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Theses and Dissertations
2014
Optimal path planning and control of quadrotor
unmanned aerial vehicle for area coverage
Jiankun Fan
University of Toledo
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A Thesis
entitled
Optimal Path Planning and Control of Quadrotor Unmanned Aerial
Vehicle for Area Coverage
by
Jiankun Fan
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Mechanical Engineering
Dr. Manish Kumar, Committee Chair
Dr. Mohammad Elahinia, Committee Member
Dr. Mehdi Pourazady, Committee Member
Dr. Patricia Komuniecki, Dean
College of Graduate Studies
The University of Toledo
December 2014
Copyright © 2014, Jiankun Fan
This document is copyrighted material. Under copyright law, no parts of this document may
be reproduced without the expressed permission of the author
An Abstract of
Optimal Path Planning and Control of Quadrotor Unmanned Aerial Vehicle for Area
Coverage
By
Jiankun Fan
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Mechanical Engineering
The University of Toledo
December 2014
An Unmanned Aerial Vehicle (UAV) is an aircraft without a human pilot on
board. Its flight is controlled either autonomously by computers onboard the vehicle, or
remotely by a pilot on the ground, or by another vehicle. In recent years, UAVs have
been used more commonly than prior years. The example includes areo-camera where a
high speed camera was attached to a UAV which can be used as an airborne camera to
obtain aerial video. It also could be used for detecting events on ground for tasks such as
surveillance and monitoring which is a common task during wars. Similarly UAVs can be
used for relaying communication signal during scenarios when regular communication
infrastructure is destroyed. The objective of this thesis is motivated from such civilian
operations such as search and rescue or wildfire detection and monitoring.
One scenario is that of search and rescue where UAV’s objective is to geo-locate
a person in a given area. The task is carried out with the help of a camera whose live feed
is provided to search and rescue personnel. For this objective, the UAV needs to carry out
iii
scanning of the entire area in the shortest time. The aim of this thesis to develop
algorithms to enable a UAV to scan an area in optimal time, a problem referred to as
“Coverage Control” in literature. The thesis focuses on a special kind of UAVs called
“quadrotor” that is propelled with the help of four rotors.
The overall objective of this thesis is achieved via solving two problems. The first
problem is to develop a dynamic control model of quadrtor. In this thesis, a proportionalintegral-derivative controller (PID) based feedback control system is developed and
implemented on MATLAB’s Simulink. The PID controller helps track any given
trajectory. The second problem is to design a trajectory that will fulfill the mission. The
planed trajectory should make sure the quadrotor will scan the whole area without
missing any part to make sure that the quadrotor will find the lost person in the area. The
generated trajectory should also be optimal. This is achieved via making some
assumptions on the form of the trajectory and solving the optimization problem to obtain
optimal parameters of the trajectory. The proposed techniques are validated with the help
of numerous simulations.
iv
Acknowledgments
At first I would like to thank my parents, who always supported and encouraged
me. They always taught me I should not give up when I am in a difficult situation.
Without their financial support, I could not finish my studies in US.
I would like to thank my advisor Dr. Manish Kumar. He always gave me most
useful suggestion when I had problem in my research. I am extremely fortunate to have
him as an advisor.
Also, I would thank my committee members, Dr. Mohammad Elahinia and Dr.
Mehdi Pourazady. They gave me a lot of valuable comments and suggestions.
Especially, I would like to thank my American parents Mr.and and Mrs. Elmer.
They have been really helpful in improving my English and helping me adapt to the life
in US.
At last but not least, a thank you to Ruoyou Tan, Alireza and Sarim who are my
labmates. Alireza gave me a lot of help at the beginning of my research for developing
my Simulink code. Ruoyou helped analyse the research paper and guided me at every
step. I am really thankful to have a very good group in our CDS lab.
v
Table of Contents
Acknowledgments............................................................................................................... v
Table of Contents ............................................................................................................... vi
List of Figures .................................................................................................................... ix
List of Symbols ................................................................................................................ xiii
1 Introduction ..................................................................................................................... 1
1.1 Unmanned Aerial Vehicles (UAVs): An Introduction .......................................... 1
1.2Problem Motivation ................................................................................................ 7
1.3 Optimal Path Planning and Control Problem ........................................................ 7
1.4 Document Outlines ................................................................................................ 8
2 Literature Review.......................................................................................................... 10
2.1 UAV control system ............................................................................................ 10
2.2 Coverage and Path Planning ................................................................................ 12
3 Problem Description ..................................................................................................... 14
3.1 Quadrotor dynamics model design ...................................................................... 14
3.2 Coverage and optimal path planning ................................................................... 16
4 Quadrotor Dynamics and Control Model ..................................................................... 18
4.1 Dynamics Analysis .............................................................................................. 18
4.1.1 Modeling ................................................................................................... 18
4.1.2 Dynamic Model ......................................................................................... 20
vi
4.1.3 Position Control ........................................................................................ 23
4.1.4 Hover Controller ....................................................................................... 23
4.1.5 Attitude Control......................................................................................... 25
4.2 Dynamics Model Simulation ............................................................................... 26
5 Trajectory Design and Optimization ............................................................................. 35
5.1 Trajectory design ................................................................................................. 35
5.2 Optimization of Trajectory Parameters ............................................................... 38
5.2.1 Segment 1: Curve Segment ....................................................................... 39
5.2.2 Segment 2: Straight line ............................................................................ 43
5.2.3 Segment 3: straight line ............................................................................. 50
5.3 Simulation Results ............................................................................................... 53
5.3.1 Segment 1: Curve Segment ....................................................................... 53
5.3.2 Segment 2: Straight line ............................................................................ 55
5.3.3 Segment 3: Straight line ............................................................................ 58
5.3.4 Overall simulation ..................................................................................... 61
6 Conclusions and Future work ....................................................................................... 64
6.1 Conclusions ......................................................................................................... 64
6.2 Contributions ....................................................................................................... 65
6.3 Future Works ....................................................................................................... 66
References ......................................................................................................................... 67
A Appendix A .................................................................................................................. 71
Parameters of the dynamics model ............................................................................ 71
B Appendix B .................................................................................................................. 72
The system for the dynamics simulation ................................................................... 72
vii
Subsystem of the dynamics simulation ..................................................................... 73
viii
List of Figures
1-1 co-axial UAV in CDS Lab...............................................................................................3
1-2 octocopter UAV model ....................................................................................................3
1-3 Parrot AR. Drone 1.0 Quadrotor .....................................................................................6
1-4 CDS Lab UAVs ...............................................................................................................6
2-1 Dubbin’s Curve................................................................................................................13
3-1 The nested control loops for position and attitude control ..............................................15
4-1 Simplified illustration of rotor motion and control of a Quadrotor ................................19
4-2 Coordinate systems and forces/moments acting on a quadrotor frame ...........................20
4-3 Diagram showing Pitch, Yaw, and Roll angle .................................................................21
4-4 Simulation result showing the trajectory of the UAV .....................................................27
4-5 Simulation result showing the x location of the quadrotor versus time ..........................28
4-6 Simulation result showing the y location of the quadrotor versus time ..........................29
4-7 Simulation result showing the z location of the quadrotor versus time ...........................30
ix
4-8 Simulation result showing the desired (shown in red) and actual trajectory (shown in
blue) of a quadrotor following the circular trajectory ............................................................31
4-9 Simulation result showing the desired x (shown in red) and actual x (shown in blue)
of the quadrotor following a circular trajectory .....................................................................32
