# Resilient Time-Varying Formation Tracking for Mobile Robot Networks under
Deception Attacks on Positioning

Yen-Chen Liu, Kai-Yuan Liu, and Zhuoyuan Song This work was supported by the
Ministry of Science and Technology (MOST), Taiwan, under grants MOST
110-2636-E-006-005 and MOST 109-2636-E-006-019. Z. Song was partially
supported by the U.S. National Science Foundation under awards
CISE/IIS–2024928 and OIA-2032522.Y.-C. Liu and K.-Y. Liu are with the
Department of Mechanical Engineering, National Cheng Kung University, Tainan,
Taiwan (e-mail: yliu@mail.ncku.edu.tw,k242424123@gmail.com).Z. Song is with
the Department of Mechanical Engineering, University of Hawai‘i at Mānoa,
Honolulu, HI, USA (e-mail: zsong@hawaii.edu).

###### Abstract

This paper investigates the resilient control, analysis, recovery, and
operation of mobile robot networks in time-varying formation tracking under
deception attacks on global positioning. Local and global tracking control
algorithms are presented to ensure redundancy of the mobile robot network and
to retain the desired functionality for better resilience. Lyapunov stability
analysis is utilized to show the boundedness of the formation tracking error
and the stability of the network under various attack modes. A performance
index is designed to compare the efficiency of the proposed formation tracking
algorithms in situations with or without positioning attacks. Subsequently, a
communication-free decentralized cooperative localization approach based on
extended information filters is presented for positioning estimate recovery
where the identification of the positioning attacks is based on
Kullback–Leibler divergence. A gain-tuning resilient operation is proposed to
strategically synthesize the formation control and cooperative localization
for accurate and rapid system recovery from positioning attacks. The proposed
methods are tested using both numerical simulation and experimental validation
with a team of quadrotors.

###### Index Terms:

Resilient control, time-varying formation tracking, deception attack, multi-
robot systems, cooperative localization, mobile sensor networks.

##  I Introduction

Formation tracking and control are significant research topics for networked
mobile robots such as unmanned ground vehicles [1, 2] and autonomous aerial
[3, 4] or underwater vehicles [5, 6]. Owing to the benefits of efficiency,
redundancy, robustness, scalability, flexibility, and reliability, maintaining
a formation during the entire mission in mobile robot networks offers many
advantages across various applications such as surveillance, drag reduction,
source seeking, environmental sampling, search and rescue operations, aerial
refueling, cooperative transportation, and closed-formation flight [7, 8, 9,
10, 11, 12, 13, 14, 15, 16]. Although several formation-control approaches
have been extensively studied including leader–follower, virtual structure,
behavior-based, and consensus-based techniques, a common assumption in such
studies has been the availability of inter-agent communication with accurate
agent position information. In real-world applications, data transmission over
cyber networks and information shared between agents could become susceptible
to not only adverse and malicious threats but also attacks that would
crucially compromise or even destroy a networked robotic system [17, 18].

Figure 1: An illustration of time-varying formation tracking in mobile robot
networks under global positioning attacks.

Although multi-agent systems are advantageous in various tasks with redundancy
and scalability, the issues of actuator faults, fallible robots, and
vulnerable communications have recently attracted significant attention [19,
20, 21, 22, 14, 23, 24]. Connectivity preservation with respect to robot
failures was addressed by [19], where a self-optimization of resilient
topologies and an experimental validation were provided. In [21], a
distributed $H_{\infty}$ controller was designed to ensure resilience and
robustness for leader–follower systems under faulty and malicious attacks on
sensors and/or actuators. The problem of multi-agent systems in achieving a
fixed formation under mismatched compasses, which would lead to distortion,
has been studied with estimation and compensation algorithms [22]. In addition
to mismatched compasses, an approach using relative position information has
been implemented in cooperative localization to improve localization accuracy
in communication among agents [20].

Accurate localization is a crucial prerequisite for the control and
coordination of multi-robot systems. Although decentralized multi-agent
coordination methods tend to alleviate the system-wide disruption caused by
localization failures of individual agents, the resulting control error
eventually leads to sub-optimal performance across the entire network. In many
applications, such as underwater robotics and indoor navigation, it is
challenging or infeasible to provide constant global positioning information
to all agents in a network [25, 26, 27]. When computational resources allow,
and the environment is structured, persistently reliable localization can be
achieved through simultaneous localization and mapping [28, 29]. Cooperative
localization is often applied to enable inter-agent observation and
information fusion, thereby reducing the localization error propagation and
maximizing the utilization of locally available global positioning information
[30, 31, 32]. Nonetheless, it is often assumed that inter-agent communication
is available, making the system susceptible to communication failures and
security complications.

In this paper, we study the time-varying formation control (TVFC) of a mobile
robot network in the absence of inter-agent communication, thereby mitigating
potential vulnerabilities to global positioning information attacks or
failures. The mobile robot network is controlled to track prescribed
formations by utilizing global positioning information and local inter-agent
relative displacements obtained from proximity and/or bearing sensors.
Although certain cyber-attacks can be avoided by adopting a communication-free
framework, attacks on global positioning systems could cause catastrophic
failures in multi-agent formation control. To retain the desired functionality
of such systems under positioning attacks or failures, a fault detection and
isolation scheme based on Kullback-Leibler (KL) divergence is proposed for
identifying positioning attacks. Subsequently, a communication-free
decentralized cooperative localization approach based on extended information
filters (EIF) is proposed to enhance the resilience of the mobile robot
network in TVFC. The system is designed using a resilient gain-tuning
formation-control approach to enhance the resilience and robustness of the
multi-agent systems in the presence of adverse effects. Systematic studies and
analyses on the proposed approach for multi-robot formation tracking were
performed. To provide a deeper understanding of the resilience of a mobile
robot network in TVFC, the resilience triangle from the performance index of
TVFC is investigated. Results from numerical simulations and experimental
implementations on a group of quadrotor flying robots are presented to
demonstrate system efficiency and performance in terms of improving the
resilience of mobile robot networks.

The main contributions of this paper are summarized as follows:

  1. 1.

TVFC of mobile robot networks is studied under a hierarchical architecture,
with the design of the performance index representing the local and global
tracking efficiency.

  2. 2.

The adversarial effects of three positioning information attacks, i.e.,
additive attack, hybrid attack, and unstable attack, on the performance of
formation control are investigated through Lyapunov stability analysis.

  3. 3.

A resilient localization system is presented that identifies positioning
attacks based on the KL divergence in fusing global positioning information
with inertial navigation, and maintains consistent localization for the
compromised agents using a communication-free, decentralized cooperative
localization method based on EIF.

  4. 4.

Resilient operation utilizing the results from cooperative localization is
demonstrated to mitigate the adverse impacts on a mobile robot network
subjected to global positioning attacks.

  5. 5.

Experimental validation using a group of quadrotor flying robots is presented
to demonstrate the efficacy and efficiency of the proposed resilient methods
for TVFC of multi-robot systems.

The remainder of this paper is organized as follows. Section II addresses the
modeling, sensory graph, and problem formulation of TVFC for a mobile robot
network. Section III presents a theoretical analysis of the mobile robot
network and the effects of malicious attacks on the performance of the TVFC.
The cooperative localization, attack detection, position recovery, and
resilient operation are discussed in Section IV. Section V presents an
experimental validation of the proposed networked robot system. Section VI
provides a discussion of the proposed resilient robotic systems and summarizes
possible future directions on TVFC for mobile robot networks.

##  II Preliminaries and Problem Formulation

###  II-A Robot Modeling and Sensory Graph

Consider a mobile robot network composed of $N\geq 3$ dynamically controlled
and fully-actuated mobile agents described as

$\displaystyle\ddot{x}_{i}=u_{i},\quad i=1,\ldots,N,$ (1)

where $x_{i}\in\mathcal{Q}$ and $u_{i}\in\mathcal{U}_{i}$ with
$\mathcal{Q}\subset\mathbb{R}^{n}$ as the compact set of the state space and
$\mathcal{U}_{i}\subset\mathbb{R}^{n}$ as the admissible control set of
$u_{i}$.

The time-varying formation tracking for mobile robot network is studied in
this paper in the absence of inter-robot communication so that we have the
next assumption.

######  Assumption 1.

The mobile agents in the robot network are equipped with range and bearing
sensors that can obtain reliable relative position with respect to their
neighbors. This so-called local displacement measurement is mutual, i.e., the
$i^{th}$ robot can obtain the measurement to the $j^{th}$ robot, and vice
versa.

