ADAPTIVE CONTROLLERS FOR QUADROTORS CARRYING UNKNOWN PAYLOADS

Thesis submitted in partial fulfillment
of the requirements for the degree of

Master of Science
in
Electronics and Communication Engineering
by Research

by

VISWA NARAYANAN S.
2019702013
<EMAIL_ADDRESS>
International Institute of Information Technology
Hyderabad - 500 032, INDIA
AUGUST 2021

Copyright © VISWA NARAYANAN S., 2021
All Rights Reserved

International Institute of Information Technology
Hyderabad, India
CERTIFICATE

It is certified that the work contained in this thesis, titled “Adaptive
controllers for quadrotors carrying unknown payloads” by VISWA NARAYANAN S.,
has been carried out under my supervision and is not submitted elsewhere for a
degree.

|
---|---
Date |  Adviser: Prof. SPANDAN ROY

To SRM Team Humanoid and the scientific community

## Acknowledgments

Like Newton famously said, ”If I have seen further, it is by standing on the
shoulders of giants” I have come so far only with the guidance and support of
all my well-wishers. I want to thank them and dedicate my thesis to each and
every one of them.

First and foremost, I am grateful to Prof. K. Madhava Krishna for giving me an
opportunity to explore and learn robotics at the Robotics Research Center,
IIIT Hyderabad. Only because of that opportunity I had got a fundamental
perspective on how to contribute to research. I am equally grateful to Dr.
Spandan Roy for guiding me and supporting me to initiate my research and carry
it on. Also, the love and support from my family in all the decisions that I
have made are incomparable.

I am fortunate to have had great friends who have pushed me beyond my comfort
zone and take research seriously. I am in debt to the trust that Hari Krishnan
and Vikram Kumar have had in me while starting a new humanoid robotics team at
SRM University. I belong to the community of roboticists only because the
doors of SRM Team Humanoid opened for me. I am glad that I have met and been
in contact with Aditya Sripada for all our technical and philosophical
discussions.

I am thankful to Robotics Research Center (RRC), which welcomed me with open
arms. The valuable mentoring by Gourav Kumar has given me a comprehensive
overview of aerial robotics. Not to forget the significant partnership with
Suraj in coursework and research that has helped me complete my degree in
time. I thank everyone in RRC for the acceptance, inspiration and motivation
you have given me from the beginning.

Finally, I have to thank the institute for providing a platform to aspiring
minds to take up the dream of becoming a researcher and developing themselves
in the process.

Thank you all.

## Abstract

With the advent of intelligent transport, quadrotors are becoming an
attractive solution while lifting or dropping payloads during emergency
evacuations, construction works, etc. During such operations, dynamic
variations in (possibly unknown) payload cause considerable changes in the
system dynamics. However, a systematic control solution to tackle such varying
dynamical behaviour is still missing. In this work, two control solutions are
proposed to solve two specific problems in aerial transportation of payload,
as mentioned below.

In the first work, we explore the tracking control problem for a six degrees-
of-freedom quadrotor carrying different unknown payloads. Due to the state-
dependent nature of the uncertainties caused by variation in the dynamics, the
state-of-the-art adaptive control solutions would be ineffective against these
uncertainties that can be completely unknown and possibly unbounded a priori.
In addition, external disturbances such as wind while following a trajectory
in all three positions and attitude angles. Hence, an adaptive control
solution for quadrotors is proposed, which does not require any a priori
knowledge of the parameters of quadrotor dynamics and external disturbances.

The second work is focused on the interchanging dynamic behaviour of a
quadrotor while loading and unloading different payloads during vertical
operations. The control problem to maintain the desired altitude is formulated
via a switched dynamical framework to capture the interchanging dynamics of
the quadrotor during such vertical operations, and a robust adaptive control
solution is proposed to tackle such dynamics when it is unknown.

The stability of the closed-loop system employing both the proposed
controllers are studied analytically via Lyapunov theory. Further, the
effectiveness of the proposed solutions is verified by deploying the
controller on a realistic simulator under various scenarios and by testing the
performance by comparing them with state-of-the-art controllers.

## Contents

Chapter Page

toc

## List of Figures

Figure Page

lof

## List of Tables

Table Page

lot

##  Chapter 1 Introduction

Robotics has been one of the significant fields of interest owing to science
fiction and fantasies. The idea of machines replacing a human in everyday
tasks is so beneficial because of two main reasons. Firstly, the inefficiency
of humans in doing mundane tasks. A free-thinking human is inefficient in
performing repetitive tasks due to constraints such as boredom, distraction
etc. The other reason is that machines (or robots in specific) can transcend
the mechanical limitations of a human. Humans somehow wanted to enslave
machines for these two reasons, and hence these machines were named robots
(meaning ‘forced labours’).

Much similar to the evolution of living beings, there has been a notable
evolution of robots too. Humans invented robots for specific operations when
humanoid fantasies were far-fetched and impractical with the technologies of
their time. Hence, robots came in numerous feasible mechanical forms with
task-specific features very similar to various living things. One significant
step of robotic evolution was the autonomy of a robotic system, which relieves
humans from the loop and drastically reduces the need for manual labour.

The autonomy of robots (or autonomous systems in general) posed a need for a
new science stream called control. A robotic system can be defined as an
integrated unit of machinery that coordinates to perform a task. In this
context, an autonomous system is a robot that can perform tasks without being
manually controlled by a human.

Autonomous vehicles/ mobile robots are fantasies brought to life with the
advancement in control techniques. Autonomous vehicles are extensively used
not only for research purposes but also for commercial applications.
Autonomous vehicles are of different forms ranging from cars to ships to
quadrotors. While the use cases of aerial and marine robots are limited due to
safety constraints, road vehicles (carrying humans) are equipped with an
autopilot feature already. So, every major automobile manufacturer is shifting
its focus towards autonomous vehicles.

Control of autonomous vehicles is a broad category ranging from a high-level
control in choosing an optimal trajectory to the low-level control in deciding
the appropriate actuator inputs. The problem statement of each level of
control is different from one another. For example, the high-level control
might be a Model Predictive Control, which would have to give desired setpoint
velocities to actuators, assuming an ideal actuator model. However, when it
comes to low-level control, the goal would be to calculate the desired forces
and torques required to follow the velocity or position trajectory that has
been given as an input to the controller while relying on prior information
about the dynamics of the system. An even lower level controller would
translate the force and torque inputs into current or fuel output.

###  1.1 Aerial Robotics

Aerial robots are one of the best examples to show how humans invented
machines to surpass their physical limitations. Aerial transportation has
allowed a species that perceives the universe in three dimensions to exploit
the third dimension for efficient travelling. As modern-day researchers on
autonomous vehicles rightfully say, “the additional dimension would solve the
exponentially increasing road traffic problems”.

Aerial vehicles are broadly classified as fixed-wing and rotary-wing
quadrotors based on their mechanical design. Unmanned aerial vehicles (UAV) is
a breakthrough in the applications of aerial vehicles. UAVs are not
necessarily autonomous. They can also be teleoperated. In both cases, control
plays an incredible role. Similar to other applications, the control problem
of UAVs is also a broad category. The focus of this particular research is on
low-level control of quadrotors. Control of a UAV is exceptionally critical
compared to that of rovers (ground robots) because there is no braking
mechanism in aerial robots when the system goes unstable.

The mechanical advantage in the structure of the fixed-wing aircraft makes it
convenient for the transport of cargo and humans. It is widely used for
personal, commercial, research, and defence purposes. However, the major
disadvantage of fixed-wing aircraft is its take-off mechanism, which needs a
runway, non-holonomic design, and its inability to hover on a position in the
air. These disadvantages make them infeasible in many applications such as
indoor goods transfer, disaster relief, surveying etc. and make room for
rotary-wing quadrotors.

###  1.2 Dynamics of Quadrotors

Modelling any robotic system can be divided into four sub-sections: model
representation, reference frames, representation of state variables, and
dynamic equations of the system. Though there are different methods to model
the dynamics in each subsection, I would focus on the method used in my works.

####  1.2.1 Model Representation

For complete dynamic representation, including non-linearities, a set of
differential equations is used. For simplified modelling to use a linear
controller, state-space representation might be convenient. However, it would
not be optimal to design nonlinear controllers. In general, the UAV dynamics
can be of a time-invariant differential form, as shown in the following
equation

$\displaystyle
a_{n}\frac{d^{n}y}{dt^{n}}+a_{n-1}\frac{d^{n-1}y}{dt^{n-1}}+a_{n-2}\frac{d^{n-2}y}{dt^{n-2}}+...+a_{1}\frac{dy}{dt}+a_{0}y=bu$

where the left hand side represents the state variable (the variable that
needs to be controlled) $y$ multiplied by gains $a_{i}$ and its derivatives,
and the right hand side represents the control variable $u$ scaled by a factor
of $b$. Depending on which degree of control is required, the differential
equation of sufficient order is chosen. For a robotic system, typically the
control inputs are forces and torques to control position and velocity of each
degree-of-freedom. Hence, a second order differential equation is chosen.

####  1.2.2 Reference Frames

A crucial step in modelling the dynamics is to identify the frames of
reference. For quadrotor applications, two reference frames are considered for
ease of representation: an inertial frame to represent the absolute position
and orientation of the quadrotor and a body-fixed frame to represent the body
forces and torques.

#####  1.2.2.1 Inertial Frame

The inertial reference frame is where a body with zero net force acting on it
is either at rest or in motion along a straight line. It is generally taken
the ground fixed plane. So the frame is stationary if we neglect the movement
of the Earth. It is also known as the World frame and is taken such that its
X-axis is aligned with the forward direction, Z-axis in the upward direction
and Y-axis is chosen based on right-hand rule. Differential equations to
represent the system dynamics is written inertial frame reference.

#####  1.2.2.2 Body-Fixed Frame

The body-fixed frame is fixed to the quadrotor with the same axes convention
as in the inertial frame. Every frame has three orthonormal axes passing
through the center-of-mass (CoM). For convenience, these axes are chosen as
the principal axes for the body-fixed frame similar to the inertial frame.
Forces from the propellers act on the body-fixed frame. Considering the
example of a quadrotor, the thrusts from the motors produce forces and torques
in the body-fixed frame. So, the angular velocities are represented in the
body-fixed frames.

Figure 1.1 shows the inertial and body-fixed frames with origins at $O_{W}$
and $O_{B}$, respectively. The position vector $p$ gives the relative position
of CoM of the quadrotor (origin of body-fixed frame) in the inertial frame.

Figure 1.1: A schematic of reference frames.

####  1.2.3 Representation of State Variables

Typically the pose of a quadrotor is represented by using the position vector
$[x,y,z]^{T}$ and the orientation $[\phi,\theta,\psi]^{T}$, where
$(\phi,\theta,\psi)$ represent roll, pitch and yaw angles respective. Though
there are other conventions for representing the pose of a quadrotor, we would
see a few methods used in the controller design.

####  1.2.4 Quadrotor Dynamics

The last part of the modelling is to derive the dynamic equations of the
system.

The forces and torques in the quadrotor are only caused by the rotation of the
propellers. Each propeller produces an upward force along the $Z_{B}$ axis in
the body-frame and two different torques: one due to the upward force acting
along $X_{B}$ and $Y_{B}$ axes, and the other is due to the spinning of each
propeller, which causes a torque along the $Z_{B}$ axis. The quadrotor is
designed symmetrically about the origin in the $X_{B}Y_{B}$ plane to reduce
complexity in dynamics. Also, the CoM, $O_{B}$ is maintained on the plane of
the propeller.

The quadrotor’s body frame is designed with its origin at the CoM of the
quadrotor and $X_{B}$ and $Y_{B}$ axes along two adjacent arms (the connecting
rod between the centre and the rotor), or 45 degrees on either side of an arm.
The first configuration is called the ”+” (plus) configuration, and the latter
is called the ”x” configuration (Fig. 1.2). Here we demonstrate the dynamics
of the quadrotor in the “x” configuration.

Figure 1.2: A picture of Q450 quadrotor with “x” configuration.

When all propellers are spinning in the same direction with equal speed, the
opposite propellers cancel out the torques along the $X_{B}$ and $Y_{B}$
directions. However, the torque along the $Z_{B}$-axis scales up, and the
quadrotor starts to spin along the $Z_{B}$-axis while holding its position.
So, the opposite propellers spin in the direction, while the adjacent ones
spin in the opposite directions.

As the rotors are always facing downward, the quadrotor’s thrust is always
along the negative $Z_{B}$-axis. This essentially causes the quadrotor to move
upwards against gravity when the body frame $Z_{B}$-axis is aligned with the
world frame $Z_{W}$-axis. The velocity in $X_{B}$ and $Y_{B}$ directions are
caused by pitching and rolling the quadrotor. To make the quadrotor pitch
forward, the motors in the forward direction have to run slower than those in
the backward direction. Similarly, to roll on the right side, the motors on
the left side have to run faster than the ones on the left side and vice
versa. Mismatch in the speeds of rotors can cause a difference between the
torques in clockwise and anti-clockwise directions. This difference would make
the quadrotor yaw. So, the sums of the squares of the speeds of the opposite
motors are maintained to be equal.

In this work we use the dynamic model of the quadrotor that is popularly used
in works such as, [1], [2] and [3]. The relationships between the thrust,
$T_{th}$, torques $\tau_{\phi},\tau_{\theta},\tau_{\psi}$ and the rotor speeds
$\omega_{i}$, where $i=1,2,3,4$ are given by,

$\displaystyle\begin{bmatrix}T_{th}\\\ \tau_{\phi}\\\ \tau_{\theta}\\\
\tau_{\psi}\end{bmatrix}=\begin{bmatrix}C_{T}&C_{T}&C_{T}&C_{T}\\\
0&lC_{T}&0&-lC_{T}\\\ -lC_{T}&0&lC_{T}&0\\\
-C_{T}C_{M}&C_{T}C_{M}&-C_{T}C_{M}&C_{T}C_{M}\end{bmatrix}\begin{bmatrix}\omega^{2}_{1}\\\
\omega^{2}_{2}\\\ \omega^{2}_{3}\\\ \omega^{2}_{4}\end{bmatrix}$ (1.1)

where $C_{T}$ and $C_{M}$ are thrust constant and moment constant
respectively, and $l$ is the arm-length (please refer to Fig 1.3).

Figure 1.3: A schematic representing a quadrotor’s dynamics.

When there are no external disturbances, these are the only forces and torques
the quadrotor would experience other than gravity. As seen from the equations,
there are only four control parameters that can be controlled, the angular
velocities of the rotors. The control of these velocities would facilitate the
control of upward thrust, roll, pitch and yaw. The linear motion in other
directions can be controlled only using the control of roll and pitch. So, the
quadrotor is an underactuated system with 4 control inputs for 6 DoF.

The linear forces acting on the system are gravitational force, $G$ along the
negative $Z_{W}$-axis in the inertial frame and the thrust $T_{th}$, acting
along the positive $Z_{B}$-axis in the body frame. Hence we use the linear
dynamics of the system as given in 1.2.

$\displaystyle m\begin{bmatrix}\ddot{x}\\\ \ddot{y}\\\
\ddot{z}\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\
mg\end{bmatrix}=\mathbf{R}^{T}\begin{bmatrix}0\\\ 0\\\ T_{th}\end{bmatrix}$
(1.2)

where $g$ is the acceleration due to gravity, $m$ is the total mass of the
quadrotor, $R$ is the rotational matrix that is used to map the forces from
the inertial frame to the body fixed frame. As per properties of orthonormal
matrices, $R^{T}$ maps vectors in body frame to the inertial frame. The
rotational dynamics are given by the form,

$\displaystyle\frac{I_{xx}}{l}\ddot{\phi}+\frac{I_{zz}-I_{yy}}{l}\dot{\varphi}\dot{\psi}=\tau_{\phi},$
$\displaystyle\frac{I_{yy}}{l}\ddot{\varphi}+\frac{I_{xx}-I_{zz}}{l}\dot{\phi}\dot{\psi}=\tau_{\varphi},$
$\displaystyle
I_{zz}\ddot{\psi}+(I_{yy}-I_{xx})\dot{\varphi}\dot{\phi}=\tau_{\psi},$ (1.3)

where $l$ is arm-length of the rotor units; $I_{xx},I_{yy},I_{zz}$ are the
inertia terms in $x,y$ and $z$ directions respectively. The rotation matrix
$R$ is given by

$\mathbf{R}=\begin{bmatrix}c_{\psi}c_{\varphi}&s_{\psi}c_{\varphi}&-s_{\varphi}\\\
c_{\psi}s_{\varphi}s_{\phi}-s_{\psi}c_{\phi}&s_{\psi}s_{\varphi}s_{\phi}+c_{\psi}c_{\phi}&s_{\phi}c_{\varphi}\\\
c_{\psi}s_{\varphi}c_{\phi}+s_{\psi}s_{\phi}&s_{\psi}s_{\varphi}c_{\phi}-c_{\psi}s_{\phi}&c_{\varphi}c_{\phi}\end{bmatrix}$

where, $c_{(\cdot)},s_{(\cdot)}$ denote $\cos_{(\cdot)},\sin_{(\cdot)}$; $m$
is the mass of the overall system.