4-10 Simulation result showing the desired y (shown in red) and actual y (shown in blue)
of the quadrotor following a circular trajectory .....................................................................33
4-11 Simulation result of the quadrotor’s motion following a set of desired locations .........34
5-1 The sketch map of Dubins Curve ....................................................................................35
5-2 Sample coverage area ......................................................................................................36
5-3 The sketch map of path segment .....................................................................................37
5-4 Path planning of the UAV for the coverage problem ......................................................38
5-5 Classification of motions in the UAV trajectory .............................................................39
5-6 Real path simulation result at the linear velocity equal to 0.1 m/s ..................................41
5-7 Real path simulation result at the linear velocity equal to 0.2 m/s ..................................41
5-8 Real path simulation result at the linear velocity equal to 0.3 m/s ..................................42
5-9 Real path simulation result at the linear velocity equal to 0.4 m/s ..................................43
5-10 Real path simulation result at the linear velocity equal to 0.5 m/s ................................43
x
5-11 Segment 2: Straight line ................................................................................................44
5-12 The rotors’ speed simulation at the acceleration equal to 1.5 m/s2 ...............................48
5-13 The rotors’ speed simulation at the acceleration equal to 2.0 m/s2 ...............................49
5-14 The rotors’ speed simulation at the acceleration equal to 2.5 m/s2 ...............................49
5-15 Simulation result of the desired velocity and actual velocity in circle segment............55
5-16 Simulation result of the desired path and actual path in circle segment........................56
5-17 Simulation result of the desired velocity and actual velocity in Segment 2: straight
line..........................................................................................................................................57
5-18 Simulation result of the desired acceleration and actual acceleration in Segment 2:
straight line.............................................................................................................................58
5-19 Simulation result of the trajectory and actual path in Segment 2: straight line .............59
5-20 Simulation result of the desired velocity and actual velocity in Segment 3: straight
line..........................................................................................................................................60
5-21 Simulation result of the desired acceleration and actual acceleration in Segment 3:
straight line.............................................................................................................................61
5-22 Simulation result of the trajectory and actual path in Segment 3: straight line .............62
5-23 The trajectory for coverage the whole area and the actual path ....................................63
5-24 Simulation result showing the area covered as the UAV tracks each of the segments
xi
................................................................................................................................................64
xii
List of Symbols
W .................................. World frame
B ................................ Body frame
g .......................................................Acceleration due to gravity
Φ .....................................................Roll Angle
θ ................................... Pitch Angle
ψ ......................................................Yaw Angle
r .......................................................Position vector of the center of mass in the world frame
m.................................Mass of Quadrotor
p, q, r ..........................................The components of angular velocity of the quadrotor
kF .....................................................Propeller force constant
kM ..................................................... Propeller moment constant
Mi.............................................. The moment produced by each rotors
Fi ...............................The force produced by each rotors
Kd, Kp ........................................... Controller gains
I ........................................................ Inertia matrix of Quadrotor
L ........................................................ The length between the axis of rotors to the center of mass
ωi...................................................... Speed of each rotor
∆ωi.................................................. Differential motor speeds
xiii
Chapter 1
Introduction
1.1 Unmanned Aerial Vehicles (UAVs): An Introduction
An Unmanned Aerial Vehicle (UAV), commonly known as a drone is an aircraft
without a human pilot on board. Its flight is controlled either autonomously by computers
in the vehicle, or via remote control by a pilot on the ground or in another vehicle [1]. In
1917, Peter Cooper and Elmer A. Sperry invented the first UAV, named Sperry Aerial
Torpedo. The size was equivalent to a normal size airplane. It took a payload of 300
pounds and flew 50 miles during its fight. There is a lot interest recently in smaller UAVs
also called miniature aerial vehicles (MAVs). The development of technology, such as
materials, electronics, sensors, and batteries has fueled the growth in the development of
MAVs that are typically between 0.1 and 0.5 m in length and 0.1–0.5 kg in mass [2].
Currently, well-known of the applications of the UAVs are all in the military. The
example include MQ-1 Predator which is an unmanned aerial vehicle (UAV) built by
General Atomics and used primarily by the United States Air Force (USAF) and Central
Intelligence Agency (CIA). Similarly, the Northrop Grumman RQ-4 Global Hawk is
primarily used for surveillance. The AeroVironment RQ-11 Raven is a hand-launched
1
remote-controlled small unmanned aerial vehicle (or SUAV) developed for the U.S.
military [3].
There can be several designs for the UAVs. These include fixed-wing, quadrotors,
co-axial, double rotors and octocopters. A fixed-wing UAV is designed based on the
fixed-wing aircraft which is an aircraft capable of flight using wings that generate lift
caused by the vehicle's forward airspeed and the shape of the wings. A quadrotor UAV
consists of two pairs of counter-rotating rotors and propellers, located at the vertices of a
square frame. It is capable of vertical take-off and landing (VTOL), but it does not
require complex mechanical linkages, such as swash plates, that commonly appear in
typical helicopters [4]. A co-axial UAV has a pair of rotors mounted on the same mast
and with the same axis of rotation, but turning in different directions as shown in Fig. 1.1.
Because of inclusion of a coaxial main rotor and exclusion of a tail rotor, which also
means a smaller footprint, co-axial rotor designs improved the co-axial UAV to become
more stable, more maneuverable, quieter and safer than traditional helicopters [5]. An
octocopters UAV consist of eight rotors and propellers as shown in Fig. 1.2. An
octocopter’s counter rotating stabilization system is similar to but more powerful and
efficiency compare with coaxial.
2
Figure 1.1 Co-axial UAV available in the Cooperative Distributed Systems (CDS)
Lab at University of Toledo
Figure 1.2 Octocopter UAV*
Copyright© XiaoXiong (2014).
3
In 2012, Andy Janetzko and his friend designed a new kind of UAV called hybrid
quadcopter. It vertically takes off and lands (VTOL) as well as derives lift like fixed win
airplanes. That way to the UAV achieves the highest possible aspect ratio and lift [6].
The UAV technology has opened sea of opportunities in its potential use in civilian
domain. Civilian agencies or departments as well as civilian commercial companies can
utilize this technology in emergency response, search and rescue, transportation, and
surveillance, monitoring or inspection of infrastructure. For example on 15th May, 2008,
the earth quake in China thoroughly destroyed the communication network in the affected
area. The Chinese government used UAVs set up as the temporary signal supplier. For
the problem being solved in this thesis, a motivating example could be search and rescue
of people or looking for a hot-spot in wildfire. In these cases, there is limited a priori
information available other than the information that the lost person or hot-spot would
likely be in a given area which needs to be spotted and located. For these situations,
UAVs provide a safe alternative where a UAV with an onboard vision or thermal camera
go through and scan this area. The images captured by the UAV can be sent back to a
ground control station whether it can be processed either manually or autonomously to
locate the object of interest.
Most of the current UAV applications are conducted using fixed-wing UAVs and
quadrotor helicopters. There is a lot of interest recently on quad-rotor UAVs. Quadrotor
is a type of UAV which consists of four rotors in total, with two pairs of counter rotating,
fixed-pitch blades located at the four corners of the aircraft [7]. Compared to a fixedwing UAV, a quadrotor has the flowing advantages:
4
1) Vertical takeoff and landing.
A fixed-wing UAV needs time to glide or taxi which generates enough thrust to fly
up. But in case of a quadrotor, there are four motors fixed on the frame. They can
provide thrust for the quadrotor to fly up vertically and work with its gravity to make
the quadrotor land vertically as well.
2) Ability to hover
When the thrust and gravitational force the same and in opposite direction, the total
force on the quadrotor is zero. So, it can hover in the air. Fixed-wing aircraft on the
other hand, cannot hover.
3) Simple structure, and easily controllable
The structure of quadrotor is very simple. It is combined with the frame and four
motors. The rotations of rotors produce forces and moments whose magnitudes
depend on the speed of roration. Hence, the only inputs to control a quadrotor are the
speeds of four rotors.
4) Low level noise
Four rotors provide the thrust for quadrotor. So the size of individual rotor as
compared single rotor helicopters is not so big. This also reduces the noise. Fig. 1.3
shows a Parrot AR Drone available in Cooperative Distributed Systems (CDS) Lab at the
University of Toledo.
5
Figure 1.3 Parrot AR. Drone 1.0 Quadrotor
The following Fig. 1.4 shows the different types of UAVs available in the CDS
lab.
Figure 1.4 The UAVs available in the CDS Lab
6
1.2Problem Motivation
The thesis' work has been motivated by the overwhelming UAV applications in
civilian and military fields, which results from the advances in communication,
computing, sensing, solid state devices, and battery technology. With the use of on-board
sensors, this technology is able to provide a feasible solution for low-altitude and highresolution applications such as scientific data gathering, homeland security, search, and
rescue, precision agriculture, forest fire monitoring, geological research and military
reconnaissance. A large class of application involves the UAV to search for a ground
event or target and communicate that information back to the ground control station.
