Content uploaded by Anam Tahir
Author content
All content in this area was uploaded by Anam Tahir on Sep 19, 2023
Content may be subject to copyright.
Anam TahirF 27ANNALES UNIVERSITATIS TURKUENSIS
TURUN YLIOPISTON JULKAISUJA – ANNALES UNIVERSITATIS TURKUENSIS
SARJA – SER. F OSA – TOM. 27 | TECHNICA – INFORMATICA | TURKU 2023
FORMATION CONTROL OF
SWARMS OF UNMANNED
AERIAL VEHICLES
Anam Tahir
TURUN YLIOPISTON JULKAISUJA – ANNALES UNIVERSITATIS TURKUENSIS
SARJA – SER. F OSA – TOM. 27 | TECHNICA – INFORMATICA| TURKU 2023
FORMATION CONTROL OF
SWARMS OF
UNMANNED AERIAL VEHICLES
Anam Tahir
University of Turku
Faculty of Technology
Department of Computing
Information and Communication Technology, Robotics and Autonomous Systems
Doctoral Programme in Technology
Supervised by
Professor Juha Plosila
University of Turku, Turku, Finland
Senior University Lecturer Jari B¨
oling
˚
Abo Akademi University, Turku, Finland
Postdoctoral Researcher
Hashem Haghbayan
University of Turku, Turku, Finland
Reviewed by
Senior Lecturer Bj¨
orn Olofsson
Lund University, Lund, Sweden
Senior Lecturer Jamshed Iqbal
University of Hull, Hull, United Kingdom
Opponent
Assistant Professor Azwirman Gusrialdi
Tampere University, Tampere, Finland
The originality of this publication has been checked in accordance with the University
of Turku quality assurance system using the Turnitin OriginalityCheck service.
ISBN 978-951-29-9412-0 (PRINT)
ISBN 978-951-29-9411-3 (PDF)
ISSN 2736-9390 (PRINT)
ISSN 2736-9684 (ONLINE)
Painosalama Oy, Turku, Finland, 2023
Dedicated to my grandparents.
UNIVERSITY OF TURKU
Faculty of Technology
Department of Computing
Information and Communication Technology, Robotics and Autonomous Systems
TAHIR, ANAM: Formation Control of Swarms of
Unmanned Aerial Vehicles
Doctoral Dissertation, 85 pp.
Doctoral Programme in Technology
September 2023
ABSTRACT
The objective of this thesis is to design a distributed formation control system
for swarms of unmanned aerial vehicles which addresses the challenges of scalabil-
ity, collision avoidance, failure recovery, energy efficiency, and control performance.
The swarms are arranged in tightly/loosely coupled architectures, which are based on
homogeneous nodes in a distributed network of leader-follower/leaderless structures.
The model of each node in the swarm formation is based on the nonlinear/linear dy-
namic model of a quadcopter, i.e. an unmanned aerial vehicle. The goal is to design
the formation control of swarms of unmanned aerial vehicles, which is divided into
high- and low-level control. From the high-level control perspective, the main con-
tribution is to propose continuous path planning which can quickly react to events.
Setpoints are generated for the swarms of unmanned aerial vehicles considering the
complex movement of a hierarchical formation, soft landing, and failure recovery.
The hierarchical formation and soft landing are executed using a fixed formation.
Reconfiguration of the formation after node failures is implemented using a short-
est path algorithm, combinatorial algorithms, and a thin plate spline. Besides this,
from the low-level control perspective, the main contribution is to manoeuvre the
nodes smoothly. The tracking of setpoints and stabilisation of each node is handled
by a nonlinear sliding mode control with proportional derivative control and a linear
quadratic regulator with integral action. The proposed strategies are evaluated using
simulations, and the obtained results are compared and analysed both qualitatively
and quantitatively using different scenario-relevant metrics.
KEYWORDS: Unmanned aerial vehicles, multi-drone systems, swarm intelligence,
formation maintenance, distributed control, hierarchical systems, soft landing, fail-
ure recovery system.
iv
TURUN YLIOPISTO
Teknillinen tiedekunta
Tietotekniikan laitos
Informaatio- ja viestint¨
ateknologia, Robotiikka ja autonomiset j¨
arjestelm¨
at
TAHIR, ANAM: Formation Control of Swarms of
Unmanned Aerial Vehicles
V¨
ait¨
oskirja, 85 s.
Teknologian tohtoriohjelma
Syyskuu 2023
TIIVISTELM ¨
A
T¨
am¨
an opinn¨
aytety¨
on tavoitteena on kehitt¨
a¨
a miehitt¨
am¨
att¨
omien ilma-alusten
parville hajautettua muodostumisen ohjausta, joka vastaa skaalautuvuuden, t¨
orm¨
ays-
ten v¨
altt¨
amisen, vikasietoisuuden, energiatehokkuuden ja ohjauksen suorituskyvyn
haasteisiin. Parvet on j¨
arjestetty tiiviisti/l¨
oyh¨
asti kytkettyihin arkkitehtuureihin, jotka
perustuvat homogeenisiin yksikk¨
oihin hajautetussa johtaja-seuraaja-/johtajattomien
rakenteiden verkostossa. Parvimuodostelman jokaisen yksik¨
on malli perustuu ne-
likopterin eli miehitt¨
am¨
att¨
om¨
an ilma-aluksen ep¨
alineaariseen dynaamiseen malliin.
Tavoitteena on ehdottaa miehitt¨
am¨
att¨
omien ilma-alusten parvien muodostelmaoh-
jausta, joka on jaettu korkean ja matalan tason ohjaukseen. Korkean tason ohjauksen
n¨
ak¨
okulmasta t¨
arkein panos on ehdottaa jatkuvaa polun suunnittelua, joka pystyy
reagoimaan nopeasti tapahtumiin. Asetusarvot luodaan miehitt¨
am¨
att¨
omien ilma-
alusten parville ottaen huomioon hierarkkisen muodostelman monimutkaisen liik-
keen, pehme¨
an laskun ja vikatilanteesta palautumisen. Hierarkkinen muodostelma ja
pehme¨
a lasku suoritetaan kiinte¨
all¨
a muodostelmalla. Muodostelman uudelleenkon-
figurointi yksik¨
oiden vikatilanteiden j¨
alkeen toteutetaan k¨
aytt¨
am¨
all¨
a lyhimm¨
an polun
algoritmia, kombinatorisia algoritmeja ja ”thin plate spline” -algoritmia. Lis¨
aksi
matalan tason ohjauksen n¨
ak¨
okulmasta t¨
arkein panos on ohjata yksik¨
oit¨
a sujuvasti.
Asetusarvojen seuranta ja kunkin yksik¨
on stabilointi hoidetaan ”sliding mode” -
s¨
a¨
atimell¨
a, jossa on suhde- ja derivoiva s¨
a¨
at¨
o, sek¨
a lineaarisella neli¨
osumman mini-
moivalla s¨
a¨
atimell¨
a, jossa on integroiva ominaisuus. Ehdotetut strategiat arvioidaan
simuloinnilla, ja saatuja tuloksia verrataan ja analysoidaan sek¨
a kvalitatiivisesti ett¨
a
kvantitatiivisesti k¨
aytt¨
am¨
all¨
a erilaisia skenaarioihin liittyvi¨
a mittareita.
ASIASANAT: Miehitt¨
am¨
att¨
om¨
at ilma-alukset, monidronej¨
arjestelm¨
at, parvi¨
alykkyys,
muodostelman yll¨
apito, hajautettu ohjaus, hierarkkiset j¨
arjestelm¨
at, pehme¨
a lasku,
vikatilannetoipumusj¨
arjestelm¨
a.
v
Acknowledgements
I would like to express my deepest gratitude to my supervisors, Professor Juha
Plosila, Senior University Lecturer Jari B¨
oling, and Postdoctoral Researcher Hashem
Haghbayan for their continued guidance, valuable support, and encouragement. Their
patience and training in all aspects of being a good scientist are greatly appreciated
and will not be forgotten. I am especially thankful to Senior University Lecturer Jari
B¨
oling for all the code/text reviews and continuous feedback, this thesis would not
have been possible without his efforts. Furthermore, I am extremely grateful to my
former supervisor Professor Hannu T. Toivonen, ˚
Abo Akademi University, Turku,
Finland, for his guidance on my studies.
I would like to thank University Lecturer Terho Jussila, Tampere University,
Tampere, Finland, for suggesting, during my exchange programme, that I collab-
orate with Professor Hannu T. Toivonen, ˚
Abo Akademi University, Turku, Finland,
from where this journey began.
I would like to acknowledge my pre-examiners, Senior Lecturer Bj¨
orn Olofsson
and Senior Lecturer Jamshed Iqbal for providing constructive comments on the the-
sis. I extend special thanks to Assistant Professor Azwirman Gusrialdi for agreeing
to be the opponent at the public thesis defence.
Many thanks to (𝑖) University of Turku, Turku, Finland, (𝑖𝑖) ADAFI (Adaptive-
Fidelity Digital Twins for Robust and Intelligent Control Systems); PI; Academy of
Finland, project no. 335513, (𝑖𝑖𝑖) Finnish Cultural Foundation, Helsinki, Finland,
decision no. 00211058, (𝑖𝑣) LARA (Learning and Assessing Risks for Enhancing
Dependability of Autonomous Socio-Technical Systems); Co-PI; Academy of Fin-
land, project no. 314048, (𝑣) Turku University Foundation, Turku, Finland, and (𝑣𝑖)
FinELib, National Library of Finland, Helsinki, Finland, for funding.
I would like to appreciate the people of the University of Turku, Turku, Finland,
who were involved from time to time or provided moral support when needed on this
long journey: Research Director Professor Hannu Tenhunen, Professor Tomi West-
erlund, Researcher Jorge Pe˜
na Queralta, Laboratory Engineer Sami Nuuttila, De-
velopment Manager Marko Lahti, Coordinator Nina Lehtim¨
aki, HR Specialist Nina
Reini, Education Manager Sanna Ranto, Project Planning Officer Sonja Kareranta,
Education Secretary Marja Luomanen, Vice Dean Professor Baoru Yang, and Facil-
ity Service Secretary Eveliina Jokinen. Also, I would like to thank Dr. Mahmood
vi
Pervaiz, COMSATS University, Islamabad, Pakistan, and Principal Lecturer Eero
Immonen, Turku University of Applied Sciences, Turku, Finland, for their guidance
on my studies when needed. Furthermore, I would like to thank Aakkosto Oy for
providing proofreading services.
Here, I would also like to acknowledge the support received from the people of
the AMO and AT-IS Master’s degree programmes of the Novia University of Applied
Sciences, Finland, Valmet Automotive EV Power Oy, Finland, and W¨
artsil¨
a Finland
Oy, with whom I got connected during the course of my doctoral degree.
I would like to close this with a few words for my family. My warmest gratitude
goes to my parents, Dr. Tahir Sharif Malik and M.A. Ghazala Malik, and my sisters,
Dr. Zona Tahir and Dr. Menahil Tahir, for all the love and support. Without their
endless encouragement, I would be nowhere near where I am.
September 2023, Turku, Finland
Anam Tahir
ANAM TAHIR
Master of Engineering, Automation Technology – Intelli-
gent Systems.
Novia University of Applied Sciences, Vaasa, Finland.
Master of Engineering, Autonomous Maritime Operations.
Novia University of Applied Sciences, Aboa Mare, Turku,
Finland.
Master of Science, Electrical Engineering.
COMSATS University, Islamabad, Pakistan.
