ) system architecture.

##  V Cooperative Simultaneous Tracking and Jamming

###  V-A Tracking Control Module

In this subsection we describe the operation of the Tracking Control Module,
as depicted in Fig. 3, which is responsible for actively controlling the
movement of a UAV in order to maintain tracking of the rogue drone. More
specifically, we seek to find the optimal mobility control action
$u^{j}_{t}\in\mathbb{U}^{j}_{t}$ that must be taken at time-step $t$ by each
UAV $j\in S$ so that the state of the rogue drone is estimated as accurately
as possible.

First, observe from the UAV sensing and measurement models (i.e., Eqn. (4) and
Eqn. (5) respectively) that the control action
$u^{j}_{t}\in\mathbb{U}^{j}_{t}$ taken by UAV $j$ at time-step $t$ affects the
probability of detecting the rogue drone and also the quality of the received
measurements $y^{j}_{t}$. The state estimation of the rogue drone depends on
the received measurements which in turn depend on the applied control actions.
For this reason we consider the UAV action which maximizes the probability of
target detection (and thus the reception of target measurements) as our
control strategy for maintaining tracking. That said, we denote the tracking
control objective for agent $j$ as $\xi^{j}(x^{j}_{t},u^{j}_{t})$, and so the
problem to solve becomes:

$\hat{u}_{t}^{j}=\underset{u^{j}_{t}\in\mathbb{U}^{j}_{t}}{\arg\max}~{}\xi^{j}(x_{t},u^{j}_{t})=\underset{u^{j}_{t}\in\mathbb{U}^{j}_{t}}{\arg\max}~{}p^{j}_{D}(x_{t},u^{j}_{t})$
(11)

where $x_{t}$ is the state the rogue drone. However, since $x_{t}$ is not
available until the control action $\hat{u}^{j}_{t}$ is applied, we
approximate it as $x^{j}_{t}\sim\tilde{x}^{j}_{t}$:

$\displaystyle\tilde{x}^{j}_{t}$ $\displaystyle=\int
x_{t}p^{j}(x_{t}|\Upsilon_{1:t-1})dx_{t}$ (12)

In essence each agent computes the predictive density
$p^{j}(x_{t}|\Upsilon_{1:t-1})$ (i.e., Eqn. (9)) and extracts the expected
predicted target state for time $t$. Note that in Eqn. (12) we have used
$\Upsilon_{t}$ to denote the UAV measurements as described by Eqn. (7). In
Sec. V-C we discuss in more detail how we use $\Upsilon_{t}$ to estimate the
state of the rogue drone.

###  V-B Jamming Control Module

The solution of the problem presented in Eqn. (11) provides the optimal
mobility control actions but only with respect to target tracking. These
mobility control actions however, are not necessarily optimal for the
objective of jamming. More specifically, in order to jam the rogue drone, the
agents need to transmit a specific amount of power towards the target while at
the same time ensuring that the power interference between them is kept below
a predefined critical threshold $\Delta$. This is necessary so that the
cooperative agents do not interfere with each other and remain operational at
all times. In order to achieve the joint objective of tracking and jamming of
the target, we utilize a cascaded control architecture in which we first find
the optimal mobility control actions which result in a satisfactory tracking
accuracy and then we perform a second optimization step where we refine
further these mobility actions to achieve jamming control. To do that, instead
of finding the mobility control actions which maximize Eqn. (11), we first
compute for each agent $j$ the set
$\tilde{\mathbb{U}}^{j}_{t}=\\{\tilde{u}^{j1}_{t},\tilde{u}^{j2}_{t},...\\}$
of all mobility control actions, which satisfy:

$\tilde{\mathbb{U}}^{j}_{t}=\\{u^{j}_{t}\in\
\mathbb{U}^{j}_{t}~{}|~{}\xi^{j}(\tilde{x}^{j}_{t},u^{j}_{t})>\vartheta\\}$
(13)

where $\vartheta\in[0,1]$ is the desired threshold for the tracking objective.
In the second step, we select the optimal mobility control actions and the
levels of transmit power for each of the cooperative UAVs which maximize the
received jamming power at the rogue drone.