These tasks require the UAV to search the target/event in an optimal and autonomous
fashion. The thesis focuses on this problem and presents a method to carry out an
exhaustive coverage of a given area in an optimal fashion.
1.3 Optimal Path Planning and Control Problem
This thesis can be divided in three parts:
The first is, obtaining the dynamic and control model of the quadrotor which consists
of the following:
1) Derivation of the dynamics.
2) Designing a suitable control system based on the dynamics.
3) Simulate the dynamics and control model using MATLAB Simulink.
7
The second part is, designing the trajectory or planning the path. Designing a
trajectory is often based on the problem at hand. For this thesis, the task is to design a
trajectory so that the quadrotor exhaustively covers the entire area. The thesis considers a
rectangular area and develops a trajectory that covers the entire area. The trajectory
consists of several segments as will be explained in the subsequent chapters.
The third part is to optimize the parameters of the path. Typically, one would like to
minimize the time required to complete the entire operation. As mentioned earlier, the
trajectory consists of several segments. The objective of this part is to obtain optimal
control actions such that the quadrotor follows the desired trajectory in order to complete
the entire operation in minimum time given the various constraints imposed by the
kinematics and dynamics of the quadrotor.
1.4 Document Outlines
The outline of this document is provided below:
Chapter 1: In this chapter, an introduction of this research is presented. The applications
of UAV in the world and the concept of UAV are stated. The difference between fixedwing UAV and quadrotor is listed.
Chapter 2: The background survey and related works are presented in this chapter. The
recent research on UAV control system, optimal path planning and coverage problem are
mentioned.
8
Chapter 3: This chapter presents details on control of UAV trajectory and optimal design
problems.
Chapter 4: This chapter presents the dynamics and control model design of the quadrotor.
A quadrotor has a complex dynamics. This chapter provides an analysis of the dynamics
of quadrotor in a step by step fashion and subsequently the model is implemented and
tested using MATLAB Simulink.
Chapter 5: This chapter provides a description of the design and selection of trajectory.
For scanning the area of interest, a trajectory is designed. Furthermore, the chapter
provides our approach to solve the optimal control problem of covering the entire area in
the shortest time.
Chapter 6: This chapter presents the conclusion and future works. The chapter lists the
advantages and disadvantages of the method utilized in the thesis. Subsequently, scope of
future work is described.
9
Chapter 2
Literature Review
This chapter presents the background survey of the existing literature on optimal control
and quadrotor control. UAVs belong to a class of non-linear systems which are inherently more
difficult to control. Due to recent interest in UAVs and their potential applications, a lot of work
has been carried out recently in the area of UAV control. In this thesis, a PID (proportionalintegral-derivative) control system was designed for controlling the UAV to follow a pre-defined
trajectory. One aspect of the problem considered in this thesis is the coverage problem, which has
been studied a lot recently. The interest is primarily governed by its several applications such as
tracking unfriendly targets, surveillance and monitoring, and urban search and rescue in the
aftermath of a natural or man-made disaster [8]. For this project, the aim is to scan the area and
find the missing person or the person in danger. To solve the coverage problem, a path planning
is needed. The last task is to optimize the path planning to minimize the time. In what follows, a
brief description of current literature in all these different aspects of UAV control and navigation
is provided.
2.1 UAV control system
With the development of materials, electronics, sensors and batteries, micro
Unmanned Aerial Vehicles’ size ranges from 0.1-0.5 meters in length and 0.1-5 kgs in
10
mass [9]. A class of UAVs with four rotors, Quadrotors, has become more and more
popular now. A lot of work has been carried out to develop new methods to control them.
The example includes HMX-4 which is a kind of quadrotor, which has a weight around
0.7 kg and is 76 cm in length between the rotor tips. The flight time is around 3 minutes
[10]. GPS or other accelerometers cannot be added onboard because of the limitation of
the weight for this kind of quadrotor [11]. The data of position and speed can be obtained
through the cameras. For feedback, the quadrotor has early Micro Electro-Mechanical
System (MEMS) gyro system for pilot-assist. The feedback linearization controller
controls the altitude and yaw angle. Due to the drift of the qaudrotor in the x-y plane
under the control of this controller, the backstepping controller is needed to help to
control the position [12]. Another commonly researched testbed uses the proportionalintegral-derivative controller (PID controller) to control the attitude and position together
[13]. AscTec Hummingbird is a typical model of this kind of quadrotor. The model has a
wood or carbon fiber frame to make it strong and have lightweight around 0.5 kg. They
have their own sensors to obtain the states. The controller adjusts the plant based on the
difference between the desired and measured values [14]. The STARMAC testbed is
another one used PID control system. The quadrotor has weight around 1.1-1.6 kg and
can carry on additional payload around 2.5 kg [15]. One capable used outdoor trajectory
tracking model is named as X4-flyer [16]. The controller was developed for bounding the
orientation of the vehicle and keeping it very small. The control strategy was based on
Lyapunow function with the application of backstepping techniques [17].In this project,
the dynamic model was obtained through Euler-Langrage method. The PID controller
was used to control the quadrotor to hover at one position or track a trajectory.
11
2.2 Coverage and Path Planning
Recently, the research on sensor coverage of a specified area has gathered much attention.
The developments of novel kids of sensors drive the research in sensor-based coverage by robots.
One of the challenges is to control the motion of robots or UAVs to drive the sensor to carry out
specific task such as exhaustive coverage of an area. The geometric motion planning techniques
allow the navigation problem come to researchers’ eyes [18]. Besides that, researchers pay more
attention to the research on competitive algorithms which will be used for the autonomous
systems [19]. These reasons promote the coverage problem to be popular. There has been a lot of
work focused on developing planning strategies recently. The example includes Potvin’s research
on delivery systems [20], autonomous UAV path planning [21], and mobile sensor network [22].
There are several new applications of coverage problem that includes snow removal, mobile
target tracking, surveillance, floor cleaning, and lawn mowing [23].
L.E. Dubins was the first person who did the research on the time-optimal paths of a
simple airplane. The airplane model developed by him is named the Dubins curve [24]. Dubins
curve always list as: LSL, RSR, RSL, LSR, RLR, and LRL. Here S means straight line
segment, L is a circular arc to the left, and R stands for the circular arc on the right side.
This is shown in Fig. 2.1 below:
12
Figure 2.1 Dubin’s Curve
He designed the shortest path by giving a characterization of time-optimal
trajectories for a car with a bounded turning radius. With the application of Dubins curve,
Reeds and Shepp finished another research to give a path with a minimum turning radius
on a special kind of model that a car goes both forwards and backwards [25]. Boissonnat,
C´er´ezo, and Leblond solved the same problem by using another kind of optimal control
theory named “Minimum principle of Pontryagin” [26]. In this work, they obtained the
shortest path between two known oriented points. After that, Summann and Tang
developed their research with the application of modern geometric optimal control theory
and classical control theory [27]. Balkcom and Mason did their research of the optimal
trajectories for a kind of differential drive vehicles [28]. Based on the same to Reeds’ cost
function, they solved other wheel-rotation trajectories. They derived 52 different
minimum wheel-rotation trajectories and obtained the shortest path to solve the problem
from
13
Chapter 3
Problem Description
In this thesis, there are two main problems to be solved: trajectory tracking and
trajectory generation. The first problem is to develop a dynamic control model of a
quadrotor. This dynamic control model will allow the UAV to track any desired
trajectory. The second problem is to obtain the desired trajectory. This involves first
designing a path that will fulfill the mission given constraints of motion of the quadrotor.
Once desired path has been obtained, then the task is to optimize the path with regard to
kinematic parameters such as position, velocity, and acceleration in order to get the
minimum time for the quadrotor to finish the coverage task. In what follows, first
description of control model design is provided followed by the proposed approach of
trajectory generation.
3.1 Quadrotor dynamics model design
The overall control architecture of the quadrotor can be represented by the block
diagram shown in Fig 3.1 [30] below:
14
Figure 3.1 The nested control loops for position and attitude control.
As Fig. 3.1 shows the nested feedback control loops for the quad rotor. The inner
control loop is used to calculate the pitch, roll and yaw in order to control the trajectory in
three dimensions by using position control in the outer loop.
The position control has to keep the position at the desired location by using the
current and desired position as an input, while the attitude control is used to calculate the
moments causing pitch, roll and yaw.
The acceleration commands along x, y and z axis in position control are
calculated from Proportional- Integral- Derivative feedback of the position error. The
Proportional-Derivative controller is also used in inner loop to calculate the moments
which are needed to be produced for four motors.
15
PID controller is a common controller which is now commonly applied in the
industrial control system. It uses the error signal, its derivative and its integral to calculate
the control action.
3.2 Coverage and optimal path planning
The quadrotor is located at (x, y, z). To completely scan the desired area, first, we
need to design the trajectory that can guide the quadrotor to go through the complete area
once. Once the trajectory is obtained, it needs to be optimized so that the quadrotor takes
minimum time to traverse it while obeying all the dynamical and kinematical constraints.
The inputs of the quadrotor are the altitude, roll, pitch and yaw angle. It can be
considered a system with the configuration or state variables denoted by:
p  ( x, y, z,  , , ) (3.1)
Where x, y, z are the 3-dimensional location and  , , are the pitch, roll, and
yaw angles. The dynamics of the quadrotor is:
p  g ( p, u (t ), t ) (3.2)
If f is the total drag force, I is the moment of inertia, the quadrotor dynamics is
represented by the equation:
16
f