Bachelor of Science, Computer Engineering.
COMSATS University, Islamabad, Pakistan.
vii
Table of Contents
Acknowledgements ............................ vi
Table of Contents ............................. viii
Abbreviations ................................ x
List of Original Publications ...................... xi
1 Introduction ............................... 1
1.1 Preliminaries ........................... 2
1.2 Problem Formulation ....................... 4
1.3 Thesis Layout ........................... 5
2 Models of Unmanned Aerial Vehicles .............. 7
2.1 Nonlinear Model ......................... 8
2.2 Linear Model ........................... 11
3 Methodology .............................. 18
3.1 High-Level Control ........................ 18
3.1.1 Thin Plate Spline .................... 19
3.2 Low-Level Control ........................ 19
3.2.1 Linear Quadratic Regulator with Integral Action . . . 19
3.2.2 Sliding Mode Control with Proportional Derivative
Control .......................... 21
4 Conclusion ................................ 26
4.1 Future Work ............................ 27
5 Overview of Publications ...................... 29
5.1 Swarms of Unmanned Aerial Vehicles – A Survey ...... 29
5.2 Comparison of Linear and Nonlinear Methods for Distributed
Control of a Hierarchical Formation of UAVs ......... 29
5.3 Navigation System for Landing a Swarm of Autonomous
Drones on a Movable Surface ................. 30
viii
TABLE OF CONTENTS
5.4 Development of a Fault-Tolerant Control System for a Swarm
of Drones ............................. 30
5.5 Energy-Efficient Post-Failure Reconfiguration of
Swarms of Unmanned Aerial Vehicles ............. 30
List of References ............................. 32
Original Publications ........................... 35
ix
Abbreviations
HLC High-Level Control
LLC Low-Level Control
LQR Linear Quadratic Regulator
LTI Linear Time-Invariant
PD Proportional Derivative
SMC Sliding Mode Control
TPS Thin Plate Spline
UAV Unmanned Aerial Vehicle
UGV Unmanned Ground Vehicle
USV Unmanned Surface Vehicle
UUV Unmanned Underwater Vehicle
UV Unmanned Vehicle
x
List of Original Publications
This thesis is based on the following original publications, which are referred to in
the text by their Roman numerals:
IAnam Tahir, Jari B¨
oling, Mohammad-Hashem Haghbayan, Hannu Toivo-
nen, and Juha Plosila. Swarms of Unmanned Aerial Vehicles – A Survey.
Journal of Industrial Information Integration, 2019; vol. 16 (100106): 1-7.
DOI: https://doi.org/10.1016/j.jii.2019.100106
II Anam Tahir, Jari B¨
oling, Mohammad-Hashem Haghbayan, and Juha Plosila.
Comparison of Linear and Nonlinear Methods for Distributed Control of a
Hierarchical Formation of UAVs. IEEE Access, 2020; vol. 8: 95667-95680.
DOI: https://doi.org/10.1109/ACCESS.2020.2988773
III Anam Tahir, Jari B¨
oling, Mohammad-Hashem Haghbayan, and Juha Plosila.
Navigation System for Landing a Swarm of Autonomous Drones on a Mov-
able Surface. Communications of the ECMS, Proceedings of the 34th Inter-
national ECMS Conference on Modelling and Simulation, Wildau, Berlin,
Germany, 2020; vol. 34, no. 1: 168–174.
DOI: https://doi.org/10.7148/2020-0168
IV Anam Tahir, Jari B¨
oling, Mohammad-Hashem Haghbayan, and Juha Plosila.
Development of a Fault-Tolerant Control System for a Swarm of Drones.
62nd International Symposium ELMAR, IEEE Proceedings, Zadar, Croatia,
2020; 79–82.
DOI: https://doi.org/10.1109/ELMAR49956.2020.9219027
VAnam Tahir, Mohammad-Hashem Haghbayan, Jari B¨
oling, and Juha Plosila.
Energy-Efficient Post-Failure Reconfiguration of Swarms of Unmanned Aerial
Vehicles. IEEE Access, 2023; vol. 11: 24768-24779.
DOI: https://doi.org/10.1109/ACCESS.2022.3181244
The original publications have been reproduced with the permission of the copyright
holders.
xi
1 Introduction
Swarms of unmanned vehicles (UVs) are sets of robots that work together to achieve
a specific goal. They include vehicles moving on the ground, UGVs, in the air,
UAVs, on the sea surface, USVs, or underwater, UUVs. UVs are increasingly getting
attention on both recreational and military groundsi[1; 2; 3]. They can shape the
future with the potential benefits of remote sensing and the elimination of human
error. For example, UVs manoeuvre autonomously in and around and are capable
of carrying out tasks in a variety of situational operations. Such vehicles can have
innovative impacts in the areas of (a) security and surveillance, (b) data collection,
(c) search and rescue, and (d) autonomous deliveries and shipping [4; 5; 6; 7].
UVs can operate in different modes depending upon their adaptation to versa-
tile environmental conditions. The operational modes can be teleoperation, remote
control, semi- or fully autonomous [8; 9].
• Teleoperation is a mode in which the human operator either directly controls
the actuators or sets incremental objectives on a continuous basis, using sen-
sory feedback from the location of the UVs [9].
• In comparison to the teleoperation mode, remote control is a mode in which the
human operator controls the UVs on a continuous basis also from the location
of the UVs but using only their direct observation [9]. In this mode, the UVs
rely on input from the human operator.
• In contrast, a mode in which the UVs enjoy relative autonomy is the semi-
autonomous mode. In this operational mode, an unmanned system and/or a
human operator plan and conduct a mission that requires different levels of
human interaction [9]. This enables the UVs to operate autonomously between
human interactions.
• Going further, the fully autonomous mode is the mode of operation where
the UVs have the most control without human intervention. It lets the UVs
accomplish the assigned mission within the defined scope while adapting to
operational and/or environmental conditions [9].
iUNMANNED AIRCRAFT SYSTEMS (UAS), DoD Purpose and Operational Use. Available at
https://dod.defense.gov/UAS/, accessed on 09.06.2023.
1
Anam Tahir
In a formation based on distributed control, i.e., a bottom-up approach, a top-
down approach is one of the main problems which lacks the perception of the overall
dynamics of a swarm in a local controller of each UV. Hence, one of the important
challenges in the formation of a swarm of UVs is the dependability of the swarm to
proceed with its mission. As a solution, the purpose of this thesis is to address the
distributed formation control problem, which enables dynamic management of the
creation, maintenance, and termination of swarms with the challenges of scalability
in terms of architecture. As a test case, unmanned aerial vehicles (UAVs) are consid-
ered, commonly known as drones. A detailed literature review of swarms of UAVs is
presented in Publication I. To achieve better scalability, a methodology is proposed
in which a high-level control (HLC) is integrated with a low-level control (LLC) so
that the swarm performs the setpoint decision-making and an individual UAV tracks
the given trajectory.
1.1 Preliminaries
The focus of this thesis is to explore multi-drone systems, i.e. swarms of UAVs, due
to their vital challenges of formation control. It is defined as organising a set of nodes
by maintaining its formation in a specific shape. To solve any formation control
problem, three main components are considered i.e., system design, its modelling,
and approaches of formation control structures [10; 11].
The system design delivers the framework upon which formation control is im-
plemented such as:
•Homogeneity and Heterogeneity
Homogeneous nodes consist of similar modules of software and/or hardware
whereas heterogeneous nodes consist of different software and/or hardware.
•Communication Structures
The communication structures of nodes can be listed with respect to their
range, topology, and bandwidth.
•Centralised, Decentralised, and Distributed Networks
In the centralised network, as illustrated in Fig. 1(a), a single node acts as a
server and holds all the information that is needed to obtain the desired ob-
jectives. Hence, these networks are prone to hacks and failures as it takes
only one node to be compromised or shut down for the entire network to
collapse. On the other hand, decentralised and distributed networks do not
rely on a central node, as described in Fig. 1(b)-(c). These networks provide
greater user control, system dependability, scalability, and privacy. Decen-
tralised and distributed networks have several characteristics and they are not
synonymous. Decentralised networks make use of a range of distinct con-
2
Introduction
(a) Centralised (b) Decentralised (c) Distributed
Figure 1. Network formations [12].
necting nodes. In decentralised networks, each node can have independent
decision-making and information processing. The functionality is distributed
across the network, which prevents total system failures or breakdowns. In
contrast, distributed networks behave in a transparent manner, for example,
location data is shared and decision-making is divided between the nodes,
which is absent in decentralised networks. Hence, their distinguishing char-
acteristic is reliance on equally powerful connecting hubs. However, unlike
decentralised networks, these can become centralised, and can also refer to
dispersed networks with a top-down approach. As a result, troubleshooting is
easier as the system failures, incursions, and crashes may be traced back to
specific nodes, making it easier to pinpoint the source of the problem. In addi-
tion, hybrid centralised/decentralised/distributed networks, in turn, use central
planners to provide HLC over autonomous robots.
The modelling of UAVs is based on their mathematical models, discussed in
Chapter 2. Furthermore, a formation structure defines how a group of UAVs can
be controlled to form and maintain the desired formation. To control the formation
of swarms, the recent literature normally classifies the different structures into the
following main categories [13], as presented in Fig. 2.
•Leader-follower [14; 15; 16; 17]
A leader aims for team objectives, while the follower(s) track the paths of the
leader with prescribed offsets. This structure is prone to error propagation and
sensitive to the leader’s failure.
•Virtual [18; 19; 20]
Letting each node have its own trajectory will solve the error propagation but
cannot avoid collisions.
3
Anam Tahir
(a) Leader-follower (b) Virtual (c) Behaviour-based
Figure 2. Formation structures.
•Behaviour-based [21; 22; 23]
To form the desired shape of a swarm, each node is assigned to one of the sev-
eral desired behaviours. The overall control is then derived from a weighting
of the relative importance of each behaviour. This structure is scalable, but
cannot ensure a fixed pattern.
The formation structures can be further categorised into position, displacement,
distance, and angle based with regards to the need for interaction topology and sens-
ing capability [24]. The formation in position-based control is defined by the de-
sired positions of nodes with respect to the global coordinate system. Also, it is
assumed that each node can sense its own position. In contrast to this, the forma-
tion in displacement-based control is defined by the desired displacements between
pairs of nodes with respect to the global coordinate system. Also, each node can
sense its own and its neighbour node(s)’ positions, which ensures maintaining the
same orientation. Furthermore, the formation in distance-based control is defined by
the desired inter-node distances that are actively controlled. Besides this, in angle-
based control, the actively controlled variable is the bearing between neighbours. In
distance- or angle-based control, distance assignment instead of position assignment
is present for pairs of nodes. The controlled variables are also the distances to the
sensed neighbours or the bearings between pairs of neighbours. Hence, all the vari-
ables can be sensed locally as there is no need for a global or aligned coordinate
system.
1.2 Problem Formulation
Inspired by the aforementioned discussions, the first key element that needs to be
adapted in most swarms of UAVs is scalability so that a swarm can be further divided
into subgroups. One of the main problems in distributed formation control of swarms
of UAVs is enabling the dynamic management of the creation, maintenance, and
termination of swarms. The following questions are addressed in this thesis:
4
Introduction
1System design for swarms of UAVs
What different set-ups are used for formation control of swarms of UAVs, and
which set-up should be used in this study?
2High-level control (HLC)
How to design a control system which can handle hierarchical formation, soft
landing, and failure recovery while maintaining its formation with collision
avoidance? This procedure is defined as the mapping problem, which de-
scribes the best shape of the desired swarm formations.