To do that, each UAV decides in a sequential fashion its mobility and power
control actions that maximize the received power at the rogue drone while
satisfying the interference constraints on the power received from all
previously decided UAVs. As discussed in Sec. III-D, we consider a predefined
discrete set of power levels $\Omega$ form which a UAV can chose its transmit
power. Let $D$ be the set of all UAVs that have made a decision on their
actions and $\bar{D}$ the set of undecided UAVs. Then the optimal decision and
control problem for UAV $j$ can be formulated according to (P1) below:

$\displaystyle\text{(P1)}~{}\max$
$\displaystyle\sum_{\omega=1}^{\Omega}\sum_{k=1}^{|\mathbb{\tilde{U}}_{t}^{j}|}b_{\omega}^{k}\mathds{1}_{\mathcal{A}^{j}}(\tilde{x}^{j}_{t})P_{t,\omega}^{j}\Lambda(\tilde{u}^{jk}_{t},\tilde{x}^{j}_{t})$
(14) $\displaystyle\mathrm{s.t.}$
$\displaystyle\>\>\sum_{\omega=1}^{\Omega}\sum_{i\in
D}b_{\omega}^{k}\mathds{1}_{\mathcal{A}^{i}}(\tilde{u}^{jk}_{t})\hat{P}_{t}^{i}\Lambda(\hat{u}^{i}_{t},\tilde{u}^{jk}_{t})<\Delta^{j}$
(15)
$\displaystyle\sum_{\omega=1}^{\Omega}\sum_{k=1}^{|\mathbb{\tilde{U}}_{t}^{j}|}b_{\omega}^{k}\mathds{1}_{\mathcal{A}^{j}}(\hat{u}^{i}_{t})P_{t,\omega}^{j}\Lambda(\tilde{u}^{jk}_{t},\hat{u}^{i}_{t})~{}~{}+$
(16) $\displaystyle\>\>\sum_{l\in D,l\neq
i}\mathds{1}_{\mathcal{A}^{l}}(\hat{u}^{i}_{t})\hat{P}_{t}^{l}\Lambda(\hat{u}^{l}_{t},\hat{u}^{i}_{t})<\Delta^{i},~{}\forall
i\in D$
$\displaystyle\>\>b_{\omega}^{k}\in\\{0,1\\},\>\>\sum_{\omega=1}^{\Omega}\sum_{k=1}^{|\mathbb{\tilde{U}}_{t}^{j}|}b_{\omega}^{k}=1$
(17)

In (P1) the best mobility control action is selected and the transmit power is
chosen for each UAV $j$ which maximizes the received power at the rogue drone
while ensuring that the interference level between the cooperating UAVs is
maintained below the critical threshold $\Delta^{j}$. The indicator function
$\mathds{1}_{\mathcal{A}^{j}}(x)$ checks if $x$ is within the conic antenna
lobe $\mathcal{A}^{j}$ of UAV $j$ and returns 1, otherwise returns 0. The
constraint in Eqn. (15) ensures that the interference caused to agent
$j\in\bar{D}$ when taking action
$\tilde{u}^{jk}_{t}\in\tilde{\mathbb{U}}^{j}_{t}$ with respect to all UAVs
($i\in D\subseteq S$) that have already made their mobility and power-level
decisions (i.e., $\hat{u}^{i}_{t}$ and $\hat{P}_{t}^{i}$) is below the
critical threshold. On the other hand, Eqn. (16) computes a) the interference
caused by UAV $j\in\bar{D}$ to UAV $i\in D$ that has already made a decision
when UAV $j$ applies mobility control $\tilde{u}^{jk}_{t}$ and power-level
$P_{t,\omega}^{j}$ and b) the interference caused to UAV $i$ by all UAVs
$l\neq i\in D$ that have made a decision and transmit in power-level
$\hat{P}_{t}^{l}$. Then, this constraint ensures that the interference caused
to UAV $i\in D$ from all the already decided UAVs $l\in D$ and the UAV
$j\in\bar{D}$ is still within the acceptable interference limit $\Delta^{i}$.
Finally, the selection of a single combination of control actions (i.e.,
mobility and power-level) is ensured by Eqn. (17) using the binary variable
$b_{\omega}^{k}$. In essence, (P1) will find $\omega\in[1,\ldots,\Omega]$ and
$k\in[1,\dots,|\tilde{\mathbb{U}}_{t}|]$ for each agent $j\in S$ such that the
applied power-level $\hat{P}_{t}^{j}$ and mobility control $\hat{u}^{j}_{t}$
maximize Eqn. (14) while satisfying the interference constraints. The
sequential optimization approach of (P1) along with the proposed cascaded
control architecture allow for the evaluation of fewer mobility control
actions at each time-step and reduces the exponential complexity of the joint
problem to linear in the number of agents and total control actions.