In this work, we have used the following quadrotor dynamics.

$\displaystyle\mathbf{M_{p}\ddot{p}}+\mathbf{G}$
$\displaystyle=\mathbf{\mathcal{T}_{p}}$
$\displaystyle\mathbf{M_{q}(q)\ddot{q}}+\mathbf{C_{q}(\dot{q})\dot{q}}$
$\displaystyle=\mathbf{\mathcal{T}_{q}}$ (1.4)

with

$\displaystyle\mathbf{M_{p}}$ $\displaystyle=\begin{bmatrix}m&0&0\\\ 0&m&0\\\
0&0&m\end{bmatrix},~{}\mathbf{M_{q}}=\begin{bmatrix}\frac{I_{xx}}{l}&0&0\\\
0&\frac{I_{yy}}{l}&0\\\
0&0&I_{zz}\end{bmatrix},~{}\mathbf{G}=\begin{bmatrix}0\\\ 0\\\
mg\end{bmatrix}$ $\displaystyle\mathbf{C_{q}}$
$\displaystyle=\begin{bmatrix}0&0&\frac{I_{zz}-I_{yy}}{l}\dot{\varphi}\\\
\frac{I_{xx}-I{zz}}{l}\dot{\psi}&0&0\\\
0&\frac{I_{yy}-I{xx}}{l}\dot{\phi}&0\end{bmatrix},~{}\mathbf{p}=\begin{bmatrix}x\\\
y\\\ z\end{bmatrix},~{}\mathbf{q}=\begin{bmatrix}\phi\\\ \varphi\\\
\psi\end{bmatrix}$ $\displaystyle\mathbf{\mathcal{T}_{p}}$
$\displaystyle=\begin{bmatrix}\tau_{x}\\\ \tau_{y}\\\
\tau_{z}\end{bmatrix},~{}\mathbf{\mathcal{T}_{q}}=\begin{bmatrix}\tau_{\phi}\\\
\tau_{\varphi}\\\ \tau_{\psi}\end{bmatrix}$

where, $\mathbf{M_{p}}$ and $\mathbf{M_{q}}$ are diagonal matrices that
represent the mass and inertia of the quadrotor respectively; $\mathbf{C_{q}}$
is the Coriolis matrix containing the cross-coupling terms; $\mathbf{p}$ and
$\mathbf{q}$ are the position and orientation vectors of the quadrotor in the
inertial frame, $\mathbf{G}$ is the gravity vector and
$\mathbf{\mathcal{T}_{p}}$ and $\mathbf{\mathcal{T}_{q}}$ are the force and
moment vectors acting on the CoM of the quadrotor.

###  1.3 Control of Quadrotors

The initial works of quadrotor control were focused on reducing the complexity
of the dynamics of the quadrotor. The simplest of all is the hovering mode
control, where the quadrotor is assumed to have no disturbances across the
$X_{W}$ and $Y_{W}$ axes. This type of control can be used to hover the
quadrotor at any given height, and thus, only controlling the altitude of the
quadrotor was sufficient. The earliest of control designs was well-proven
Proportional-Integral-Derivative (PID) controller [4, 5, 6].

Due to the limited applications of hovering mode, the next step in the
development of quadrotor control was feedback linearization [7, 8, 9]: this
approach is also known as small-angle variation control as it assumes that the
variation in roll and pitch to be minimal. This method is ideal for near
hovering flying modes, and the adaptive controller is also designed for this
specific condition [10]. Though small-angle approximation based control is
easy to implement, it might be unstable when the quadrotors need to perform
aggressive manoeuvres.

One of the significant pieces of research on the control of quadrotors is the
geometric controller introduced by [1]. This work proposed a partially
decoupled dynamic model of the quadrotor (cf. Chapter 2 for details) and a
dual-loop control system for the same. The geometric control in the manifold
of rotation matrices makes it suitable for aggressive manoeuvres. Another
notable work in this line came from [11] and their group thereafter. The
completely decoupled dynamic model of quadrotors, however is still in the
theoretical phase and needs additional mechanical systems to realize it in
practice (cf. tiltrotor mechanism [12, 13]. An exceptional survey on the
existing control strategies for autonomous quadrotors can be found in [14].

Nevertheless, the aforementioned works (and references therein) depends
heavily on the accuracy on system modelling and dynamics. In reality, it is
tedious and almost impossible to model the exact dynamics of the quadrotor due
to factors such as nonlinearities, inaccurate parametric knowledge, imbalance
in the mass distribution, external disturbances etc. When quadrotors are
deployed for applications such as surveying, manipulation, or payload
transport [15, 16, 17], their dynamics keep changing due to payload
variations. In this thesis, we particularly look into two types of aerial
transportation and the control problems and solutions for them. We discuss
these two scenarios below:

####  1.3.1 Adaptive Path Tracking of Quadrotors for Aerial Transportation of
Unknown Payloads

Over the past two decades, quadrotors have been a source of considerable
research interest owing to their advantages such as simple structure, vertical
taking off and landing, rapid manoeuvring etc. [18, 19, 20]. Such advantages
are crucial in various military and civil applications such as surveillance,
fire fighting, environmental monitoring, to name a few [21, 22]. Most
recently, global research is more and more interested in smart transport
systems [16, 17].

One of the most common ways to attach a payload to a quadrotor is via a
suspended cable, which allows the quadrotor to retain most of its agility [16,
23, 17]. However, in indoor scenarios, especially in disaster sites, the
quadrotor might need to manoeuvre through constrained altitudes, where using a
suspended payload may not be optimal/desirable. In such scenarios, rigidly
attaching a payload would be a preferred mode, which also provides the
flexibility to autonomously pick up and drop the payload. From a research
point of view, crucial control challenges arise when the mass and inertia of
the quadrotor vary due to unknown payload, imprecise knowledge of the
quadrotor parameters and unknown external disturbances: these uncertainties
make the control design for a payload-carrying quadrotor challenging.

####  1.3.2 Dynamic Payload Lifting Operations: In-flight Payload Variation

Figure 1.4: A schematic representing a quadrotor ascending with dynamic
payload.

In the previous scenario, the payload mass does not vary during the flight.
However, in a typical construction assignment or emergency evacuation, a
quadrotor might be required to lift or drop varying payloads (construction and
disaster relief materials), emergency evacuation from a high rise building,
where humans are rescued from various floors. Dynamic variations in payload
are orchestrated from these situations. One such lifting operation is sketched
in Fig. 1.4 via three phases: (i) in the first phase, the quadrotor rests on
the ground without any payload; (ii) in the second phase, it starts to ascend
with an initial payload; (iii) and in the third phase, a new payload is
attached to the quadrotor as it ascends to its desired height. Similar phases
can be observed during a construction scenario. In a reverse scenario to Fig.
1.4, a quadrotor may descend from a height while releasing payloads at
different heights (e.g., dropping/throwing fire extinguishing materials at
different floors). Clearly, the overall mass/inertia of the quadrotor changes
(switches) during the interchange of these phases.

Though robust controllers [24, 25, 26, 27, 28] might solve the problem of
variation in dynamics to an extent, they also suffer when bounds of
uncertainties are unknown. Hence, an adaptive controller is typically employed
to solve these issues [29, 30, 31, 32, 33, 34, 35, 17, 36, 37]. However,
adaptive control typically requires structural knowledge of the system and
cannot handle unmodelled dynamics, and such prerequisites are difficult to
meet in the cases of unknown payload transport. Further, it is well-known that
under a switched dynamics, typically arising during dynamic payload variation,
conventional controllers are not suitable [38, 39, 40, 41, 42, 43]. In
addition, to the best of our knowledge, no control solution exists to tackle
the interchanging/switched dynamics of a quadrotor in an adaptive setting.
This thesis aims to solve these control challenges.

###  1.4 Preliminaries

####  1.4.1 Stability Notions

An autonomous nonlinear system represented by the following dynamic equation

$\displaystyle{\dot{x}}=f(x(t)),\;\;\;\;x(0)=x_{0}$

where ${\displaystyle x(t)\in\mathbb{R}^{n}}$ represents the system state
vector is said to be stable, the function $f$ attains an equilibrium at the
point $x=x_{e}$, i.e, $f(x_{e})=0$ and $f^{\prime}(x_{e})=0$.

The equilibrium point is said to be:

  * •

Stable, if there exists a bound $\epsilon>0,\delta>0$ such that
$|x(0)-x_{e}|<\delta\implies|x(t)-x_{e}|<\epsilon,\quad\forall t\geq 0$.
Stability makes sure the states starting from close enough to a bound $\delta$
will remain within the bound $\epsilon$.

  * •

Asymptotically stable, if in addition to being stable,
$\lim_{t\rightarrow\infty}\|x(t)-x_{e}\|=0$. Therefore, the system not only
remains within the bound $\epsilon$ but also converges to the equilibrium
point.

  * •

Exponentially stable, if in addition with asymptotic stability there exists
$\alpha>0,\beta>0,\delta>0$ such that
$\|x(t)-x_{e}\|\leq\alpha\|x(0)-x_{e}\|e^{-\beta t},\quad\forall t\geq 0$.
Hence, system converges to the equilibrium at an exponential rate.

  * •

Uniformly Ultimately Bound (UUB): The solutions of $\dot{x}=f(x(t))$ is said
to be uniformly ultimately bound with ultimate bound $b$ if $\exists b>0,$
$c>0$ and for every $0<a<c,$ $\exists T=T(a,b)>0$ such that

$\displaystyle||x(t_{0})||\leq a\implies||x(t)||\leq b,\quad\forall t\geq
t_{o}+T.$ (1.6)

###  1.5 Contributions of the Thesis

Two different problems were highlighted earlier while a quadrotor is carrying
a payload of unknown mass and inertia. While suspended payloads retain the
dynamic properties of the quadrotor, they cannot be used in indoor scenarios
and safety-critical operations. So, the following contributions are made in
this thesis to solve the dynamic problems in quadrotors with rigidly attached
payloads for their respective applications:

  * •

Based on the partly decoupled six degrees-of-freedom dynamic model, an
adaptive sliding mode controller is proposed that tackles variation in
quadrotor dynamics caused by uncertainties while carrying payloads of unknown
mass and inertia, and external disturbances. The proposed controller does not
require any a priori knowledge of the system dynamics and of uncertainty
bound. The closed-loop system stability is analysed via the Lyapunov-based
method. The performance of the proposed scheme is verified using a simulated
quadrotor on the Gazebo framework and in comparison with the state-of-the-art
controllers.

  * •

In the previous case, the case of payload variation during the flight (e.g.,
payload drop during construction, disaster management scenario etc.) was not
considered. Such in-flight payload variations give rise to the switched
dynamics in the quadrotor system. A suitable switched adaptive controller,
which does not require any a priori knowledge of the system dynamics and of
uncertainties, is introduced for tracking any arbitrary altitude trajectory of
the quadrotor. The closed-loop stability is analysed using multiple Lyapunov
functions, and the effectiveness of the proposed controller is verified in
simulation.

###  1.6 Organization of the Thesis

The thesis is organized into four chapters. A brief summary of each chapter is
mentioned below.

  * •

Chapter 1: This introductory chapter gives an overview of aerial robotics,
modelling the dynamics of a quadrotor and state-of-the-art control strategies.
It briefly describes the motivation for this research, the problem
orientation, the pertaining gaps in the literature, the main contributions and
an outline of the thesis.

  * •

Chapter 2: This chapter introduces the adaptive sliding mode controller for a
quadrotor to address the problems involved while carrying payloads of unknown
masses. The quadrotor dynamics are partly decoupled into two loops, with
position tracking as the outer loop and attitude tracking as the inner loop.
The uncertainties in both the loops are modelled and tackled using the
adaptive sliding mode controller strategies. The stability of the controller
is proved using the Lyapunov stability method and verified using a simulated
quadrotor. The results are compared with the state-of-the-art controllers to
show the superiority in the performance.

  * •

Chapter 3: This chapter introduces the fundamentals of the switched mode
controller for in-flight switched dynamics stemming from the addition and
removal of unknown payloads. The quadrotor is modelled using hovering dynamics
with control only on roll, pitch, yaw and altitude. An swtiched adaptive
controller is designed to tackle the uncertainty. The closed-loop system is
proved using dwell-time based multiple Lyapunov method and verified via
simulation settings.

  * •

Chapter 4: This chapter concludes the thesis by summarizing the various
contributions brought out by this thesis. It also provides a brief discussion
about the future work in this line of research.

###  1.7 Symbols and Notations:

The symbols and notations used in the following chapters are as shown in Table
1.1

$\mathbb{R}$ | Real line
---|---
$\mathbb{R}^{+}$ | Real line of positive numbers
$\mathbb{R}^{n}$ | Real space of dimension $n$
$\mathbb{R}^{n}$ | Real matrix of dimension $n\times n$
$\exists$ | there exists
$\forall$ | for all
$\mathbf{I}$ | Identity matrix
$\mathbf{\Xi}>0(<0)$ | Positive (negative) definite matrix
$\lambda_{\min}(\mathbf{\Xi})$ | Minimum eigen value of the matrix $\mathbf{\Xi}$
$\lambda_{\max}(\mathbf{\Xi})$ | Maximum eigen value of the matrix $\mathbf{\Xi}$
$||\mathbf{\Xi}||$ | Euclidean norm of the matrix $\mathbf{\Xi}$
$\operatorname{sgn}(x)$ | Signum of $x$ = $x/||x||$
$\operatorname{sat}(x,\varpi)$ | Saturation of $x$ = $\begin{cases}x/||x||,~{}if~{}||x||\geq\varpi\\\ x/\varpi,~{}otherwise\\\ \end{cases}$
Table 1.1: Nomenclature of various symbols and notations used in the thesis.

##  Chapter 2 Adaptive Path Tracking of Quadrotors for Aerial Transportation
of Unknown Payloads

###  2.1 Introduction

In the quest to operate a quadrotor under various sources of uncertainties,
the control community has inevitably looked into robust control [27, 44, 45,
46] and adaptive control methods [29, 30, 31, 32, 47, 33, 34, 35, 17].
However, robust control methods [27, 44, 45, 46] rely on a priori knowledge of
bounds on uncertainties; on the other hand, the adaptive control designs [29,
30, 31, 32, 33, 34, 35, 17] require a priori knowledge of system structures.
Such constraints are often difficult to be satisfied in practice due to
unknown parametric variations and external disturbances.

To avoid a priori knowledge of bounds on uncertainties, researchers have
recently applied adaptive sliding mode designs [47, 48, 49, 50] considering
uncertainties to be a priori bounded. Nevertheless, when such assumption is
not satisfied (e.g. state-dependent uncertainty), instability cannot be ruled
out (cf. [37, 51]): in the attitude dynamics of a quadrotor, state-dependent
uncertainty naturally occurs owing to the Coriolis terms.

In view of the above discussions, an adaptive control solution for the
quadrotor is still missing in the presence of completely unknown state-
dependent uncertainty and external disturbances. Toward this direction, the
proposed adaptive solution has the following major contributions:

  * •

An adaptive controller for quadrotor is formulated, which does not require any
a priori knowledge of the system dynamic paramters, payload and of
disturbances.

  * •

Differently from [29, 30, 31, 32, 47], the control framework considers six
degrees-of-freedom (DoF) dynamics with unknown external perturbations
affecting both actuated and non-actuated sub-dynamics.

  * •

The closed-loop stability of the system is analysed via Lyapunov-based method
and comparative simulation results suggest significant improvement in tracking
accuracy of the proposed scheme compared to the state of the art.

The rest of the chapter is organised as follows: Section 2.2 describes the
dynamics of quadrotor; Section 2.3 details the proposed control framework,
while corresponding stability analysis is provided in Section 2.4; comparative
simulation results are provided in Section 2.5.