The graph theory [33] is utilized to describe the displacement measurement
topologies (sensory graph) among $N$ agents in the network. For an
interconnected graph $\mathcal{G}(\mathcal{W})$, the vertex set is given as
$\mathcal{V}(\mathcal{G})$, the edge set is denoted by
$\mathcal{E}(\mathcal{G})\in\mathcal{V}(\mathcal{G})\times\mathcal{V}(\mathcal{G})$,
and the weight matrix $\mathcal{W}=\\{w_{ij}\\}$ denotes the weighting for
each of the edges in $\mathcal{E}(\mathcal{G})$. With Assumption 1 and the
graph describing the sensory topology, $\mathcal{G}(\mathcal{W})$ is
undirected such that $w_{ij}=w_{ji}$. The interconnection between the $N$
mobile robots in the network can be described by the weighted Laplacian matrix
$L(\mathcal{G})\in\mathbb{R}^{N\times N}$ [34, 19] defined as
$L(\mathcal{G})=D(\mathcal{G})-A(\mathcal{G}),$ where $D(\mathcal{G})$ is the
degree matrix of $\mathcal{G}$, and $A(\mathcal{G})$ is the corresponding
adjacency matrix with entries $a_{ij}=w_{ij}$. The diagonal terms of the
Laplacian matrix are given as $[L(\mathcal{G})]_{ii}=\sum_{j=1}^{N}w_{ij}$,
and the off-diagonal elements of the Laplacain matrx are given as
$[L(\mathcal{G})]_{ij}=-w_{ij}$ for $i\neq j$.

To achieve formation tracking for the mobile robot network, the graph
$\mathcal{G}$ should be connected so that the robots can obtain the inter-
robot displacement. The Laplacian matrix of an undirected graph $\mathcal{G}$
exhibits the following property:

######  Property 1.

For an undirected graph $\mathcal{G}$, the eigenvalues of its Laplacian
matrix, $\lambda_{i}$ $(i=1,\ldots,N)$, are real and can be ordered such that
$0=\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{N}$. Additionally, if
$\mathcal{G}$ is connected, then $\lambda_{2}$ is positive and called as the
algebraic connectivity of the graph.

Given a graph $\mathcal{G}(\mathcal{W})$, let us denote $\mathcal{N}_{i}$ as
the set of the neighbors of the $i^{th}$ robot, which has a direct edge to the
$j^{th}$ robot for $j\in\mathcal{N}_{i}$. Thus, the $i^{th}$ robot is able to
obtain the displacement to $j\in\mathcal{N}_{i}$ by using the proximity
sensors on the $i^{th}$ robot. By denoting the displacement vector
$r_{ji}(t)=x_{i}(t)-x_{j}(t)$, the distance between the $i^{th}$ and $j^{th}$
robots are given as $d_{ji}(t)=\|r_{ji}(t)\|$, where $\|\cdot\|$ denotes the
Euclidean norm of the enclosed vector. Since the local displacement
measurement is mutual, the sensory topology $\mathcal{G}(\mathcal{W})$ being
undirected leads to $r_{ji}(t)=-r_{ij}(t)$ and $d_{ji}(t)=d_{ij}(t)$. The
formation of the mobile robot network is constructed by utilizing $r_{ji}$
according to the inter-agent measurement topologies without data
exchangeaaaThe argument of time dependent signals is omitted, for example
$r_{ji}\equiv r_{ji}(t)$, unless otherwise required for the sake of clarity.
Thus, we have the next assumption for the proposed networked system:

######  Assumption 2.

There is no inter-agent communication or data exchange in the robot network.

###  II-B Global and Local Formation Tracking

The time-varying global trajectory of the desired formation with respect to
$\Sigma_{W}$, the world coordinates, is predefined and denoted by
$h^{d}(t):\left[0,\infty\right)\rightarrow\mathcal{Q}$, which is a twice
differentiable continuous function of time. As illustrated in Fig. 1, the
desired formation of the mobile robots are described by
$\bar{h}_{i}^{d}(t):\left[0,\infty\right)\rightarrow\mathcal{Q}$, which are
time-varying continuous vectors, with respect to $h^{d}(t)$. Subsequently, the
time-varying desired trajectories for each of the mobile robots,
$h_{i}^{d}(t)\in\mathcal{Q}$, are given as
$h_{i}^{d}(t)=h^{d}(t)+\bar{h}_{i}^{d}(t)$, $i=1,\ldots,N$. For the mobile
agents to achieve time-varying formation, the desired relative displacement
between the $i^{th}$ robot and its neighbors, $j\in\mathcal{N}_{i}$, are
expressed by $h_{ji}^{d}(t)=\bar{h}_{i}^{d}(t)-\bar{h}_{j}^{d}(t)$, which
further leads to $h_{ji}^{d}(t)=h_{i}^{d}(t)-h_{j}^{d}(t)$ and
$h_{ij}^{d}(t)=-h_{ji}^{d}(t)$.

Let $\mathcal{X}_{0}$ be a subset of the compact set $\mathcal{Q}$ such that
$\mathcal{X}_{0}\subseteq\mathcal{Q}$ is closed and bounded. The time-varying
formation tracking can be defined for local formation tracking and global
formation tracking, respectively.

######  Definition 1.

(Local Formation Tracking) The networked robot system (1) is said to achieve
local formation tracking if for any given initial states
$x_{i}(0)\in\mathcal{X}_{0}$, $i=1,\ldots,N$, the states satisfy that
$\lim_{t\rightarrow\infty}\left[(x_{i}(t)-x_{j}(t))-(\bar{h}_{i}^{d}(t)-\bar{h}_{j}^{d}(t))\right]=0,$
for all $j\in\mathcal{N}_{i}$, which implies that
$\lim_{t\rightarrow\infty}r_{ji}(t)=\lim_{t\rightarrow\infty}h_{ji}^{d}(t).$

######  Definition 2.

(Global Formation Tracking) The networked robot system (1) is said to achieve
global formation tracking if for any given initial states
$x_{i}(0)\in\mathcal{X}_{0}$, $i=1,\ldots,N$, the states satisfy that
$\lim_{t\rightarrow\infty}\left(x_{i}(t)-h_{i}^{d}(t)\right)=0.$

The problem can be considered as a hierarchical framework consisting of the
local and global formation tracking systems. Thus, in the proposed mobile
robot network, global formation tracking (Definition 2) is the sufficient
condition for local formation tracking, and the local formation tracking
(Definition 1) is necessary for global formation tracking.

###  II-C Problem Formulation

The time-varying formation tracking for a mobile robot network is studied in
this paper, as illustrated in Figure 1. Based on the dynamics (1), the $N$
mobile agents are controlled to maintain a time-varying formation based on the
global positioning and local relative displacement measurements. The global
positioning measurements can be obtained from the GPS (global positioning
system) outdoors or other indoor/outdoor positioning and tracking systems. It
is noted that we use the term ‘GPS’ generally to refer to any global
positioning systems, e.g. location based service in mobile devices, WiFi
positioning system, or optical localization system.In the proposed system,
each robot can obtain global positioning data for the global formation
tracking as addressed in Definition 2.

The local measurement is implemented by extrinsic sensors on each of the
mobile agents to obtain the relative displacement to all its neighbors.
Moreover, the relative velocity between two adjacent robots can also be
obtained accordingly. The local positioning information is utilized to control
the robot network for achieving local formation tracking as stated in
Definition 1. Since there is no communication between the mobile agents, the
desired trajectories of an agent’s neighbors are important in achieving the
time-varying formation tracking. Thus, we have the next assumption:

######  Assumption 3.

The desired formation trajectories $h^{d}(t)$, $h_{i}^{d}(t)$, and
$h_{j}^{d}(t)$ for $j\in\mathcal{N}_{i}$ are available to the $i^{th}$ robot.

We consider the situation that the signals transmitted from the global
positioning systems to the mobile robots may go under deception attacks[35,
21, 36]. Let us define the position of the $i^{th}$ robot from the global
positioning system as $x_{i}^{g}(t)$, which might be different from the actual
position $x_{i}(t)$ depending on condition of the positioning system. If the
global positioning signal is accurate, we have $x_{i}^{g}(t)=x_{i}(t)$.
However, if the global positioning signal is under deception attacks, then the
erroneous position data $\hat{x}_{i}^{g}(t)$ is described as

$\displaystyle\hat{x}_{i}^{g}(t)=\delta_{xi}(t)x_{i}(t)+\Delta_{xi}(t),$ (2)

where $\delta_{xi}(t)\in\mathbb{R}$ is a time-varying function, and
$\Delta_{xi}(t)\in\mathbb{R}^{n}$ is a continuous function.

##  III Time-varying Formation Tracking

###  III-A Formation Tracking Control Design and Analysis

For dynamically controlled mobile agents, let us consider the time-varying
formation tracking control

$\displaystyle
u_{i}(t)=\ddot{h}_{i}^{d}-\sigma_{fi}(\dot{x}_{i}-\dot{h}_{i}^{d})-\kappa_{gi}f^{g}_{i}(t)-\kappa_{f}f_{i}^{l}(t),$
(3)

where $\sigma_{fi}{\in\mathbb{R}^{n\times n}}$ is a positive-definite matrix,
$\kappa_{f},\kappa_{gi}\in\mathbb{R}$ are positive gains for local and global
formation tracking, respectively, and

$\displaystyle
f_{i}^{l}(t)=\sum_{j\in\mathcal{N}_{i}}w_{ji}(\dot{r}_{ji}-\dot{h}_{ji}^{d})+\sigma_{fi}\sum_{j\in\mathcal{N}_{i}}w_{ji}\left(r_{ji}-h_{ji}^{d}\right),$
(4) $\displaystyle
f_{i}^{g}(t)=(\dot{x}_{i}-\dot{h}_{i}^{d})+\sigma_{fi}\left(x^{g}_{i}-h^{d}_{i}\right)$
(5)

are the formation tracking control commands. In the control input (3), the
relative displacement $r_{ji}$ and velocity $\dot{r}_{ji}$ are obtained by the
sensors mounted on each of the mobile robots as mentioned in Section II. The
velocity of the mobile robot $\dot{x}_{i}$ with respect to $\Sigma_{W}$ can be
obtained from inertial navigation systems, e.g. an inertial measurement unit
(IMU), or an odometry system such as visual odometry. It is noted that the
global positioning data, $x_{i}^{g}(t)$, are required in formation tracking
because the absolute position obtained from the integration of the velocity
data are not reliable or accurate enough to be implemented in the formation
control due to the accumulative drifting errors.