3Low-level control (LLC)
How do UAVs smoothly manoeuvre in the formations, arranged in tightly and
loosely coupled architectures? This problem addresses the optimal tracking
movement of each node from its initial to final position in a swarm formation.
1.3 Thesis Layout
This thesis is divided into five chapters. The introduction and the problem formu-
lation are stated in Chapter 1. The models of UAVs are outlined in Chapter 2. The
methodology for designing formation control of swarms of UAVs is elaborated in
Chapter 3. The concluding remarks are discussed in Chapter 4. Lastly, the overview
of publications is presented in Chapter 5.
In a swarm formation, most of the state-of-the-art is focused on the distributed
controllers i.e., the local controller of each UAV. It only manipulates its actuators
based on the observations of the local dynamic model of each node i.e., a bottom-
up approach. Hence, it is unaware of the behaviour and constraints of the swarm’s
higher levels of hierarchy i.e., the top-down approach. Both approaches try to refine
and optimise the partial behaviour of the systems, which eventually leads to global
emergent behaviour. This optimal or near-optimal behaviour of the system is the final
goal of control designs. This work addresses the problem of a distributed formation
control that is based on the dynamics of the overall swarm and the local dynamics
of each node via implementing the partial controllers i.e., the top-down, bottom-up
approach.
As a preliminary study, a comprehensive literature review of the general research
field of UAVs, with a particular emphasis on swarms is presented in Publication I.
In addition, an online survey among the general public was performed to investi-
gate the public awareness of the technology field. The model of each node in the
swarm formation is based on the nonlinear (Publications II, IV, and V) and linear
(Publication III) dynamic models of a quadcopter, i.e. a UAV. This work is based
on the formation control of swarms of homogeneous UAVs in a distributed network
of leader-follower and leaderless structures in Publications II-IV and Publication V
respectively. It sheds light on the challenges of scalability, collision avoidance, fail-
5
Anam Tahir
Figure 3. Structure of the thesis.
ure recovery, energy efficiency, and control performance. This addresses Research
Question 1. The goal is to design the formation control of swarms of UAVs, which
is divided into HLC and LLC, as explained in Fig.3.
In the HLC, to avoid collisions among UAVs, setpoints are generated for the
swarms of UAVs considering the complex movement of a hierarchical formation,
soft landing, and failure recovery. The hierarchical formation and soft landing are
executed using a fixed formation in Publications II and III respectively. The recon-
figuration of the formation after a single failed node is implemented using a short-
est path algorithm in Publication IV. For multiple failed nodes, the reconfiguration
of the formation is implemented using combinatorial algorithms (i.e., distance- and
time-optimal algorithms) and thin plate spline (TPS) method in Publication V. This
addresses Research Question 2.
Besides this, in the LLC, the tracking of setpoints and stabilisation of each node
for smooth manoeuvre is handled by a nonlinear sliding mode control (SMC) with
proportional derivative (PD) control in Publication II and a linear quadratic regulator
(LQR) with integral action in Publications II-V. This addresses Research Question 3.
6
2 Models of Unmanned Aerial Vehicles
The model of each node in the swarm formation consists of the nonlinear (Publica-
tions II, IV, and V) and linear (Publication III) dynamic models of a quadcopter, i.e.
UAV. Exhibiting the concept of the proposed control designs, this model presents
itself as a case study for swarms of UAVs.
Each UAV in a swarm is responsible for tracking the desired trajectories as well
as for hovering at desired positions for given time periods. In this section, the dy-
namic model of a quadcopter is examined. Consider a quadcopter as shown in Fig.
4(a), commonly known as a drone, an under-actuated system having four input en-
gines and propellers enabling six degrees of freedom including roll 𝜑, pitch 𝜃, yaw
𝜓, and thrust for movement and manoeuvre. For most quadcopter designs, there are
two possible configurations as illustrated in Fig. 4(b), i.e. plus and cross [25; 26; 27].
In each configuration, the two rotors on the opposite ends always rotate in the same
direction while the other two rotate in the opposite direction, whereas all the thrusts
have the same direction.
(a) (b)
Figure 4. (a) Quadcopter’s movement about the axis. (b) Configuration of a quadcopter.
The direction of movement, i.e. rotations, is the roll 𝜑, pitch 𝜃, and yaw 𝜓, which
are affected by thrusts from the propeller. The roll 𝜑, pitch 𝜃, and yaw 𝜓movements
are controlled to ensure the stability of rotation around the 𝑥-axis (longitudinal axis),
𝑦-axis (lateral or transverse axis), and 𝑧-axis (vertical axis) respectively. In other
words, the roll 𝜑is the measure of side-to-side tilting, i.e., it causes the vehicle
to move to one side or the other depending on the tilt. The pitch 𝜃determines the
7
Anam Tahir
rotation of the vehicle fixed between the side-to-side axis, i.e., it would tilt the vehicle
up and down from front to back causing the vehicle to move forwards or backward
depending on which way it is tilted. The yaw 𝜓moves the vehicle around in a
clockwise/anticlockwise rotation as it stays level to the ground, i.e., it changes the
direction of the vehicle accordingly. However, the thrust is not a directional element
like roll 𝜑, pitch 𝜃, and yaw 𝜓, but controls the altitude of the vehicle.
2.1 Nonlinear Model
Consider a quadcopter that has four propellers with fixed pitch mechanically movable
blades, as described in Fig. 5.
Figure 5. Kinematics of the quadcopter.
The major forces acting on the quadcopter are the gravity 𝑔and the thrust 𝑇𝑖,𝑖∈
{1,2,3,4}, of the propellers. In this model, the inertial reference is the earth shown
as (𝑥, 𝑦, 𝑧)which is the origin of the reference frame. A quadcopter is assumed to
be a rigid body that has a constant mass symmetrically distributed with respect to the
planes (𝑥, 𝑦),(𝑦, 𝑧), and (𝑥, 𝑧).
The position of a quadcopter reference frame (𝑥, 𝑦, 𝑧)with respect to an inertial
frame (𝑥, 𝑦, 𝑧)0can be expressed mathematically in a state variable form [28; 29]
where translational and angular accelerations are given by
8
Models of Unmanned Aerial Vehicles
˙𝑣𝑥=−𝑣𝑧𝑤𝑦+𝑣𝑦𝑤𝑧−𝑔sin 𝜃
˙𝑣𝑦=−𝑣𝑥𝑤𝑧+𝑣𝑧𝑤𝑥+𝑔cos 𝜃sin 𝜑
˙𝑣𝑧=−𝑣𝑦𝑤𝑥+𝑣𝑥𝑤𝑦+𝑔cos 𝜃cos 𝜑−𝑇
𝑚
(1)
and
˙𝑤𝑥=1
𝐽𝑥
(−𝑤𝑦𝑤𝑧(𝐽𝑧−𝐽𝑦) + 𝑀𝑥−𝑘𝑤𝑇
𝑘𝑀𝑇
𝐽𝑚𝑝𝑀𝑧𝑤𝑦)
˙𝑤𝑦=1
𝐽𝑦
(−𝑤𝑥𝑤𝑧(𝐽𝑥−𝐽𝑧) + 𝑀𝑦−𝑘𝑤𝑇
𝑘𝑀𝑇
𝐽𝑚𝑝𝑀𝑧𝑤𝑥)
˙𝑤𝑧=𝑀𝑧
𝐽𝑧
(2)
respectively. The thrust produced by each propeller 𝑇𝑖is translated into a total thrust
𝑇and the reactive torques 𝑀𝑖,𝑖∈ {𝑥, 𝑦, 𝑧}, affecting the rotations along the cor-
responding axis. 𝐽𝑖,𝑖∈ {𝑥, 𝑦 , 𝑧}, is known as the moment of inertia along the
corresponding axis, and 𝐽𝑚𝑝 is the moment of inertia of a motor with a propeller.
The angular velocities of propellers are assumed to be proportional to the thrusts of
propellers i.e., 𝑤𝑗=𝑘𝑤𝑇 𝑇𝑖,𝑗∈ {𝑥, 𝑦, 𝑧 }. Similarly, the reactive moments of pro-
pellers are assumed to be proportional to the thrust of propellers i.e., 𝑀𝑖=𝑘𝑀𝑇 𝑇𝑖.
Depending on the chosen configuration, the propeller thrusts 𝑇𝑖will generate differ-
ent thrust 𝑇and torques 𝑀𝑖namely
⎡
⎢
⎢
⎣
𝑇
𝑀𝑥
𝑀𝑦
𝑀𝑧
⎤
⎥
⎥
⎦
=⎡
⎢
⎢
⎣
1 1 1 1
0−ℓ0ℓ
ℓ0−ℓ0
𝑘𝑀𝑇 −𝑘𝑀 𝑇 𝑘𝑀 𝑇 −𝑘𝑀 𝑇
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
𝑇1
𝑇2
𝑇3
𝑇4
⎤
⎥
⎥
⎦
(3)
for a plus configuration and
⎡
⎢
⎢
⎣
𝑇
𝑀𝑥
𝑀𝑦
𝑀𝑧
⎤
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 1 1 1
√2
2ℓ−√2
2ℓ−√2
2ℓ√2
2ℓ
√2
2ℓ√2
2ℓ−√2
2ℓ−√2
2ℓ
𝑘𝑀𝑇 −𝑘𝑀 𝑇 𝑘𝑀 𝑇 −𝑘𝑀 𝑇
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎣
𝑇1
𝑇2
𝑇3
𝑇4
⎤
⎥
⎥
⎦
(4)
for a cross configuration where ℓis the length of the fixed pitch to mechanically
movable blades [30]. The velocities corresponding to Equations (1) and (2) are
9
Anam Tahir
˙𝑥=𝑣𝑥cos 𝜓cos 𝜃+𝑣𝑦(−sin 𝜓cos 𝜑+ cos 𝜓sin 𝜃sin 𝜑) +
𝑣𝑧(sin 𝜓sin 𝜑+ cos 𝜓sin 𝜃cos 𝜑)
˙𝑦=𝑣𝑥sin 𝜓cos 𝜃+𝑣𝑦(cos 𝜓cos 𝜑+ sin 𝜓sin 𝜃sin 𝜑) +
𝑣𝑧(−cos 𝜓sin 𝜑+ sin 𝜓sin 𝜃cos 𝜑)
˙𝑧=𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑
(5)
and
˙
𝜃=𝑤𝑦cos 𝜑−𝑤𝑧sin 𝜑
˙
𝜑=𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃
˙
𝜓=𝑤𝑦
sin 𝜑
cos 𝜃+𝑤𝑧
cos 𝜑
cos 𝜃
(6)
respectively. Equations (1) to (6) represent the complete nonlinear model of a quad-
copter, composed of twelve states, four inputs, and twelve outputs. More precisely,
x=𝑣𝑥𝑣𝑦𝑣𝑧𝑤𝑥𝑤𝑦𝑤𝑧𝜃 𝜑 𝜓 𝑥 𝑦 𝑧 T(7)
is the state or system vector,
u=𝑇 𝑀𝑥𝑀𝑦𝑀𝑧T(8)
is the input or control vector, and
y=x(9)
is the output (measured) vector. In any practical setting, some of the states would
be estimated using numerical derivation or integration with respect to time. The
nonlinear equations can be expressed in a compact form
˙x =𝑓(𝑥, 𝑢).(10)
Furthermore, the reduced state vector
xs=𝑣𝑥𝑣𝑦𝑣𝑧𝑤𝑥𝑤𝑦𝑤𝑧T(11)
and the performance outputs
𝑦𝑝1∈ {𝑥, 𝑦, 𝑧},(12)
𝑦𝑝2∈ {𝜃, 𝜑, 𝜓},(13)
and
10
Models of Unmanned Aerial Vehicles
yp=𝑥 𝑦 𝑧 T(14)
are defined for future use. {𝑣𝑥, 𝑣𝑦, 𝑣𝑧}and {𝑤𝑥, 𝑤𝑦, 𝑤𝑧}are defined as translational
and angular velocities respectively. Furthermore, 𝑦𝑝1and 𝑦𝑝2are defined as the trans-
lational and angular positions respectively. The values assigned to each quadcopter
are illustrated in Table 1, which are used in analytical solutions and simulations.