 sin 


m


f


sin  cos 


m
x
 


 y
f

cos  cos   g 
 


m
z

 (3.3)
2
2
p 
b(3  1 ) L



 
I


xx
 


2
2
 
b(2  4 ) L

  


I yy


 d (12  22  32  42 ) 


I zz


Where b, d are the thrust and drag moment constants respectively. L is the length
of the frame of qudrotor. In the above equation, the control variable u(t) represent the
speed of the rotors wi. The cost function is the total time required to finish the complete
operation:
T
J (u )   dt (3.4)
0
The optimization problem is:
min
𝑇
J (u) subject to p  g ( p, u (t ), t )  0 and ∫0 𝐴(𝑥, 𝑦, 𝑡)𝑑𝑡 ≥ Ω
where 𝐴(𝑥, 𝑦, 𝑡) is the footprint of the UAV and Ω is the desired area to be
covered.
17
Chapter 4
Quadrotor Dynamics and Control Model
4.1 Dynamics Analysis
4.1.1 Modeling
The rotors of the quadrotor are designed to provide the only force on the top of
quadrotor. The rotation of rotors also introduce moments. Both the force and moment
generated are dependent on the rotational speed of the rotor. The main challenge is to
control the respective speeds of the four rotors in order to generate stable flight along a
desired trajectory.
18
Figure 4.1 Simplified illustration of rotor motion and control of a Quadrotor
Figure 4.1 is a simplified illustration of how the rotors control the motion of
Quadrotor. The rotors1 and 3 spin in the anticlockwise direction and give a momentum
in clockwise direction. The rotors 2 and 4 spin in the clockwise direction and give a
momentum in anticlockwise direction. All rotors provide force in the upward direction
perpendicular the plane of rotation. For the left figure in Fig. 4.1, the speeds of rotors 2
and4 are bigger than the speeds of rotors 1 and 3. So, the effective momentum will be in
the anticlockwise direction. The Quadrotor will yaw anticlockwise. For the right figure in
Fig. 4.1, four rotors have the same speed. The momentum in anticlockwise direction will
balance the momentum in clockwise direction. The quadrotor will go upside when the
sum of the lift forces is bigger than the gravity.
19
4.1.2 Dynamic Model
Figure 4.2 Coordinate systems and forces/moments acting on a quadrotor frame
In order to develop a dynamic model of the quadrotor, let us set the body frame to
be on the quardrotor. As shown in Fig. 4.2, the origin of the body frame is set on the
center at the point O of the quadrotor. One arm is selected as the X axis, and the other
one is selected as the Y axis.
20
Figure 4.3 Diagram showing Pitch, Yaw, and Roll angle
The world frame is marked as W and determined by Xw, Yw, Zw as shown in the
Fig. 4.3. The body frame is marked as B. The red chain lines in the Fig. 4.3 are combined
as a simple schematic of quarotor. Each line stands for the rotors of quadrotor. The body
frame is attached at the point O. Point O is the center of the mass of the quadrotor. The
direction pointing from O to rotor No. 1 is defined as positive XB direction. While the
direction forward to No. 3 is negative XB direction. Similar, the direction pointing from O
to rotor No. 2 is defined as positive YB direction and to rotor No. 4 is the negative YB
direction. ZB direction is the direction from the point O to the opposite direction of the
gravity. In Fig. 4.3, rotate about zW by the yaw angle, ψ, rotate about the intermediate x
21
axis by the roll angle, Φ, and rotate about the yB axis by the pitch angle, θ. The rotation
matrix for transforming the coordinates from B to W is given by
 cos cos   sin  sin sin 
R  cos  sin  cos sin  sin 