######  Assumption 4.

All robots start from initial positions that are known or measurable by its
neighbors.

By denoting $\tilde{x}_{i}:=x_{i}-h^{d}_{i}$ as the global tracking error and
$e_{i}=\sum_{j\in\mathcal{N}_{i}}w_{ji}(\tilde{x}_{i}-\tilde{x}_{j})$ as the
accumulated inter-agent errors with respect to the weighted Laplacian
$L(\mathcal{G})$, we have
$r_{ji}-h_{ji}^{d}=\left(x_{i}-x_{j}\right)-(h_{i}^{d}-h_{j}^{d})={\tilde{x}}_{i}-{\tilde{x}}_{j}$.
If the position of the mobile agents in the robot network can be obtained
accurately such that $x_{i}^{g}(t)=x_{i}(t)$, the closed-loop dynamics is
given as
$\ddot{\tilde{x}}_{i}+\kappa_{gi}\dot{\tilde{x}}_{i}+\kappa_{f}\dot{e}_{i}=-\sigma_{fi}\dot{\tilde{x}}_{i}-\sigma_{fi}\kappa_{gi}\tilde{x}_{i}-\sigma_{fi}\kappa_{f}e_{i}.$
Given $\xi_{i}=\dot{\tilde{x}}_{i}+\kappa_{gi}\tilde{x}_{i}+\kappa_{f}e_{i}$,
the above equation becomes $\dot{\xi}_{i}=-\sigma_{fi}\xi_{i}.$ Hence, under
Assumptions 1 through 4, we have the following result for the mobile robot
network with perfect global positioning information.

######  Theorem 1.

For the mobile robot network described by (1) under an undirected and
connected displacement sensory graph $\mathcal{G}$ with weighted Laplacian
matrix $L(\mathcal{G})$, if the global positioning data are accurate such that
$x_{i}^{g}(t)=x_{i}(t)$ for $i=1,\ldots,N$, then the control input (3)
guarantees that the mobile robot network achieves global formation tracking.

A proof of Theorem 1 is provided in AppendixA. This theorem demonstrates that
guarantee of the global formation tracking (Definition 2) can also ensure
local formation tracking (Definition 1) as addressed in Section II. The
tracking performance of the local and global formation depends heavily on the
values of $\kappa_{f}$ and $\kappa_{gi}$ in (A). Let us first consider the
case where there is only global formation tracking such that $\kappa_{f}=0$,
then the closed-loop dynamics becomes
$\ddot{\tilde{x}}_{i}+\kappa_{gi}\dot{\tilde{x}}_{i}=-\sigma_{fi}\dot{\tilde{x}}_{i}-\sigma_{fi}\kappa_{gi}\tilde{x}_{i}$.
By denoting $\xi_{i}^{g}=\dot{\tilde{x}}_{i}+\kappa_{gi}\tilde{x}_{i}$, the
given closed-loop system becomes $\dot{\xi}_{i}^{g}=-\sigma_{fi}\xi_{i}^{g}$.
It is noted that there is no inter-agent term in the closed-loop dynamics
because this case considers only global positioning signals; thus, the mobile
robot network becomes a decoupled system for trajectory tracking. Hence, by
considering
$V_{i}^{g}(\xi_{i}^{g})=\frac{1}{2}(\xi_{i}^{g})^{T}\sigma_{fi}^{-1}\xi_{i}^{g}$,
we have $\dot{V}_{i}^{g}=-(\xi_{i}^{g})^{T}\xi_{i}^{g}$, which is negative
definite. Therefore, we conclude that
$\lim_{t\rightarrow\infty}\xi_{i}^{g}(t)=0$ by following the proof of Theorem
1. From the closed-loop control system, we further have
$\dot{\tilde{x}}_{i}=-\kappa_{gi}\tilde{x}_{i}+\xi_{i}^{g}$. As shown in [37,
38], if $\xi_{i}^{g}(t)$ is a signal that asymptotically converges to zero,
and $\tilde{x}_{i}$ is bounded, then
$\lim_{t\rightarrow\infty}\tilde{x}_{i}(t)=0$, which implies that
$\lim_{t\rightarrow\infty}\left(x_{i}(t)-h_{i}^{d}(t)\right)=0$,
$i=1,\ldots,N$. Consequently, the mobile robot network achieves global
formation tracking as stated in Definition 2.

Next, let us consider the case of formation tracking with only local
measurements such that $\kappa_{gi}=0$. For the mobile robot network with an
undirected and connected displacement sensory graph $\mathcal{G}$, the closed-
loop dynamics is described as
$\ddot{\tilde{x}}_{i}+\kappa_{f}\dot{e}_{i}=-\sigma_{fi}\dot{\tilde{x}}_{i}-\sigma_{fi}\kappa_{f}e_{i}$.
From $\xi_{i}^{l}=\dot{\tilde{x}}_{i}+\kappa_{f}e_{i}$, the stacked dynamics
$\dot{\xi}_{i}^{l}=-\sigma_{fi}\xi_{i}^{l}$ can be proved to be asymptotically
stable by considering
$V^{l}(\xi_{i}^{l})=\frac{1}{2}(\xi^{l})^{T}\sigma_{fi}^{-1}\xi^{l}$, where
$\xi^{l}$ is the stacked vector of $\xi_{i}^{l}$. With
$\dot{V}^{l}=-(\xi^{l})^{T}\xi^{l}$, we obtain that
$\xi^{l}\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}$ and
$\lim_{t\rightarrow\infty}\xi^{l}(t)=0$. By following (A) in the proof of
Theorem 1, the stacked form is given as
$\dot{\tilde{x}}=-[(\kappa_{f}L)\otimes I_{n}]\tilde{x}+\xi^{l}$ with
$\kappa_{gi}=0$. For $\xi^{l}(t)$ converging to the origin, the mobile robot
network achieves consensus to the agreement set that is the subspace of
$span\\{1\\}$ such that $\tilde{x}_{i}(t)=\tilde{x}_{j}(t)$, $i=1,\ldots,N$,
$j\in\mathcal{N}_{i}$ when $t\rightarrow\infty$ [34]. Therefore, from the
definition of $\tilde{x}_{i}(t)$, we obtain that
$\lim_{t\rightarrow\infty}r_{ji}(t)=\lim_{t\rightarrow\infty}h_{ji}^{d}(t)$ so
that the local formation tracking is guaranteed.

###  III-B Malicious Attacks on Global Positioning Signal

By following the analysis in Section III-A, we obtain that the local formation
tracking can be ensured with or without global positioning information. In the
proposed system, the resilience of the networked robot system is guaranteed
based on the inter-agent position information. If the formation tracking
errors are abnormally large due to initial conditions, erroneous judgement
could occur and significantly influence the resilient efficacy. Therefore, in
this system, we consider the following assumption:

######  Assumption 5.

The mobile robot network has achieved asymptotic stability before any agents
are subject to adversarial attacks on global positioning signals such that
attacks occur at $t=t_{ai}\geq 0$ and $q_{i}\in\mathcal{Q}_{as}$, $\forall
i=1,\ldots,N$, where
$\mathcal{Q}_{as}=\\{q_{i}:=(\tilde{x}_{i}^{T},\dot{\tilde{x}}_{i}^{T})^{T}~{}|~{}\tilde{x}_{i}=\dot{\tilde{x}}_{i}=0\\}$.

The influence on the system due to malicious attack depends on its features
[36]. For a deep understanding of the deception attack defined in (2), it is
classified into three types, named additive attack, hybrid attack, and
unstable attack, to investigate the effect of each parameter in (2). The
deception attack with $\delta_{xi}=1$ is an additive attack; otherwise it’s a
hybrid attack. Compared with additive attack, hybrid attack is more related to
original data $x_{i}$. Additionally, unstable attack is defined as
$\hat{x}_{i}^{g}(t)=x_{i}(t)-c_{ai}\xi_{i}(t),$ where $c_{ai}$ is a
coefficient of the attack model. By choosing the same Lyapunov function
candidate in Theorem 1, it is derived as
$\dot{V}_{i}=(\kappa_{gi}c_{ai}-1)\xi_{i}^{T}\xi_{i}$. Obviously, if
$\kappa_{gi}c_{ai}>1$, $\dot{V}_{i}$ is positive definite, which means the
system is unstable.

###  III-C Performance Index

In this section, we design a global performance index to quantify the tracking
performance of the mobile robot network and evaluate the system’s performance
under various global positioning attacks through simulation. It is noted that,
to the best of the authors’ knowledge, this is the first performance index
proposed to mobile robot network on time-varying formation tracking.