Table 1. System parameters [28].
Symbol Quantity Value
𝑔gravitational force 9.81 m/s2
ℓlength of the blades 0.2m
𝑚mass of the quadcopter 0.8kg
𝐽𝑚𝑝 moment of inertia of motor with propeller ≈0*
𝐽𝑥, 𝐽𝑦moment of inertia with respect to axis (𝑥, 𝑦) 1.8×10−3kgm2
𝐽𝑧moment of inertia with respect to axis 𝑧1.5×10−3kgm2
𝑘𝑀𝑇 ratio of reactive moment and thrust of used propellers 0.1m
*Assuming zero means that the rpm of a propeller can be directly manipulated.
2.2 Linear Model
In reconnaissance missions, the prevailing state of a quadcopter is either to have
a slow flight or to hover in equilibrium with all the state derivatives equal to zero
such as ˙x = 0, known as an equilibrium point. Putting all the derivatives to zero in
nonlinear Equations (1) to (6) leads to equilibrium values,
𝑇0=𝑚𝑔 and
𝑀𝑥0=𝑀𝑦0=𝑀𝑧0= 0,(15)
for the manipulated variables.
Consequently, nonlinear dynamic equations can be converted into standard equa-
tions of a linear system by assuming small values of state variables and small dif-
ferences of manipulated variables from equilibrium using a standard linearisation
method given by
˙x =Ax +Bu (16)
where A= [𝐴𝑖𝑗 ]and B= [𝐵𝑖𝑗 ], defined as
𝐴𝑖𝑗 =𝜕𝑓𝑖
𝜕𝑥𝑗|𝑥=𝑥0, 𝑢 =𝑢0(17)
and
11
Anam Tahir
𝐵𝑖𝑗 =𝜕𝑓𝑖
𝜕𝑢𝑗|𝑥=𝑥0, 𝑢 =𝑢0,(18)
are the matrices with functions as constant elements for the state vector xfrom Equa-
tion (7) and the input or control vector ufrom Equation (8) respectively. The matri-
ces’ components are established systemically term by term at the equilibrium point
(𝑥0, 𝑢0) = (0,0).
Consider Equation (10) and Equation (16) where xdoes not depend on uso the
derivative with respect to u= 0. Differentiating 𝑓1=−𝑣𝑧𝑤𝑦+𝑣𝑦𝑤𝑧−𝑔sin 𝜃with
respect to xleads to
𝜕𝑓1
𝜕𝑥2
=𝜕𝑓1
𝜕𝑣𝑦
=𝑤𝑧|0= 0,
𝜕𝑓1
𝜕𝑥3
=𝜕𝑓1
𝜕𝑣𝑧
=−𝑤𝑦|0= 0,
𝜕𝑓1
𝜕𝑥5
=𝜕𝑓1
𝜕𝑤𝑦
=−𝑣𝑧|0= 0,
𝜕𝑓1
𝜕𝑥6
=𝜕𝑓1
𝜕𝑤𝑧
=𝑣𝑦|0= 0,and
𝜕𝑓1
𝜕𝑥7
=𝜕𝑓1
𝜕𝜃 =−𝑔cos 𝜃|0=−𝑔.
(19)
Differentiating 𝑓2=−𝑣𝑥𝑤𝑧+𝑣𝑧𝑤𝑥+𝑔cos 𝜃sin 𝜑with respect to xleads to
𝜕𝑓2
𝜕𝑥1
=𝜕𝑓2
𝜕𝑣𝑥
=−𝑤𝑧|0= 0,
𝜕𝑓2
𝜕𝑥3
=𝜕𝑓2
𝜕𝑣𝑧
=𝑤𝑥|0= 0,
𝜕𝑓2
𝜕𝑥4
=𝜕𝑓2
𝜕𝑤𝑥
=𝑣𝑧|0= 0,
𝜕𝑓2
𝜕𝑥6
=𝜕𝑓2
𝜕𝑤𝑧
=−𝑣𝑥|0= 0,
𝜕𝑓2
𝜕𝑥7
=𝜕𝑓2
𝜕𝜃 =−𝑔sin 𝜃sin 𝜑|0= 0,and
𝜕𝑓2
𝜕𝑥8
=𝜕𝑓2
𝜕𝜑 =𝑔cos 𝜃cos 𝜑|0=𝑔.
(20)
Differentiating 𝑓3=−𝑣𝑦𝑤𝑥+𝑣𝑥𝑤𝑦+𝑔cos 𝜃cos 𝜑with respect to xleads to
12
Models of Unmanned Aerial Vehicles
𝜕𝑓3
𝜕𝑥1
=𝜕𝑓3
𝜕𝑣𝑥
=𝑤𝑦|0= 0,
𝜕𝑓3
𝜕𝑥2
=𝜕𝑓3
𝜕𝑣𝑦
=−𝑤𝑥|0= 0,
𝜕𝑓3
𝜕𝑥4
=𝜕𝑓3
𝜕𝑤𝑥
=−𝑣𝑦|0= 0,
𝜕𝑓3
𝜕𝑥5
=𝜕𝑓3
𝜕𝑤𝑦
=𝑣𝑥|0= 0,
𝜕𝑓3
𝜕𝑥7
=𝜕𝑓3
𝜕𝜃 =−𝑔sin 𝜃cos 𝜑|0= 0,and
𝜕𝑓3
𝜕𝑥8
=𝜕𝑓3
𝜕𝜑 =−𝑔cos 𝜃sin 𝜑|0= 0.
(21)
Differentiating 𝑓4=1
𝐽𝑥(−𝑤𝑦𝑤𝑧(𝐽𝑧−𝐽𝑦)) with respect to xleads to
𝜕𝑓4
𝜕𝑥5
=𝜕𝑓4
𝜕𝑤𝑦
=−𝑤𝑧(𝐽𝑧−𝐽𝑦)
𝐽𝑥|0= 0 and
𝜕𝑓4
𝜕𝑥6
=𝜕𝑓4
𝜕𝑤𝑧
=−𝑤𝑦(𝐽𝑧−𝐽𝑦)
𝐽𝑥|0= 0.
(22)
Differentiating 𝑓5=1
𝐽𝑦(−𝑤𝑥𝑤𝑧(𝐽𝑥−𝐽𝑧)) with respect to xleads to
𝜕𝑓5
𝜕𝑥4
=𝜕𝑓5
𝜕𝑤𝑥
=−𝑤𝑧(𝐽𝑥−𝐽𝑧)
𝐽𝑦|0= 0 and
𝜕𝑓5
𝜕𝑥6
=𝜕𝑓5
𝜕𝑤𝑧
=−𝑤𝑥(𝐽𝑥−𝐽𝑧)
𝐽𝑦|0= 0.
(23)
Differentiating 𝑓6= 0 with respect to xleads to 0. Differentiating 𝑓7=𝑤𝑦cos 𝜑−
𝑤𝑧sin 𝜑with respect to xleads to
𝜕𝑓7
𝜕𝑥5
=𝜕𝑓7
𝜕𝑤𝑦
= cos 𝜑|0= 1,
𝜕𝑓7
𝜕𝑥6
=𝜕𝑓7
𝜕𝑤𝑧
=−sin 𝜑|0= 0,and
𝜕𝑓7
𝜕𝑥8
=𝜕𝑓7
𝜕𝜑 = (−𝑤𝑦sin 𝜑−𝑤𝑧cos 𝜑)|0= 0.
(24)
Differentiating 𝑓8=𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃with respect to xleads to
13
Anam Tahir
𝜕𝑓8
𝜕𝑥4
=𝜕𝑓8
𝜕𝑤𝑥
= 1,
𝜕𝑓8
𝜕𝑥5
=𝜕𝑓8
𝜕𝑤𝑦
= sin 𝜑tan 𝜃|0= 0,
𝜕𝑓8
𝜕𝑥6
=𝜕𝑓8
𝜕𝑤𝑧
= cos 𝜑tan 𝜃|0= 0,
𝜕𝑓8
𝜕𝑥7
=𝜕𝑓8
𝜕𝜃 = (𝑤𝑦sin 𝜑sec2𝜃+𝑤𝑧cos 𝜑sec2𝜃)|0= 0,and
𝜕𝑓8
𝜕𝑥8
=𝜕𝑓8
𝜕𝜑 = (𝑤𝑦cos 𝜑tan 𝜃−𝑤𝑧sin 𝜑tan 𝜃)|0= 0.
(25)
Differentiating 𝑓9=𝑤𝑦sin 𝜑
cos 𝜃+𝑤𝑧cos 𝜑
cos 𝜃with respect to xleads to
𝜕𝑓9
𝜕𝑥5
=𝜕𝑓9
𝜕𝑤𝑦
=sin 𝜑
cos 𝜃|0= 0,
𝜕𝑓9
𝜕𝑥6
=𝜕𝑓9
𝜕𝑤𝑧
=cos 𝜑
cos 𝜃|0= 1,
𝜕𝑓9
𝜕𝑥7
=𝜕𝑓9
𝜕𝜃 = (𝑤𝑦sin 𝜑(−1
√1−𝜃2) + 𝑤𝑧cos 𝜑(−1
√1−𝜃2))|0= 0,and
𝜕𝑓9
𝜕𝑥8
=𝜕𝑓9
𝜕𝜑 = (𝑤𝑦
cos 𝜑
cos 𝜃−𝑤𝑧
sin 𝜑
cos 𝜃)|0= 0.
(26)
Differentiating 𝑓10 =𝑣𝑥cos 𝜓cos 𝜃+𝑣𝑦(−sin 𝜓cos 𝜑+cos 𝜓sin 𝜃sin 𝜑)+𝑣𝑧(sin 𝜓sin 𝜑+
cos 𝜓sin 𝜃cos 𝜑)with respect to xleads to
𝜕𝑓10
𝜕𝑥1
=𝜕𝑓10
𝜕𝑣𝑥
= cos 𝜓cos 𝜃|0= 1,
𝜕𝑓10
𝜕𝑥2
=𝜕𝑓10
𝜕𝑣𝑦
= (−sin 𝜓cos 𝜑+ cos 𝜓sin 𝜃sin 𝜑)|0= 0,
𝜕𝑓10
𝜕𝑥3
=𝜕𝑓10
𝜕𝑣𝑧
= (sin 𝜓sin 𝜑+ cos 𝜓sin 𝜃cos 𝜑)|0= 0,
𝜕𝑓10
𝜕𝑥7
=𝜕𝑓10
𝜕𝜃 = (−𝑣𝑥cos 𝜓sin 𝜃+𝑣𝑦cos 𝜓cos 𝜃sin 𝜑+
𝑣𝑧cos 𝜓cos 𝜃cos 𝜑)|0= 0,
𝜕𝑓10
𝜕𝑥8
=𝜕𝑓10
𝜕𝜑 = (𝑣𝑦sin 𝜓sin 𝜑+𝑣𝑦cos 𝜓sin 𝜃cos 𝜑+𝑣𝑧sin 𝜓cos 𝜑−
𝑣𝑧cos 𝜓sin 𝜃sin 𝜑)|0= 0,and
𝜕𝑓10
𝜕𝑥9
=𝜕𝑓10
𝜕𝜓 = (−𝑣𝑥sin 𝜓cos 𝜃−𝑣𝑦cos 𝜓cos 𝜑−𝑣𝑦sin 𝜓sin 𝜃sin 𝜑+
𝑣𝑧cos 𝜓sin 𝜑−𝑣𝑧sin 𝜓sin 𝜃cos 𝜑)|0= 0.