###  V-C Rogue Drone State Estimation

The objective here is to use stochastic filtering in order estimate the state
of the rogue drone given its uncertain dynamics, noisy measurements and false-
alarms as described in Sec. III. In order to do that we need to compute the
expressions for the transitional density and measurement likelihood function
as discussed in Sec. IV. From our modeling assumptions we already know that
$p(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\Phi
x_{t-1},\Gamma\Sigma_{\text{v}}\Gamma^{\top})$ is a multivariate Gaussian
transitional density according to the target dynamics of Eqn. (1). What is
left is to compute the expression for the likelihood function $p(z_{t}|x_{t})$
(see Eqn. (10)) which in our case becomes $p(\Upsilon_{t}|x_{t},s_{t})$. This
can be computed as:

$\displaystyle p(\Upsilon_{t}|x_{t},s_{t})$
$\displaystyle=\left[1-p_{D}(x_{t},s_{t})\right]e^{-\lambda_{c}}\prod_{\ell\in\Upsilon_{t}}\lambda_{c}p_{c}(\ell)~{}+$
(18) $\displaystyle
e^{-\lambda_{c}}p_{D}(x_{t},s_{t})\underset{\ell\in\Upsilon_{t}}{\sum}g_{y}(\ell|x_{t},s_{t})\underset{\begin{subarray}{l}\varepsilon\in\Upsilon_{t}\\\
\varepsilon\neq\ell\end{subarray}}{\prod}\lambda_{c}p_{c}(\varepsilon)$

where the function
$g_{y}(y_{t}|x_{t},s_{t})=\mathcal{N}(y_{t};h(x_{t},s_{t}),\Sigma_{w})$ is due
to Eqn. (5). To obtain the above expression we first use the fact that if the
UAV receives only false alarm measurements (i.e., no measurement from the
rogue drone) then $\Upsilon_{t}=\\{c^{1}_{t},...,c^{|\Upsilon_{t}|}\\}$ and
$p(\Upsilon_{t}|x_{t},s_{t})$ is given by:

$|\Upsilon_{t}|!\left[1-p_{D}(x_{t},s_{t})\right]\text{Pois}(|\Upsilon_{t}|)\prod_{\ell\in\Upsilon_{t}}p_{c}(\ell)$
(19)

where the function
$\text{Pois}(|\Upsilon_{t}|)=\frac{\lambda_{c}^{|\Upsilon_{t}|}e^{-\lambda_{c}}}{|\Upsilon_{t}|!}$
is the Poisson distribution with rate parameter $\lambda_{c}$ and gives the
probability of obtaining exactly $|\Upsilon_{t}|$ false alarm measurements at
time $t$ and $\prod_{\ell\in\Upsilon_{t}}p_{c}(\ell)$ is the joint spatial
density of the false alarms. The above expression is multiplied by
$|\Upsilon_{t}|![1-p_{D}(x_{t},s_{t})]$ to account for all possible
permutations of the measurement sequence and for the fact that the target has
not been detected, which explains the first line in Eqn. (18).

The second line of Eqn. (18) is due to the more general case in which the UAV
receives measurements from the rogue drone $y_{t}$ in addition to false alarms
e.g., $\Upsilon_{t}=\\{y_{t},c^{1}_{t},c^{2}_{t},\ldots\\}$. In this case the
joint likelihood accounts for the detection of the rogue drone and its
measurement i.e., $p_{D}(x_{t},s_{t})g_{y}(y_{t}|x_{t},s_{t})$, multiplied by
the joint density of the remaining $|\Upsilon_{t}|-1$ measurements which are
considered false alarms (i.e.,
$(|\Upsilon_{t}|-1)!\cdot\text{Pois}(|\Upsilon_{t}|-1)~{}\prod_{\ell~{}\in~{}\Upsilon_{t},~{}\ell~{}\neq~{}y_{t}}~{}p_{c}(l)$).
Finally, we have to sum over all possible $|\Upsilon_{t}|$ ways in which we
can select $y_{t}$ from $\Upsilon_{t}$.