The design of the performance index depends on both local and global formation
tracking. The global formation tracking errors for the $i^{th}$ agent has been
defined previously as $\tilde{x}_{i}=x_{i}-h_{i}^{d}$. For local tracking, we
define $e_{i}^{1}=\sum_{j\in\mathcal{N}_{i}}(\tilde{x}_{i}-\tilde{x}_{j})$ as
the summation of local formation tracking error with a unity weightbbbThe
design of $e_{i}^{1}$ is similar to the inter-agent errors $e_{i}$ defined in
Section III and multi-agent consensus, but this term, without the weights for
the graph Laplacian, is considered in the evaluation of the local tracking
performance.. Furthermore, the average of local formation errors is given as
$\bar{e}_{i}^{1}=e_{i}^{1}/n_{i}$, where $n_{i}$ is the number of the
neighbors of the $i^{th}$ mobile agent. Hence, the performance index is
defined as

$\displaystyle\mathcal{I}_{f}^{g}=\left(\vartheta+\sum_{i=1}^{N}\|\bar{e}_{i}^{1}\|\right)\\!/\\!\left(\vartheta+\sum_{i=1}^{N}\|\bar{e}_{i}^{1}\|+\alpha_{f}^{g}\sum_{i=1}^{N}\|\tilde{x}_{i}\|\right)\\!\\!,$
(6)

where $\vartheta$ and $\alpha_{f}^{g}$ are positive constants. It is noted
that $\mathcal{I}_{f}^{g}$ is defined globally for the formation tracking
control.

Based on $\mathcal{I}_{f}^{g}$, if the mobile robot network achieves global
formation tracking, then the local formation tracking is also guaranteed;
therefore, $\tilde{x}_{i}$ and $\bar{e}_{i}^{1}$ all converging to zero gives
that $\mathcal{I}_{f}^{g}=1$. If the mobile robots only achieve local
formation tracking but fail to track the global formation, then
$\bar{e}_{i}^{1}$ converges to zero but $\tilde{x}_{i}$ could be non-zero.
Thus, $\mathcal{I}_{f}^{g}$ is less than one, and its actual value depends on
the tracking errors of the global formation. If the system becomes unstable,
then $\tilde{x}_{i}$ diverges so that $\mathcal{I}_{f}^{g}$ goes to zero.
Hence, the index $\mathcal{I}_{f}^{g}$ tells the possibility of the mobile
robots being under deception attack on the global positioning data.

Figure 2: Sensory topology of the mobile agents in the networked robotic
system considered in the simulation analysis.

###  III-D Numerical Examples - Formation Tracking and Attacks

We consider a mobile robot network composed of six agents under the sensory
topology illustrated in Fig. 2. The control algorithm (3) is utilized to
control the mobile robot network to form a hexagon, rectangle, and triangle at
$t=0\sim 15$ sec, $t=15\sim 30$ sec, and after $t=45$ sec, respectively.
Moreover, the formation is considered to track a lemniscate time-varying
trajectory described by $h^{d}(t)=\big{[}-5\sin(2\pi t/15)/\big{(}2(\cos(2\pi
t/15)-3)\big{)},-12\cos(\pi t/15)/\big{(}5(\cos(2\pi
t/15)-3)\big{)}\big{]}^{T}$. The control gains are selected as $\kappa_{f}=2$,
$\sigma_{fi}=I_{2}$, $\kappa_{gi}=2$ with an identical sensory weights of
$w_{ji}=1$, where $I_{n}\in\mathbb{R}^{n\times n}$ denotes an identity matrix.
The performance index (6) is utilized to evaluate the formation tracking in
the following simulation with $\vartheta=10$ and $\alpha_{f}^{g}=5$.

(a)

(b)

Figure 3: Performance index ($\mathcal{I}_{f}^{g}$) and trajectory snapshots
of the mobile robot network (a) without (b) with global positioning attack.
The vertical dashed lines in the sub-figure for $\mathcal{I}_{f}^{g}$
represent the time of snapshots in the other three sub-figures.

For the mobile robot network that is not under positioning attack, the agents’
trajectories and performance index are shown in Fig. 3a. With the unaffected
global positioning signals and local displacement measurement, the mobile
agents are able to stably keep a time-varying formation while tracking a
trajectory. The performance index converges to one if the system is
asymptotically stable with guaranteed local and global formation tracking.
Next, we consider the case where Agent 1 is under hybrid attack
($\delta_{x1}=2$ and $\Delta_{x1}=[-2,-2]^{T}$) starting at $t_{a1}=15$ sec,
which is after the networked robotic system achieving asymptotic stability as
stated in Assumption 5. The results in Fig. 3b show that the hybrid attack
affected the motion of Agent 1, which is marked in blue, and the adjacent
agents of Agent 1 also deviated from their positions in the desired local
formation. Due to malicious positioning attack, the performance index
decreased far from one after $t=t_{a1}$, the starting time of the attacks.

To compare the evolution of the performance index with respect to different
types of attacks designed in Section III-B, we have conducted various analyses
shown in Fig. 4. If the additive attack is with a constant bias signal, then
the performance index decreases to a smaller value without time-dependent
variation; however, under the hybrid attack, resulting from the multiplicative
type $\delta_{xi}$, the variation of the performance index is dependent on the
formation and desired trajectory. From the case where both Agent 1 and Agent 4
were under attack, we can observed that multi-agent attacks would degrade the
performance index. Moreover, the unstable attack, due to the unstable term
$-c_{ai}\xi_{i}$, would cause oscillations in the performance index, which
results from the constant switching between the local and global formation
tracking control.

Figure 4: Performance indices of mobile robot network subjected to different
malicious attacks on global positioning. Left: Only Agent 1 is under attack at
$t_{a1}=15$ sec under either an hybrid attack (as in Fig. 3b) or additive
attack. Right: Agent 1 is under additive attack at $t_{a1}=15$ sec, and
subsequently Agent 4 is under either hybrid or unstable attacks at $t_{a4}=25$
sec. ($\delta_{x1}=\delta_{x4}=2$, $\Delta_{x1}=\Delta_{x4}=[-2,-2]^{T}$, and
$c_{a4}=5$)

##  IV Cooperative Localization and Resilient Operation

The main focus here is to improve the localization resilience toward
potentially faulty global positioning information due to system-originated
errors or malicious attacks. For the rest of the paper, we generally refer to
the external positioning system that is subject to attacks as GPS. By default,
the global tracking of the robot network relies on the localization
information from the GPS. Since inter-agent observations are available, we
introduce a cooperative localization scheme as the secondary localization
approach in the case of GPS failures or attacks. In this section, we first
present a communication-free cooperative localization scheme based on the EIF
and then discuss the self-identification and exclusion of GPS attacks.

###  IV-A Positioning Recovery

Localization resilience is achieved through redundancy, i.e., a sensor fusion
mechanism consisting of proprioceptive sensor measurements, GPS measurements,
and inter-agent relative measurements with respect to neighboring agents. It
is assumed that only the GPS measurements may be subject to attacks. The EIF
was chosen as the information fusion method in favor of its efficiency in
incorporating multi-modal sensor measurements in multi-step filter updates.
Here we only include the details necessary to understand the propose
positioning recovery method. For a comprehensive treatment of EIF, the readers
are referred to [39].

For applications in a three-dimensional space, the state vector for the
$i^{th}$ agent at time $k$ consists of the location of the agent defined as
$x_{i}(k):=[x(k);y(k);z(k)]^{T}\in\mathbb{R}^{3}$. The agent’s motion model
can be found as

$\displaystyle x_{i}(k)=x_{i}(k-1)+[\dot{x}_{i}(k)+\nu_{i}(k)]\Delta t,$ (7)

where $\dot{x}_{i}(k)\in\mathbb{R}^{3}$ denotes the agent velocity that can be
measured directly, and $\nu_{i}(k)\sim\mathcal{N}\big{(}0,Q(k)\big{)}$ is the
random process noise that follows a zero-mean Gaussian distribution with
covariance matrix $Q(k)\in\mathbb{R}^{3\times 3}$. The state vector prediction
at time $k$, $\hat{x}_{i}(k|k-1)$, can be calculated as

$\hat{x}_{i}(k|k-1)=\hat{x}_{i}(k-1|k-1)+\dot{x}_{i}(k)\Delta t.$ (8)

The covariance of the location estimate for the $i^{th}$ agent,
$P_{i}\in\mathbb{R}^{3\times 3}$, is propagated accordingly as

$\displaystyle P_{i}(k|k-1)=F_{i}(k)P_{i}(k-1|$ $\displaystyle
k-1)F_{i}(k)^{T}$ $\displaystyle+G_{i}(k)Q(k)G_{i}(k)^{T},$ (9)

where $F_{i}(k)$ and $G_{i}(k)$ are the Jacobian matrices calculated based on
the state estimate from (8). Here, $F_{i}(k)=I_{3}$ and $G_{i}(k)=\Delta
t\cdot I_{3}$. The information matrix, $\Phi_{i}\in\mathbb{R}^{3\times 3}$,
and information vector, $\varphi_{i}\in\mathbb{R}^{3}$, can be found as
$\Phi_{i}(k|k-1)=P_{i}(k|k-1)^{-1}$ and
$\varphi_{i}(k|k-1)=\Phi_{i}(k|k-1)\,\hat{x}_{i}(k|k-1)$, respectively.