 cos  sin 
 cos  sin
cos  cos
sin 
cos sin   cos  sin  sin 
sin sin   cos cos  sin   (4.1)

cos  cos 
If r denotes the position vector of the center of mass in the world frame, the
acceleration of the center of mass is given by the equation:
0

 0 




mr   0   R 0



 mg 
  Fi 
(4.2)
In the body frame, the components of angular velocity of the robot are p, q, and r.
The relationship between these values and the derivatives of the roll, pitch, and yaw
angles is:
 p  cos 
q    0
  
 r   sin 
0  cos  sin    
 
1
sin    
0 cos  cos    
 
(4.3)
Each rotor i produces a moment Mi which is perpendicular to the plane of rotation
of the blade. Rotors 1 and 3 produce the moments in the - zB direction. The moments
produced by Rotors 2 and 4 are in the opposite direction: zB direction. The moment
22
produced on the quadrotor is opposite to the direction of the rotation of the blades so M1
and M3 act in the zB direction while M2 and M4 act in the opposite direction. The distance
from the axis of rotation of the rotors to the center of the quadrotor is marked as L.
Through weighing the individual components of the quadrotor, the moment of inertia
matrix, I which is referenced to the center of mass along xB- yB- zB axes is obtained [18].
The angular acceleration is obtained by the Euler equations as shown below:
  p  p
 p   L  F2  F4 

    
I  q    L  F3  F1 
  q   I q 
 r   M1  M 2  M 3  M 4   r   r 


(4.4)
4.1.3 Position Control
In this thesis a proportional-integral-derivative controller (PID controller) is designed
to control the quadrotor. The controller adjusts the process control inputs to minimize the
error which is the value of the difference between a measured process variable and a desired
set-point [19]. The roll and pitch angle were put as the inputs. With the use of a method
similar to back stepping approach, two kinds of position control methods will be obtained.
One is hover controller which will control the quadrotor to keep staying in a desired position.
The other one is to control the quadrotor to track any trajectory in 3-D [20].
4.1.4 Hover Controller
The quadrotor will hover at one point when the nominal thrusts from the propellers
are equal to the gravity, so:
23
Fi ,0 
mg
4
(4.5)
And the motor speed will be:
i ,0  h 
mg
4k F
(4.6)
Let rT (t) be the trajectory position and ψT (t) be the yaw angles that are being
tracked. Note that ψT (t) = 0 for the hover controller.
The position error is given by:
ei   ri ,T  ri 
(4.7)
des
The commanded acceleration: ri can be calculated using the PID law as:
r
i ,T
 ri des   kd ,i  ri ,T  ri   k p ,i  ri ,T  ri   ki ,i   ri ,T  ri   0
(4.8)
For hover, we have:
ri ,T  ri ,T  0
(4.9)
The relationship between the desired accelerations and roll, pitch angles as shown:
24
r1des  g  des cos T   des sin T 
r2des  g  des sin T   des cos T 
r3des 
(4.10)
8k F h
F
m
1 des
r1 sin T  r2des cos T 

g
1
 des   r1des cos T  r2des sin T 
g
m
F 
r3des
8k F h
 des 
(4.11)
4.1.5 Attitude Control
For hover state,   p,   q, and   r Φdot≈p, proportional-derivative control
laws are used that take the form
  k p ,  des     kd ,  p des  p 
  k p ,  des     kd ,  q des  q 
(4.12)
  k p ,  des    kd ,  r des  r 
I xx p  Lk F 22  42   qr  I zz  I yy 
I yy q  Lk F 32  12   pr  I xx  I zz 
I zz r  kM 12  22  32  42 
25
(4.13)
From the equation above the result of the ω desire are obtainable.
1des  1 0 1 1  h  F 
 des  