###  2.2 Quadrotor System Dynamics and Problem Formulation

Figure 2.1: Schematic of quadrotor with coordinate frames

Let us consider the Euler-Lagrange system dynamics of a quadrotor model (cf.
Fig. 2.1), widely used in literature ([30])

$\displaystyle m\ddot{p}+G+d_{p}$ $\displaystyle=\tau_{p},$ (2.1)
$\displaystyle J(q)\ddot{q}+C_{q}(q,\dot{q})\dot{q}+d_{q}$
$\displaystyle=\tau_{q},$ (2.2) $\displaystyle\tau_{p}$
$\displaystyle=R_{B}^{W}U,$ (2.3)

where (2.1) and (2.2) are the position and attitude dynamics of the quadrotor,
respectively. Various symbols in the dynamics (2.1) and (2.2) are described as
follows: $m\in\mathbb{R}^{+}$ and $J(q)\in\mathbb{R}^{3\times 3}$ represent
the mass and inertia matrix, respectively;
$p(t)\triangleq\begin{bmatrix}x(t)&y(t)&z(t)\end{bmatrix}^{T}\in\mathbb{R}^{3}$
is the position vector of the center of mass of the quadrotor;
$q(t)\triangleq\begin{bmatrix}\phi(t)&\theta(t)&\psi(t)\end{bmatrix}^{T}\in\mathbb{R}^{3}$
is the orientation/attitude (roll, pitch, yaw angles respectively);
$G\triangleq\begin{bmatrix}0&0&mg\end{bmatrix}^{T}\in\mathbb{R}^{3}$ is the
gravitational force; $C_{q}(q,\dot{q})\in\mathbb{R}^{3\times 3}$ represents
the Coriolis matrix; $d_{p}(t)$ and $d_{q}(t)$ denote the unknown disturbances
in position and attitude dynamics, respectively;
$\tau_{q}\triangleq\begin{bmatrix}u_{2}(t)&u_{3}(t)&u_{4}(t)\end{bmatrix}^{T}\in\mathbb{R}^{3}$
denotes the control inputs for roll, pitch and yaw;
$\tau_{p}(t)\in\mathbb{R}^{3}$ is the generalized control input for position
tracking in Earth-fixed frame, with
$U(t)\triangleq\begin{bmatrix}0&0&u_{1}(t)\end{bmatrix}^{T}\in\mathbb{R}^{3}$
being the force vector in body-fixed frame and
$R_{B}^{W}\in\mathbb{R}^{3\times 3}$ being the $Z-Y-X$ Euler angle rotation
matrix describing the rotation from the body-fixed coordinate frame to the
Earth-fixed frame, given by

$\displaystyle
R_{B}^{W}=\begin{bmatrix}c_{\psi}c_{\theta}&c_{\psi}s_{\theta}s_{\phi}-s_{\psi}c_{\phi}&c_{\psi}s_{\theta}c_{\phi}+s_{\psi}s_{\phi}\\\
s_{\psi}c_{\theta}&s_{\psi}s_{\theta}s_{\phi}+c_{\psi}c_{\phi}&s_{\psi}s_{\theta}c_{\phi}-c_{\psi}s_{\phi}\\\
-s_{\theta}&s_{\phi}c_{\theta}&c_{\theta}c_{\phi}\end{bmatrix},$ (2.4)

where $c_{(\cdot)},s_{(\cdot)}$ and denote $\cos{(\cdot)},\sin{(\cdot)}$
respectively.

As per the celebrated Euler-Lagrange dynamics, the attitude dynamics of
quadrotor satisfy the following properties [30]:

Property 1. The inertia matrix $J(q)$ is uniformly positive definite $\forall
q$ and there exist $\underline{j},\overline{j}\in{R}^{+}$ such that
$\underline{j}I\leq J(q)\leq\overline{j}I$.
Property 2.
$\exists\overline{c}_{q},\overline{d}_{p},\overline{d}_{q}\in\mathbb{R}^{+}$
such that $||C_{q}(q,\dot{q})||\leq\overline{c}_{q}||\dot{q}||$,
$||d_{q}||\leq\overline{d}_{q}$, $||d_{p}||\leq\overline{d}_{p}$.
Property 3. The matrix $(\dot{J}-2C_{q})$ is skew symmetric, i.e., for any
non-zero vector $r$, we have $r^{T}(\dot{J}-2C_{q})r=0$.

The following assumption highlights the available knowledge of various system
parameters for control design:

######  Assumption 2.1.

The system dynamics terms $m,J,C_{q},d_{p},d_{q}$ and their bounds
$\overline{c}_{q},\overline{d}_{p},\overline{d}_{q}$ are unknown for control
design.

######  Remark 2.1 (Knowledge of system dynamics).

Assumption 1 is indeed a design challenge: the objective is to formulate a
control framework without any knowledge of system parametric uncertainties and
external disturbances. Assumption 1 overcomes the need for either precise
knowledge of mass/inertia matrix (cf. [27, 44, 45, 46]), or a priori knowledge
of system structures and of bounds on external disturbances (cf. [29, 30, 31,
32, 33, 34, 35, 17]).

######  Remark 2.2 (Position and attitude tracking co-design).

Tracking control problem of quadrotors can be broadly classified under two
categories: (i) reduced-order model based design (cf. [29, 31, 32, 47]) and
(ii) position and attitude tracking co-design (cf. [30, 11]). Control designs
relying on the first category ignore the non-actuated $(x,y)$ dynamics and
define the control problem as tracking of only the actuated degrees-of-freedom
(DoF) i.e. of $z$ and $(\phi,\theta,\psi)$. Such approach is generally termed
as ‘collocated design’ in the literature of underactuated systems (cf. [51]).
The second category relies on co-designing position and attitude tracking
controller for the six DoF dynamics (2.1)-(2.2).

In this work, we shall follow the co-design approach as the reduced-order
based approach is conservative for relying on a priori boundedness of $(x,y)$
dynamics. Therefore, we take the following standard assumption:

######  Assumption 2.2 ([30, 11, 16]).

Let
$p_{d}(t)\triangleq\begin{bmatrix}x_{d}(t)&y_{d}(t)&z_{d}(t)\end{bmatrix}^{T}$
and $\psi_{d}(t)$ be the desired position and yaw trajectories to be tracked,
which are designed to be sufficiently smooth and bounded. Further, it is
assumed that the following inequality holds for all time:
$-\frac{\pi}{2}<(\phi_{d},\theta_{d})<\frac{\pi}{2}$ radians.

######  Remark 2.3 (Desired roll and pitch).

In position and attitude tracking co-design, the desired roll ($\phi_{d}$) and
pitch $(\theta_{d}$) trajectories are derived based on the computed position
control input $\tau_{p}$ and the desired yaw trajectory $\psi_{d}$ (cf. [11])
and it is discussed later.

Control Problem: Under Properties 1-3, to design an adaptive controller to
track a desired trajectory (cf. Assumption 2) without any knowledge of system
dynamics parameters (cf. Assumption 1).

The following section provides a solution to this control problem.

###  2.3 Controller Co-design

The position and attitude tracking co-design approach consists of simultaneous
design of an outer loop controller for position dynamics (2.1) and of an inner
loop controller for attitude dynamics (2.2). Following this approach, the
proposed control solution is elaborated in the following subsections.

####  2.3.1 Outer Loop Controller

Let us define the position tracking error as $e_{p}\triangleq p-p_{d}$ and a
sliding variable as

$\displaystyle s_{p}$ $\displaystyle=\dot{e}_{p}+\Phi_{p}e_{p}$ (2.5)

where $\Phi_{p}$ is a positive definite gain matrix. Multiplying the time
derivative of (2.5) by $m$ and using (2.1) yields

$\displaystyle
m\dot{s}_{p}=m(\ddot{p}-\ddot{p}_{d}+\Phi_{p}\dot{e}_{p})=\tau_{p}+\varphi_{p}-G$
(2.6)

where $\varphi_{p}\triangleq-(d_{p}+m\ddot{p}_{d}-m\Phi_{p}\dot{e}_{p})$.
Using Property 2, we have

$\displaystyle||\varphi_{p}||$
$\displaystyle\leq\overline{d}_{p}+{m}(||\ddot{p}_{d}||+||\Phi_{p}||||\dot{e}_{p}||).$
(2.7)

Further, let us define
$\xi_{p}\triangleq\begin{bmatrix}e_{p}^{T}&\dot{e}_{p}^{T}\end{bmatrix}^{T}$.
Then using the inequality $||\xi_{p}||\geq||\dot{e}_{p}||$ and boundedness of
the desired trajectories, we have

$\displaystyle||\varphi_{p}||$ $\displaystyle\leq
K_{p0}^{*}+K_{p1}^{*}||\xi_{p}||$ (2.8)

where $K_{p0}^{*}\triangleq{\overline{d}_{p}+m||\ddot{p}_{d}||}$ ,
$K_{p1}^{*}\triangleq m||\Phi_{p}||$ are unknown finite scalars. The outer
loop control law is designed as

$\displaystyle\tau_{p}(t)$
$\displaystyle=-\Lambda_{p}s_{p}(t)-\rho_{p}(t)\operatorname{sgn}(s_{p}(t))+\hat{m}(t)g_{p}$
(2.9a) $\displaystyle\rho_{p}(t)$
$\displaystyle=\hat{K}_{p0}(t)+\hat{K}_{p1}||\xi_{p}||$ (2.9b)

where $\Lambda_{p}$ is a positive definite user-defined gain matrix;
$g_{p}=\begin{bmatrix}0&0&g\end{bmatrix}^{T}$ is the gravity vector;
$\hat{K}_{pi}$ and $\hat{m}$ are the estimates of $K_{pi}^{*}$ and ${m}$,
$i=0,1$ respectively and they are evaluated via the following adaptive laws

$\displaystyle\dot{\hat{K}}_{pi}(t)$
$\displaystyle=||s_{p}(t)||||\xi_{p}(t)||^{i}-\alpha_{pi}\hat{K}_{pi}(t),~{}\hat{K}_{pi}(0)>0$
(2.10a) $\displaystyle\dot{\hat{m}}(t)$
$\displaystyle=-s_{p}(t)^{T}g_{p}-\alpha_{m}\hat{m}(t),~{}\hat{m}(0)>0$
(2.10b)

where $\alpha_{pi},\alpha_{m}\in\mathbb{R}^{+}$ are user-defined design
scalars. Note that eventually $U$ is applied to the system by transforming
$\tau_{p}$ via the relation (2.3) ($R^{W}_{B}$ is invertible rotational
matrix), giving non-zero input only in $z$ direction.

####  2.3.2 Inner Loop Controller

To realize the inner loop controller, it is important to generate the desired
roll ($\phi_{d}$) and pitch ($\theta_{d}$) angles. This process involves
defining an intermediate coordinate frame as the first step (cf. [11]):

$\displaystyle z_{B}$ $\displaystyle=\frac{\tau_{p}}{||\tau_{p}||}$ (2.11a)
$\displaystyle y_{A}$
$\displaystyle=\begin{bmatrix}-s_{\psi_{d}}&c_{\psi_{d}}&0\end{bmatrix}^{T}$
(2.11b) $\displaystyle x_{B}$ $\displaystyle=\frac{y_{A}\times
z_{B}}{||y_{A}\times z_{B}||}$ (2.11c) $\displaystyle y_{B}$
$\displaystyle=z_{B}\times x_{B}$ (2.11d)

where $y_{A}$ is the $y$-axis of the intermediate coordinate frame $A$;
$x_{B}$, $y_{B}$ and $z_{B}$ are the $x$-axis, $y$-axis and $z$-axis of the
body fixed coordinate frame. Given the desired yaw angle $\psi_{d}(t)$ and
based on the computed intermediate axes as in (2.11), $\phi_{d}(t)$ and
$\theta_{d}(t)$ can be determined using the desired body frame axes as
described in [11].

Further, to achieve the attitude tracking control objective, the error in
orientation/attitude is defined as [11]

$\displaystyle e_{q}$
$\displaystyle={((R_{d})^{T}R_{B}^{W}-(R_{B}^{W})^{T}R_{d})}^{v}$ (2.12)
$\displaystyle\dot{e}_{q}$
$\displaystyle=\dot{q}-R_{d}^{T}R_{B}^{W}\dot{q}_{d}$ (2.13)

where $(.)^{v}$ represents vee map, which converts elements of $SO(3)$ to
$\in{\mathbb{R}^{3}}$ [11] and $R_{d}$ is the rotation matrix as in (2.4)
evaluated at ($\phi_{d},\theta_{d},\psi_{d}$).

The sliding variable $s_{q}\in{\mathbb{R}^{3}}$ for inner loop control is
defined as

$\displaystyle s_{q}=\dot{e}_{q}+\Phi_{q}e_{q}$ (2.14)

where $\Phi_{q}$ is a positive definite gain matrix. Multiplying the
derivative of (2.14) by $J$ and using (2.2) yield

$\displaystyle
J\dot{s}_{q}=J(\ddot{q}-\ddot{q}_{d}+\Phi_{q}{\dot{e}_{q}})=\tau_{q}-C_{q}s_{q}+\varphi_{q}$
(2.15)

where
$\varphi_{q}\triangleq-(C_{q}\dot{q}+d_{q}+J\ddot{q}_{d}-J\Phi_{q}\dot{e}_{q}-C_{q}s_{q})$
represents the overall uncertainties in attitude dynamics. Using (28) and
Properties 1 and 2 we have

$\displaystyle||\varphi_{q}||$
$\displaystyle\leq\overline{c}_{q}||\dot{q}||^{2}+\overline{d_{q}}+\overline{j}(||\ddot{q}_{d}||+||\Phi_{q}||||\dot{e}_{q}||)+\overline{c}_{q}||\dot{q}||(||\dot{e}_{q}||+||\Phi_{q}||||q||).$
(2.16)

Further, let us define
$\xi_{q}\triangleq\begin{bmatrix}e_{q}^{T}&\dot{e}_{q}^{T}\end{bmatrix}^{T}$.
Then using the inequalities $||\xi_{q}||\geq||e_{q}||$,
$||\xi_{q}||\geq||\dot{e}_{q}||$, boundedness of the desired trajectories, and
substituting $\dot{q}=\dot{e}_{q}+\dot{q}_{d}$ in $(\ref{q_phi_bound})$ yield

$\displaystyle||\varphi_{q}||$ $\displaystyle\leq
K_{q0}^{*}+K_{q1}^{*}||\xi_{q}||+K_{q2}^{*}||\xi_{q}||^{2},$ (2.17)

where
$K_{q0}^{*}\triangleq\overline{c}_{q}||\dot{q}_{d}||^{2}+\overline{d}_{q}+\overline{j}||\ddot{q}_{d}||$,
$K_{q1}^{*}\triangleq\overline{c}_{q}||\dot{q}_{d}||(3+||\Phi_{q}||)+\overline{j}||\Phi_{q}||$,
$K_{q2}^{*}\triangleq{\overline{c}_{q}||\dot{q}_{d}||(2+||\Phi_{q}||)}$ are
unknown finite scalars. Hence, a state-dependent upper bound occurs naturally
to the attitude tracking system. So, the inner loop control law is designed as

$\displaystyle\tau_{q}(t)$
$\displaystyle=-\Lambda_{q}s_{q}(t)-\rho_{q}(t)\operatorname{sgn}(s_{q}(t)),$
(2.18a) $\displaystyle\rho_{q}(t)$
$\displaystyle=\hat{K}_{q0}(t)+\hat{K}_{q1}(t)||\xi_{q}(t)||+\hat{K}_{q2}(t)||\xi_{q}(t)||^{2},$
(2.18b)

where $\Lambda_{q}$ is a positive definite user-defined gain matrix and
$\hat{K}_{qi}$, $i=0,1,2$ are the estimates of ${K}_{qi}^{*}$ adapted via the
following law:

$\displaystyle\dot{\hat{K}}_{qi}(t)=||s_{q}(t)||||\xi_{q}||^{i}-\alpha_{qi}\hat{K}_{qi}(t),~{}\hat{K}_{qi}(0)>0,$
(2.19)

where $\alpha_{qi}\in\mathbb{R}^{+}$, $i=0,1,2$ are user-defined scalars.

######  Remark 2.4 (On state-dependent uncertainty).

The inequalities (2.8) and (2.17) reveal that the state-dependencies occur
inherently in the upper bound structures of uncertainties via $\xi_{p}$ and
$\xi_{q}$. Therefore, conventional adaptive sliding mode designs such as [47,
48, 49, 50] are not feasible as state-dependent uncertainty cannot be bounded
a priori. On the other hand, the gains $\rho_{p}$ in (2.9b) and $\rho_{q}$ in
(2.18b) are designed according to the state-dependent uncertainty structures
(2.8) and (2.17) respectively.

######  Remark 2.5 (On co-design of outer and inner loop).

Note that the outer and inner loop designs are not fully decoupled process:
rather, the inner loop design process relies on $\tau_{p}$ from the outer
loop. Therefore, these loop designs are coupled and simultaneously affect the
quadrotor system: this makes the proposed design a more realistic one compared
to decoupled dynamics based approaches (cf. [29, 31, 32, 47]) or autopilot
based designs where often linearized or decouple dynamics are considered (cf.
[52, 53]).

###  2.4 Stability Analysis of The Proposed Controller

######  Theorem 2.1.

Under Properties 1-3 and Assumptions 1-2, the trajectories of the closed-loop
systems (2.6) and (2.15) using the control laws (2.9) and (2.18), in
conjunction with the adaptive laws (2.10) and (2.19) are Uniformly Ultimately
Bounded (UUB).