During normal operations, each agent updates its location estimate with the
GPS measurement. In the event that the GPS measurement is deemed unreliable or
faulty, the agent performs location update through cooperative localization
instead. The measurement models for these two types of updates are

$\displaystyle s^{\text{GPS}}_{i}(k)$
$\displaystyle=x_{i}(k)+\omega^{\text{GPS}}_{i}(k),$ (10) $\displaystyle
s^{r_{ij}}_{i}(k)$ $\displaystyle=x_{j}(k)-x_{i}(k)+\omega^{r_{ij}}_{i}(k),$
(11)

where $\omega^{\text{GPS}}_{i}(k)$ and $\omega^{r_{ij}}_{i}(k)$ are the random
noise signals associated with the GPS or relative position measurements, both
of which are assumed to follow zero-mean Gaussian distributions with
covariance matrices $R^{\text{GPS}}(k)$ and $R^{r_{ij}}(k)$, respectively, and
$j\in\mathcal{N}_{i}$ denotes the index of the neighboring agents. Note that
(11) does not require information to be sent from the neighbors because the
neighbors’ desired trajectories are known to others. For either update mode,
denoted by a superscript ${\square}\in\\{\text{GPS},\,r_{ij}\\}$, the updates
of the information matrix and vector follow the same EIF procedure such that

$\displaystyle\Phi_{i}(k|k)$
$\displaystyle=\Phi_{i}(k|k-1)+H^{\square}_{i}(k)^{T}R^{\square}_{i}(k)^{-1}H^{\square}_{i}(k),$
(12) $\displaystyle\varphi_{i}(k|k)$
$\displaystyle=\varphi_{i}(k|k-1)+H^{\square}_{i}(k)^{T}R^{\square}_{i}(k)^{-1}[s_{i}^{\square}(k)-\hat{s}_{i}^{\square}(k)$
$\displaystyle\hskip 10.84006pt+H^{\square}_{i}(k)\hat{x}_{i}(k|k-1)],$ (13)

where $H_{i}^{\square}$ is the Jacobian matrix calculated from either (10) or
(11) with $R^{\square}_{i}(k)$ being the covariance matrix of the
corresponding measurement. Here, $H^{\text{GPS}}_{i}=I_{3}$ and
$H^{r_{ij}}_{i}=-I_{3}$. The measurement predication,
$\hat{s}_{i}^{\square}(k)$, can be found based on either of the measurement
models (10) and (11) as

$\displaystyle\hat{s}^{\text{GPS}}_{i}(k)$ $\displaystyle=\hat{x}_{i}(k|k-1),$
(14) $\displaystyle\hat{s}^{r_{ij}}_{i}(k)$
$\displaystyle=h^{d}_{j}(k)-\hat{x}_{i}(k|k-1).$ (15)

Using the updated information matrix and vector, the state estimation vector
and covariance matrix can be recovered as

$\displaystyle\hat{x}_{i}(k|k)$
$\displaystyle=\Phi_{i}(k|k)^{-1}\varphi_{i}(k|k),$ (16) $\displaystyle
P_{i}(k|k)$ $\displaystyle=\Phi_{i}(k|k)^{-1}.$ (17)

######  Theorem 2.

If the control input (3) guarantees that the mobile robot network achieves
global formation tracking as stated in Definition 2, the expectation of the
position estimation errors, $\tilde{x}_{i}(k|k):=x_{i}(k)-\hat{x}_{i}(k|k)$,
$i=1,\cdots,N$, strictly decreases after the update with inter-agent
measurements using measurement model (11).

A proof of Theorem 2 is provided in the AppendixB. This theorem provides a
theoretical guarantee that, when global tracking is achieved asymptotically,
the positioning error will decrease through inter-agent measurement updates
using only the desired positions of the neighbors. When the robot network
remains connected and as least one of the agent tracks its desired trajectory
faithfully, converging localization performance across the entire agent
network can be achieved.

###  IV-B Attack Detection and Isolation

With the communication-free cooperative localization as the secondary
positioning approach in the event of GPS attacks, proper detection and
isolation of attacks are necessary. Several fault detection and isolation
solutions have been proposed [40]. We favor a residual-based method that
utilizes the results of EIF-based sensor fusion as introduced previously. More
specifically, the detection of the GPS attack is achieved by comparing the
residual at the EIF update step with new GPS measurements. This is realized
through comparing the Kullback-Leibler (KL) divergence between the state
estimate distributions before and after the update against a predefined
threshold.

The KL divergence is a non-symmetric measure that quantifies the distance
between two probability distributions. For two probability density functions,
$p(x)$ and $q(x)$, defined on the same probability space, the KL divergence is
defined as

$D_{KL}(p\,\|\,q)=\int^{+\infty}_{-\infty}p(x)\log\left(\dfrac{p(x)}{q(x)}\right)\text{d}x.$
(18)

In the case where both $p$ and $q$ are multivariate Gaussian with means
$\mu_{p}$, $\mu_{q}$ and covariance matrices $\Sigma_{p}$, $\Sigma_{q}$,
respectively, the KL divergence can be found as

$\displaystyle D_{KL}(\mathcal{N}_{p}\,\|\,\mathcal{N}_{q})$
$\displaystyle=\dfrac{1}{2}\Bigg{[}(\mu_{q}-\mu_{p})^{T}\Sigma^{-1}_{q}(\mu_{q}-\mu_{p})$
$\displaystyle\hskip
7.22743pt\;+\text{tr}(\Sigma_{q}^{-1}\Sigma_{p})-n+\ln\left(\dfrac{\det(\Sigma_{q})}{\det(\Sigma_{p})}\right)\Bigg{]},$
(19)

where $n$ is the dimension of $x$.

Before each GPS measurement is used in the update step, the KL divergence
between the following two probability density functions is calculated:

$\displaystyle p_{1}$
$\displaystyle=p\big{(}\hat{x}_{i}(k|k-1)\,|\,\hat{x}_{i}(k-1|k-1),\dot{x}_{i}(k)\big{)},$
(20) $\displaystyle p_{2}$
$\displaystyle=p\big{(}\hat{x}_{i}(k|k)\,|\,s^{\text{GPS}}_{i}(k),\dot{x}_{i}(k)\big{)}.$
(21)

Since both probability density functions are Gaussian distributions, the KL
divergence at time $k$ can be calculated as

$\displaystyle D^{k}_{KL}(p_{1}\,\|\,p_{2})$ (22)
$\displaystyle=\dfrac{1}{2}\Bigg{\\{}\big{[}\hat{x}(k|k)-\hat{x}(k|k-1)\big{]}\Phi(k|k)\big{[}\hat{x}(k|k)-\hat{x}(k|k-1)\big{]}^{T}$
$\displaystyle+\text{tr}\left(\Phi(k|k)\cdot\Phi_{i}(k|k-1)^{-1}\right)+\ln\left(\dfrac{\det\left(\Phi(k|k-1)\right)}{\det\left(\Phi(k|k)\right)}\right)-n\Bigg{\\}}.$

A GPS measurement is detected as a faulty signal when the value of
$D^{k}_{KL}(p_{1}\,\|\,p_{2})$ is above a predefined threshold $\chi$. In
addition, a quality measure is defined for the global positioning information
to inform the formation control:

$\beta_{i}(k)=1-\text{sat}\left(\dfrac{D^{k}_{KL}(p_{1}\,\|\,p_{2})}{\chi}\right),$
(23)

where $\text{sat}:\mathbb{R}\rightarrow\mathbb{R}$ is the standard saturation
function defined as $\text{sat}(x):=\text{sign}(x)\min(1,|x|)$.

Algorithm 1 summarizes the resilient state estimator with GPS attack detection
and isolation using the criterion based on the KL divergence. By default, the
agents prioritize the global positioning information for localization. In the
event of a global positioning failure or attack, the resilient state estimator
switches to cooperative localization. This switch occurs instantaneously such
that the positioning of the $i^{th}$ agent is not affected by the failure of
the global positioning system even if one or multiple neighbors are
experiencing similar failures as long as their global tracking is faithful.

Algorithm 1 Resilient State Estimator for the $i^{th}$ Agent

1:$\hat{x}_{i}(k-1|k-1)$, $P_{i}(k-1\,|\,k-1)$, $\dot{x}_{i}(k-1)$,
$s^{\text{GPS}}_{i}(k)$, $\\{s^{r_{ij}}_{i}(k)\,|\,j\in\mathcal{N}_{i}\\}$,
$\chi$

2:Predict $\hat{x}_{i}(k|k-1)$ and $P_{i}(k\,|\,k-1)$ $\triangleright$ (7)–(9)

3:Try $\hat{x}_{i}(k|k)$ and $P_{i}(k\,|\,k)$ with $s^{\text{GPS}}_{i}(k)$
- $\triangleright$ (10), (12)–(13), (17)–(16)

4:Compute KL divergence $D^{i,k}_{DL}$ $\triangleright$ (22)

5:Compute $\beta_{i}(k)$ $\triangleright$ (23)

6:if $D^{i,k}_{KL}<\chi$ then

7: Accept $\hat{x}_{i}(k|k)$ and $P_{i}(k\,|\,k)$

8:else

9: Reject $\hat{x}_{i}(k|k)$ and $P_{i}(k\,|\,k)$

10: for $j$ in $\mathcal{N}_{i}$ do

11: Compute $\hat{x}_{i}(k|k)$ and $P_{i}(k\,|\,k)$ with $s^{r_{ij}}_{i}(k)$
- $\triangleright$ (11), (12)–(13)

12:Return: $\hat{x}_{i}(k|k)$, $P_{i}(k\,|\,k)$, $\beta_{i}(k)$

###  IV-C Gain-Tuning Resilient Operation

In addition to positioning recovery from cooperative localization, the mobile
robot network in time-varying formation tracking is a redundant system with
both the local and global formation tracking. As mentioned in Section III-A,
if there is no global tracking for a portion of the mobile agents, the
formation tracking can still be guaranteed. Therefore, in this section, we
propose an autonomous gain-tuning technique to strengthen the resilience of
the mobile robot network by taking the advantage of redundancy.