(27)
14
Models of Unmanned Aerial Vehicles
Differentiating 𝑓11 =𝑣𝑥sin 𝜓cos 𝜃+𝑣𝑦(cos 𝜓cos 𝜑+sin 𝜓sin 𝜃sin 𝜑)+𝑣𝑧(−cos 𝜓sin 𝜑+
sin 𝜓sin 𝜃cos 𝜑)with respect to xleads to
𝜕𝑓11
𝜕𝑥1
=𝜕𝑓11
𝜕𝑣𝑥
= sin 𝜓cos 𝜃|0= 0,
𝜕𝑓11
𝜕𝑥2
=𝜕𝑓11
𝜕𝑣𝑦
= (cos 𝜓cos 𝜑+ sin 𝜓sin 𝜃sin 𝜑)|0= 1,
𝜕𝑓11
𝜕𝑥3
=𝜕𝑓11
𝜕𝑣𝑧
= (−cos 𝜓sin 𝜑+ sin 𝜓sin 𝜃cos 𝜑)|0= 0,
𝜕𝑓11
𝜕𝑥7
=𝜕𝑓11
𝜕𝜃 = (−𝑣𝑥sin 𝜓sin 𝜃+𝑣𝑦sin 𝜓cos 𝜃sin 𝜑+
𝑣𝑧sin 𝜓cos 𝜃cos 𝜑)|0= 0,
𝜕𝑓11
𝜕𝑥8
=𝜕𝑓11
𝜕𝜑 = (−𝑣𝑦cos 𝜓sin 𝜑+𝑣𝑦sin 𝜓sin 𝜃cos 𝜑−𝑣𝑧cos 𝜓cos 𝜑−
𝑣𝑧sin 𝜓sin 𝜃sin 𝜑)|0= 0,and
𝜕𝑓11
𝜕𝑥9
=𝜕𝑓11
𝜕𝜓 = (𝑣𝑥cos 𝜓cos 𝜃−𝑣𝑦sin 𝜓cos 𝜑+𝑣𝑦cos 𝜓sin 𝜃sin 𝜑+
𝑣𝑧sin 𝜓sin 𝜑+𝑣𝑧cos 𝜓sin 𝜃cos 𝜑)|0= 0.
(28)
Differentiating 𝑓12 =𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑with respect to xleads
to
𝜕𝑓12
𝜕𝑥1
=𝜕𝑓12
𝜕𝑣𝑥
= sin 𝜃|0= 0,
𝜕𝑓12
𝜕𝑥2
=𝜕𝑓12
𝜕𝑣𝑦
=−cos 𝜃sin 𝜑|0= 0,
𝜕𝑓12
𝜕𝑥3
=𝜕𝑓12
𝜕𝑣𝑧
=−cos 𝜃cos 𝜑|0=−1,
𝜕𝑓12
𝜕𝑥7
=𝜕𝑓12
𝜕𝜃 = (𝑣𝑥cos 𝜃+𝑣𝑦sin 𝜃sin 𝜑+𝑣𝑧sin 𝜃cos 𝜑)|0= 0,and
𝜕𝑓12
𝜕𝑥8
=𝜕𝑓12
𝜕𝜑 = (−𝑣𝑦cos 𝜃cos 𝜑+𝑣𝑧cos 𝜃sin 𝜑)|0= 0.
(29)
Keeping all the attained differential values from Equations (19) to (29) into Equa-
tion (17), the matrix
15
Anam Tahir
𝐴𝑖𝑗 =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0 0 0 −𝑔00000
0 0 0 0 0 0 0 𝑔0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0 −1 0 0 0 0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(30)
with functions as constant elements is obtained for the state vector x.
Consider Equation (10) and Equation (16) where udoes not depend on xso the
derivative with respect to x= 0. Differentiating 𝑓1=−𝑇
𝑚with respect to uleads
to
𝜕𝑓1
𝜕𝑢1
=𝜕𝑓1
𝜕𝑇 =−1
𝑚|0=−1
𝑚.(31)
Differentiating 𝑓2=𝑀𝑥
𝐽𝑥(where 𝐽𝑚𝑝 = 0, see Table 1) with respect to uleads to
𝜕𝑓2
𝜕𝑢2
=𝜕𝑓2
𝜕𝑀𝑥
=1
𝐽𝑥|0=1
𝐽𝑥
.(32)
Differentiating 𝑓3=𝑀𝑦
𝐽𝑦(where 𝐽𝑚𝑝 = 0, see Table 1) with respect to uleads to
𝜕𝑓3
𝜕𝑢3
=𝜕𝑓3
𝜕𝑀𝑦
=1
𝐽𝑦|0=1
𝐽𝑦
.(33)
Differentiating 𝑓4=𝑀𝑧
𝐽𝑧with respect to uleads to
𝜕𝑓4
𝜕𝑢4
=𝜕𝑓4
𝜕𝑀𝑧
=1
𝐽𝑧|0=1
𝐽𝑧
.(34)
Keeping all the attained differential values from Equations (31) to (34) into Equa-
tion (18), the matrix
𝐵𝑖𝑗 =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 −1
𝑚0 0 0 0 0 0 0 0 0
0 0 0 1
𝐽𝑥0 0 0 0 0 0 0 0
0 0 0 0 1
𝐽𝑦0 000000
0 0 0 0 0 1
𝐽𝑧000000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
T
(35)
16
Models of Unmanned Aerial Vehicles
with functions as constant elements is obtained for the input or control vector u.
Hence, the complete dynamics of a quadcopter can be represented in a standard
linear time-invariant (LTI) form
˙
x=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0 0 0 −𝑔00000
0 0 0 0 0 0 0 𝑔0000
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0 −1000 0 00000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
𝑣𝑥
𝑣𝑦
𝑣𝑧
𝑤𝑥
𝑤𝑦
𝑤𝑧
𝜃
𝜑
𝜓
𝑥
𝑦
𝑧
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0
0 0 0 0
−1
𝑚0 0 0
01
𝐽𝑥0 0
0 0 1
𝐽𝑦0
0 0 0 1
𝐽𝑧
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
𝑇
𝑀𝑥
𝑀𝑦
𝑀𝑧
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(36)
using Equations (16), (30) and (35). More precisely, it can be written as
˙x =Ax +Bu
=−𝑔𝜃 𝑔𝜑 −𝑇
𝑚𝑀𝑥
𝐽𝑥
𝑀𝑦
𝐽𝑦
𝑀𝑧
𝐽𝑧𝑤𝑦𝑤𝑥𝑤𝑧𝑣𝑥𝑣𝑦−𝑣𝑧T(37)
and
y=Cx +Du =x(38)
where Cand Dare defined as output (measured) matrix and feedthrough matrix
respectively. The state or system vector, input or control vector, output (measured)
vector, reduced state vector, and the performance outputs are given in Equations (7)
to (9) and (11) to (14) respectively.
17
3 Methodology
In this thesis, the model of each node in the swarm formation is based on the non-
linear (Publications II, IV, and V) and linear (Publication III) dynamic models of a
quadcopter, i.e. a UAV. The goal is to design the formation control of swarms of
UAVs, which is divided into HLC and LLC, as presented in Fig.3. The HLC(s) relies
on continuous monitoring, which communicates with the LLC(s), as explained in
Fig. 6. In particular, the HLC integrates a continuous path planning and execution
cycle which can quickly react to events. In contrast, the LLC decides the control
modes and the associated trajectories taking into account the mission, path, and con-
trol constraints and requirements of each UAV in a swarm.
Figure 6. Overall control scheme.
3.1 High-Level Control
In the HLC, the setpoints are generated using offsets in Publication II to maintain the
hierarchical formation of UAVs in the swarms. In contrast, the setpoints are gener-
ated using offsets in Publication III to guarantee the smoothness, i.e. soft landing, of
desired landing missions on a movable surface while minimising the possibilities for
landing errors. The hierarchical formation and soft landing are executed using a fixed
formation. Besides this, the failure recovery method is included, which reconfigures
the swarm formation after single and multiple failed nodes in Publications IV and V
respectively. For reconfigurations, the setpoints are generated using the shortest path
algorithm in Publication IV. In contrast, the setpoints are generated using combina-
torial algorithms (i.e., distance- and time-optimal algorithms) and the TPS method
in Publication V. Only the TPS method is described in the following subsection.
18
Methodology
3.1.1 Thin Plate Spline
For robust nonrigid 2D transformation, consider a point set registration [31; 32; 33]
using TPS, which is known for solving data interpolation and smoothing problems
[34]. A spline is a function defined by polynomials in a piecewise manner. For
the approximation of complicated shapes, spline curves are used via curve fitting
that is adaptable and has noncomplicated construction [34]. To make it simpler, this
method is analysed in 2D in Publication V. Consequently, two sets of correspondence
data points, 𝑋=𝑥𝑖and 𝑉=𝑣𝑖where 𝑖= 1,2,3, ..., 𝑛 are considered. Here,
𝑥𝑖and 𝑣𝑖are defined as the locations of a point in the desired formation (scene)
and the initial formation (model) respectively. Keeping the shape of the disturbed
formation/function under consideration, a mapping function 𝑓(𝑣𝑖)can be obtained
by minimising the energy function
𝐸𝑇 𝑃 𝑆 (𝑓) =
𝑛
𝑖=1 ||𝑥𝑖−𝑓(𝑣𝑖)||2+𝜆 (𝜕2𝑓
𝜕𝑥2)2+ 2( 𝜕2𝑓
𝜕𝑥𝜕 𝑦 )2+ ( 𝜕2𝑓
𝜕𝑦2)𝑑𝑥𝑑𝑦 (39)
that evaluates the amount of formation disturbance. The data points of 𝑉are mapped
as closely as possible to the data points of 𝑋by minimising the first error measure-
ment term. The second regularisation term is a penalty on the smoothness of 𝑓, and it
is in a general case needed to make the mapping unique. In the cases that are studied
in this thesis, one can put 𝜆= 0, and still get unique solutions.
3.2 Low-Level Control
In the LLC, a feedback controller is designed to track the setpoints and stabilise the
positions of UAVs for smooth manoeuvre in a swarm formation. The tracking of
setpoints and stabilisation of each UAV is handled by LQR with integral action in
Publications II-V and SMC PD in Publication II. This is further described in the
following subsections.
3.2.1 Linear Quadratic Regulator with Integral Action
LQR is a simple control design method that provides good control performance in
the full-state feedback case and can handle a multivariable system [35]. For the for-
mation flight, a standard LQR augmented with integral action is tested as illustrated
in Fig. 7, similar to what was done in [36] and [37]. Each UAV has its local control
system, with the setpoints coming from the above level. LQR provides an optimal
full-state feedback controller using linear model Equations (37) and (38), which is
tested on the nonlinear model Equations (1) to (6) of each UAV in Publications II,
IV, and V whereas it is tested on the linear model Equations (37) and (38) of each
UAV in Publication III.