Proof. Note that the solutions of the adaptive gains $(\ref{p_adap_K})$ and
$(\ref{q_adap_K})$ are given by [37]

$\displaystyle\vspace{-0.8cm}\hat{K}_{pi}(t)$
$\displaystyle=\underbrace{\exp(-\alpha_{pi}t)\hat{K}_{pi}(0)}_{\geq
0}+\underbrace{\int_{0}^{t}\exp(-\alpha_{pi}(t-\vartheta))(||{s_{p}}(\vartheta)||||\xi_{p}(\vartheta)||^{i})\mathrm{d}\vartheta}_{\geq
0}$ $\displaystyle\Rightarrow~{}\hat{K}_{pi}(t)$ $\displaystyle\geq
0,~{}i=0,1~{}\forall t\geq 0$ (2.20) $\displaystyle\hat{K}_{qi}(t)$
$\displaystyle=\underbrace{\exp(-\alpha_{qi}t)\hat{K}_{qi}(0)}_{\geq
0}+\underbrace{\int_{0}^{t}\exp(-\alpha_{qi}(t-\vartheta))(||{s_{q}}(\vartheta)||||\xi_{q}(\vartheta)||^{i})\mathrm{d}\vartheta}_{\geq
0}$ $\displaystyle\Rightarrow~{}\hat{K}_{qi}(t)$ $\displaystyle\geq
0,~{}i=0,1,2~{}\forall t\geq 0.$ (2.21)

Closed-loop stability is analysed using the following Lyapunov function

$\displaystyle V$ $\displaystyle=V_{p}+V_{q},$ (2.22)
$\displaystyle\text{where}~{}V_{p}$
$\displaystyle=\frac{1}{2}s_{p}^{T}ms_{p}+\frac{1}{2}\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})^{2}+\frac{1}{2}(\hat{m}-m)^{2}$
(2.23) $\displaystyle V_{q}$
$\displaystyle=\frac{1}{2}s_{q}^{T}Js_{q}+\frac{1}{2}\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})^{2}.\vspace{-0.5cm}$
(2.24)

Using $(\ref{dot_s_p})$ and $(\ref{dot_s_q})$, the time derivatives of
$(\ref{p_lyap})$ and $(\ref{q_lyap})$ yield

$\displaystyle\dot{V}_{p}$
$\displaystyle=s_{p}^{T}m\dot{s}_{p}+\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})\dot{\hat{K}}_{pi}+(\hat{m}-m)\dot{\hat{m}}$
$\displaystyle=s_{p}^{T}(\tau_{p}+\varphi_{p}+G)+\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})\dot{\hat{K}}_{pi}+(\hat{m}-m)\dot{\hat{m}}$
(2.25) $\displaystyle\dot{V}_{q}$
$\displaystyle=s_{q}^{T}J\dot{s}_{q}+\frac{1}{2}s_{q}^{T}\dot{J}s_{q}+\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})\dot{\hat{K}}_{qi}$
$\displaystyle=s_{q}^{T}(\tau_{q}-C_{q}s_{q}+\varphi_{q})+\frac{1}{2}s_{q}^{T}\dot{J}s_{q}+\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})\dot{\hat{K}}_{qi}.$
(2.26)

Utilizing the upper bound structure given by (2.8) and Property 2, and the
control laws (2.9), (2.18), the terms $\dot{V}_{p}$ in (2.25) and
$\dot{V}_{q}$ in (2.26) are simplified to

$\displaystyle\dot{V}_{p}$
$\displaystyle=s_{p}^{T}(-\Lambda_{p}s_{p}-\rho_{p}\operatorname{sgn}(s_{p})+\hat{m}g_{p}+\varphi_{p}-G)+\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})\dot{\hat{K}}_{pi}+(\hat{m}-m)\dot{\hat{m}}$
(2.27) $\displaystyle\dot{V}_{q}$
$\displaystyle=s_{q}^{T}(-\Lambda_{p}s_{p}-\rho_{p}\operatorname{sgn}(s_{p})+\varphi_{q})+\frac{1}{2}s_{q}^{T}(\dot{J}-2C_{q})s_{q}+\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})\dot{\hat{K}}_{qi}.$
(2.28)

Property 3 implies that $s_{q}^{T}(J-2C_{q})s_{q}=0$. Then utilizing the upper
bound structure (2.8) and (2.17), and the facts that $G=mg_{p}$, $\rho_{p}\geq
0,\rho_{q}\geq 0$ from (2.20) and (2.21) we have

$\displaystyle\dot{V}_{p}$
$\displaystyle=-s_{p}^{T}\Lambda_{p}s_{p}-\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})(||\xi_{p}||^{i}||s_{p}||-\dot{\hat{K}}_{pi})+(\hat{m}-m)(s_{p}^{T}g_{p}+\dot{\hat{m}})$
(2.29) $\displaystyle\dot{V}_{q}$
$\displaystyle=-s_{q}^{T}\Lambda_{q}s_{q}-\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})(||\xi_{q}||^{i}||s_{q}||-\dot{\hat{K}}_{qi}).$
(2.30)

Using (2.10) and (2.19), we have

$\displaystyle(\hat{K}_{pi}-K_{pi}^{*})\dot{\hat{K}}_{pi}$
$\displaystyle=||s_{p}||(\hat{K}_{pi}-K_{pi}^{*})||\xi_{p}||^{i}+\alpha_{pi}\hat{K}_{pi}K_{pi}^{*}-\alpha_{pi}\hat{K}_{pi}^{2}$
(2.31) $\displaystyle(\hat{m}-m)\dot{\hat{m}}$
$\displaystyle=-(\hat{m}-m)s_{p}^{T}g_{p}+\alpha_{m}\hat{m}m-\alpha_{m}\hat{m}^{2}$
(2.32) $\displaystyle(\hat{K}_{qi}-K_{qi}^{*})\dot{\hat{K}}_{qi}$
$\displaystyle=||s_{q}||(\hat{K}_{qi}-K_{qi}^{*})||\xi_{q}||^{i}+\alpha_{qi}\hat{K}_{qi}K_{qi}^{*}-\alpha_{qi}\hat{K}_{qi}^{2}$
(2.33)

Substituting (2.31)-(2.33) into (2.29) and (2.30) yield

$\displaystyle\dot{V}_{p}$
$\displaystyle\leq-\frac{\lambda_{\min}(\Lambda_{p})||s_{p}||^{2}}{3}+\sum\limits_{i=0}^{1}(\alpha_{pi}\hat{K}_{pi}K_{pi}^{*}-\alpha_{pi}\hat{K}_{pi}^{2})+\alpha_{m}\hat{m}m-\alpha_{m}\hat{m}^{2}$
$\displaystyle\leq-\frac{\lambda_{\min}(\Lambda_{p})||s_{p}||^{2}}{3}-\sum\limits_{i=0}^{1}\left(\frac{\alpha_{pi}(\hat{K}_{pi}-K_{pi}^{*})^{2}}{2}-\frac{\alpha_{pi}{K_{pi}^{*}}^{2}}{2}\right)$
$\displaystyle\quad-\left(\frac{\alpha_{m}(\hat{m}-m)^{2}}{2}-\frac{\alpha_{m}{m}^{2}}{2}\right)$
(2.34) $\displaystyle\dot{V}_{q}$
$\displaystyle\leq-\frac{\lambda_{\min}(\Lambda_{q})||s_{q}||^{2}}{3}+\sum\limits_{i=0}^{2}(\alpha_{qi}\hat{K}_{qi}K_{qi}^{*}-\alpha_{qi}\hat{K}_{qi}^{2})$
$\displaystyle\leq-\frac{\lambda_{\min}(\Lambda_{q})||s_{q}||^{2}}{3}-\sum\limits_{i=0}^{2}\left(\frac{\alpha_{qi}(\hat{K}_{qi}-K_{qi}^{*})^{2}}{2}-\frac{\alpha_{qi}{K_{qi}^{*}}^{2}}{2}\right).$
(2.35)

Further the definition of Lyapunov function yields

$\displaystyle V_{p}$
$\displaystyle\leq\frac{{m}}{2}||s_{p}||^{2}+\frac{1}{2}\sum\limits_{i=0}^{1}(\hat{K}_{pi}-K_{pi}^{*})^{2}+\frac{1}{2}(\hat{m}-m)^{2}$
(2.36) $\displaystyle V_{q}$
$\displaystyle\leq\frac{\overline{j}}{2}||s_{q}||^{2}+\frac{1}{2}\sum\limits_{i=0}^{2}(\hat{K}_{qi}-K_{qi}^{*})^{2}.$
(2.37)

From $(\ref{p_lyap_upbound})$ and $(\ref{q_lyap_upbound})$, the conditions
$(\ref{p_lyap_dot_5})$ and $(\ref{q_lyap_dot_5})$ can be further simplified to

$\displaystyle\dot{V}_{p}$
$\displaystyle\leq-\varrho_{p}V_{p}+\frac{1}{2}\left(\sum\limits_{i=0}^{1}\alpha_{pi}{K^{*}_{pi}}^{2}+\alpha_{m}{m}^{2}\right)$
(2.38) $\displaystyle\dot{V}_{q}$
$\displaystyle\leq-\varrho_{q}V_{q}+\frac{1}{2}\left(\sum\limits_{i=0}^{2}\alpha_{qi}{K^{*}_{qi}}^{2}\right)$
(2.39)

where
$\varrho_{p}\triangleq\frac{\min_{i}\\{\lambda_{\min}(\Lambda_{p})/3,\alpha_{pi},\alpha_{m}\\}}{\max\\{{m}/2,1/2\\}}$,
$\varrho_{q}\triangleq\frac{\min_{i}\\{\lambda_{min}(\Lambda_{q})/3,\alpha_{qi}\\}}{\max\\{\overline{j}/2,1/2\\}}>0$
can be designed via (2.9a), (2.10), (2.18a), and (2.19). From (2.38) and
(2.39), the upper bound for the time derivative of the overall Lyapunov
function $\dot{V}$ can be obtained as

$\displaystyle\dot{V}$ $\displaystyle\leq-\varrho
V+\frac{1}{2}\left(\sum\limits_{i=0}^{1}\alpha_{pi}{K^{*}_{pi}}^{2}\right)+\frac{1}{2}\left(\alpha_{m}{m}^{2}\right)+\frac{1}{2}\left(\sum\limits_{i=0}^{2}\alpha_{qi}{K^{*}_{qi}}^{2}\right)$
(2.40)

where $\varrho=\min\\{\varrho_{p},\,\varrho_{q}\\}$. Defining a scalar
$\kappa$ such that $0<\kappa<\varrho$, $(\ref{overall_V_dot_1})$ is simplified
to

$\displaystyle\dot{V}$ $\displaystyle=-\kappa
V-(\varrho-\kappa)V+\frac{1}{2}\left(\sum\limits_{i=0}^{1}\alpha_{pi}{K^{*}_{pi}}^{2}+\alpha_{m}{m}^{2}+\sum\limits_{i=0}^{2}\alpha_{qi}{K^{*}_{qi}}^{2}\right).$
(2.41)

Defining a scalar
$\mathcal{\bar{B}}\triangleq\frac{\left(\sum\limits_{i=0}^{1}\alpha_{pi}{K^{*}_{pi}}^{2}+\alpha_{m}{m}^{2}+\sum\limits_{i=0}^{2}\alpha_{qi}{K^{*}_{qi}}^{2}\right)}{2(\varrho-\kappa)}$,
it can be seen that $\dot{V}(t)<-\kappa V(t)$ when
$V(t)\geq\mathcal{\bar{B}}$, so that

$\displaystyle V$ $\displaystyle\leq max\\{V(0),\mathcal{\bar{B}}\\},\forall
t\geq 0,$ (2.42)

and the closed-loop system remains UUB.

######  Remark 2.6.

Control laws (2.9a) and (2.18a) are discontinuous in nature, which may cause
chattering. Therefore, as standard for sliding mode designs, the control laws
can be made continuous by replacing the ‘signum’ function by a ‘saturation’
function defined as $\operatorname{sat}(s_{p},\varpi_{p})=s_{p}/||s_{p}||$
(resp. $s_{p}/\varpi_{p}$) if $||s_{p}||\geq\varpi_{p}$ (resp.
$||s_{p}||<\varpi_{p}$) where $\varpi_{p}\in\mathbb{R}^{+}$ is a small scalar
used to avoid chattering. Similarly, $\operatorname{sgn}(s_{q})$ can be
modified. Such modifications do not change the closed-loop stability result
albeit some minor modifications in stability analysis, which can be carried
out in the same line as in [54] and thus, repetition is avoided.

###  2.5 Simulation Results and Analysis

The performance of the proposed controller is tested on a Gazebo simulation
platform using the RotorS Simulator framework [2] for ROS with the Pelican
quadrotor model (mass of the quadrotor without any payload is set to be $2$
kg). The results of the proposed adaptive sliding mode controller (ASMC) is
compared with the geometric controller [1] (referred to as PD control
hereafter) available in the RotorS framework, and sliding mode controller
(SMC) [27] (note that existing ASMC designs such as [47] cannot be applied as,
aside considering the reduced-order model, they are designed for a priori
bounded uncertainty, which is not valid for a quadrotor).

The Gazebo model of the quadrotor can be actuated by commanding the angular
velocities for the rotors. The angular velocities are calculated by solving
the following relationship between thrust, moments and angular velocities
[11]:

$\displaystyle\begin{bmatrix}\omega_{1}\\\ \omega_{2}\\\ \omega_{3}\\\
\omega_{4}\end{bmatrix}$ $\displaystyle=K^{-1}\begin{bmatrix}u_{1}\\\ u_{2}\\\
u_{3}\\\ u_{4}\end{bmatrix}$ (2.43) $\displaystyle K$
$\displaystyle=\begin{bmatrix}k_{f}&k_{f}&k_{f}&k_{f}\\\ -lk_{f}&0&lk_{f}&0\\\
0&-lk_{f}&0&lk_{f}\\\ k_{\tau}&-k_{\tau}&k_{\tau}&-k_{\tau}\end{bmatrix}$
(2.44)

where $k_{f}$ and $k_{\tau}$ are force and moment constants respectively of
the motors; $l$ is the arm length of the quadrotor and $\omega_{i}$ is the
angular velocity of $i^{th}$ motor.

To properly judge the performance of the proposed design, two different
simulation scenarios are created in the following subsections. For both the
scenarios, the control parameters of the proposed ASMC are selected to be:
$\Phi_{p}=diag\\{2.4,2.4,16\\}$, $\Phi_{q}=diag\\{0.22,0.22,0.01\\}$,
$\Lambda_{p}=diag\\{1.2,1.2,0.8\\}$, $\Lambda_{q}=diag\\{4.5,4.5,3.5\\}$,
$\hat{K}_{p0}(0)=\hat{K}_{p1}(0)=0.01$,
$\hat{K}_{q0}(0)=\hat{K}_{q1}(0)=\hat{K}_{q2}(0)=0.0001$,
$\alpha_{p0}=\alpha_{p1}=3$, $\alpha_{q0}=\alpha_{q1}=\alpha_{q2}=50$,
$\hat{m}(0)=0.01$ and $\alpha_{m}=0.1$, $\varpi_{p}=0.1$, $\varpi_{q}=1$. For
a fair comparison, similar sliding surfaces are selected for SMC as in (2.5)
and (2.14). The gains for geometric controller [1] is selected to be same as
in the RotorS package, which are optimized for the Pelican model. Initial
position and attitude for the quadrotor are selected to be $x(0)=y(0)=0$,
$z(0)=0.1$ and $\phi(0)=\theta(0)=\psi(0)=0$.

To perform the experiments, initial values of positions, angles and their
derivatives are taken to be zero. The control parameters chosen for the
geometric controller is their default parameters for Pelican drone, i.e.,
position gain $k_{p}=4$, linear velocity gain $k_{d}=2.7$; for pitch and roll,
angular gain and angular rate gain are chosen $1$ and $0.22$ respectively, and
$0.035,0.01$ are chosen as angular gain and angular rate gain for yaw,
respectively.

####  2.5.1 Scenario 1: Aggressive Manoeuvre

The objective of this scenario is to test the performance of various
controllers when a quadrotor is to make aggressive manoeuvres. To this end, a
set of way-points $(x_{d},y_{d},z_{d})$ and $\psi_{d}$ are commanded as input
to the controllers (cf. Fig. 2.2). The list of waypoints provided is mentioned
in the table 2.1. Once the last waypoint is reached, the quadrotor altitude is
reduced to the height of the payload, and the payload is attached. Once again,
the set of waypoints is given in the same time interval. These abrupt
transitions between the positions demand aggressive manoeuvres for a fast
response. For time duration $0\leq t<48$s, the set of waypoints are commanded
without any payload attached to the quadrotor. Then, at $t=48$s, a payload of
$0.5$ kg is added to the quadrotor, and the same set of waypoints are
repeated.