For the proposed networked robot system under global positioning attacks, the
analyses in Section III-B demonstrate that the tracking gain of global
formation $\kappa_{gi}$ plays an important role in tracking stability. Even
though the networked system is under unstable attack, it would keep stable
with larger attack gains $c_{ai}$ wit very small $\kappa_{gi}$. Therefore, the
manipulation of $\kappa_{gi}$ with respect to the attack identification is a
useful resilient feature for the system.

The gain tuning can be designed as a function of $\beta_{i}\in[0,1]$, the
quality measure of the global positioning signals defined in Section IV-B. If
$\beta_{i}=1$, there is a higher confidence on the quality of the global
positioning signals; whereas for $\beta_{i}=0$, the $i^{th}$ agent is
considered under attack while the value of $D^{k}_{KL}(p_{1}\,\|\,p_{2})$ is
above the threshold $\chi$. Therefore, we can set

$\displaystyle\dot{\kappa}_{gi}=\gamma_{i}\tanh\big{(}\sigma_{\beta_{i}}(\beta_{i}-\chi_{\beta_{i}})\big{)}.$
(24)

where $\chi_{\beta_{i}}\in(0,1]$ is the triggering threshold and
$\sigma_{\beta_{i}}$ is the tuning gain for $\gamma_{i}$. It is noted that
$\beta_{i}$ encodes the confidence in the global positioning signals. This
design of resilience operation utilizes the GPS quality measure $\beta_{i}$
from cooperative localization, and therefore is called CL-based (cooperative-
localization-based) gain-tuning approach.

###  IV-D Numerical Examples - Recovery and Resilient Operation

Figure 5: Performance indices under global positioning attacks with the
proposed resilience operations. Agent 1 is under additive attack at
$t_{a1}=15\sim 30$ sec, and Agent 4 is under U-hybrid attack at $t_{a4}=25\sim
40$ sec with $\delta_{x1}=\delta_{x4}=2$,
$\Delta_{x1}=\Delta_{x4}=[-2,-2]^{T}$, and $c_{a4}=5$. Figure 6: KL divergence
of the mobile agents under global positioning attacks with the threshold
$\chi=5$.

The effect of the proposed resilient operations is verified through numerical
simulations. The sensory topology and control gains in the following
simulation are identical to Section III-D. The attack scenarios are satisfied
of Assumption 5 and similar to the case with multiple attacks in Fig. 4,
including the unstable attack, but both attacks are removed after 15 sec to
show the resilience and recovery during and after the attacks. It is noted
that the resilient state estimator is active from the start of the simulation,
and CL-based gain-tuning starts to regulate the global tracking gains
$\kappa_{gi}$ at $t=10$ sec. The performance indices of the resilient
operations are illustrated in Fig. 5.

####  IV-D1 CL Recovery Approach

The cooperative localization and resilient state estimator are implemented in
this section with $\chi=5$ to recover global positioning. From Fig. 5, it can
be seen that the CL recovery approach can maintain the performance index on a
constantly high level during the positioning attacks. It shows the
invulnerability of mobile robot network with resilient state estimator is
evident. The evolution of KL divergence is shown in Fig. 6, where $D_{KL}^{1}$
and $D_{KL}^{4}$ increase drastically to a relatively higher value when Agents
1 and 4 are under attacks. Therefore, it is beneficial to use the state
estimation results to inform the controller about the presence of malicious
attacks. Although the performance index is insensitive to attacks, it never
closely approaches one even when there is no attacks due to the inferior state
estimation accuracy than external positioning systems.

Figure 7: Evolution of the GPS quality measure $\beta_{i}$ of the mobile robot
network under malicious attacks. Figure 8: Evolution of the global tracking
gains by using the proposed CL-based gain-tuning approach.

####  IV-D2 CL-Based Gain-Tuning Approach

To enhance the resilience and recovery performance of mobile robot network
before, during, and after attacks, the CL recovery and gain-tuning approaches
are combined to form the CL-based gain-tuning approach. The GPS quality
measure (23), obtained from KL divergence, is utilized to design a mechanism
for regulating the global tracking gains $\kappa_{gi}$ in an adaptive fashion.
For $\sigma_{\beta_{i}}=3$ and $\chi_{\beta_{i}}=0.5$ with $\gamma_{i}=1$, the
simulation results are illustrated in Figs. 5, 7, and 8. It is noted that the
CL-based gain-tuning approach is applied for all agents starting at $t=10$ sec
after the system achieves asymptotic stability. Fig. 7 shows that the GPS
quality measure decreases when the agent becomes under attack. Therefore, the
GPS quality measure can be used to indicate the health condition of the global
positioning information. For $\beta_{i}$ less than $\chi_{\beta_{i}}$,
$\kappa_{gi}$ starts decreasing to zero so that the global positioning
information subjected to malicious attacks can be excluded from the control
system. At $t=25$ sec, $\kappa_{g4}$ also decreases to zero because of the
unstable attack on Agent 4 while other normal agents keep their own value.
Moreover, after the attacks are removed at $t=30$ sec and $t=40$ sec,
respectively for Agent 1 and Agent 4, $\kappa_{g1}$ and $\kappa_{g24}$ can be
recovered to the original value before attacks.

####  IV-D3 Restoration

During the simulation, the performance drop is an important indication for
resilience. We introduce the concept of restoration, $R_{s}$, as the reference
to evaluate the recovery performance. The restoration is defined by

$\displaystyle
R_{s}=\int_{t_{a}}^{t_{r}}\big{(}1-\mathcal{I}_{f}^{g}(\tau)\big{)}d\tau,$
(25)

where $\mathcal{I}_{f}^{g}$ is the performance index defined in (6), $t_{a}$
is the time when an attack occurs, and $t_{r}$ is the instant when the
performance index recovers to asymptotic stability that is
$\mathcal{I}_{f}^{g}=1$. The restoration can be considered as the size of the
resilience triangle for the proposed networked robot systems for performance
evaluations and comparisons. In addition to $R_{s}$, the time-to-recover,
$t_{r}-t_{a}$, and minimum $\mathcal{I}_{g}^{f}$ are also considered to
evaluate the resilience of the system.

##  V Experimental Results

In this section, we discuss the implementation of the proposed scheme on a
group of six quadrotors that are subject to global positioning attacks. It is
noted that our general approach is not limited to flying robots. The
experimental setup of the quadrotor network system is illustrated in Fig. 9.
Six Crazyflie 2.1, designed by Bitcraze, are used as the mobile agents in the
mobile robot network, and the system is built upon the Robot Operating System
(ROS) and its package “crazyswarm” [41]. The motion of agents is captured by a
motion capture (MOCAP) system consisted of IR cameras at 120 Hz, and then the
commands from the proposed design are sent to agents at 100 Hz asynchronously.

Figure 9: Experimental setup of the multi-quadrotor system for the
implementation of the resilient TVFC in the presence of positioning attacks.

###  V-A Scenario 1: Time-Varying Formation with Stationary Global Trajectory

To demonstrate the capability of the proposed resilient operations under
positioning attacks, the time-varying formation tracking with stationary
(time-invariant) global trajectory, $h^{d}=[0,0,0.9]^{T}$ m, is considered as
the first scenario. By considering a stationary global trajectory, the
influence from positioning attacks and recovery can be accessed separately
with time-invariant trajectories. Six mobile agents arrive at horizontal
formations of hexagon, rectangle, and triangle at $t=10$, $20$, and $30$ sec,
respectively, as shown in Fig. 10. The sensory topology of local positioning
measures is represented by the dashed lines in each of the formations in Fig.
10. Moreover, the vertical formations of the mobile agents are formed by
trajectories of $\bar{h}_{i}^{d}$ that are given as
$\bar{h}_{iz}^{d}(t)=0.15\sin(0.05\pi t+i\pi)$ m for $i=\\{1,\ldots,6\\}$.

TABLE I: Modified restoration ($\bar{R}_{s}^{\text{exp}}$) in experiments for the cases in Scenario 1 ($\int_{15}^{40}Nor(\mathcal{I}_{f}^{g}(\tau))d\tau=24.61$). Cases in Scenario 1 |  | Resilient Operation
---
$\bar{R}_{s}^{\text{exp}}$
Normal (attack-free) | - | $0\%$
Additive attack at $t_{a1}=15$ s | No | $23.78\%$
Additive attack at $t_{a1}=15$ s | Yes | $8.48\%$
|  Additive attack at $t_{a1}=15$ s
---
Unstable attack at $t_{a4}=20$ s
Yes | $8.99\%$
Figure 10: Desired formation (top) and performance indices (bottom) for the
time-varying formation tracking with stationary global trajectory with/without
global positioning attacks. For the cases with additive attack, Agent 1 is
subject to attack for $t\geq t_{a1}=15$ sec. For the case with additional
unstable attack, Agent 4 is under attack for $t\geq t_{a4}=20$ sec.