19
Anam Tahir
Figure 7. Block diagram of LQR integral control design.
Considering the linear model Equations (37) and (38), the control input umin-
imises the quadratic cost function
𝐽(u) = ∞
0
(˙xT
a𝑄˙xa+˙uT𝑅˙u)𝑑𝑡 (40)
where
xa=𝑡
0
e(𝜏)T𝑑𝜏 xTT
,(41)
the states x, and the control vector uare defined in Equations (7) and (8) respectively.
𝑄is a positive semi-definite matrix that defines the weights of states, whereas 𝑅is a
positive definite matrix that weights the control inputs. To get the desired response,
the controller can be tuned by changing the (diagonal) elements in the 𝑄and 𝑅
matrices. Positive-definite and semi-definite criteria for the diagonal matrices mean
that the diagonal elements are greater than zero and greater than or equal to zero
respectively. A larger diagonal element in 𝑄or 𝑅means fewer variations in the
corresponding state or control input respectively. Furthermore,
e=r−yp(42)
is the error term and
r=𝑥𝑟𝑦𝑟𝑧𝑟T(43)
is the reference for yp, defined in Equation (14). The state feedback law
u=𝐾𝑖
𝑠e−𝐾𝑝(x−^x)(44)
gives the four control inputs i.e., thrust 𝑇and torques 𝑀𝑖;𝑀𝑥(along 𝑥-axis i.e. roll
𝜑), 𝑀𝑦(along 𝑦-axis i.e. pitch 𝜃), and 𝑀𝑧(along 𝑧-axis i.e. yaw 𝜓). The controller
uses two separate gain matrices 𝐾𝑝and 𝐾𝑖where 𝐾𝑝is the state feedback gain and
𝐾𝑖is the integral gain. ^x is the setpoints of the states xand 𝐺𝑟is the reference gain.
20
Methodology
3.2.2 Sliding Mode Control with Proportional Derivative Control
The SMC method reduces the order of the state equations and provides a quick re-
sponse, which leads to the simplification of a design procedure. It may reduce the
system’s sensitivity with respect to parameter variation and disturbances when it en-
ters the sliding surface. However, this may lead to a chattering phenomenon that
can be minimised by using proper switching gains in the saturation function. Fur-
thermore, PD control is a closed-loop system in which the proportional controller
reduces the rise time and steady-state error, and the derivative controller reduces
overshoot, increases the stability of the system, and improves the transient response.
For the formation flight, an SMC PD control method from [38] is adapted and
tested on the nonlinear model Equations (1) to (6) of each UAV in Publication II. Fig.
8 shows the control design of each UAV, which is divided into two feedback control
loops, i.e., outer and inner, where xs,𝑦𝑝1, and 𝑦𝑝2are defined in Equations (11)
to (13). Furthermore,
𝑟1∈ {𝑥𝑟, 𝑦𝑟, 𝑧𝑟}(45)
is the reference for 𝑦𝑝1. In this control design, SMC is tied to the PD control and the
gain parameters can be adjusted based on the overall dynamics of the swarm, which
eventually produces input or control vector u, defined in Equation (8). A UAV’s
translational position 𝑦𝑝1is handled by the outer feedback control loop using PD
control whereas angular position is handled by the inner feedback control loop using
SMC. This is further described in the following subsections.
Figure 8. Block diagram of SMC PD control design.
Translational Position Control Design
The control design of each UAV’s translational positions 𝑦𝑝1is described in Fig. 9.
Using three PD controllers, the angular position errors 𝑒2∈ {𝜃𝑒, 𝜑𝑒, 𝜓𝑒}are gener-
ated, which are fed to the angular position control design as inputs. The feedback
law
𝑒2=𝑦𝑝2−𝑟2=𝑦𝑝2−(𝐾𝑃·𝑒1+𝐾𝐷
𝑑
𝑑𝑡𝑒1)(46)
21
Anam Tahir
is obtained where 𝑒1=𝑟1−𝑦𝑝1, and the parameters 𝐾𝑃and 𝐾𝐷are the propor-
tional and derivative gains respectively. 𝑦𝑝1and 𝑦𝑝2are the translational and angular
positions respectively, see Equations (12) and (13).
Figure 9. The translational position control design.
Furthermore, to control the hierarchical formation of tightly coupled architec-
tures, a lead compensator is connected in series with the PD control block, which
reduces the overshoots of translational positions 𝑦𝑝1. Mathematically, the transfer
function
𝐺(𝑠) = 𝑠−𝑧𝑧
𝑠−𝑧𝑝
(47)
is defined as the lead compensator where 𝑧𝑧is the zero and 𝑧𝑝is the pole, satisfying
0< 𝑧𝑧< 𝑧𝑝.
Angular Position Control Design
For each UAV, the input or control vector u, see Equation (8), is obtained using
four SMC controllers. The actuator thrust 𝑇is generated to drive each UAV, and
the torques 𝑀𝑖∈ {𝑀𝑥, 𝑀𝑦, 𝑀𝑧}are generated for the stabilisation of the position
angles, as presented in Fig. 10.
Figure 10. The angular position control design.
Consider a time-varying surface 𝑠𝑎(𝑡)in the state space R𝑛by the scalar equation
𝑠𝑎(𝑥, 𝑡)=0. The sliding surface
𝑠𝛼=𝑐𝛼𝑒𝛼+ ˙𝑒𝛼(48)
22
Methodology
is chosen where 𝛼∈ {𝑇, 𝑀𝑥, 𝑀𝑦, 𝑀𝑧},𝑐𝛼is a strictly positive constant, and 𝑒𝛼is
the tracking error [39]. The chattering free control law
𝛼= ^𝛼−𝐾𝛼sat(𝑠𝛼
𝛽𝛼
)(49)
for thrust 𝑇and torques 𝑀𝑖;𝑀𝑥(along 𝑥-axis i.e. roll 𝜑), 𝑀𝑦(along 𝑦-axis i.e. pitch
𝜃), and 𝑀𝑧(along 𝑧-axis i.e. yaw 𝜓) is obtained. The control design parameters ^𝛼,
defined as equivalent control, and 𝐾𝛼indicate the motion of the state trajectory along
the sliding surface 𝑠𝛼and the maximum controller output respectively. To eliminate
the chattering phenomena, the saturation function
sat(𝑠𝛼
𝛽𝛼
) = ⎧
⎪
⎨
⎪
⎩
𝑠𝛼
𝛽𝛼,|𝑠𝛼
𝛽𝛼| ≤ 1
sgn(𝑠𝛼
𝛽𝛼),otherwise
(50)
allocates a low pass filter to the local dynamics of the variable 𝑠𝛼, which approxi-
mates the sgn(.)term in a boundary layer of the manifold 𝑠𝑎(𝑡)=0. This bound-
ary layer solution avoids control discontinuities and switching actions in the control
loop. The 𝛽𝛼is a constant that defines the thickness of the boundary layer. The
control design is more efficient when a minimum amount of boundary layer is used.
The method of obtaining the equivalent control ^𝛼is described below.
Differentiating the Equation (48) with respect to 𝑒leads to
˙𝑠𝛼=𝑐𝛼˙𝑒𝛼+ ¨𝑒𝛼(51)
for thrust 𝑇and torques 𝑀𝑖. The sliding surfaces
˙𝑠𝑇=𝑐𝑇˙𝑧+ ¨𝑧,
˙𝑠𝑀𝑥=𝑐𝑀𝑥˙
𝜑+¨
𝜑,
˙𝑠𝑀𝑦=𝑐𝑀𝑦˙
𝜃+¨
𝜃, and
˙𝑠𝑀𝑧=𝑐𝑀𝑧˙
𝜓+¨
𝜓
(52)
are obtained by letting {˙𝑒𝑇,˙𝑒𝑀𝑥,˙𝑒𝑀𝑦,˙𝑒𝑀𝑧}={˙𝑧, ˙
𝜑, ˙
𝜃, ˙
𝜓}respectively. Keeping
the ˙𝑧from Equation (5) and ˙𝑣𝑧from Equation (1) in Equation (52),
˙𝑠𝑇=𝑐𝑇(𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑) + 𝑑
𝑑𝑧 ˙𝑧
=𝑐𝑇(𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑)−˙𝑣𝑧cos 𝜃cos 𝜑
=𝑐𝑇(𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑)−
(−𝑣𝑦𝑤𝑥+𝑣𝑥𝑤𝑦+𝑔cos 𝜃cos 𝜑−𝑇
𝑚) cos 𝜃cos 𝜑
(53)
23
Anam Tahir
is obtained for the thrust 𝑇. Keeping the ˙
𝜑from Equation (6) and ˙𝑤𝑥from Equa-
tion (2) where 𝐽𝑚𝑝 = 0, see Table 1, in Equation (52),
˙𝑠𝑀𝑥=𝑐𝑀𝑥(𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃) + 𝑑
𝑑𝑥 ˙
𝜑
=𝑐𝑀𝑥(𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃) + ˙𝑤𝑥
=𝑐𝑀𝑥(𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃) +
1
𝐽𝑥
(−𝑤𝑦𝑤𝑧(𝐽𝑧−𝐽𝑦) + 𝑀𝑥)
(54)
is obtained for the torque 𝑀𝑥. Keeping the ˙
𝜃from Equation (6) and ˙𝑤𝑦from Equa-
tion (2) where 𝐽𝑚𝑝 = 0, see Table 1, in Equation (52),
˙𝑠𝑀𝑦=𝑐𝑀𝑦(𝑤𝑦cos 𝜑−𝑤𝑧sin 𝜑) + 𝑑
𝑑𝑦 ˙
𝜃
=𝑐𝑀𝑦(𝑤𝑦cos 𝜑−𝑤𝑧sin 𝜑) + ˙𝑤𝑦cos 𝜑
=𝑐𝑀𝑦(𝑤𝑦cos 𝜑−𝑤𝑧sin 𝜑)+( 1
𝐽𝑦
(−𝑤𝑥𝑤𝑧(𝐽𝑥−𝐽𝑧) + 𝑀𝑦)) cos 𝜑
(55)
is obtained for the torque 𝑀𝑦. Keeping the ˙
𝜓from Equation (6) and ˙𝑤𝑧from Equa-
tion (2) in Equation (52),
˙𝑠𝑀𝑧=𝑐𝑀𝑧(𝑤𝑦
sin 𝜑
cos 𝜃+𝑤𝑧
cos 𝜑
cos 𝜃) + 𝑑
𝑑𝑧 ˙
𝜓
=𝑐𝑀𝑧(𝑤𝑦
sin 𝜑
cos 𝜃+𝑤𝑧
cos 𝜑
cos 𝜃) + ˙𝑤𝑧
cos 𝜑
cos 𝜃
=𝑐𝑀𝑧(𝑤𝑦
sin 𝜑
cos 𝜃+𝑤𝑧
cos 𝜑
cos 𝜃) + 𝑀𝑧
𝐽𝑧
cos 𝜑
cos 𝜃
(56)
is obtained for the torque 𝑀𝑧. Let {˙𝑠𝑇,˙𝑠𝑀𝑥,˙𝑠𝑀𝑦,˙𝑠𝑀𝑧}= 0 in Equations (53)
to (56), the equivalent control ^𝛼;
^
𝑇=−𝑚𝑐𝑇
cos 𝜃cos 𝜑(𝑣𝑥sin 𝜃−𝑣𝑦cos 𝜃sin 𝜑−𝑣𝑧cos 𝜃cos 𝜑) +
𝑚(−𝑣𝑦𝑤𝑥+𝑣𝑥𝑤𝑦+𝑔cos 𝜃cos 𝜑),
^
𝑀𝑥=−𝑐𝑀𝑥𝐽𝑥(𝑤𝑥+𝑤𝑦sin 𝜑tan 𝜃+𝑤𝑧cos 𝜑tan 𝜃) + 𝑤𝑦𝑤𝑧(𝐽𝑧−𝐽𝑦),
^
𝑀𝑦=−𝑐𝑀𝑦𝐽𝑦
cos 𝜑(𝑤𝑦cos 𝜑−𝑤𝑧sin 𝜑) + 𝑤𝑥𝑤𝑧(𝐽𝑥−𝐽𝑧),and
^
𝑀𝑧=−𝑐𝑀𝑧𝐽𝑧(𝑤𝑦tan 𝜑+𝑤𝑧)
(57)
is obtained for the Equation (49).