Table 2.1: List of waypoints for Scenario 1 time (s) |  $x$ (m) |  $y$ (m) |  $z$ (m) |  $\psi$ (deg)
---|---|---|---|---
0.0 | 2.0 | 2.0 | 2.0 | 0.0
6.0 | 0.0 | 0.0 | 3.0 | 0.0
12.0 | 0.0 | 2.0 | 2.0 | 0.0
18.0 | 2.0 | 0.0 | 2.0 | 90.0
24.0 | 2.0 | 2.0 | 4.0 | 0.0

Figures 2.2-2.4 depict the position tracking with various controllers and the
corresponding errors incurred by them in position and attitude. It can be
noted at $t\geq 72$s, the altitude ($z$ position) of the quadrotor using the
geometric controller (PD) has become zero, i.e., the quadrotor has crashed on
the floor during the manoeuvres after adding the payload. Reduction in the $z$
position error for the PD controller at $t>82$s should not be confused to be
performance improvement: it is a result of reduction in the desired altitude
$z_{d}$ (cf. Fig. 2.2), whereas the quadrotor still remains crashed on the
ground.

To demonstrate the performance of the controllers while carrying a payload, a
performance comparison via root-mean-squared (RMS) error between SMC and ASMC
is provided in Table 2.2 after the payload is attached, i.e., for $t\geq 48$s
(since the quadrotor crashed during the scenario for the geometric controller,
its RMS error data is not tabulated). It clearly demonstrates the superior
position tracking performance of the proposed scheme, where it has delivered a
minimum performance improvement of $14.5\%$. Despite position tracking is of
more importance for transportation, it is also worth mentioning that attitude
tracking of both the controllers is, however, almost similar. This is caused
by the higher transients for ASMC compared to SMC during the sharp changes in
the desired trajectory. This happens because the gains of ASMC need to re-
adapt itself every time with the sharp changes in trajectories, while gains
for SMC are fixed and less sensitive to abrupt changes. These transients are
inherent for any adaptive design, but it helps them to adjust with unknown
changes, while fixed-gain designs can fail when uncertainties lie beyond the a
priori knowledge: such a situation is studied in the next scenario.

Figure 2.2: Position tracking comparison for aggressive manoeuvre Figure 2.3:
Position tracking error comparison for aggressive manoeuvre Figure 2.4:
Attitude tracking error comparison for aggressive manoeuvre

Table shows the Root Mean Squared Error in the states $(x,y,z)$ and the
overall position for the controllers. It can be observed that the proposed
controller performs better than the other controllers.

Table 2.2: Performance comparison for Scenario 1 | RMS error (m) | RMS error (degree)
---|---|---
position | $x$ | $y$ | $z$ | $\phi$ | $\theta$ | $\psi$
SMC | 0.47 | 0.69 | 0.65 | 3.70 | 4.06 | 9.90
ASMC (proposed) | 0.40 | 0.48 | 0.42 | 3.80 | 2.67 | 10.06

####  2.5.2 Scenario 2: Dynamic payload

Figure 2.5: Snapshots from experimental scenario 2 (with ASMC): quadrotor (a)
starting from initial position (b) picking up first payload ($0.4$ kg) (c)
moving over a wall with first payload (d) dropping the first payload (e)
picking up second payload ($1$ kg) (f) moving over the wall with second
payload (g) dropping the second payload at initial position.

In a real-life aerial transport scenario, payloads can vary according to
application requirements (e.g., human evacuations in an emergency).
Accordingly, a second experimental scenario is created in the ‘indoor
environment’ available in the RotorS simulator, wherein performance of the
various controllers under dynamic payload variation is tested while following
a minimum snap trajectory [11]. In this scenario, the quadrotor is required to
perform the following actions sequentially (cf. Fig. 2.5):

  * •

From an initial position the quadrotor (unloaded) moves to the first payload
location;

  * •

it picks up a payload of $0.4$ kg (at approx $t=32$s) and an altitude of 1.0 m
is given as the setpoint;

  * •

it moves over a wall to the other side;

  * •

it delivers the payload (at approx $t=58$s) and holds its altitude of 0.6 m;

  * •

then, it moves towards a second payload of $1.0$ kg;

  * •

it picks up the payload (at approx $t=83$s) and holds an altitude of 1.0 m;

  * •

then, it moves over the wall back to the initial position;

  * •

it delivers the payload (at approx $t=123$s) and holds its altitude of 0.5 m.

Moving over the walls with payload is an important phase for this experiment
as it tests the capability of the controllers to maintain a designated
altitude: such test cases are important for aerial transport in urban
scenarios. Masses of the first and second payload are chosen to be $0.4$ kg
and $1.0$ kg. Figures 2.6, 2.7 show the states $(x,y,z)$ for different
controllers with respect to time along with the commanded trajectory and their
tracking errors respectively. The first payload is loaded at $t=32$s and
unloaded at $t=58$s, while the second payload is loaded at $t=83$s and dropped
at $t=123$s.

Table 2.3: Performance of ASMC for Scenario 2 | RMS error (m) | RMS error (degree)
---|---|---
position | $x$ | $y$ | $z$ | $\phi$ | $\theta$ | $\psi$
| 0.159 | 0.162 | 0.159 | 1.278 | 1.072 | 0.311

Figures 2.6-2.8 show the position responses with various controllers and their
corresponding tracking errors. The position responses in the geometric
controller become constant for $t>45$s, the quadrotor topples while moving
with the first payload. A similar incident can be observed with SMC for
$t>92$s: though the quadrotor could manage to lift the first payload, it could
not lift the second payload high enough to cross the wall. This happened
because SMC relies on a priori fixed gains based on predefined uncertainty
bounds; when the payload becomes heavier, the controller cannot adjust its
gain to adapt with higher uncertainty. In such a scenario, thanks to the
adaptive gains as in (2.10) and (2.19), the proposed ASMC can tackle
uncertainties without any a priori knowledge and thus, outperforms other
controllers. The performance of ASMC is compiled in Table 2.3. The desired
trajectories in this scenario being a smoother one (cf. [11]) compared to the
one in Scenario 1, transients in the attitude response for ASMC are visibly
absent except at the instants when payloads are added. Similar to Scenario 1,
since the quadrotor crashed during the scenario for SMC and PD, their
performances are not tabulated.

Figure 2.6: Position tracking by various controllers with variable payloads
Figure 2.7: Position Tracking error comparison with variable payloads Figure
2.8: Attitude tracking error comparison with variable payloads

##  Chapter 3 Dynamic Payload Lifting Operation

###  3.1 Introduction

In the previous chapter, it was considered that the payloads do not change
during the flight. However, in many practical applications, payload mass
varies abruptly: for example, dropping relief material, adding construction
material during vertical movement (cf. Fig. 3.1). Such abrupt changes give
rise to switched dynamics: switched systems are an important class of hybrid
systems that consists of subsystems with continuous dynamics, where at any
given time instant, one subsystem remains active while others being inactive.
Switching between active and inactive subsystems is governed by a switching
law [55].

In the quest to operate a quadrotor under various sources of parametric
uncertainty, the control regime, after initial model based designs (cf. the
survey paper [56] for the evolutions of various designs), has inevitably moved
to some notable adaptive control works [29, 30, 31, 32, 47, 33, 34, 35, 17,
57]. However, no adaptive design addresses the issue of switched dynamics as
discussed earlier. On the other hand, extending these works to a switched
dynamics scenario would rely on finding a common Lyapunov function, which
again is difficult (if at all possible) to achieve in practical systems [38].

Therefore, a relevant question arises whether the existing switched adaptive
control designs [38, 39, 40, 41, 42, 58, 59, 60, 61, 62] (and references
therein) can be applied to a switched quadrotor system while carrying an
uncertain (and possibly unknown) payload. Unfortunately, existing literature
does not present a positive answer to this: though [60, 61] do not rely on the
structure of system dynamics in contrast to state of the art, they still
require bound knowledge of the uncertain mass/inertia terms to design either
control law [61] or switching law [60].

In light of the above discussions, an adaptive switched framework that can
tackle completely unknown dynamics of quadrotor in a switched dynamics setting
is still missing. Toward this direction, the proposed switched adaptive
solution has the following major contributions:

  * •

A switched dynamics for quadrotor is formulated which suitably represents the
changing dynamics during a vertical (lifting/dropping) operation under dynamic
variation of payload.

  * •

Compared to the state-of-the-art, the proposed adaptive law and switching law
do not require any a priori knowledge of the system dynamics parameters and
payload. Further, differently from [29, 47], the effects of cross-coupling
terms arising from the lateral motion are considered as state-dependent
uncertainty.

The rest of the chapter is organized as follows: Section 3.2 describes the
switched quadrotor systems and highlights various issues in the state of the
art; Section 3.3 details the proposed control framework, while the
corresponding stability analysis is carried out in Section 3.4 and a
simulation study is provided in Section 3.5.

###  3.2 System Dynamics and Problem Formulation

Figure 3.1: Schematic depicting vertical operation with dynamic payload.

Let us consider the dynamics of quadrotor as given in Ch. 1:

$\displaystyle m\begin{bmatrix}\ddot{x}\\\ \ddot{y}\\\
\ddot{z}\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\
mg\end{bmatrix}=\mathbf{R}^{T}\begin{bmatrix}0\\\ 0\\\ T_{th}\end{bmatrix}$
(3.1)
$\displaystyle\frac{I_{xx}}{l}\ddot{\phi}+\frac{I_{zz}-I_{yy}}{l}\dot{\varphi}\dot{\psi}=\tau_{\phi},$
$\displaystyle\frac{I_{yy}}{l}\ddot{\varphi}+\frac{I_{xx}-I_{zz}}{l}\dot{\phi}\dot{\psi}=\tau_{\varphi},$
$\displaystyle
I_{zz}\ddot{\psi}+(I_{yy}-I_{xx})\dot{\varphi}\dot{\phi}=\tau_{\psi},$ (3.2)

with the rotation matrix $\mathbf{R}$ is given by

$\mathbf{R}=\begin{bmatrix}c_{\psi}c_{\varphi}&s_{\psi}c_{\varphi}&-s_{\varphi}\\\
c_{\psi}s_{\varphi}s_{\phi}-s_{\psi}c_{\phi}&s_{\psi}s_{\varphi}s_{\phi}+c_{\psi}c_{\phi}&s_{\phi}c_{\varphi}\\\
c_{\psi}s_{\varphi}c_{\phi}+s_{\psi}s_{\phi}&s_{\psi}s_{\varphi}c_{\phi}-c_{\psi}s_{\phi}&c_{\varphi}c_{\phi}\end{bmatrix}$

where, $c_{(\cdot)},s_{(\cdot)}$ denote $\cos_{(\cdot)},\sin_{(\cdot)}$; $m$
is the mass of the overall system.

Multiplying both side of the position subsystem (3.1) with the rotation
matrix, $\mathbf{R}$ yields

$\displaystyle
m\ddot{x}c_{\psi}c_{\varphi}+m\ddot{y}s_{\psi}c_{\varphi}-m\ddot{z}s_{\varphi}-mgs_{\varphi}=0,$
(3.3a) $\displaystyle
m\ddot{x}(c_{\psi}s_{\varphi}s_{\phi}-s_{\psi}c_{\phi})+m\ddot{y}(s_{\psi}s_{\varphi}s_{\phi}+c_{\psi}c_{\phi})+m\ddot{z}s_{\phi}c_{\varphi}+mgs_{\phi}c_{\varphi}=0,$
(3.3b) $\displaystyle
m\ddot{x}(c_{\psi}s_{\varphi}c_{\phi}+s_{\psi}s_{\phi})+m\ddot{y}(s_{\psi}s_{\varphi}c_{\phi}-c_{\psi}s_{\phi})+m\ddot{z}c_{\varphi}c_{\phi}+mgc_{\varphi}c_{\phi}=T_{th}.$
(3.3c)

After such rearrangements of the dynamics, we are now ready to define the
(non-switched) four degrees-of-freedom dynamics structure suitable for
collocated design as

$\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})+\mathbf{H}(\mathbf{q})\ddot{\mathbf{q}}_{\mathbf{u}}=\boldsymbol{\tau},$
(3.4)

where

$\displaystyle\mathbf{M}=\begin{bmatrix}m(c_{\varphi}c_{\phi})&0&0&0\\\
0&\frac{I_{xx}}{l}&0&0\\\ 0&0&\frac{I_{yy}}{l}&0\\\ 0&0&0&I_{zz}\\\
\end{bmatrix},~{}\mathbf{G}=\begin{bmatrix}mg(c_{\varphi}c_{\phi})\\\ 0\\\
0\\\ 0\end{bmatrix}$ (3.5) $\displaystyle\mathbf{C}=\begin{bmatrix}0&0&0&0\\\
0&0&0&\frac{(I_{zz}-I_{yy})}{l}\dot{\varphi}\\\
0&\frac{(I_{xx}-I_{zz})}{l}\dot{\psi}&0&0\\\
0&0&{(I_{yy}-I_{xx})}\dot{\phi}&0\end{bmatrix},$ (3.6)
$\displaystyle\mathbf{H}=\begin{bmatrix}m(c_{\psi}s_{\varphi}c_{\phi}+s_{\psi}s_{\phi})&m(s_{\psi}s_{\varphi}c_{\phi}-c_{\psi}s_{\phi})\\\
0&0\\\ 0&0\\\ 0&0\end{bmatrix},$ (3.7)
$\displaystyle\boldsymbol{\tau}=\begin{bmatrix}T_{th}\\\ \tau_{\phi}\\\
\tau_{\varphi}\\\ \tau_{\psi}\end{bmatrix},~{}\mathbf{q}=\begin{bmatrix}z\\\
\phi\\\ \varphi\\\ \psi\end{bmatrix},~{}\mathbf{q_{u}}=\begin{bmatrix}x\\\
y\end{bmatrix}.$ (3.8)

During a vertical operation, a quadrotor needs to maintain a given (possibly
time-varying) height ($z$) and attitude $(\phi,\varphi,\psi)$, at a fixed
$(x,y)$ position. Under such scenario, and based on the underactuated dynamics
model as in (3.4), a switched quadrotor dynamics (cf. Fig. 3.1) for vertical
operations can be represented as

$\mathbf{M}_{\sigma}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}_{\sigma}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+\mathbf{G}_{\sigma}(\mathbf{q})+\mathbf{H}_{\sigma}(\mathbf{q})\ddot{\mathbf{q}}_{\mathbf{u}}+\mathbf{d}_{\sigma}=\boldsymbol{\tau}_{\sigma},$
(3.9)

where $\mathbf{q}=\\{z,\phi,\varphi,\psi\\}$ and $\mathbf{q_{u}}=\\{x,y\\}$;
for each subsystem $\sigma$, $\mathbf{M_{\sigma}(q)}\in\mathbb{R}^{4\times 4}$
is the mass/inertia matrix;
$\mathbf{C}_{\sigma}(\mathbf{q},\dot{\mathbf{q}})\in\mathbb{R}^{4\times 4}$
denote the Coriolis, centripetal terms;
$\mathbf{G}_{\sigma}(\mathbf{q})\in\mathbb{R}^{4}$ denotes the gravity vector;
$\mathbf{H}_{\sigma}(\mathbf{q})\ddot{\mathbf{q}}_{\mathbf{u}}\in\mathbb{R}^{4}$
represents the vector of cross-coupling terms stemming from $(x,y)$ sub-
dynamics; $\mathbf{d_{\sigma}}(t)\in\mathbb{R}^{4}$ denotes bounded external
disturbance and $\boldsymbol{\tau}_{\sigma}\in\mathbb{R}^{4}$ is the control
input. The mapping from $\mathbf{\tau}$ to thrust and torques of individual
motors are discussed in Sect. 2.5.

Here $\sigma(t):[0~{}\infty)\mapsto\Omega$ is a piecewise constant function of
time, called the switching signal, taking values in
$\Omega=\\{1,2,\cdots,N\\}$. The following class of slowly-switching signals
is considered:

######  Definition 3.1.

Average Dwell Time (ADT) [63]: For a switching signal $\sigma(t)$ and each
$t_{2}\geq t_{1}\geq 0$, let $N_{\sigma}(t_{1},t_{2})$ denote the number of
discontinuities in the interval $[t_{1},t_{2})$. Then $\sigma(t)$ has an
average dwell time $\vartheta$ if for a given scalar $N_{0}>0$

$\displaystyle N_{\sigma}(t_{1},t_{2})\leq
N_{0}+(t_{2}-t_{1})/\vartheta,~{}~{}\forall t_{2}\geq t_{1}\geq 0$

where $N_{0}$ is termed as chatter bound.

As the quadrotor dynamics (3.9) follows the celebrated Euler-Lagrange dynamics
formulation [16], each subsystem in (3.9) presents a few interesting
properties (cf. [64]), which are later exploited for control design as well as
stability analysis:

Property 1:
$\exists\overline{c}_{\sigma},\overline{g}_{\sigma},\overline{h}_{\sigma},\overline{d}_{\sigma}\in\mathbb{R}^{+}$
such that
$||\mathbf{C}_{\sigma}(\mathbf{q},\dot{\mathbf{q}})||\leq\overline{c}_{\sigma}||\dot{\mathbf{q}}||$,
$||\mathbf{G_{\sigma}(q)}||\leq\overline{g}_{\sigma}$,
$||\mathbf{H}_{\sigma}(\mathbf{q})||\leq\overline{h}_{\sigma}$ and
$||\mathbf{d_{\sigma}}(t)||\leq\overline{d}_{\sigma}$.
Property 2: The matrix $\mathbf{M_{\sigma}(q)}$ is symmetric and uniformly
positive definite $\forall\mathbf{q}$, implying that
$\exists\underline{m}_{\sigma},\overline{m}_{\sigma}\in\mathbb{R}^{+}$ such
that

$0<\underline{m}_{\sigma}\mathbf{I}\leq\mathbf{M_{\sigma}(q)}\leq\overline{m}_{\sigma}\mathbf{I}.$
(3.10)

A few remarks regarding system (3.9) follow, which aid in designing the
proposed switching control.