Figure 4: The figure shows the trajectories of 4 agents for the task of
cooperative simultaneous tracking and jamming of a rogue drone.

To summarize each UAV $j$ propagates in time the filtering density
$p^{j}(x_{t}|\Upsilon_{1:t})$ and computes the EAP state for the rogue drone
as $\hat{x}^{j}_{t}=\int x_{t}p^{j}(x_{t}|\Upsilon_{1:t})dx_{t}$ and its
covariance matrix
$K^{j}_{\hat{x}^{j}_{t}}=\mathbb{E}[(x-\hat{x}^{j}_{t})(x-\hat{x}^{j}_{t})^{\top}]$.
Finally, the UAVs exchange their local estimates with each other and apply
sequential covariance intersection [23] to compute the final fused state
$\hat{x}_{t}$ of the rogue drone. The resulting CSTJ algorithm is shown below
in Alg. 1.

Algorithm 1 Proposed Cooperative Simultaneous Tracking and Jamming approach

0:
$p^{j}(x_{t-1}|\Upsilon_{1:t-1}),\mathbb{U}^{j}_{t},P^{j}_{t,\omega}~{}\forall~{}j,\omega,D\subseteq
S$

1: Compute the predictive density $p^{j}(x_{t}|\Upsilon_{1:t-1})$ using Eq.
(9).

2: Compute the predicted target state at time $t$, $\tilde{x}^{j}_{t}$ using
Eqn. (12)

3: Obtain the set of optimized tracking control actions
$\tilde{\mathbb{U}}^{j}_{t}$ by solving Eqn. (13)

4: Solve P1 to obtain the optimal mobility control action $\hat{u}^{j}_{t}$
and power-level $\hat{P}^{j}_{t}$.

5: Execute the optimal mobility and power control actions.

6: Receive the target measurement set $\Upsilon^{j}_{t}$.

7: Compute posterior density $p^{j}(x_{t}|\Upsilon_{1:t})$ by plugging in Eqn.
(18) into Eqn. (10)

8: Estimate target state $\hat{x}^{j}_{t}$ and its covariance matrix
$K^{j}_{\hat{x}^{j}_{t}}$.

9: Produce the final state estimate $\hat{x}_{t}$ by fusing together the
information from line 8 using sequential covariance intersection [23].

##  VI Evaluation

###  VI-A Simulation Setup

In order to evaluate the performance of the proposed approach we have
conducted several numerical experiments. For these experiments we have used
the following setup: The UAV agents and the rogue drone maneuver in a 3D space
with dimensions $100\text{m}\times 100\text{m}\times 100\text{m}$. The target
dynamics are according to Eqn. (1) with
$\Sigma_{\text{v}}=\text{diag}[2,2,2]\text{m}/\text{s}^{2}$ and $\Delta T=1$s.
The agent dynamics are according to Eqn. (3) with $\Delta_{R}=[1,3,5]$,
$N_{\phi}=2$ and $N_{\theta}=4$. The agent sensing model is given by Eqn. (4)
with $p_{D}^{\text{max}}=0.99$, $\eta=0.02\text{m}^{-1}$ and $R_{0}=2$m. The
agent’s measurement model is given by Eqn. (5) with parameters
$\sigma_{\theta}=~{}\sigma_{\phi}=\pi/50$rad, $\sigma_{\rho_{0}}=2$m and
$\beta_{\rho}=0.05\text{m}^{-1}$ and finally $\lambda_{c}=15$. For the antenna
model we have $h_{a}=100$m and $\theta_{a}=80$deg. Additionally, the UAV
agents transmit in the following power levels $[\text{off},-10,0,7,10]$dB, the
agent interference level $\Delta$ is set at $-50$dB, the path-loss exponent is
$2.5$, the near field loss effects $EF$ are set to 32.4dB and the attenuation
effects $AF$ are computed based on a $2$GHz carrier frequency as
$AF=20\text{log}_{10}(2)=6.0206$dB. Finally, the tracking threshold
$\vartheta$ is set to $\vartheta=0.8$ and the filtering approach described in
Sec. V-C is implemented as a SIR particle filter [24] in order to handle the
non-linear sensing model described in Sec. III-C.