2   1 1 0 1   
3des  1 0 1 1    
 des  


4  1 1 0 1   
(4.14)
4.2 Dynamics Model Simulation
In this section, numerical simulation results for validation of the dynamic and
control model discussed in the previous section are presented. The parameters used for
the simulation are shown below in the Table 4.2.1.
Kpx=1
Kpy=1
Kpz=1
Kdx=1
Kdy=1
Kdz=1
kF=6.11*10-8 N/(r/min2).
KM=1.5*10-9 N*m/(r/min2)
Km=20
m=1.08 kg
g=9.8 m/s2
L=0.22m
Table 4.2.1 Parameters of the dynamics model
Based on the dynamic model of the quadrotor, the control model was developed
in the MATLAB Simulink. The AppendixⅡhows details of the Simulink model of
quadrotor. For the simulation, the quadrotor flies from initial location (0,0,0) to goal
location (10,10,10) and hovers at the point (10,10,10).The distance units mentioned here
26
are in meters. The Fig. 4.4 shows the actual path taken by the quadrotor as it moves from
the intial to the goal location.
12
10
z(m)
8
6
4
2
0
10
5
y(m)
0
0
2
4
6
8
10
x(m)
Figure 4.4 Simulation result showing the trajectory of the UAV
Fig. 4.5, Fig 4.6 and Fig 4.7 repectively show the plots of x, y, z locations of the
quadrotor versus time as the quadrotor moves from the intial to the desired location.
27
12
10
x(m)
8
6
4
2
0
0
10
20
30
40
50
t(s)
60
70
80
90
100
Figure 4.5 Simulation result showing the x location of the quadrotor versus time
28
12
10
8
y(m)
6
4
2
0
-2
0
10
20
30
40
50
t(s)
60
70
80
90
100
Figure 4.6 Simulation result showing the y location of the quadrotor versus time
29
12
10
8
y(m)
6
4
2
0
-2
0
10
20
30
40
50
t(s)
60
70
80
90
100
Figure 4.7 Simulation result showing the z location of the quadrotor versus time
In order to further verify the dynamics and control model with respect to tracking
a desired trajectory (rather goal location as above), the quadrotor was commanded to
move in a circle with center at (5, 0) and radius 5 m. The Fig. 4.8 shows the simulation
result of the desired and actual trajectory of the quadrotor following a circular trajectory.
The Fig. 4.9 shows the simulation result show the desired x and actual x for this circular
tracking. The Fig 4.10 shows the simulation result show the desired y and actual y for this
circular tracking.
30
6
4
y(m)
2
0
-2
-4
-6
-2
0
2
4
x(m)
6
8
10
12
Figure 4.8 Simulation result showing the desired (shown in red) and actual trajectory
(shown in blue) of a quadrotor following the circular trajectory
31
Figure 4.9 Simulation result showing the desired x (shown in red) and actual x (shown in
blue) of the quadrotor following a circular trajectory
32
Figure 4.10 Simulation result showing the desired y (shown in red) and actual y (shown
in blue) of the quadrotor following a circular trajectory
The control model was further verified showing quadrotor move from rest to a set
of desired locations and land again. The Fig. 4.11 shows the simulation of quadrotor’s
motion. The quadtor took off at the point (0, 0, 0) flew up to the point (0, 0, 10) then
turned and moved to the point (10, 10, 10) and then landed at the point (10, 10, 0), The
unit used is meter.
33
12
10
z(m)
8
6
4
2
0
15
15
10
10
5
y(m)
5
0
0
x(m)
Figure 4.11 Simulation result of the quadrotor’s motion following a set of desired
locations
The above numerical simulations demonstrated that the quadrotor was able to
navigate to any desired way-point locations and follow any desired trajectory.
34
Chapter 5
Trajectory Design and Optimization
5.1 Trajectory design
As previously stated in the Chapter 4, the Dubins curve is composed of S straight
line segment, circular arc to the left, L, and circular arc on the right side, R as shown in
Fig. 5.1 shown:
Figure 5.1 The sketch of Dubins Curve
35
It may be noted that the unit for distance considered in this thesis is meters. For
this thesis, the desired area is identified as a square area L (length) * W (width) =50 m*50
m as shown in Fig. 5.2:
Figure 5.2 Sample coverage area
To plan path for such a coverage area, the each path segment is designed to be
composed of a straight line segment, a semi-circular arc, followed by a straight line
segment as shown in the Fig. 5.3:
36
Figure 5.3 The sketch of a segment of the path
The overall path is obtained as shown in Fig. 5.4. Initial location of the UAV is
(0, 5). The quadrotor is required to initiate from this point into the desired area. The
quadrotor flies up in a straight line to reach the point (0, 55). Total distance for the linear
segment is 50. Then the UAV turns right to follow a semi-circular arc with radius equal
to 5. It may be noted that the sensor coverage/footprint area for the UAV is considered to
be 10 meters in diameter. After the quadrotor reachs the point (10, 55), it then flies in a
straight line. These motions are repeated to completely finish the coverage task.
37
Figure 5.4 Path planning of the UAV for the coverage problem
5.2 Optimization of Trajectory Parameters
Once the path or trajectory has been planned, the next step is to obtain trajectory
parameters such as velocity and acceleration so as to minimize the overall time required
to finish the mission. It may be noted that the trajectory parameters need to be optimized
while keeping in view the constraints of the quadrotor dynamics. The overall trajectory is
comprised of three kinds of movements as shown in Fig.5.5:
38
Figure 5.5 Classification of motions in the UAV trajectory
5.2.1 Segment 1: Curve Segment
For the curve segment, a semi-circular trajectory is designed. As previously
discussed in Chapter 4, a simulation was designed to track the circular trajectory. The
circular trajectory would be followed at a constant speed. The challenge is to determine
the maximum velocity that can be allowed in presence of the dynamic of quadrotor. This
maximum velocity can be obtained through the experiment with the Simulation model.
The designed curve is a semi-circular with radius equal to 5 m. The quadrotor will
track this circle with a constant linear speed Vcircle. The maximum speed that can track
39
this circle can be found via performing simulations with increasing magnitudes of
velocities. The results from these simulation experiments are shown in Fig.5.6 to Fig.5.10
which shows the trajectory followed by the UAV. The velocities were set to 0.1 m/s, 0.2
m/s, 0.3 m/s, 0.4 m/s and 0.5 m/s.
Figure 5.6 Trajectory followed by the UAV for magnitude of velocity equal to 0.1 m/s
40
Figure 5.7 Trajectory followed by the UAV for magnitude of velocity equal to 0.2 m/s
Figure 5.8 Trajectory followed by the UAV for magnitude of velocity equal to 0.3 m/s
41
Figure 5.9 Trajectory followed by the UAV for magnitude of velocity equal to 0.4 m/s
Figure 5.10 Trajectory followed by the UAV for magnitude of velocity equal to 0.5 m/s
42
The maximum errors between these real path and trajectory are 0.0005, 0.0139,
0.0539, 0.1318 and 0.32. We can see, as expected, that the error increases the speed
increases. Because the maximum acceptable error is 0.2, so 0.4 m/s is the maximum
linear velocity chosen for the quadrotor to follow the semi-circular trajectory.
The radius is 5 m. The constant magnitude of velocity to move on this trajectory
is 0.4 m/s. For the half circle, the minimum time needed is:
tcircle 
 *r
Vcircle

3.14*5
 39.25s
0.4
5.2.2 Segment 2: Straight line
Figure 5.11 Segment 2: Straight line
43
(5.4)
These two straight line movements shown in the above Fig. 5.11 consists of two
segments: acceleration motion and deceleration motion. For motion labeled 1 in the
figure, the quadrotor flies into the desired area from the initial point (0, 5). First, the
quadrotor flies with the acceleration a1 for t1 time. At time t1 the velocity is equal to v1.
Then, the quadrotor switches to fly with the deceleration a2 for t2 time and reaches the
point (0, 55). At time t2, the velocity is v2. This is the same velocity with which the
quadrotor flies during the curved segment of motion. The quadrotor then accelerates and
decelerates (as discussed in next Section: Segment 3). The final segment, shown as
Number 2 in the above figure, is exactly inverse of the first segment. Finally, the
quadrotor stops at the point (50, 5).
Now, the question arises what should these acceleration and deceleration be so as
to minimize the time take to travel this segment of path while ensuring that the UAV
satisfies all its dynamics constraints. For this thesis, the method based on Langrage
multiplier is used to obtain the maximum or minimum value of a mathematical function
in presence of constraints. With the set of Langrage multiplier, a new equation set or
objective function is obtained. For No. 1 straight line, the objective is to minimize the
time, i.e.,:
f (t1 , t2 )  t1  t2
(5.5)
The dynamics of the motion is obtained using kinematic equations for motion of
particles, as follows:
44
v1  a1t1



 s1  1 a1t12

2




v2  a1t1  a2t2



1
 s2  a1t1t2  a2t2 2 
2


 s1  s2  L

(5.6)
1
1 2

2
 a1t1  a1t1t2  a2t2  L 
2
2

v2  a1t1  a2t2

(5.7)
So:
And, also, the maximum acceleration constraint due to dynamics of the aircraft
poses the following constraints:
a1  amax
a2  amax
(5.8)
To get the limited function:
1 2
1