The tracking gains for the proposed TVFC are chosen as $\kappa_{gi}=0.8$,
$\kappa_{fi}=0.4$, $\sigma_{fi}=0.05I_{3}$, and $w_{ij}=1$. The performance
indices of the robot network under different attack settings are illustrated
in Fig. 10, where $\vartheta=20$ and $\alpha_{f}^{g}=3$ in
$\mathcal{I}_{f}^{g}$. With various positioning attacks, the quadrotors would
deviate from their desired positions and cause degradation of the performance
indices. In the absence of malicious attack, the performance index
$\mathcal{I}_{f}^{g}$ stably stays around $0.985$ providing satisfactory
formation tracking. When an additive attack is applied on Agent 1 with
$\Delta_{xi}=[-1.5,-2.5,-1.0]^{T}$ m at $t_{a1}=15$ sec, the performance
index, $\mathcal{I}_{f}^{g}$, degrades significantly without resilient
operation as the tracking error of the compromised agent propagates through
the network and causes its neighbors to drift away from their desired
trajectories. We then consider the same attack situation but now apply the
proposed CL-based gain-tuning resilient operation (24) with $\gamma_{i}=1$,
$\sigma_{\beta_{i}}=3$, and $\chi_{\beta_{i}}=0.5$. It can be seen from Fig.
10 that $\mathcal{I}_{f}^{g}$ would gradually increase and the performance
improves significantly comparing to last case. Here we chose
$Q=0.02^{2}I_{n}$, $R^{\text{GPS}}=(5\times 10^{-4})^{2}I_{n}$,
$R^{r_{ij}}=(5\times 10^{-4})^{2}I_{n}$, and $\chi=4000$ based on the typical
accuracy of the MOCAP system. Furthermore, under an additional unstable attack
on Agent 4 with $\delta_{x4}=1$ and $c_{a4}=5$ at $t_{a4}=20$ sec, the robot
team maintains a better performance, and $\mathcal{I}_{f}^{g}$ is larger than
$0.85$ and keeps increasing for better formation tracking.

To quantitatively assess the performance of the system in experiments, we
define a new measure modified from the restoration $R_{s}$ in (25) as

$\displaystyle\bar{R}_{s}^{\text{exp}}=\frac{\int_{t_{a}}^{t_{s}}\big{(}Nor(\mathcal{I}_{f}^{g})-Att(\mathcal{I}_{f}^{g})\big{)}d\tau}{\int_{t_{a}}^{t_{s}}Nor(\mathcal{I}_{f}^{g})d\tau},$
(26)

where $t_{s}$ is the terminal time of the experiment,
$Nor(\mathcal{I}_{f}^{g}(t))$ is the reference of the performance index
without attacks, and $Att(\mathcal{I}_{f}^{g}(t))$ is the performance index
under attacks with/without resilience. Since $Nor(\mathcal{I}_{f}^{g}(t))$ is
not equal to one, the modified restoration, $\bar{R}_{s}^{\text{exp}}$,
provides a way to compare the resilience performance of the system in
experiments to the cases without attacks. It can be seen as the degree of
performance degradation from the attack-free case, with $0\%$ representing no
degradation and $100\%$ being full degradation. Table I summarizes the
modified restoration of different cases in Scenario 1. Obviously, the proposed
resilient operation holds better better performance even if there is unstable
attack.

###  V-B Scenario 2: Time-Varying Formation and Global Trajectory

Figure 11: Snapshots of the time-varying formation and global trajectory
tracking in the absence of global positioning attacks.

To further investigate the effect of the proposed TVFC for mobile robot
networks and the resilient operation, two advanced cases are presented where
the mobile robot network tracks a predefined global trajectory. The quadrotor
team reaches hexagon, rectangle, and triangle horizontal formations at $t=5$,
20, and 35 sec, respectively. The vertical formation is the same as in the
previous case. Additionally, the global trajectory of the mobile agents is
defined as $h^{d}=[1.3\sin(2\pi t/15)/(\cos(2\pi t/15)-3),-1.2\cos(\pi
t/15)/(\cos(2\pi t/15)-3),0.2\cos(2\pi t/15)+0.7]^{T}$ m, which is a 3D
lemniscate trajectory with a time-varying $Z$-component. The control gains are
chosen as $\kappa_{gi}=0.4$, $\kappa_{fi}=0.4$, and $\sigma_{fi}=0.1I_{3}$.

The snapshots of the formation tracking without attacks are shown in Fig. 11,
where the dashed lines represent the local measurement network among the
mobile agents. The performance indices of the proposed control algorithm
tracking the global and local trajectories with and without global positioning
attacks are shown in Fig. 12. In the absence of attacks, the performance index
varies around $0.9$, indicating that the mobile robot network can closely
track the global trajectory and time-varying formation.

Figure 12: Performance indices of the experiments with healthy global
positioning, and under additive and unstable attacks with resilient operation.
For the case with attacks, Agent 1 is subject to additive attack for $t\geq
t_{a1}=15$ sec, and Agent 4 is under unstable attack for $t\geq t_{a4}=20$
sec.

When Agent 1 and Agent 4 are under additive and unstable attacks,
respectively, the performance index slightly degrades due to the attacks but
the tracking performance remains as satisfactory as the case without attack in
Fig. 12. The parameters used in cooperative localization are identical to the
previous section with the stationary global trajectory. This result implies
that the proposed control algorithm is capable of eliminating severe influence
of malicious attacks on agents to protect the systems from attacks and
recovery to attack-free performance. The evolution of $\beta_{i}$ and
$\kappa_{gi}$ is shown in Fig. 13, where the identification of attacks is via
$\beta_{i}$ and the corresponding global tracking gains decrease to eliminate
the adversarial effect on the mobile robot network.

Figure 13: Evolution of $\beta$ and $\kappa_{g}$ in the experimental case with
both additive attack on Agent 1 and unstable attack on Agent 4. Parameter
projection [42, 43] is utilized in (24) such that $0\leq\kappa_{gi}\leq 0.4$.

##  VI Conclusion

The persistency of maintaining a time-varying formation to track the reference
trajectory of a mobile robot network under global positioning attacks is
investigated in this paper. Based on the local and global positioning
information, control algorithms are presented and studied to ensure formation
tracking performance. Subsequently, Lyapunov-based stability analyses are
provided in the presence of three types of malicious attacks on the global
positioning signals that are additive attack, hybrid attack, and unstable
attack. Moreover, a performance index is presented to evaluate the efficiency
of mobile robot networks under time-varying formation tracking control
with/without compromised agents subjected to adverse impacts. Simulation
results and experiments validate the control system and the influence of
adversarial attacks on global positioning. Future work will address the
resilience in coordinating heterogeneous robotic systems, reaction to
sensory/actuator faults, and handling unreliable sensory topologies.

##  Appendix A Proof for Theorem 1

###### Proof.

Without loss of generality, let us consider the Lyapunov function candidate
for the $i^{th}$ robot in the the mobile robot network as
$V_{i}(\xi_{i})=\frac{1}{2}\xi_{i}^{T}\sigma_{fi}^{-1}\xi_{i}$. By taking the
time-derivative of $V_{i}$ along the trajectories of (III-A), we obtain
$\dot{V}_{i}=-\xi_{i}^{T}\xi_{i}$, which is negative definite. Since $V_{i}$
is positive definite and $\dot{V}_{i}$ is negative definite, we obtain
$V_{i}(\xi_{i}(t))\leq V_{i}(\xi_{i}(0))$ so that
$\xi_{i}\in\mathcal{L}_{\infty}$ because $\sigma_{fi}$ is a positive-definite
matrix. Next, by integrating $\dot{V}_{i}(t)$ with respect to time from $0$ to
$\infty$, we have
$V_{i}(t)-V_{i}(0)=-\int_{0}^{t}\xi_{i}^{T}(\tau)\xi_{i}(\tau)d(\tau)$, which
results in
$\int_{0}^{t}\xi_{i}^{T}(\tau)\xi_{i}(\tau)d(\tau)+V_{i}(t)=V_{i}(0)$. Since
$V_{i}(\xi_{i})$ is positive definite, we can further obtain that
$\int_{0}^{t}\xi_{i}^{T}(\tau)\xi_{i}(\tau)d(\tau)\leq V_{i}(0)<\infty$;
consequently, we have $\xi_{i}\in\mathcal{L}_{2}$. From the closed-loop
dynamics (III-A), we further obtain that
$\dot{\xi}_{i}\in\mathcal{L}_{\infty}$. Since
$\xi_{i}\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}$ and
$\dot{\xi}_{i}\in\mathcal{L}_{\infty}$, we get from Barbalat’s lemma [44] that
$\lim_{t\rightarrow\infty}\xi_{i}(t)=0$. Therefore, we conclude that $\xi_{i}$
is asymptotically stable.