###  VI-B Results

Figure 4a shows a simulated scenario over 10 time-steps in which 4 agents
cooperate in order to simultaneously actively track and jam a single target.
More specifically, agents 1, 2, 3 and 4 (shown in different colors) are
spawned from $[29,30,23],[24,25,24],[23,28,21]$ and $[23,27,37]$ respectively.
The target initial state is $[21,21,30,1.8,1.6,1.1]$ as indicated by the black
circle and moves along the black line with final position marked with $\times$
as shown in the figure. The agents maneuver around the target in order to
achieve their CSTJ objective. More specifically, the agents cooperatively
choose their control actions in each time step which result in the optimal
tracking and jamming performance while at the same time accommodating their
in-between interference constraints. Interestingly, to achieve this the agents
follow interleaving trajectories to avoid the interference between them.
Moreover, it is shown that these optimized control decisions also allow for a
satisfactory tracking performance. As time progresses the estimated target
trajectory (i.e., cyan line) converges to the ground truth (i.e., black line).
This is also evident by the tracking error (i.e., the Euclidian distance
between the estimated and the ground truth target position) shown in Fig. 4e.
In more detail, the agents try to maximize the target received power while at
the same time keeping the interference between them below the critical
threshold $\Delta$. In order to meet the interference constraints the agents
can either (a) optimize their mobility controls and move to a non-interfering
position or (b) optimize the antenna power-level and possibly transmit at a
lower power-level. At the same time the agents need to accurately track the
target over time in order to be able to steer their antenna towards the
direction of the target. Figure 4b shows the agents being jammed (due to
interference), Fig. 4c shows at which power-levels the agents transmit power
and finally Fig. 4d shows the target received power and the amount of received
power for each agent due to interference. As we can observe at time-step
$t=1$, agent 1 is jamming agent 2, and transmits with full power at 10dB.
Agent 2 however, manages to move to a position at $t=1$ in which the
interference caused by agent 1 remains below the critical threshold as shown
in Fig. 4d. Moreover, at the beginning of this experiment (i.e., $t=1..3$) the
target received power is relatively low compared to later time-steps (Fig.
4d). This is because, at the beginning of the experiment the target estimated
position deviates significantly from the true position (due to the high
initial uncertainty on the target position) and thus the agents are having
difficulty steering their antennas at the direction of the target. This
however, changes as the agents improve their cooperative estimation over time.
Moreover, it is shown that between time-steps $t=3..5$ agent 3 is jamming
agent 2 and for this reason agent 3 adjusts its power-levels accordingly in
order to keep the received power at agent 2 below the critical threshold as
shown in Fig. 4c and Fig. 4d. What we have discussed so far briefly
illustrates the operation of the proposed system. We should point out here
that the parameters of the agent dynamical model i.e., Eqn. (3), used for this
simulation have been chosen in order to produce the illustrated Manhattan-like
trajectories in order to aid visual inspection and analysis.

Figure 5: Performance comparison between CSTJ and CT.

Next, we compare the performance of the proposed system (CSTJ) against a
baseline method which performs cooperative tracking (CT) with jamming
capabilities. In CT the UAVs are capable of jamming the rogue drone, however
the interference between the UAVs is not taken into account. Additionally, in
CT each agent maximizes its own tracking objective as discussed in Sec. V-A
and then their local results are fused using covariance intersection in order
to compute the final target state. This experiment aims to investigate the
pros and cons of the proposed system in terms of tracking performance, jamming
performance and the ability of the system to maintain the UAVs below the
specified interference threshold. Since, in CT the UAVs do not posses the
capability of switching/deciding between the different transmit power levels,
for this experiment we assume that in both methods (CSTJ and CT) the UAV
antenna is always on and transmits with constant power at 7db. For this test
we used the following procedure: First, we randomly initialize a target inside
the surveillance area. Then we spawn randomly 4 agents inside a sphere with
radius 5m centered at the target birth location and we run CSTJ and CT for 50
time-steps monitoring the target received power, the tracking error and the
interference between the UAVs. The above procedure is repeated 50 times.
Figure 5 shows the average values of (a) target received power, (b) tracking
error and (c) UAV interference for the 50 trials. More specifically, Fig. 5a
shows that the target received power obtained with CT is on average 1.7dB
higher in each time-step compared to what is obtained with CSTJ. Moreover,
with CT the tracking error shown in Fig. 5b is on average 0.3m lower compared
to the error achieved with CSTJ. These results are quite reasonable since with
the addition of the jamming objective in CSTJ, the UAVs do not always take the
optimal control actions that result in the best possible tracking performance.
In contrast, they consider the joint tracking and jamming objective which
might result in control decisions that deviate for the optimal in terms of
tracking accuracy. In addition since CT does not considers any interference
constraints between the agents, it usually achieves higher target received
power as the agents can get closer to the target which explains the results
obtained. However, the advantage of the proposed CSTJ framework is depicted in
Fig. 5c. More specifically, the proposed system can maintain the interference
between the UAVs below the critical threshold of -50dB while achieving
reasonable tracking and jamming performance. On the other hand the CT does not
considers these constraints and as a result the system becomes unstable and
fails as the UAVs receive very large amounts of power. This is because in CT
the UAVs can get extremely close to each other.