24
Methodology
The control law 𝛼from Equation (49) ensures both the reachability condition
𝑠𝛼˙𝑠𝛼<0and the sliding condition ˙
𝑉𝛼=1
2
𝑑
𝑑𝑡 𝑠2
𝛼using Lyapunov’s stability analysis,
given below. Consider a positive definite scalar function for the thrust 𝑇and torques
𝑀𝑖as
𝑉𝛼=1
2𝑠2
𝛼(58)
and its derivative leads to
˙
𝑉𝛼=𝑠𝛼˙𝑠𝛼(59)
where ˙𝑠𝛼is given in Equations (53) to (56). Keeping the control law 𝛼from Equa-
tion (49) and the equivalent control ^𝛼from Equation (57) in Equation (59),
˙
𝑉𝛼=−𝜂|𝑠𝛼|<0(60)
is obtained by letting 𝐾𝛼;
𝐾𝑇=𝑚
cos 𝜃cos 𝜑,
𝐾𝑀𝑥=𝐽𝑥,
𝐾𝑀𝑦=𝐽𝑦
cos 𝜑, and
𝐾𝑀𝑧=𝐽𝑧cos 𝜃
cos 𝜑,
(61)
where 𝜂= 1, a strictly positive constant, and |𝑠𝛼|=𝑠𝛼sat(𝑠𝛼
𝛽𝛼). Hence, Equa-
tion (60) states that all the system trajectories point towards the sliding surface 𝑠𝛼in
a finite time. To be specific, once on the surface, the system trajectories stay on the
surface. The conditions are verified by 𝑠𝛼and therefore the inner closed-loop system
is guaranteed to be stable.
25
4 Conclusion
The objective of this thesis is to design the distributed formation control for swarms
of UAVs, arranged in tightly and loosely coupled architectures, which addresses the
challenges of scalability, collision avoidance, failure recovery, energy efficiency, and
control performance. A comprehensive literature review of the general research field
of UAVs with a particular emphasis on swarms is presented in Publication I.
The overall system design is based on the formation control of swarms of homo-
geneous UAVs in a distributed network of leader-follower and leaderless structures
in Publications II-IV and Publication V respectively. The leader-follower structure
is prone to error propagation and sensitive to the leader’s failure. Furthermore, the
distributed networks provide user control, system dependability, scalability, and pri-
vacy. They behave in a transparent manner, for example, location data is shared and
decision-making is divided between the nodes. However, these can become cen-
tralised, and can also refer to dispersed networks with a top-down approach. As a
result, troubleshooting is easier as system failures, incursions, and crashes may be
traced back to specific nodes, making it easier to pinpoint the source of the problem.
The model of each node in the swarm formation is based on the nonlinear (Publica-
tions II, IV, and V) and linear (Publication III) dynamic models of a quadcopter, i.e.
a UAV. Exhibiting the concept of the proposed control designs, this model presents
itself as a case study for swarms of UAVs. This addresses Research Question 1.
The overall control scheme of the formation control of swarms of UAVs is di-
vided into high- and low-level control. In the proposed method, an HLC is integrated
with an LLC. In the HLC, the setpoints, i.e. desired points of tracking, are generated
for the swarms of UAVs considering the complex movement of a hierarchical forma-
tion, soft landing, and failure recovery. This addresses Research Question 2. In the
LLC, a feedback controller is designed to track the setpoints and stabilise the posi-
tions of UAVs for smooth manoeuvre in a swarm formation. This addresses Research
Question 3.
Due to the hybrid nature of the whole system architecture (system design and
control methods), the dependability of each node in terms of its location gives rise
to problems in the adjustment of the controller’s design parameters. In other words,
deviations of a node might cause significant unwanted changes in the other nodes’ lo-
cations, especially in cases of leader-follower structures. From the HLC perspective,
the main contribution is to propose continuous path planning which can quickly re-
26
Conclusion
act to events. This procedure is defined as the mapping problem, which describes the
best shape of the desired swarm formations. Besides this, from the LLC perspective,
the main contribution is to manoeuvre the nodes smoothly. This problem addresses
the optimal tracking movement of each node from its initial to final position in a
swarm formation. Hence, the overall contribution of this thesis can be outlined as
follows:
• This work addresses the problem of a distributed formation control that is
based on the dynamics of the overall swarm and the local dynamics of each
node via implementing the partial controllers i.e., the top-down, bottom-up
approach. The formation complexity of a large system of tightly and loosely
coupled swarms of UAVs is reduced. To achieve better scalability, a methodol-
ogy is proposed in which an HLC is integrated with an LLC so that the swarm
performs the setpoint decision-making and an individual node tracks the given
trajectory.
• To avoid collisions among nodes, continuous path planning is proposed in the
HLC. The hierarchical formation and soft landing are executed using a fixed
formation in Publications II and III respectively. The reconfiguration of the
formation after a single failed node is implemented using a shortest path algo-
rithm in Publication IV. For multiple failed nodes, the reconfiguration of the
formation is implemented using combinatorial algorithms (i.e., distance- and
time-optimal algorithms) and the TPS method in Publication V. The objec-
tives of the post-failure reconfiguration are to provide collision avoidance and
smooth energy-efficient movement.
• To manoeuvre the nodes smoothly, optimal tracking movement is proposed in
the LLC. The tracking of setpoints and stabilisation of each node for smooth
manoeuvre is handled by a nonlinear SMC with PD control in Publication II
and a linear LQR with integral action in Publications II-V. The control per-
formance of each node is improved by reducing the system’s settling time as
much as possible without introducing oscillations in the response. The param-
eters of the controllers are determined through testing of the overall dynamics
of the swarm, and convergence towards the setpoints is guaranteed.
4.1 Future Work
All the work in this thesis is based on simulations, and it would be valuable to con-
sider experimental studies on small-scale UAVs. Also, potential challenges are dis-
cussed in [3].
There can be various directions in which this work can be extended. For in-
stance, variations of system design can be implemented to achieve scalability such
27
Anam Tahir
as (1) Heterogeneous models of UAVs or the hybrid combination of homogeneous
and heterogeneous UAVs can be considered. (2) Centralised, decentralised or cou-
pling of centralised/decentralised distributed networks can be studied. (3) Similarly,
virtual, behaviour-based or their combination with leader-follower structures can be
examined.
In a leader-follower structure, the further the follower(s) node is from its re-
spective leader(s)/sub-leader(s) node, the larger oscillations can be observed in a
formation. This effect can be compensated/cancelled using other methods such as
model predictive control and reinforcement learning [40; 41; 42]. Another possible
extension is to improve the proposed solutions to gain better overall system control
performance such as robust stability, collision avoidance, and energy efficiency.
28
5 Overview of Publications
5.1 Swarms of Unmanned Aerial Vehicles – A Survey
UAVs, in the market, come with diversity in the number of propellers. They can
also be grouped based on their size, range, and equipment. The sizes can be either
nano, mini, regular, or large while the range can be either very close, close, short,
mid, or endurance. In this Publication, a survey-type study on UAVs and swarms
of UAVs is presented, with a special focus on quadcopters. The mechanics, func-
tionality, organisation, modelling, applications, and autonomy aspects of such UAVs
and their swarms are discussed. In addition, the Publication includes the result of
an online survey in order to get a picture of public awareness regarding the use of
UAV technology. The participants of this survey were from different countries and
associated with several professional fields. The results showed that although a large
proportion of the responders were concerned about the privacy and security concerns
of swarms of UAVs, on the other hand, it can be quite useful in disaster management,
environmental mapping, and search and rescue activities.
5.2 Comparison of Linear and Nonlinear Methods for
Distributed Control of a Hierarchical Formation of
UAVs
A key problem in cooperative robotics is the maintenance of a geometric configura-
tion during movement. As a solution for this, a multi-layered and distributed control
system is proposed for the swarm of UAVs in the formation of hierarchical levels
based on the leader-follower structure. The complexity of developing a large system
can be reduced in this way. To ensure the tracking performance and response time
of the ensemble system, nonlinear and linear control designs are presented; (a) SMC
connected with a PD controller and (b) LQR with integral action respectively. The
safe travel distance strategy for collision avoidance is introduced and integrated into
the control designs for maintaining the hierarchical states in the formation. Both de-
signs provide rapid adoption with respect to their settling time without introducing
oscillations for the dynamic flight movement of vehicles in the cases of (1) nominal,
(2) plant-model mismatch, and (3) external disturbance inputs. Also, the nominal
settling time of the swarm is improved by 44% on average when using the nonlinear
29
Anam Tahir
method as compared to the linear method. Furthermore, the proposed methods are
fully distributed so that each UAV autonomously performs the feedback laws in order
to achieve better modularity and scalability.
5.3 Navigation System for Landing a Swarm of Auto-
nomous Drones on a Movable Surface
The development of a navigation system for the landing of a swarm of UAVs on a
movable surface is one of the major challenges in building a fully autonomous plat-
form. Hence, the purpose of this study is to investigate the behaviour of a swarm
of ten UAVs under the mission of soft landing on a movable surface that has con-
stant speed with oscillations. This swarm, arranged in a leader-follower hierarchical
manner, has distributed control units based on LQR with an integral action method.
Furthermore, to prevent UAVs from landing arbitrarily, the leader node takes the
feedback of translational coordinates from the movable surface and adjusts its posi-
tion accordingly. Hence, each follower tracks the leader’s trail with offsets, taking
collision avoidance into account. The design parameters of controllers are mapped
in such a way that the simulations demonstrate the feasibility and great potential of
the proposed method.
5.4 Development of a Fault-Tolerant Control System for
a Swarm of Drones
One of the important challenges in an autonomous swarm of UAVs is the dependabil-
ity of the swarm to continue its mission. Engine failure or propeller disintegration
poses a significant risk to the operation of each node of the swarm, and if it happens
the system should be able to tolerate such malfunction by reconfiguring the swarm
and reforming if it is necessary. In this Publication, a fault-tolerant control system
for an autonomous leader-follower-based swarm of UAVs is presented. For defining
the fault model, the full failure of an engine is considered as an emergency situation,
and the controller of each node is facilitated to reconfigure the swarm by imposing
a bottom-up reformation to bypass the faulty node, which keeps the formation intact
as much as possible. The simulation results show the effectiveness of the proposed
method with respect to reliability and robust stability.