Next, let us rewrite
$\xi_{i}=\dot{\tilde{x}}_{i}+\kappa_{gi}\tilde{x}_{i}+\kappa_{f}e_{i}$ as
$\dot{\tilde{x}}_{i}=-\kappa_{gi}\tilde{x}_{i}-\kappa_{f}e_{i}+\xi_{i}$, which
can be considered as the state equation of $\tilde{x}_{i}$ with $\xi_{i}$ as
the input. From the definition of $e_{i}$ with the weighted Laplacian matrix,
we have $e=[L\otimes I_{n}]\tilde{x}$, where
$e=[e_{1}^{T},\ldots,e_{N}^{T}]^{T}\in\mathbb{R}^{Nn}$ and
$\tilde{x}=[\tilde{x}_{1}^{T},\ldots,\tilde{x}_{N}^{T}]^{T}\in\mathbb{R}^{Nn}$.
By considering $\xi\in\mathbb{R}^{Nn}$ as the stacked vector of $\xi_{i}$ such
that $\xi=[\xi_{1}^{T},\xi_{2}^{T},\ldots,\xi_{N}^{T}]^{T}$, the
interconnected system becomes

$\displaystyle\dot{\tilde{x}}=$ $\displaystyle-D_{\kappa_{g}}\otimes
I_{n}\tilde{x}-\kappa_{f}[L\otimes I_{n}]\tilde{x}+\xi$ $\displaystyle=$
$\displaystyle-[(D_{\kappa_{g}}+\kappa_{f}L)\otimes I_{n}]\tilde{x}+\xi,$
(A.1)

where $D_{\kappa_{g}}\in\mathbb{R}^{N\times N}$ denotes a diagonal matrix with
the $i^{th}$ diagonal term as $\kappa_{g}$. Since $\kappa_{f}$, $\kappa_{gi}$
are all positive, we can show that $-(D_{\kappa_{g}}+\kappa_{f}L)$ is a
Hurwitz matrix [44] because $L(\mathcal{G})$ has an isolated eigenvalue of
zero and all others with a positive real part (Property 1). As $\xi$ converges
to the origin as time goes to infinity, the linear time-invariant (LTI) system
(A) with $\xi$ as the input is asymptotically stable. Consequently,
$\tilde{x},\dot{\tilde{x}}\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}$ so that
$\tilde{x}\rightarrow 0$ as $t\rightarrow\infty$. Furthermore, by observing
(A), the convergence of $\tilde{x}$ and $\xi$ to the origin lead to
$\dot{\tilde{x}}\rightarrow 0$ asymptotically. Consequently,
$\lim_{t\rightarrow\infty}\tilde{x}_{i}(t)=0$ and
$\lim_{t\rightarrow\infty}\dot{\tilde{x}}_{i}(t)=0$ demonstrate that
$\lim_{t\rightarrow\infty}\left(x_{i}(t)-h_{i}^{d}(t)\right)=0$ and
$\lim_{t\rightarrow\infty}\dot{x}_{i}(t)=\lim_{t\rightarrow\infty}\dot{h}_{i}^{d}(t)$.
The mobile robot network achieves global formation tracking.∎

##  Appendix B Proof for Theorem 2

###### Proof.

Based on measurement model (11), the measurement error, or innovation, of the
$i^{th}$ agent relative to the $j^{th}$ agent can be calculated as

$\displaystyle\tilde{s}^{r_{ij}}_{i}(k)=$ $\displaystyle
s^{r_{ij}}_{i}(k)-\hat{s}^{r_{ij}}_{i}(k)$ $\displaystyle=$ $\displaystyle
x_{j}(k)-x_{i}(k)+\omega^{r_{ij}}_{i}(k)-\left[h^{d}_{j}(k)-\hat{x}_{i}(k|k-1)\right]$
$\displaystyle=$
$\displaystyle\tilde{x}_{j}(k)-\tilde{x}_{i}(k|k-1)+\omega^{r_{ij}}_{i}(k).$
(B.1)

Recall that $\tilde{x}_{j}(k)$ denotes the global tracking error. Based on the
definition for information vector, $H^{r_{ij}}_{i}=-I_{3}$, and (13), the
state estimation error after update with inter-agent measurement can be
calculated as

$\displaystyle\tilde{x}_{i}(k|k)=$ $\displaystyle x_{i}(k)-\hat{x}_{i}(k|k)$
$\displaystyle=$ $\displaystyle x_{i}(k)-\Phi_{i}(k|k)^{-1}\varphi_{i}(k|k)$
$\displaystyle=$ $\displaystyle
x_{i}(k)-\Phi_{i}(k|k)^{-1}\Big{\\{}\varphi_{i}(k|k-1)$ $\displaystyle\qquad-
R_{i}(k)^{-1}\big{[}\tilde{s}_{i}^{r}(k)-\hat{x}_{i}(k|k-1)\big{]}\Big{\\}}.$
(B.2)

In this proof, we use a simplified notation for the inter-agent measurement
covariance matrix: $R_{i}(k)=R^{r_{ij}}_{i}(k)$. Left multiplying
$\Phi_{i}(k|k)$ on both sides of (B.2) and rearranging yield

$\displaystyle\Phi_{i}(k|k)\tilde{x}_{i}(k|k)$
$\displaystyle=\Phi_{i}(k|k-1)\left[x_{i}(k)-\hat{x}_{i}(k|k-1)\right]$
$\displaystyle\qquad\qquad\qquad\quad+R_{i}(k)^{-1}\left[\tilde{s}_{i}^{r_{ij}}(k)+x_{i}(k)-\hat{x}_{i}(k|k-1)\right]$
$\displaystyle=\Phi_{i}(k|k-1)\tilde{x}_{i}(k|k-1)+R_{i}(k)^{-1}\left[\tilde{s}_{i}^{r_{ij}}(k)+\tilde{x}_{i}(k|k-1)\right]$
$\displaystyle=\Phi_{i}(k|k-1)\tilde{x}_{i}(k|k-1)+R_{i}(k)^{-1}\left[\tilde{x}_{j}(k)+\omega^{r_{ij}}_{i}(k)\right],$
(B.3)

which leads to
$\tilde{x}_{i}(k|k)=\Phi_{i}(k|k)^{-1}\Big{\\{}\Phi_{i}(k|k-1)\tilde{x}_{i}(k|k-1)$
$+R_{i}(k)^{-1}\left[\tilde{x}_{j}(k)+\omega^{r_{ij}}_{i}(k)\right]\Big{\\}}$.
Taking the expectation of both sides and the limit when $k\rightarrow\infty$
leads to

$\displaystyle
E\left[\tilde{x}_{i}(k|k)\right]=\Phi_{i}(k|k)^{-1}\Phi_{i}(k|k-1)E\left[\tilde{x}_{i}(k|k-1)\right].$
(B.4)

Here we applied $\lim_{k\rightarrow\infty}\tilde{x}_{j}(k)=0$ and
$E[\omega^{r_{ij}}_{i}(k)]=0$.

To show that $\tilde{x}_{i}(k|k)$ strictly decreases after the inter-agent
measurement update, we show that the scalar-valued quadratic function
$V(k|k):=E[\tilde{x}_{i}(k|k)]^{T}\Phi_{i}(k|k)E[\tilde{x}_{i}(k|k)]$
decreases from $V(k|k-1)$. Using the result from (B.4), we can find that

$\displaystyle V(k|k)=$ $\displaystyle
E[\tilde{x}_{i}(k|k-1)]^{T}\Phi_{i}(k|k-1)^{T}\Phi_{i}(k|k)^{-1}$
$\displaystyle\qquad\qquad\qquad\Phi_{i}(k|k-1)E[\tilde{x}_{i}(k|k-1)].$ (B.5)

By applying the Woodbury matrix identity, we can find that

$\displaystyle\Phi_{i}(k|k)^{-1}=$
$\displaystyle[\Phi_{i}(k|k-1)+R_{i}(k)^{-1}]^{-1}$ $\displaystyle=$
$\displaystyle\Phi_{i}(k|k-1)^{-1}-[\Phi_{i}(k|k-1)$
$\displaystyle+\Phi_{i}(k|k-1)R_{i}(k)\Phi_{i}(k|k-1)]^{-1}.$ (B.6)

Substituting (B.6) into (B.5) and rearranging lead to

$\displaystyle V(k|k)=$ $\displaystyle
E[\tilde{x}_{i}(k|k-1)]^{T}\Phi_{i}(k|k-1)^{T}E[\tilde{x}_{i}(k|k-1)]$
$\displaystyle-E[\tilde{x}_{i}(k|k-1)]^{T}\Phi_{i}(k|k-1)^{T}[\Phi_{i}(k|k-1)$
$\displaystyle+\Phi_{i}(k|k-1)R_{i}(k)\Phi_{i}(k|k-1)]^{-1}$
$\displaystyle\Phi_{i}(k|k-1)E[\tilde{x}_{i}(k|k-1)].$ (B.7)

Note that the first term on the right-hand side is $V(k|k-1)$, and the second
term is non-positive, i.e., $V(k|k)\leq V(k|k-1)$. This proves that $V(k|k)$
decreases from $V(k|k-1)$ after inter-agent measurement update.∎

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