Figure 6: The figure shows the effect of the number of UAVs on the performance
of the proposed system.

Finally, we investigate how the number of agents affects the performance of
the proposed approach. In order to do that we follow a similar setup with the
one described in the previous paragraph. We randomly initialize a target
inside the surveillance area and we vary the number of agents that are
randomly spawned around the target (within a 5m radius). We let our system to
run for 50 time-steps and we measure the average power received at the target
and the average tracking error. Figure 6 shows the averaged results of this
experiment over 50 Monte-Carlo trials. For this test we set the agent power-
levels as $[\text{off},0,7,10]$dB and their mobility control parameters as
$\Delta_{R}=[1,3]$, $N_{\phi}=2$ and $N_{\theta}=4$.

As we can observe from Fig. 6a the average received power at the target
increases significantly between 2 and 8 agents and then slows down. For more
than 8 agents we can say that the target received power almost reaches a
plateau. A similar pattern is also shown for the tracking error in Fig. 6b.
Interestingly, this behavior is due to the following reason: as the number of
agent increases the number of available mobility control actions that the
agents can find which result in good tracking performance and satisfy the
interference constraints decreases. We have observed that as the number of
agent increases, the agents either a) switch off their antenna and transmit no
power since they cannot find a position to move into in which the interference
constraints are met or b) move to a distant position (with respect to other
UAVs) and transmit with minimum power. This limitation can be alleviated (with
extra computational cost) by increasing the the number of degrees of freedom
of the system i.e., increasing the number of mobility control actions, the
power-levels and reducing the antenna opening angle $\theta_{a}$.

##  VII Conclusion

In this paper we have studied the problem of simultaneous tracking and jamming
of a rogue drone with a team of cooperative UAV agents. We have presented a
novel estimation, decision and control framework that allows the UAVs to
reconfigure their positions at each time-step so that the tracking-and-jamming
objective is achieved while taking into account the interference induced
between them. Finally, we have demonstrated the effectiveness of the proposed
technique through extensive simulation experiments. Future work, will focus on
the real-world implementation of the proposed system, and its extension to
multiple targets.

## Acknowledgments

This work has been supported by the European Union’s H2020 research and
innovation programme under grant agreement No 739551 (KIOS CoE) and from the
Government of the Republic of Cyprus through the Directorate General for
European Programmes, Coordination and Development.