5.5 Energy-Efficient Post-Failure Reconfiguration of
Swarms of Unmanned Aerial Vehicles
In this Publication, the reconfiguration of swarms of UAVs after simultaneous fail-
ures of multiple nodes is considered. The objectives of the post-failure reconfig-
uration are to provide collision avoidance and smooth energy-efficient movement.
30
Overview of Publications
To incorporate such a mechanism, three different failure recovery algorithms are
proposed, namely TPS, distance- and time-optimal algorithms. These methods are
tested on six swarms, with two variations on failing nodes for each swarm. The sim-
ulation results of reconfiguration show that the execution of such algorithms main-
tains the desired formations with respect to avoiding collisions at run-time. Also,
the results show the effectiveness of the proposed methods concerning the distance
travelled, kinetic energy, and energy efficiency. As expected, the distance-optimal
algorithm gives the shortest movements, and the time-optimal algorithm gives the
most energy-efficient movements. The TPS is also found to be energy-efficient and
has less computational cost than the other two proposed methods. Despite the sug-
gested heuristics, these are combinatorial in nature and might be difficult to use in
practice. Furthermore, the use of the regularisation parameter 𝜆in TPS is also in-
vestigated, and it is found that too large values of 𝜆can lead to incorrect locations,
including multiple nodes in the same location. In fact, it is found that using 𝜆= 0
works well in all cases.
31
List of References
[1] Gaemus Collins, David Twining, and Joshua Wells. Using Vessel-Based Drones to Aid Commer-
cial Fishing Operations. In OCEANS 2017, pages 1–5, Aberdeen, UK, 2017.
[2] Małgorzata Polkowska. Technical and Legal Problems of Space Drones. Zeszyty Naukowe SGSP,
85:233–249, 2023.
[3] Syed Agha Hassnain Mohsan, Nawaf Qasem Hamood Othman, Yanlong Li, Mohammed H. Al-
sharif, and Muhammad Asghar Khan. Unmanned Aerial Vehicles (UAVs): Practical Aspects,
Applications, Open Challenges, Security Issues, and Future Trends. Intelligent Service Robotics,
16:109–137, 2023.
[4] Marianne Harbo Frederiksen and Mette Præst Knudsen. Drones for Offshore and Maritime Mis-
sions: Opportunities and Barriers. SDU Centre for Integrative Innovation Management, Den-
mark, 2018.
[5] Christopher S. Tang and Lucas P. Veelenturf. The strategic role of logistics in the industry 4.0 era.
Transportation Research Part E: Logistics and Transportation Review, 129:1–11, 2019.
[6] Manel Khelifi and Ismail Butun. Swarm Unmanned Aerial Vehicles (SUAVs): A Comprehensive
Analysis of Localization, Recent Aspects, and Future Trends. Journal of Sensors, 8600674:1–10,
2022.
[7] Ursula K. Verfuss, Ana Sofia Aniceto, Danielle V. Harris, Douglas Gillespie, Sophie Fielding,
Guillermo Jim´
enez, Phil Johnston, Rachael R. Sinclair, Agnar Sivertsen, Stian A. Solbø, Rune
Storvold, Martin Biuw, and Roy Wyatt. A Review of Unmanned Vehicles for the Detection and
Monitoring of Marine Fauna. Marine Pollution Bulletin, 140:17–29, 2019.
[8] Farha Jahan, Weiqing Sun, Quamar Niyaz, and Mansoor Alam. Security Modeling of Au-
tonomous Systems: A Survey. ACM Computing Surveys, 52(5, Article 91):1–34, 2019.
[9] Hui-Min Huang. Autonomy Levels for Unmanned Systems (ALFUS) Framework, volume I: Ter-
minology version 2.0. National Institute of Standards and Technology, Special Publication 1011-
I-2.0, 2008.
[10] Kiattisin Kanjanawanishkul. Formation Control of Mobile Robots: Survey. UBU Engineering
Journal, 4(1):50–64, 2011.
[11] Bassem Hichri, Abir Gallala, Francesco Giovannini, and Slawomir Kedziora. Mobile Robots Path
Planning and Mobile Multirobots Control: A Review. Robotica, Cambridge University Press, 40
(12):4257–4270, 2022.
[12] John Thomas and Pam Mantri. Complex Adaptive Blockchain Governance. MATEC Web of
Conferences, 223(01010):1–23, 2018.
[13] Payam Nourizadeh, Aghil Yousefi-Koma, and Moosa Ayati. Design and Implementation of a
Fuzzy Adaptive Controller for Time-Varying Formation Leader-Follower Configuration of Non-
holonomic Mobile Robots. arXiv,cs.RO, 2205.11174:1–11, 2022.
[14] Paul Wang and Fred Y. Hadaegh. Coordination and Control of Multiple Microspacecraft Moving
in Formation. Journal of the Astronautical Sciences, 44(3):315–355, 1996.
[15] D. Galzi and Yuri Shtessel. UAV Formations Control Using High Order Sliding Modes. In 2006
American Control Conference, pages 4249–4254, Minneapolis, MN, USA, 2006.
[16] Ben Yun, Ben M. Chen, Kai Yew Lum, and Tong H. Lee. Design and Implementation of a Leader-
Follower Cooperative Control System for Unmanned Helicopters. Journal of Control Theory and
Applications, 8(1):61–68, 2010.
32
LIST OF REFERENCES
[17] Omar Mechali, Limei Xu, Xiaomei Xie, and Jamshed Iqbal. Theory and Practice for Autonomous
Formation Flight of Quadrotors via Distributed Robust Sliding Mode Control Protocol with Fixed-
Time Stability Guarantee. Control Engineering Practice, 123(105150):1–26, 2022.
[18] M. Anthony Lewis and Kar-Han Tan. High Precision Formation Control of Mobile Robots Using
Virtual Structures. Autonomous Robots, 4(4):387–403, 1997.
[19] Tobias Paul, Thomas R. Krogstad, and Jan Tommy Gravdah. Modelling of UAV Formation Flight
using 3D Potential Field. Simulation Modelling Practice and Theory, 16(9):1453–1462, 2008.
[20] Zhou Chao, Shao-Lei Zhou, Lei Ming, and Wen-Guang Zhang. UAV Formation Flight Based on
Nonlinear Model Predictive Control. Mathematical Problems in Engineering, 2012(Article ID
261367):1–15, 2012.
[21] Tucker Balch and Ronald C. Arkin. Behavior-Based Formation Control for Multirobot Teams.
IEEE Transactions on Robotics and Automation, 14(6):926–939, 1998.
[22] Jonathan R. T. Lawton, Randal W. Beard, and Brett J. Young. A Decentralized Approach to
Formation Maneuvers. IEEE Transactions on Robotics and Automation, 19(6):933–941, 2003.
[23] Derek Bennet and C. R. McInnes. Verifiable Control of a Swarm of Unmanned Aerial Vehicles.
Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineer-
ing, 223(7):939–953, 2009.
[24] Kwang-Kyo Oh, Myoung-Chul Park, and Hyo-Sung Ahn. A Survey of Multi-Agent Formation
Control. Automatica, 53:424–440, 2015.
[25] Samir Bouabdallah and Roland Siegwart. Full Control of a Quadrotor. In 2007 IEEE/RSJ Inter-
national Conference on Intelligent Robots and Systems, pages 153–158, San Diego, CA, USA,
2007.
[26] Robert Niemiec and Farhan Gandhi. A Comparison Between Quadrotor Flight Configurations. In
42nd Annual European Rotorcraft Forum, Lille, France, 2016.
[27] Mohammad Fatin Fatihur Rahman, Shurui Fan, Yan Zhang, and Lei Chen. A Comparative Study
on Application of Unmanned Aerial Vehicle Systems in Agriculture. Agriculture, 11(1):1–26,
2021.
[28] FRANTIˇ
SEK ˇ
Solc. Modelling and Control of a Quadrocopter. Advances in Military Technology,
5(2):29–38, 2010.
[29] Petr Gabrlik, Vlastimil Kriz, and Ludek Zalud. Reconnaissance Micro UAV System. Acta Poly-
technica CTU Proceedings, 2:15–21, 2015.
[30] Quan Quan. Introduction to Multicopter Design and Control. Springer, Singapore, 2017.
[31] Changcai Yang, Yizhang Liu, Xingyu Jiang, Zejun Zhang, Lifang Wei, Taotao Lai, and Riqing
Chen. Non-Rigid Point Set Registration via Adaptive Weighted Objective Function. IEEE Access,
6:75947–75960, 2018.
[32] Ping Guo, Wei Hu, Haibing Ren, and Yimin Zhang. PCAOT: A Manhattan Point Cloud Reg-
istration Method Towards Large Rotation and Small Overlap. In 2018 IEEE/RSJ International
Conference on Intelligent Robots and Systems (IROS), pages 7912–7917, Madrid, Spain, 2018.
[33] Andriy Myronenko and Xubo Song. Point Set Registration: Coherent Point Drift. IEEE Transac-
tions on Pattern Analysis and Machine Intelligence, 32(12):2262–2275, 2010.
[34] Haili Chui and Anand Rangarajan. A New Algorithm for Non-Rigid Point Matching. In Pro-
ceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat.
No.PR00662), pages 44–51 vol.2, Hilton Head, SC, USA, 2000.
[35] Michael Safonov and Michael Athans. Gain and Phase Margin for Multiloop LQG Regulators.
IEEE Transactions on Automatic Control, 22(2):173–179, 1977.
[36] Qasim Ali and Sergio Montenegro. Explicit Model Following Distributed Control Scheme for
Formation Flying of Mini UAVs. IEEE Access, 4:397–406, 2016.
[37] Catherine Mass´
e, Olivier Gougeon, Duc-Tien Nguyen, and David Saussi´
e. Modeling and Control
of a Quadcopter Flying in a Wind Field: A Comparison Between LQR and Structured H∞
Control Techniques. In 2018 International Conference on Unmanned Aircraft Systems (ICUAS),
pages 1408–1417, Dallas, TX, USA, 2018.
33
[38] Diego A. Mercado, Rafael S. Castro, and Rogelio Lozano. Quadrotors Flight Formation Control
Using a Leader-Follower Approach. In 2013 European Control Conference (ECC), pages 3858–
3863, Zurich, Switzerland, 2013.
[39] Jean-Jacques E. Slotine and Weiping Li. Applied Nonlinear Control. Englewood Cliffs, N.J:
Prentice-Hall, 1991.
[40] Soyeon Koo, Seungkeun Kim, and Jinyoung Suk. Model Predictive Control for UAV Automatic
Landing on Moving Carrier Deck with Heave Motion. IFAC-PapersOnLine, 3rd IFAC Workshop
on Multivehicle Systems, 48(5):59–64, 2015.
[41] Emil Bostr¨
om and Erik B¨
orjesson. Autonomous Landing of an Unmanned Aerial Vehicle on an
Unmanned Ground Vehicle using Model Predictive Control. Master of Science Thesis in Electrical
Engineering, Department of Electrical Engineering, Link¨
oping University, Link¨
oping, Sweden,
2022.
[42] Fadi AlMahamid and Katarina Grolinger. Autonomous Unmanned Aerial Vehicle Navigation
using Reinforcement Learning: A Systematic Review. Engineering Applications of Artificial
Intelligence, 115(105321):1–20, 2022.
Anam TahirF 27ANNALES UNIVERSITATIS TURKUENSIS
ISBN 978-951-29-9412-0 (PRINT)
ISBN 978-951-29-9411-3 (PDF)
ISSN 2736-9390 (PRINT)
ISSN 2736-9684 (ONLINE)
Painosalama, Turku, Finland 2023