2
 g1 (t1 , t2 , a1 , a2 )  2 a1t1  a1t1t2  2 a2t2  L 


 g 2 (t1 , t2 , a1 , a2 )  a1t1  a2t2  v2

 g3 (t1 , t2 , a1 , a2 )  a1  amax



 g 4 (t1 , t2 , a1 , a2 )  a2  amax

45
(5.9)
So, the augmented cost function is:
F (t1, t2 , a1, a2 )  f (t1, t2 , a1, a2 )  1g1 (t1, t2 , a1, a2 )  2 g2 (t1, t2 , a1, a2 )  3 g3 (t1, t2 , a1, a2 )  4 g4 (t1, t2 , a1, a2 )(5.10)
The optimization is carried by simultaneously solving:
F / t1  0 
F / t  0 
2


F / a1  0 


F / a2  0 


F / 1  0 
F / 2  0


F / 3  0 
F /   0
4


(5.11)
In order to obtain the maximum value of the acceleration the quadrotor can
provide, tests were done using the simulation model. The results are shown in the
following figures. The values of the acceleration were set as 1.5 m/s2, 2.0 m/s2 and 2.5
m/s2. The maximum rotor speed is set to 130 rad/s. As the Fig. 5.12-5.14 shown, the rotor
speed will increase as the acceleration increase. The rotor speed required is bigger than
130 rad/s when the acceleration equal to 2.5 m/s2. So the maximum acceleration the
quadrotor can provide is 2.0 m/s2.
46
Figure 5.12 The rotors’ speed simulation at the acceleration equal to 1.5 m/s2
47
Figure 5.13 The rotors’ speed simulation at the acceleration equal to 2.0 m/s2
Figure 5.14 The rotors’ speed simulation at the acceleration equal to 2.5 m/s2
48
So, the maximum acceleration constraint is chosen as:
amax  2m / s2
(5.12)
Also set:
L  50m, v2  0.4m / s
(5.13)
The v2 is set to 0.4m/s from the circular segment of the motion. Substituting the
above values, a set of function is obtained:
1  1 (a1t1  a1t2 )  2 a1  0
1   (a t  a t )   a  0
1 11
2 2
2 2

2
1 (0.5t1  t1t2 )  2t1  3  0

1 (0.5t2 2 )  2t2  4  0

2
2
0.5a1t1  a1t1t2  0.5a2t2  L  0
a t  a t  v  0
 11 2 2 2
a1  amax  0

a2  amax  0
(5.14)
So:
t1  5.0

t 2  4.8
(5.15)
The minimum time to cover the segment No. 1 straight line is t1+t2=5.0+4.8=9.8 s.
Because the motion of segment No.2 is the inverse motion of No.1, so the shortest time to
cover the same is 9.8 s.
49
5.2.3 Segment 3: straight line
The segment 3 is similar as to segment 2, the motion of quadrotor also comprises
of an acceleration motion followed by a deceeration motion. However, in segment 3, the
initial and final velocities both are 0.4 m/s.
The objective function is:
f (t3 , t4 )  t3  t4
(5.16)
The motion function is obtained using kinematic equations for motion of particles,
as follows:
v3  0.4
v  v  a t
 4 3 33

1
2
 s3  v3t3  a3t3
2

v5  v4  a4t4

1
2
 s4  v4t4  a4t4
2

 s3  s4  L
(5.17)
So:
a3t3  a4t4  0


1
1
0.4t3  a3t32  (0.4  a3t3 )t4  a4t4 2  L


2
2
50
(5.18)
And also:
 a1  amax

 a2  amax
(5.19)
The limited function:
1
1

2
2
 g1 (t3 , t4 , a3 , a4 )  0.4t3  2 a3t3  (0.4  a3t3 )t4  2 a4t4  L

 g (t , t , a , a )  a t  a t
3 3
4 4
 2 3 4 3 4
 g3 (t3 , t4 , a3 , a4 )  a3  amax


 g 4 (t3 , t4 , a3 , a4 )  a4  amax
(5.20)
So, the augmented cost function is:
F t3 , t4 , a3 , a4   f (t3 , t4 , a3 , a4 )  1g1 (t3 , t4 , a3 , a4 )  2 g2 (t3 , t4 , a3 , a4 )  3 g3 (t3 , t4 , a3 , a4 )  4 g4 (t3, t4 , a3, a4 ) 5.21
The optimization is carried by simultaneously solving:
F / t3  0
F / t  0
4

F / a3  0

F / a4  0

F / 1  0
F / 2  0

F / 3  0
F /   0

4
51
(5.22)
From the simulation model:
amax  2m / s2
(5.23)
Also set:
L  50m, v3  0.4m / s, v5  0.4m / s
(5.24)
Substituting the above values, a set of function is obtained:
1  1 (0.4  a3t3  a3t4 )  2 a3  0
1   (0.4  a t  a t )   a  0
1
3 3
4 4
2 4