## References

  * [1] Federal Aviation Administration, “FAA Aerospace Forecast: Fiscal Years 2019-2039,” 2019.
  * [2] D. Schneider, “Regulators seek ways to down rogue drones: Growing anti-drone industry offers radar, remote id, and other tools,” _IEEE Spectrum_ , vol. 56, no. 4, pp. 10–11, April 2019.
  * [3] I. Guvenc, F. Koohifar, S. Singh, M. L. Sichitiu, and D. Matolak, “Detection, tracking, and interdiction for amateur drones,” _IEEE Communications Magazine_ , vol. 56, no. 4, pp. 75–81, April 2018.
  * [4] J. Loeb, “Exclusive: Anti-drone technology to be tested in uk amid terror fears,” _Engineering Technology_ , vol. 12, no. 3, pp. 9–9, April 2017.
  * [5] BBC. (2016) Eagles trained to take down drones. [Online]. Available: https://www.bbc.com/news/av/world-europe-35750816/eagles-trained-to-take-down-drones (Last accessed: 30-05-2019)
  * [6] K. Wesson and T. Humphreys, “Hacking drones,” _Scientific American_ , vol. 309, no. 5, pp. 54–59, 2013.
  * [7] M. B. Revill, “Uav swarm behavior modeling for early exposure of failure modes,” Naval Postgraduate School Monterey United States, Tech. Rep., 2016.
  * [8] Seong-Taek Park and Jang Gyu Lee, “Improved kalman filter design for three-dimensional radar tracking,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 37, no. 2, pp. 727–739, April 2001.
  * [9] Humphreys, Todd, “Statement on the security threat posed by unmanned aerial systems and possible countermeasures,” _Oversight and Management Efficiency Subcommittee, Homeland Security Committee, Washington, DC, US House_ , 2015.
  * [10] S. R. Ganti and Y. Kim, “Implementation of detection and tracking mechanism for small uas,” in _2016 International Conference on Unmanned Aircraft Systems (ICUAS)_ , June 2016, pp. 1254–1260.
  * [11] X. Shi, C. Yang, W. Xie, C. Liang, Z. Shi, and J. Chen, “Anti-drone system with multiple surveillance technologies: Architecture, implementation, and challenges,” _IEEE Communications Magazine_ , vol. 56, no. 4, pp. 68–74, April 2018.
  * [12] S. Papaioannou, P. Kolios, T. Theocharides, C. G. Panayiotou, and M. M. Polycarpou, “Decentralized search and track with multiple autonomous agents,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_ , 2019, pp. 909–915.
  * [13] G. M. Hoffmann, S. L. Wasl, and C. J. Tomlin, “Distributed cooperative search using information-theoretic costs for particle filters with quadrotor applications,” in _Proceedings of the AIAA Guidance, Navigation, and Control Conference_ , 2006, pp. 21–24.
  * [14] S. Papaioannou, P. Kolios, T. Theocharides, C. G. Panayiotou, and M. Polycarpou, “Jointly-optimized searching and tracking with random finite sets,” _IEEE Transactions on Mobile Computing_ , 2019.
  * [15] S. Papaioannou, P. Kolios, T. Theocharides, C. G. Panayiotou, and M. M. Polycarpou, “Probabilistic search and track with multiple mobile agents,” in _2019 International Conference on Unmanned Aircraft Systems (ICUAS)_ , June 2019, pp. 253–262.
  * [16] A. Perkins, L. Dressel, S. Lo, and P. Enge, “Antenna characterization for uav based gps jammer localization,” in _Proceedings of the 28th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2015)_ , 2015.
  * [17] T. Multerer, A. Ganis, U. Prechtel, E. Miralles, A. Meusling, J. Mietzner, M. Vossiek, M. Loghi, and V. Ziegler, “Low-cost jamming system against small drones using a 3d mimo radar based tracking,” in _2017 European Radar Conference (EURAD)_ , Oct 2017, pp. 299–302.
  * [18] L. Dressel and M. J. Kochenderfer, “Hunting drones with other drones: Tracking a moving radio target,” in _2019 International Conference on Robotics and Automation (ICRA)_ , May 2019, pp. 1905–1912.
  * [19] M. R. Brust and G. Danoy and P. Bouvry and D. Gashi and H. Pathak and M. P. Gon alves, “Defending Against Intrusion of Malicious UAVs with Networked UAV Defense Swarms,” in _2017 IEEE 42nd Conference on Local Computer Networks Workshops (LCN Workshops)_ , Oct 2017, pp. 103–111.
  * [20] I. Jang, J. Jeong, H.-S. Shin, S. Kim, A. Tsourdos, and J. Suk, “Cooperative control for a flight array of uavs and an application in radar jamming,” _IFAC-PapersOnLine_ , vol. 50, no. 1, pp. 8011 – 8018, 2017, 20th IFAC World Congress.
  * [21] D. Simon, _Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches_. Wiley-Interscience, 2006\.
  * [22] Z. Chen, “Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond,” _Technical report. Hamilton, ON, Canada: Adaptive Syst. Lab., McMaster University_ , 2003.
  * [23] Z. Deng, P. Zhang, W. Qi, J. Liu, and Y. Gao, “Sequential covariance intersection fusion kalman filter,” _Inf. Sci._ , vol. 189, pp. 293–309, Apr. 2012.
  * [24] F. Gustafsson, “Particle filter theory and practice with positioning applications,” _IEEE Aerospace and Electronic Systems Magazine_ , vol. 25, no. 7, pp. 53–82, July 2010.