######  Remark 3.1 ((Uncertainty)).

As a design challenge, the switched system (3.9) is considered to be unknown
in the sense that the knowledge of
$\mathbf{M}_{\sigma},\mathbf{C_{\sigma},H_{\sigma},G_{\sigma},d}_{\sigma}$ and
their corresponding bounds, i.e.,
$\underline{m}_{\sigma},\overline{m}_{\sigma},\overline{c}_{\sigma},\overline{g}_{\sigma},\overline{h}_{\sigma},\overline{d}_{\sigma}$
are completely unknown.

######  Remark 3.2 ((Collocated Design)).

Note that a quadrotor dynamics is essentially underactuated, where $x$ and $y$
positions represent the non actuated coordinates. During a vertical operation
(i.e., a near hovering condition), there exist time-scale separation between
$(z,\phi,\varphi,\psi)$ and $(x,y)$ dynamics, where the former evolves faster
than the later [31]. Therefore, it is standard to track only the actuated
coordinates $(z,\phi,\varphi,\psi)$ of a quadrotor during such operation [47,
29]. In the literature of underactuated systems, such control design, i.e.,
tracking of only actuated coordinates, is called collocated design (cf. [65,
66]).

######  Remark 3.3 ((Cross-coupling terms)).

Notwithstanding slow evolution of the lateral (i.e., $(x,y)$) motion compared
to the vertical ($z$) and attitude $(\phi,\varphi,\psi)$ motions due to time-
scale separation, ignoring its effect is conservative: especially, variation
in payload and perturbation in attitude may cause some lateral motion,
affecting $\mathbf{q}$. Therefore, differently form [47, 29], the cross-
coupling term $\mathbf{H}_{\sigma}(\mathbf{q})\ddot{\mathbf{q}}_{\mathbf{u}}$
is considered in (3.9), where $\mathbf{H}_{\sigma}(\mathbf{q})$ captures the
effects of variations in payload and attitude (cf. Appendix).

Control Problem: Under Properties 1 and 2, to design a switched adaptive
control framework for a vertical operation of a quadrotor following the
dynamics (3.9), without any knowledge of system parameters (in line with
Remark 1).

The following section solves this control problem.

###  3.3 Switched Controller Design

######  Assumption 3.1 ([67, 68]).

The desired trajectories satisfy
$\mathbf{q}^{d},\dot{\mathbf{q}}^{d},\ddot{\mathbf{q}}^{d}\in\mathcal{L}_{\infty}$.
Furthermore, $\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}$ are available for
feedback.

For control design purposes, the dynamics (3.9) is re-arranged as

$\mathbf{D}_{\sigma}\ddot{\mathbf{q}}+\mathbf{E}_{\sigma}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}},\ddot{\mathbf{q}}_{\mathbf{u}},t)=\boldsymbol{\tau}_{\sigma},$
(3.11)

which has been obtained by adding and subtracting
$\mathbf{D}_{\sigma}\ddot{\mathbf{q}}$ to (3.9), where $\mathbf{D}_{\sigma}$
is a user-defined constant positive definite matrix and
$\mathbf{E}_{\sigma}\triangleq(\mathbf{M}_{\sigma}-\mathbf{D}_{\sigma})\ddot{\mathbf{q}}+\mathbf{C}_{\sigma}\dot{\mathbf{q}}+\mathbf{G}_{\sigma}+\mathbf{H}_{\sigma}\ddot{\mathbf{q}}_{\mathbf{u}}+\mathbf{d}_{\sigma}$.
The selection of $\mathbf{D}_{\sigma}$ would be discussed later (cf. Remark
3.5).

Let $\mathbf{e}(t)\triangleq\mathbf{q}(t)-\mathbf{q}^{d}(t)$ be the tracking
error, $\boldsymbol{\xi}(t)\triangleq[\mathbf{e}(t),~{}\dot{\mathbf{e}}(t)]$
and $\mathbf{r}_{\sigma}$ be the filtered tracking error variable defined as

$\displaystyle\mathbf{r}_{\sigma}\triangleq\mathbf{B}^{T}\mathbf{P}_{\sigma}\boldsymbol{\xi},\qquad\sigma\in\Omega$
(3.12)

where $\mathbf{P}_{\sigma}>\mathbf{0}$ is the solution to the Lyapunov
equation
$\mathbf{A}_{\sigma}^{T}\mathbf{P}_{\sigma}+\mathbf{P_{\sigma}A_{\sigma}}=-\mathbf{Q}_{\sigma}$
for some $\mathbf{Q}_{\sigma}>\mathbf{0}$,
$\mathbf{A}_{\sigma}\triangleq\begin{bmatrix}\mathbf{0}&\mathbf{I}\\\
-\mathbf{K}_{1\sigma}&-\mathbf{K}_{2\sigma}\end{bmatrix}$ and
$\mathbf{B}\triangleq\begin{bmatrix}\mathbf{0}&\mathbf{I}\end{bmatrix}^{T}$.
Here, $\mathbf{K}_{1\sigma}$ and $\mathbf{K}_{2\sigma}$, are two user-defined
positive definite gain matrices and their positive definiteness guarantees
that $\mathbf{A}_{\sigma}$ is Hurwitz.

The control law is designed as

$\displaystyle\boldsymbol{\tau}_{\sigma}$
$\displaystyle=\mathbf{D}_{\sigma}(-\boldsymbol{\Lambda}_{\sigma}\boldsymbol{\xi}-\Delta\boldsymbol{\tau}_{\sigma}+\ddot{\mathbf{q}}^{d}),$
(3.13a) $\displaystyle~{}\Delta\boldsymbol{\tau}_{\sigma}$
$\displaystyle=\begin{cases}{\rho_{\sigma}}\frac{\mathbf{r}_{\sigma}}{||\mathbf{r}_{\sigma}||}&~{}\text{if
}||\mathbf{r}_{\sigma}||\geq\varpi\\\
{\rho_{\sigma}}\frac{\mathbf{r}_{\sigma}}{\varpi}&~{}\text{if
}||\mathbf{r}_{\sigma}||<\varpi\\\ \end{cases},$ (3.13b)

where
$\boldsymbol{\Lambda}_{\sigma}\triangleq[\mathbf{K}_{1\sigma}~{}\mathbf{K}_{2\sigma}]$;
$\varpi>0$ is a user-defined scalar to avoid chattering and the design of
$\rho_{\sigma}$ will be discussed later. Substituting (3.13a) in (3.11) yields

$\displaystyle\ddot{\mathbf{e}}=-\boldsymbol{\Lambda}_{\sigma}\boldsymbol{\xi}-\Delta\boldsymbol{\tau}_{\sigma}+\boldsymbol{\chi}_{\sigma},$
(3.14)

where
$\boldsymbol{\chi}_{\sigma}\triangleq-\mathbf{D}_{\sigma}^{-1}\mathbf{E}_{\sigma}$
is defined as the _overall uncertainty_. Using Properties 1, 2 and Assumption
1 one can verify that

$||\boldsymbol{\chi}_{\sigma}||\leq\theta_{0\sigma}^{*}+\theta_{1\sigma}^{*}||\boldsymbol{\xi}||+\theta_{2\sigma}^{*}||\boldsymbol{\xi}||^{2}+\theta_{3\sigma}^{*}||\ddot{\overline{\mathbf{q}}}||\triangleq\mathbf{Y}_{\sigma}^{T}\boldsymbol{\Theta}_{\sigma}^{*},$
(3.15)

where
$\overline{\mathbf{q}}=[\mathbf{q}^{T}~{}\mathbf{q}_{\mathbf{u}}^{T}]^{T}$;
$\mathbf{Y}_{\sigma}=[1~{}||\boldsymbol{\xi}||~{}||\boldsymbol{\xi}||^{2}~{}||\ddot{\overline{\mathbf{q}}}||]^{T}$
and
$\boldsymbol{\Theta}_{\sigma}^{*}=[\theta_{0\sigma}^{*}~{}\theta_{1\sigma}^{*}~{}\theta_{2\sigma}^{*}~{}\theta_{3\sigma}^{*}]^{T}$.
The scalars $\theta_{i\sigma}^{*}\in\mathbb{R}^{+}$ $i=0,\cdots,3$ are unknown
$\forall\sigma\in\Omega$ and they are defined as

$\displaystyle\theta_{0\sigma}^{*}$
$\displaystyle=||\mathbf{D}_{\sigma}^{-1}||(\overline{g}_{\sigma}+\overline{d}_{\sigma}+\overline{c}_{\sigma}||\dot{\mathbf{q}}^{d}||^{2}),$
$\displaystyle\theta_{1\sigma}^{*}$
$\displaystyle=2\overline{c}_{\sigma}||\mathbf{D}_{\sigma}^{-1}||||\dot{\mathbf{q}}^{d}||,$
$\displaystyle\theta_{2\sigma}^{*}$
$\displaystyle=\overline{c}_{\sigma}||\mathbf{D}_{\sigma}^{-1}||,$
$\displaystyle\theta_{3\sigma}^{*}$
$\displaystyle=||\mathbf{D}_{\sigma}^{-1}\mathbf{M}_{\sigma}-\mathbf{I}||+||\mathbf{D}_{\sigma}^{-1}||\overline{h}.$

The upper bound structure (3.15) is obtained by using the following relations
$\mathbf{q}=\mathbf{e}+\mathbf{q}^{d}$,
$\dot{\mathbf{q}}=\dot{\mathbf{e}}+\dot{\mathbf{q}}^{d}$,
$||\boldsymbol{\xi}||\geq||\mathbf{e}||$ and
$||\boldsymbol{\xi}||\geq||\dot{\mathbf{e}}||$ (cf. [36, 37] for details).

Based on the upper bound structure in (3.15), the gain $\rho_{\sigma}$ in
(3.13a) is designed as

$\displaystyle\rho_{\sigma}$
$\displaystyle=\hat{\theta}_{0\sigma}+\hat{\theta}_{1\sigma}||\boldsymbol{\xi}||+\hat{\theta}_{2\sigma}||\boldsymbol{\xi}||^{2}+\hat{\theta}_{3\sigma}||\ddot{\overline{\mathbf{q}}}||+\zeta_{\sigma}+\gamma_{\sigma}$
$\displaystyle\triangleq\mathbf{Y}^{T}_{\sigma}\hat{\boldsymbol{\Theta}}_{\sigma}+\zeta_{\sigma}+\gamma_{\sigma},$
(3.16)

where
$\hat{\boldsymbol{\Theta}}_{\sigma}\triangleq[\hat{\theta}_{0\sigma}~{}\hat{\theta}_{1\sigma}~{}\hat{\theta}_{2\sigma}~{}\hat{\theta}_{3\sigma}]^{T}$
and $\zeta_{\sigma},\gamma_{\sigma}$ are auxiliary gains needed for closed-
loop stabilization. Defining
$\varrho_{\sigma}\triangleq({\lambda_{\min}(\mathbf{Q}_{\sigma})}/{\lambda_{\max}(\mathbf{P}_{\sigma})})$,
the gains $\hat{\theta}_{i\sigma},\zeta_{\sigma}$ and $\gamma_{\sigma}$ are
adapted using the following laws:

$\displaystyle\dot{\hat{\theta}}_{jp}=||\mathbf{r}_{p}||||\boldsymbol{\xi}||^{j}-\alpha_{jp}{\hat{\theta}}_{jp},~{}\dot{\hat{\theta}}_{j\overline{p}}=0,~{}j=0,1,2$
(3.17a)
$\displaystyle\dot{\hat{\theta}}_{3p}=||\mathbf{r}_{p}||||\ddot{\overline{\mathbf{q}}}||-\alpha_{3p}{\hat{\theta}}_{3p},~{}\dot{\hat{\theta}}_{3\overline{p}}=0$
(3.17b)
$\displaystyle\dot{\zeta}_{p}=-\left(1+\hat{\theta}_{3p}||\ddot{\overline{\mathbf{q}}}|||\mathbf{r}_{p}||\right){\zeta}_{p}+\bar{\epsilon}_{p},~{}\dot{\zeta}_{\overline{p}}=0,$
(3.17c)
$\displaystyle\dot{\gamma}_{p}=0,~{}\dot{\gamma}_{\overline{p}}=-\left(1+\frac{\varrho_{\overline{p}}}{2}\sum_{i=0}^{3}{\hat{\theta}}_{i\overline{p}}^{2}\right){\gamma}_{\overline{p}}+{\epsilon}_{\overline{p}},$
(3.17d)
$\displaystyle\text{with}~{}\alpha_{i\sigma}>\varrho_{\sigma}/2,~{}i=0,1,2,3,$
(3.17e)
$\displaystyle\hat{\theta}_{i\sigma}(t_{0})>0,~{}\zeta_{\sigma}(t_{0})=\bar{\zeta}_{\sigma}>{\bar{\epsilon}}_{\sigma},~{}\gamma_{\sigma}(t_{0})=\bar{\gamma}_{\sigma}>{\epsilon}_{\sigma},$
(3.17f)

where $p$ and $\overline{p}\in\Omega\backslash\\{p\\}$ denote the active and
inactive subsystems respectively;
$\alpha_{ip},\bar{\epsilon}_{p},\epsilon_{\overline{p}}\in\mathbb{R}^{+}$ are
static design scalars and $t_{0}$ is the initial time.

We define
$\varrho_{M\sigma}\triangleq\lambda_{\max}(\mathbf{P}_{\sigma}),~{}\varrho_{m\sigma}\triangleq\lambda_{\min}(\mathbf{P}_{\sigma})$,
${\bar{\varrho}_{M}}\triangleq\max_{\sigma\in\Omega}(\varrho_{M\sigma})$ and
${\underline{\varrho}_{m}}\triangleq\min_{\sigma\in\Omega}(\varrho_{m\sigma})$.
Following Definition 1 of ADT [63], the switching law is proposed as

$\displaystyle\vartheta>\vartheta^{*}=\ln\mu/{\kappa},$ (3.18)

where $\mu\triangleq{\bar{\varrho}_{M}}/\underline{\varrho}_{m}$; $\kappa$ is
a scalar defined as $0<\kappa<\varrho$ where
$\varrho\triangleq\min_{\sigma\in\Omega}({\lambda_{\min}(\mathbf{Q}_{\sigma})}/{\lambda_{\max}(\mathbf{P}_{\sigma})})$.

######  Remark 3.4 (On the use of acceleration measurements).

Being primarily intended for outdoor applications, use of inertial navigation
systems with accelerometers are quite common for quadrotor systems [32, 69].
Intial pioneering adaptive control designs for (non-switched) systems made use
of acceleration measurements [68], which was later avoided due to want of
sensors. Nevertheless, with the technological advancements and with cheaper
prices, usage of acceleration feedback can be found nowadays in many (non-
switched) robust and adaptive designs (cf. [70, 67, 37, 71] and references
therein).

###  3.4 Stability Analysis of The Proposed Controller

######  Theorem 3.1.

Under Assumption 1 and Properties 1-2, the closed-loop trajectories of system
(3.14) employing the control laws (3.13) and (3.16) with adaptive law (3.17)
and switching law (3.18) are Uniformly Ultimately Bounded (UUB). An ultimate
bound $b$ on the tracking error $\boldsymbol{\xi}$ can be found as

$\displaystyle
b=\sqrt{\frac{{2\bar{\varrho}_{M}^{(N_{0}+1)}}\left(\delta+\varpi\delta_{1}\right)}{{{\underline{\varrho}_{m}^{(N_{0}+2)}\left(\varrho-\kappa\right)}}}},$
(3.19)

where the scalars $\delta$ and $\delta_{1}$ are defined during the proof.