 1 2
1 ( t3  t3t4 )  2t3  3  0
 2
 1 2
1 ( t4 )  2t4  4  0
 2

1
1
2
2
0.4t3  a3t3  (0.4  a3t3 )t4  a4t4  50  0
2
2

a3t3  a4t4  0

a3  2  0
a4  2  0
(5.25)
So:
t3  4.8s

t4  4.8s
The minimum time to scan one of the straight lines in segment 3 is
t3+t4=4.8+4.8=9.6 s.
52
(5.26)
The desired area is set as L*W=50*50 square area. The trajectory for coverage
scanning is as shown before in the Fig.5.4. The trajectory of this quadrotor is known. It
made 5 half circle curves. It needed time tcurve:
tcurve  39.25*5  196.25s
(5.27)
The quadrotor that flied into the area from (0, 5) to (0, 55) and moved out from
(50, 55) to (50, 5), it needed time tsegment2:
tsegment 2  9.8*2  19.6s
(5.28)
Besides those above, the quadrotor also moves 4 times in a straight line same as
the segment 3 as previously discussed. For this segment movement it need time tsegment3:
tsegment 2  9.6*4  38.4s
(5.29)
So the total time for the quadrotor covering L*W=50*50 square area is:
t=236.61 s
(5.30)
5.3 Simulation Results
5.3.1 Segment 1: Curve Segment
Fig. 5.15 shows the simulation result of the desired magnitude of velocity and
actual magnitude of velocity in the circular segment of the trajectory. The purple line
shows the desired magnitude of velocity which is 0.4 m/s. The yellow line shows the
actual magnitude of velocity. The errors are less than 0.05 m/s. Fig. 5.16 shows the
simulation result of the desired path and the actual path in circular segment of the
53
trajectory. Red line is the desired trajectory and the blue line shows the actual path. The
actual time to track this segment is 38.89 s.
Figure 5.15 Simulation result of the desired velocity and the actual velocity in the
circular segment
54
Figure 5.16 Simulation result of the desired path and the actual path in the circular
segment
5.3.2 Segment 2: Straight line
Fig. 5.17 shows the simulation result of the desired magnitude of velocity (or
speed) and the actual speed in tracking Segment 2 that represents a straight line. The red
line shows the desired speed. The blue line shows the actual speed. The errors are less
than 0.5 m/s. Fig.5.18 shows the simulation result of the desired acceleration and the
actual acceleration in tracking Segment 2 of the trajectory. The red line shows the desired
acceleration. The blue line shows the actual acceleration. Fig. 5.19 shows the simulation
55
result of the desired path and actual path in tracking Segment2: straight line. Red line is
the desired trajectory and the blue line shows the actual path. The errors are less than
0.1m. The actual time to track this segment is 98 s.
Figure 5.17 Simulation result of the desired velocity and the actual velocity in
Segment 2 (straight line motion)
56
Figure 5.18 Simulation result of the desired acceleration and the actual
acceleration in Segment 2 (straight line motion)
57
Figure 5.19 Simulation result of the desired trajectory and the actual path in
Segment 2 ( straight line motion)
5.3.3 Segment 3: Straight line
Fig. 5.20 shows the simulation result of the speed and the actual speed in tracking
the Segment 3 of the motion which is again a straight line. The red line shows the desired
speed. The blue line shows the actual speed. The errors are less than 0.5 m/s. Fig.5.21
shows the simulation result of the desired acceleration and the actual acceleration. The
red line shows the desired acceleration. The blue line shows the actual acceleration. Fig.
5.22 shows the simulation result of the desired path and actual path in tracking the
58
desired trajectory in Segment3 of the motion. Red line is the desired trajectory and the
blue line show the actual path. The errors are less than 0.09 m. The actual time to track
this segment is 96 s.
Figure 5.20 Simulation result of the desired velocity and the actual velocity in
Segment 3 ( straight line motion)
59
Figure 5.21 Simulation result of the desired acceleration and the actual
acceleration in Segment 3 (straight line motion)
60
Figure 5.22 Simulation result of the desired trajectory and the actual path in
Segment 3 (straight line motion)
5.3.4 Overall simulation
Fig. 5.23 shows the trajectory and the actual path for coverage the complete area.
The blue line shows the desired trajectory and the red line show the real path. As shown
in the figure, the real path closely tracked the trajectory.
61
Fig. 5.23 The desired and actual trajectory for coverage of the complete area
The following Figs. 5.24 show the area covered as the UAV tracks each of the
segments. The total time used for covering this area is 234.81 s.
62
Figure 5.24 Simulation result showing the area covered as the UAV tracks each of
the segments
63
Chapter 6
Conclusions and Future work
6.1 Conclusions
The thesis focused on solving two problems: i) obtaining a dynamic model of the
quadrotor, and ii) the planning of the path, and the optimiastion of the path planning.
In the first problem, the dynamics of the quadrotor was analyzed and then a
controller based on PID method was developed. In this section, the parameters were
chosen to make this simulation model to be nearly identical to the real quadrotor. Then,
point-to-point navigation as well as trajectory tracking experiments were performed using
the simulation model developed based on the dynamics of the quadrotor. The results
showed that this model worked well.
For second problem, the trajectory for exhaustive coverage of a specified area was
designed. The parameters of the trajectory were identified, and the Langrage multiplier
algorithm was utilized to obtain those parameters that minimized the total time taken to
traverse the entire trajectory. The trajectory was divided into three main parts, which
were calculated individually. During this calculation, the experiment on the simulation
model was used to determine the maximum velocity within the dynamic constraints of
64
the quadrotor (that limits the acceleration) to follow a half circle with radius equal to 5
meters. This velocity was found to be 0.4 m/s. The maximum acceleration of the
quadrotor, based on the dynamical constraints, was assumed to be 2.0 m/s2. Based on
those data, the minimum time for the quadrotor to finish each part was calculated based
on a Lagranian formulation. From this, an example scenario that required the quadrotor to
cover an area of L*H=50 m*50 m was designed. The minimum time to finish this area
was found, which t=236.61 s. From the simulation we know the simulation model to scan
this area need time 234.81 s. There were some errors between model and real quadrotor
because of the choice of parameters. This solution can be used in the future for other
develop techniques for other kinds coverage control problems.
6.2 Contributions
The contributions of this thesis are twofold: i) the thesis developed a MATLB
Simulink based simulation platform for control and navigation of a quadrotor UAV. The
model makes use of realistic dynamic model of the UAV and a controller based on PID
method that helps the UAV track a desired trajectory. ii) the thesis proposed a trajectory
to exhaustively cover a rectangular search area, and then developed a mechanism to
optimize the parameters of the trajectory so that the entire trajectory is traversed in
minimum time.
65
6.3 Future Works
There is a lot of interest recently on use of quadrotors for carrying out a number of
tasks such as package delivery, search and rescue, and monitoring. A lot of research still
needs to be carried out before the UAVs can be used in operational environment that
involve path planning and collision avoidance.
There has been a lot of research recently on designing the control system for a single
UAV and extending that to multiple UAVs. This thesis used a single quadrotor to finish
the coverage task. In the future, multiple quadrotors working together to accomplish the
task can be considered which will result in more time-efficient operation.
A lot of work has been carried out recently on development of sensors recently. These
sensors are have better performance, more efficient, cheaper, and lighter. The sensor can
now to be used to measure the light (vision) and the temperature (infra-red). These
advanced sensors can be placed on the quadrotor, so that the area can be patrolled for
monitoring hot-spots. Similarly, infra-red sensors can be used to find a lost person in vast
area where manual ground search is difficult. .
66
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70
Appendix A
Parameters of the dynamics model
kp_x=1;kp_y=1;kp_z=1
kd_x=1;kd_y=1;kd_z=0.3
kf=6.11*10^(-4)*3600
km=1.5*10^(-9)*3600
m=1.08
wH=sqrt(m*9.8/(kf*4))
g=9.8
kp_pitch=0.01;kd_pitch=1;kp_roll=0.01;kd_roll=1;kp_yaw=1;kd_yaw=0.01
dt=0.1
L=0.22
i_xx=m*2*(L^2);i_yy=m*2*(L^2);i_zz=m*4*(L^2)
kM=20
71
Appendix B
The system for the dynamics simulation
The UAV dynamics simulation model
72
Subsystem of the dynamics simulation
PD-Position Subsystem
73
Position Control Subsystem
Subsystem 5
74
Subsystem 6
Subsystem 2
75
Subsystem 3
Subsystem4
76
Subsystem 7
77