###### Proof.

From (3.17a)-(3.17d) and the initial conditions (3.17f), it can be verified
that
$\exists\underline{\zeta}_{\sigma},\underline{\gamma}_{\sigma}\in\mathbb{R}^{+}$
such that

$\displaystyle\hat{\theta}_{i\sigma}(t)\geq
0,~{}0<\underline{\zeta}_{\sigma}\leq\zeta_{\sigma}(t)\leq\bar{\zeta}_{\sigma},$
and $\displaystyle
0<\underline{\gamma}_{\sigma}\leq\gamma_{\sigma}(t)\leq\bar{\gamma}_{\sigma}~{}~{}\forall
t\geq t_{0}.$ (3.20)

Stability analysis is carried out based on the multiple Lyapunov candidate:

$\displaystyle V$
$\displaystyle=\frac{1}{2}\boldsymbol{\xi}^{T}\mathbf{P}_{\sigma}\boldsymbol{\xi}+\sum_{s=1}^{N}\sum_{i=0}^{3}\frac{(\hat{\theta}_{is}-{\theta}_{is}^{*})^{2}}{2}+\frac{\gamma_{s}}{\underline{\gamma}}+\frac{\zeta_{s}}{\underline{\zeta}},$
(3.21)

where $\underline{\gamma}=\min_{s\in\Omega}(\underline{\gamma}_{s})$ and
$\underline{\zeta}=\min_{s\in\Omega}(\underline{\zeta}_{s})$. Observing that
$\boldsymbol{\Lambda}_{\sigma}\boldsymbol{\xi}=\mathbf{K}_{1\sigma}\mathbf{e}+\mathbf{K}_{2\sigma}\dot{\mathbf{e}}$,
the error dynamics obtained in (3.14) becomes

$\displaystyle\dot{\boldsymbol{\xi}}=\mathbf{A}_{\sigma}\boldsymbol{\xi}+\mathbf{B}\left(\boldsymbol{\chi}_{\sigma}-\Delta\boldsymbol{\tau}_{\sigma}\right).$
(3.22)

Note that $V(t)$ might be discontinuous at the switching instants and only
remains continuous during the time interval between two consecutive
switchings. Without loss of generality, the behaviour of $V$ is studied at the
switching instant $t_{l+1},~{}l\in\mathbb{N}^{+}$. Let an active subsystem be
${\sigma({t_{l+1}^{-}})}$ when $t\in[t_{l}~{}~{}t_{l+1})$ and
${\sigma({t_{l+1}})}$ when $t\in[t_{l+1}~{}~{}t_{l+2})$. We have before and
after switching

$\displaystyle V({t_{l+1}^{-}})$
$\displaystyle=\frac{1}{2}\boldsymbol{\xi}^{T}({t_{l+1}^{-}})\mathbf{P}_{\sigma({t_{l+1}^{-}})}\boldsymbol{\xi}({t_{l+1}^{-}})+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{\frac{(\hat{\theta}_{is}({t_{l+1}^{-}})-{\theta}_{is}^{*})^{2}}{2}+\frac{\gamma_{s}({t_{l+1}^{-}})}{\underline{\gamma}}+\frac{\zeta_{s}({t_{l+1}^{-}})}{\underline{\zeta}}\right\\},$
$\displaystyle V({t_{l+1}})$
$\displaystyle=\frac{1}{2}\boldsymbol{\xi}^{T}({t_{l+1}})\mathbf{P}_{\sigma({t_{l+1}})}\boldsymbol{\xi}({t_{l+1}})+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{\frac{(\hat{\theta}_{is}({t_{l+1}})-{\theta}_{is}^{*})^{2}}{2}+\frac{\gamma_{s}({t_{l+1}})}{\underline{\gamma}}+\frac{\zeta_{s}({t_{l+1}})}{\underline{\zeta}}\right\\},$

respectively. Thanks to the continuity of the tracking error
$\boldsymbol{\xi}$ in (3.22) and of the gains
$\hat{\theta}_{i\sigma},\zeta_{\sigma}$ and $\gamma_{\sigma}$ in (3.17), we
have $\boldsymbol{\xi}({t_{l+1}^{-}})=\boldsymbol{\xi}({t_{l+1}})$,
$(\hat{\theta}_{is}({t_{l+1}^{-}})-{\theta}_{is}^{*})=(\hat{\theta}_{is}({t_{l+1}})-{\theta}_{is}^{*})$,
$\gamma_{s}({t_{l+1}^{-}})=\gamma_{s}({t_{l+1}})$ and
$\zeta_{s}({t_{l+1}^{-}})=\zeta_{s}({t_{l+1}})$. Further, owing to the facts
$\boldsymbol{\xi}^{T}({t})\mathbf{P}_{\sigma(t)}\boldsymbol{\xi}({t})\leq{\bar{\varrho}_{M}}\boldsymbol{\xi}^{T}({t})\boldsymbol{\xi}({t})$
and
$\boldsymbol{\xi}^{T}({t})\mathbf{P}_{\sigma(t)}\boldsymbol{\xi}({t})\geq{\underline{\varrho}_{m}}\boldsymbol{\xi}^{T}({t})\boldsymbol{\xi}({t})$,
one has

$\displaystyle V({t_{l+1}})-V({t_{l+1}^{-}})$
$\displaystyle=\frac{1}{2}\boldsymbol{\xi}^{T}({t_{l+1}})(\mathbf{P}_{\sigma({t_{l+1}})}-\mathbf{P}_{\sigma({t_{l+1}^{-}})})\boldsymbol{\xi}({t_{l+1}})$
$\displaystyle\leq\frac{{\bar{\varrho}_{M}}-{\underline{\varrho}_{m}}}{{2\underline{\varrho}_{m}}}\boldsymbol{\xi}^{T}({t_{l+1}})\mathbf{P}_{\sigma({t_{l+1}^{-}})}\boldsymbol{\xi}({t_{l+1}})$
$\displaystyle\leq\frac{{\bar{\varrho}_{M}}-{\underline{\varrho}_{m}}}{{\underline{\varrho}_{m}}}V(t_{l+1}^{-})$
$\displaystyle\Rightarrow V({t_{l+1}})$ $\displaystyle\leq\mu V(t_{l+1}^{-}),$
(3.23)

with $\mu={\bar{\varrho}_{M}}/{\underline{\varrho}_{m}}\geq 1$. At this point,
the behaviour of $V(t)$ between two consecutive switching instants, i.e., when
$t\in[t_{l}~{}~{}t_{l+1})$ can be studied.

We shall proceed the stability analysis for the two cases (i)
$||\mathbf{r}_{\sigma}||\geq\varpi$ and (ii) $||\mathbf{r}_{\sigma}||<\varpi$
using the Lyapunov function (3.21). For convenience of notation, let us denote
the active subsystem ${\sigma({t_{l+1}^{-}})}$ with $p$ and any inactive
subsystem with $\overline{p}$.

Case (i) $||\mathbf{r}_{\sigma}||\geq\varpi$

Using (3.15), (3.22), (3.17) and the Lyapunov equation
$\mathbf{A}_{p}^{T}\mathbf{P}_{p}+\mathbf{P}_{p}\mathbf{A}_{p}=-\mathbf{Q}_{p}$,
the time derivative of (3.21) yields

$\displaystyle\dot{V}$
$\displaystyle=\frac{1}{2}\boldsymbol{\xi}^{T}(\mathbf{A}_{p}^{T}\mathbf{P}_{p}+\mathbf{P}_{p}\mathbf{A}_{p})\boldsymbol{\xi}+\boldsymbol{\xi}^{T}\mathbf{P}_{p}\mathbf{B}\left(\boldsymbol{\chi}_{p}-{\rho_{p}}\frac{\mathbf{r}_{p}}{||\mathbf{r}_{p}||}\right)+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{(\hat{\theta}_{is}-{\theta}_{is}^{*})\dot{\hat{\theta}}_{is}+\frac{\dot{\gamma}_{s}}{\underline{\gamma}}+\frac{\dot{\zeta}_{s}}{\underline{\zeta}}\right\\}$
$\displaystyle\leq-\frac{1}{2}\boldsymbol{\xi}^{T}\mathbf{Q}_{p}\boldsymbol{\xi}+||\boldsymbol{\chi}_{p}||||\mathbf{r}_{p}||-\rho_{p}{||\mathbf{r}_{p}||}+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{(\hat{\theta}_{is}-{\theta}_{is}^{*})\dot{\hat{\theta}}_{is}+\frac{\dot{\gamma}_{s}}{\underline{\gamma}}+\frac{\dot{\zeta}_{s}}{\underline{\zeta}}\right\\}$
(3.24)
$\displaystyle\leq-\frac{1}{2}\boldsymbol{\xi}^{T}\mathbf{Q}_{p}\boldsymbol{\xi}-\mathbf{Y}^{T}_{p}(\hat{\boldsymbol{\Theta}}_{p}-\boldsymbol{\Theta}_{p}^{*})||\mathbf{r}_{p}||+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{(\hat{\theta}_{is}-{\theta}_{is}^{*})\dot{\hat{\theta}}_{is}+\frac{\dot{\gamma}_{s}}{\underline{\gamma}}+\frac{\dot{\zeta}_{s}}{\underline{\zeta}}\right\\}.$
(3.25)

Using (3.17a)-(3.17b) we have

$\displaystyle\sum_{i=0}^{3}(\hat{\theta}_{ip}-{\theta}_{ip}^{*})\dot{\hat{\theta}}_{ip}$
$\displaystyle=\sum_{j=0}^{2}(\hat{\theta}_{jp}-{\theta}_{jp}^{*})(||\mathbf{r}_{p}||||\boldsymbol{\xi}||^{j}-\alpha_{jp}{\hat{\theta}}_{jp})+(\hat{\theta}_{3p}-{\theta}_{3p}^{*})(||\mathbf{r}_{p}||||\ddot{\overline{\mathbf{q}}}||-\alpha_{3p}{\hat{\theta}}_{3p})$
$\displaystyle=\mathbf{Y}^{T}_{p}(\hat{\boldsymbol{\Theta}}_{p}-\boldsymbol{\Theta}_{p}^{*})||\mathbf{r}_{p}||+\sum_{i=0}^{3}\left\\{\alpha_{ip}{\hat{\theta}}_{ip}{\theta}_{ip}^{*}-\alpha_{ip}{\hat{\theta}}_{ip}^{2}\right\\}.$
(3.26)

Similarly using the facts $\hat{\theta}_{is}\geq 0$,
$0<\underline{\zeta}_{s}\leq\zeta_{s}(t),~{}0<\underline{\gamma}_{s}\leq\gamma_{s}(t)$
from (3.20) and
$\underline{\zeta}=\min_{s\in\Omega}(\underline{\zeta}_{s}),~{}\underline{\gamma}=\min_{s\in\Omega}(\underline{\gamma}_{s})$,
(3.17c) and (3.17d) lead to

$\displaystyle\frac{\dot{\zeta}_{p}}{\underline{\zeta}}$
$\displaystyle=-\left(1+\hat{\theta}_{3p}||\ddot{\overline{\mathbf{q}}}|||\mathbf{r}_{p}||\right)\frac{{\zeta}_{p}}{\underline{\zeta}}+\frac{\bar{\epsilon}_{p}}{\underline{\zeta}}$
$\displaystyle\leq-\hat{\theta}_{3p}||\ddot{\overline{\mathbf{q}}}|||\mathbf{r}_{p}||+\frac{\bar{\epsilon}_{p}}{\underline{\zeta}},$
(3.27) $\displaystyle\frac{\dot{\gamma}_{\overline{p}}}{\underline{\gamma}}$
$\displaystyle=-\left(1+(\varrho_{\overline{p}}/2)\sum_{i=0}^{3}{\hat{\theta}}_{i\overline{p}}^{2}\right)\frac{{\gamma}_{\overline{p}}}{\underline{\gamma}}+\frac{{\epsilon}_{\overline{p}}}{\underline{\gamma}}$
$\displaystyle\leq-\frac{\varrho_{\overline{p}}}{2}\sum_{i=0}^{3}{\hat{\theta}}_{i\overline{p}}^{2}+\frac{{\epsilon}_{\overline{p}}}{\underline{\gamma}},$
(3.28)

Substituting (3.26)-(3.28) in (3.25) yields

$\displaystyle\dot{V}\leq$
$\displaystyle-\frac{1}{2}\lambda_{\min}(\mathbf{Q}_{p})||\boldsymbol{\xi}||^{2}+\sum_{i=0}^{3}\left\\{\alpha_{ip}{\hat{\theta}}_{ip}{\theta}_{ip}^{*}-\bar{\alpha}_{ip}{\hat{\theta}}_{ip}^{2}\right\\}$
$\displaystyle+\frac{\bar{\epsilon}_{p}}{\underline{\zeta}}-\left\\{\sum_{\forall\overline{p}\in\Omega\backslash\\{p\\}}\sum_{i=0}^{3}\varrho_{\overline{p}}{\hat{\theta}}_{i\overline{p}}^{2}-\frac{{\epsilon}_{\overline{p}}}{\underline{\gamma}}\right\\}.$
(3.29)

Since $\hat{\theta}_{is}\geq 0$, $\zeta_{s}(t)\leq\bar{\zeta}_{s}$ and
$\gamma_{s}(t)\leq\bar{\gamma}_{s}$ by design (3.20), one obtains

$\displaystyle
V\leq\frac{1}{2}\lambda_{\max}(\mathbf{P}_{p})||\boldsymbol{\xi}||^{2}+\sum_{s=1}^{N}\sum_{i=0}^{3}\frac{(\hat{\theta}_{is}^{2}+{\theta_{is}^{*}}^{2})}{2}+\frac{\bar{\gamma}_{s}}{\underline{\gamma}}+\frac{\bar{\zeta}_{s}}{\underline{\zeta}}.$
(3.30)

Hence, using (3.30), the condition (3.29) is further simplified to

$\displaystyle\dot{V}\leq$ $\displaystyle-\varrho
V+\frac{\bar{\epsilon}_{p}}{\underline{\zeta}}+\sum_{i=0}^{3}\left\\{\alpha_{ip}{\hat{\theta}}_{ip}{\theta}_{ip}^{*}-\bar{\alpha}_{ip}{\hat{\theta}}_{ip}^{2}\right\\}+\sum_{s=1}^{N}\sum_{i=0}^{3}\frac{\varrho_{s}{\theta_{is}^{*}}^{2}}{2}+\varrho_{s}\left(\frac{\bar{\gamma}_{s}}{\underline{\gamma}}+\frac{\bar{\zeta}_{s}}{\underline{\zeta}}\right)+\frac{{\epsilon}_{s}}{\underline{\gamma}},$
(3.31)

where $\varrho=\min_{p\in\Omega}\\{\varrho_{p}\\}$;
$\bar{\alpha}_{ip}=({\alpha}_{ip}-(\varrho_{p}/2))>0$ by design from (3.17e).
Again, the following rearrangement can be made

$\displaystyle\alpha_{ip}{\hat{\theta}}_{ip}{\theta}_{ip}^{*}-\bar{\alpha}_{ip}\hat{\theta}_{ip}^{2}$
$\displaystyle=-\bar{\alpha}_{ip}\left(\hat{\theta}_{ip}-\frac{\alpha_{ip}{\theta}_{ip}^{*}}{2\bar{\alpha}_{ip}}\right)^{2}+\frac{\left(\alpha_{ip}{\theta}_{ip}^{*}\right)^{2}}{4\bar{\alpha}_{ip}}.$
(3.32)

We had defined earlier $0<\kappa<\varrho$. Then, using (3.32), $\dot{V}(t)$
from (3.31) gets simplified to

$\displaystyle\dot{V}(t)\leq-\kappa V(t)-(\varrho-\kappa)V(t)+\delta,$ (3.33)

where
$\delta\triangleq\max_{p\in\Omega}\left(\sum_{i=0}^{3}\left(\alpha_{ip}{\theta}_{ip}^{*}\right)^{2}/({4\bar{\alpha}_{ip}})+(\bar{\epsilon}_{p}/\underline{\zeta})\right)+\sum_{s=1}^{N}\sum_{i=0}^{3}(\varrho_{s}/2){\theta_{is}^{*}}^{2}+(\varrho_{s}\bar{\gamma}_{s})/\underline{\gamma}+(\varrho_{s}\bar{\zeta}_{s})/\underline{\zeta}+({\epsilon}_{s}/\underline{\gamma})$.

Case (ii) $||\mathbf{r}_{\sigma}||<\varpi$

In this case, the time derivative of (3.21) yields

$\displaystyle\dot{V}$
$\displaystyle\leq-\frac{1}{2}\boldsymbol{\xi}^{T}\mathbf{Q}_{p}\boldsymbol{\xi}+||\boldsymbol{\chi}_{p}||||\mathbf{r}_{p}||-\rho_{p}\frac{{||\mathbf{r}_{p}||}^{2}}{\varpi}+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{(\hat{\theta}_{is}-{\theta}_{is}^{*})\dot{\hat{\theta}}_{is}+\frac{\dot{\gamma}_{s}}{\underline{\gamma}}+\frac{\dot{\zeta}_{s}}{\underline{\zeta}}\right\\}$
(3.34)
$\displaystyle\leq-\frac{1}{2}\boldsymbol{\xi}^{T}\mathbf{Q}_{p}\boldsymbol{\xi}+||\boldsymbol{\chi}_{p}||||\mathbf{r}_{p}||+\sum_{s=1}^{N}\sum_{i=0}^{3}\left\\{(\hat{\theta}_{is}-{\theta}_{is}^{*})\dot{\hat{\theta}}_{is}+\frac{\dot{\gamma}_{s}}{\underline{\gamma}}+\frac{\dot{\zeta}_{s}}{\underline{\zeta}}\right\\}.$
(3.35)

Following similar lines of proof as in Case (i) we have

$\displaystyle\dot{V}$ $\displaystyle\leq-\kappa
V-(\varrho-\kappa)V+\delta+(\mathbf{Y}^{T}_{p}\hat{\boldsymbol{\Theta}}_{p}-\hat{\theta}_{3p}||\ddot{\overline{\mathbf{q}}}||)||\mathbf{r}_{p}||$
$\displaystyle=-\kappa
V-(\varrho-\kappa)V+\delta+\sum_{j=0}^{2}\hat{\theta}_{jp}||\boldsymbol{\xi}||^{j}||\mathbf{r}_{p}||.$
(3.36)

From (3.12) one can verify
$||\mathbf{r}||<\varphi\Rightarrow||\boldsymbol{\xi}||\in\mathcal{L}_{\infty}$
and consequently, the adaptive law (3.17a) implies
$||\mathbf{r}||,||\boldsymbol{\xi}||\in\mathcal{L}_{\infty}\Rightarrow\hat{\theta}_{jp}(t)\in\mathcal{L}_{\infty}$,
$j=0,1,2$. Therefore, $\exists\delta_{1}\in\mathbb{R}^{+}$ such that
$\sum_{j=0}^{2}\hat{\theta}_{jp}||\boldsymbol{\xi}||^{j}\leq\delta_{1}$
$\forall p\in\Omega$ when $||\mathbf{r}_{p}||<\varphi$. Hence, replacing this
relation in (3.36) yields

$\displaystyle\dot{V}(t)\leq-\kappa
V(t)-(\varrho-\kappa)V(t)+\delta+\varpi\delta_{1}.$ (3.37)

Therefore, investigating the stability results of Cases (i) and (ii), it can
be concluded that $\dot{V}(t)\leq-\varrho V(t)$ when

$\displaystyle
V(t)\geq\mathcal{B}\triangleq\frac{\delta+\varpi\delta_{1}}{(\varrho-\kappa)}.$
(3.38)

In light of this, further analysis is needed to observe the behaviour of
$V(t)$ between the two consecutive switching instants, i.e.,
$t\in[t_{l}~{}t_{l+1})$, for two possible scenarios:

  * (i)

when $V(t)\geq\mathcal{B}$, we have $\dot{V}(t)\leq-\varrho V(t)$ implying
exponential decrease of $V(t)$;

  * (ii)

when $V(t)<\mathcal{B}$, no exponential decrease can be derived.

Behaviour of $V(t)$ is discussed below individually for these two scenarios.

Scenario (i): There exists a time, call it $T_{1}$, when $V(t)$ enters into
the bound $\mathcal{B}$ and $N_{\sigma}(t)$ denotes the number of all
switching intervals for $t\in[t_{0}~{}~{}t_{0}+T_{1})$. Accordingly, for
$t\in[t_{0}~{}~{}t_{0}+T_{1})$, using (3.23) and $N_{\sigma}(t_{0},t)$ from
Definition 1 we have

$\displaystyle V(t)$
$\displaystyle\leq\exp\left(-\kappa(t-t_{N_{\sigma}(t)-1})\right)V(t_{N_{\sigma}(t)-1})$
$\displaystyle\leq\mu\exp\left(-\kappa(t-t_{N_{\sigma}(t)-1})\right)V(t_{N_{\sigma}(t)-1}^{-})$
$\displaystyle\leq\mu\exp\left(-\kappa(t-t_{N_{\sigma}(t)-1})\right)$
$\displaystyle~{}~{}~{}\cdot\mu\exp\left(-\kappa(t_{N_{\sigma}(t)-1}-t_{N_{\sigma}(t)-2})\right)V(t_{N_{\sigma}(t)-2}^{-})$
$\displaystyle\qquad\qquad\qquad\qquad\vdots$
$\displaystyle\leq\mu\exp\left(-\kappa(t-t_{N_{\sigma}(t)-1})\right)\mu\exp\left(-\kappa(t_{N_{\sigma}(t)-1}-t_{N_{\sigma}(t)-2})\right)$
$\displaystyle~{}~{}~{}~{}\cdots\mu\exp\left(-\kappa(t_{1}-t_{0})\right)V(t_{0})$
$\displaystyle=\mu^{N_{\sigma}(t_{0},t)}\exp\left(-\kappa(t-t_{0})\right)V(t_{0})$
$\displaystyle=c\left(\exp\left(-\kappa+({\ln\mu}/{\vartheta})\right)\right)V(t_{0}),$
(3.39)

where $c\triangleq\exp\left(N_{0}\ln\mu\right)$ is a constant. Substituting
the ADT condition $\vartheta>\ln\mu/\varrho$ in (3.39) yields $V(t)<cV(t_{0})$
for $t\in[t_{0}~{}~{}t_{0}+T_{1})$. Moreover, as $V(t_{0}+T_{1})<\mathcal{B}$,
one has $V(t_{N_{\sigma}(t)+1})<\mu\mathcal{B}$ from (3.23) at the next
switching instant $t_{N_{\sigma}(t)+1}$ after $t_{0}+T_{1}$. This implies that
$V(t)$ may be larger than $\mathcal{B}$ from the instant
$t_{N_{\sigma}(t)+1}$. This necessitates further analysis.

We assume $V(t)\geq\mathcal{B}$ for
$t\in[t_{N_{\sigma}(t)+1}~{}~{}t_{0}+T_{2})$, where $T_{2}$ denotes the time
before next switching. Let $\bar{{N}}_{\sigma}(t)$ represents the number of
all switching intervals for $t\in[t_{N_{\sigma}(t)+1}~{}~{}t_{0}+T_{2})$.
Then, substituting $V(t_{0})$ with $V(t_{N_{\sigma}(t)+1})$ in (3.39) and
following the similar procedure for analysis as (3.39), we have $V(t)\leq
cV(t_{N_{\sigma}(t)+1})<c\mu\mathcal{B}$ for
$t\in[t_{N_{\sigma}(t)+1}~{}~{}t_{0}+T_{2})$. Since
$V(t_{0}+T_{2})<\mathcal{B}$, we have
$V(t_{N_{\sigma}(t)+\bar{N}_{\sigma}(t)+2})<\mu\mathcal{B}$ at the next
switching instant $t_{N_{\sigma}(t)+\bar{N}_{\sigma}(t)+2}$ after
$t_{0}+T_{2}$. If we follow similar lines of proof recursively, we can come to
the conclusion that $V(t)<c\mu\mathcal{B}$ for
$t\in[t_{0}+T_{1}~{}~{}\infty)$. This confirms that once $V(t)$ enters the
interval $[0,\mathcal{B}]$, it cannot exceed the bound $c\mu\mathcal{B}$ any
time later with the ADT switching law (3.18).

Scenario (ii): It can be verified that the same argument below (3.39) also
holds for Scenario (ii).

Thus, observing the stability arguments of the Scenarios (i) and (ii), it can
be concluded that the closed-loop system remains UUB with the control laws
(3.13) and (3.16) with the adaptive law (3.17) and switching law (3.18)
implying

$\displaystyle
V(t)\leq\max\left(cV(t_{0}),c\mu\mathcal{B}\right),~{}~{}\forall t\geq t_{0}.$
(3.40)

Again, the definition of the Lyapunov function (3.21) yields

$\displaystyle
V(t)\geq({1}/{2})\lambda_{\min}(\mathbf{P}_{\sigma(t)})||\boldsymbol{\xi}||^{2}\geq\left({\underline{\varrho}_{m}}/{2}\right)||\boldsymbol{\xi}||^{2}.$
(3.41)

Using (3.40) and (3.41) we have

$\displaystyle||\boldsymbol{\xi}||^{2}\leq\left({2}/{\underline{\varrho}_{m}}\right)\max\left(cV(t_{0}),c\mu\mathcal{B}\right),~{}~{}\forall
t\geq t_{0}.$ (3.42)

Therefore, using the expression of $\mathcal{B}$ from (3.30), an ultimate
bound $b$ on the tracking error $\boldsymbol{\xi}$ can be found as (3.19). ∎

Note that in collocated design, it is standard to consider the internal
dynamics to be stable/bounded a priori [65, 66]. In the following, a few
design aspects of the proposed mechanism are highlighted.

######  Remark 3.5 (The role of gain $\mathbf{D}_{\sigma}$ and its selection).

The control law (3.13a) relies on the inverse dynamics-type design: in its
absence, $\mathbf{D}_{\sigma}$ would have been replaced by some nominal value
of mass matrix following the conventional design (cf. [61, 54]). However,
according to Remark 1, such knowledge is unavailable. Again, avoiding inverse
dynamics-type design and involving mass matrix in the Lyapunov function
eventually lead to a switching law that relies on bound knowledge of mass
matrix [60]. These observations highlight the role of $\mathbf{D}_{\sigma}$ to
realize a switched controller without any knowledge of mass matrix.

Aside being positive definite, the proposed design does not put any
restriction on the choice of $\mathbf{D}_{\sigma}$. It can be observed from
(3.14) that higher values of $\mathbf{D}_{\sigma}$ reduces the effect of
uncertainty $\boldsymbol{\chi}_{\sigma}$ on the error dynamics, albeit at the
cost of higher control input (cf. (3.13a)). Therefore, $\mathbf{D}_{\sigma}$
needs to be designed according to the application requirements.

###  3.5 Simulation Results

Figure 3.2: The switching signal. Figure 3.3: Pulse type disturbance. Figure
3.4: Tracking performance of the proposed controller.

We test the proposed controller for a switched dynamics scenario as in Fig.
3.1 having three phases (a.k.a subsystems), denoted by $\sigma=1,2,3$ as

  * •

$\sigma=1$: the quadrotor rests on the ground without any payload;

  * •

$\sigma=2$: it starts to ascend with an initial payload;

  * •

$\sigma=3$: a new payload is attached to the quadrotor as it ascends to its
desired height.

For fruitful verification and as found in many applications (cf. [72, 73]), we
verify the controller on the six degrees-of-freedom quadrotor model. The
following parametric variations (due to addition of payload) are selected for
different subsystems

$\displaystyle\sigma=1:~{}m$ $\displaystyle=1.5,~{}I_{xx}=I_{yy}=1.69\cdot
10^{-5},~{}I_{zz}=3.38\cdot 10^{-5},$ $\displaystyle\sigma=2:~{}m$
$\displaystyle=1.6,~{}I_{xx}=0.011,~{}I_{yy}=0.010,~{}I_{zz}=1.27\cdot
10^{-4},$ $\displaystyle\sigma=3:~{}m$
$\displaystyle=1.7,~{}I_{xx}=0.032,~{}I_{yy}=0.030,~{}I_{zz}=2.20\cdot{10}^{-4}.$

The objective for the quadrotor is to lift and drop various payloads at
different heights while starting from the ground position. To achieve this
task, the following desired trajectories are defined
$x^{d}=y^{d}=0,z^{d}=2+\sin(0.1t),\phi^{d}=\varphi^{d}=\psi^{d}=0$ with an
initial position of $x(0)=y(0)=0.1,z(0)=0,\phi(0)=\varphi(0),\psi(0)=0.1$.
Selection of
$\mathbf{K}_{11}=120\mathbf{I},\mathbf{K}_{21}=100\mathbf{I},\mathbf{K}_{12}=150\mathbf{I},\mathbf{K}_{22}=120\mathbf{I},\mathbf{K}_{13}=200\mathbf{I},\mathbf{K}_{23}=140\mathbf{I},\mathbf{Q}_{1}=\mathbf{Q}_{2}=\mathbf{Q}_{3}=2\mathbf{I}$,
$\kappa=0.9\varrho$ yields the ADT $\vartheta^{*}=6.57$sec according to
(3.18). Therefore, a switching law $\sigma(t)$ is designed as in Fig. 3.2
(note that the initial fast switchings are compensated by slower switchings
later on).

Other control parameters are designed as

$\mathbf{D}_{\sigma}=\begin{bmatrix}2&0&0&0\\\ 0&1\cdot 10^{-4}&0&0\\\
0&0&1\cdot 10^{-4}&0\\\ 0&0&0&1\cdot 10^{-4}\end{bmatrix},$

$\alpha_{i\sigma}=0.6,\epsilon_{\sigma}=\bar{\epsilon}_{\sigma}=0.005$ with
$i=0,1,2,3$ for all $\sigma$. The initial conditions are selected as
$\hat{\theta}_{0\sigma}(0)=1.2,\hat{\theta}_{1\sigma}(0)=1.3,\hat{\theta}_{2\sigma}(0)=1.4,\hat{\theta}_{3\sigma}(0)=1.5$,
$\zeta_{\sigma}(0)=\gamma_{\sigma}(0)=1$ $\forall\sigma$. The external
disturbance is selected as
$\mathbf{d}_{\sigma}=\\{0.05\sin(0.5t),0,0,d_{p}\\}$ $\forall\sigma$, where
$d_{p}$ represents a pulse type disturbance (cf. Fig. 3.3) emulating sudden
disturbances like gust of wind. It is worth remarking here that, the proposed
mechanism being a collocated one, external disturbances should be selected
such that the nonactuated dynamics does not become unstable (cf. [65, 66]).
Therefore, no external disturbances are considered in $(\phi,\varphi)$ as well
as in $(x,y)$.

Figure 3.5: Gains for subsystem 1. Figure 3.6: Gains for subsystem 2. Figure
3.7: Gains for subsystem 3.

The performance of the proposed controller is depicted in Fig. 3.4 in terms of
tracking errors, where the attitude errors are reported in degree for better
comprehension. In line with the proposed adaptive law (3.17a)-(3.17d), Figs.
3.5-3.7 reveal that: (i) for an active subsystem, only the gains
$\hat{\theta}_{i{p}}$ and $\zeta_{p}$ are updated while $\gamma_{p}$ remains
constant; (ii) for an inactive subsystem, while $\gamma_{\overline{p}}$ is
updated, $\hat{\theta}_{i\overline{p}}$ and $\zeta_{\overline{p}}$ remain
constant.

##  Chapter 4 Conclusions

In this thesis, two types of adaptive control was proposed for two operational
scenarios of a quadrotor:

  * •

Aerial Transportation of Unknown Payloads: In this scenario, the payload does
not change during the flight. Under such a case, an adaptive controller for
quadrotors was proposed, which can tackle parametric uncertainties and
external disturbances without their a priori knowledge for the application of
carrying unknown payloads. The control framework was built to negotiate
possibly a priori unbounded state-dependent uncertainties. Closed-loop system
stability was established via the notion of uniformly ultimately boundedness.
The performance of the proposed controller was verified using Gazebo
simulation using RotorS Simulator framework, and comparative simulations
confirmed the effectiveness of the proposed scheme against state-of-the-art
methods under various scenarios.

  * •

Dynamic Payload Lifting Operations: Contrary to the first scenario, the
payload mass was considered to change for this scenario. Accordingly, a new
concept of adaptive control design for quadrotors was introduced for vertical
operations, using the framework of average dwell time based switched dynamics.
The proposed design did not require any a priori knowledge of the structure
and bound of uncertainty. The effectiveness of the concept was validated via
simulations with dynamically varying payloads.

###  4.1 Future Works

  * •

The current work on dynamic payload lifting operations uses the hovering
control dynamics. A more realistic scenario would be to operate in six
degrees-of-freedom. In order to solve this problem, a future controller needs
to utilise the partly decoupled dynamics for the switched mode control
approach.

  * •

The adaptive controllers developed in this thesis do not tackle the situations
of manoeuvring under various motion constraints (e.g., moving through a
window, pipeline interior inspection). Therefore, an interesting future work
would be to extend the proposed adaptive designs under position and velocity
constraints.

  * •

Throughout the thesis, the external payloads are considered to be rigidly
attached to the quadrotor platform. The case of suspended payload thus needs
to be considered in future.

Related Publications

### Relevant publications

  1. 1.

V. N. Sankaranarayanan, S. Roy and S. Baldi, “Aerial Transportation of Unknown
Payloads: Adaptive Path Tracking for Quadrotors,” 2020 IEEE/RSJ International
Conference on Intelligent Robots and Systems (IROS), pp. 7710-7715, 2020.

  2. 2.

V. N. Sankaranarayanan and S. Roy, “Introducing switched adaptive control for
quadrotors for vertical operations”, Optimal Control Applications and Methods,
vol. 41, no. 6, pp. 1875-1888, 2020.

### Other publications during MS

  1. 1.

S. Ganguly, *V. N. Sankaranarayanan, B. V. S. G. Suraj, R. Yadav and S. Roy
“Efficient Manoeuvring of Quadrotor under Constrained Space and Predefined
Accuracy,” 2021 IEEE/RSJ International Conference on Intelligent Robots and
Systems (IROS), 2021, accepted for publication.
* joint first authorship.

  2. 2.

S. Roy, S. Baldi, P. Li and V. N. Sankaranarayanan, “Artificial-Delay Adaptive
Control for Under-actuated Euler-Lagrange Robotics,” IEEE/ASME Transactions on
Mechatronics, 2021.

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