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DYNAMICS, MODELLING AND DESIGN OF A
QUADCOPTER
BY
AHIANTE STEPHEN ORIASOTIE,
MAT NO: 1487/2013
DEPARTMENT OF PHYSICS,
COLL...
DYNAMICS, MODELLING AND DESIGN OF A
QUADCOPTER
BY
NAME: AHIANTE STEPHEN ORIASOTIE
MAT NO: 1587/2013
LEVEL: 400L
1487/2013
...
i
Certification and Approval
This is to certify that this project work entitled “Dynamics, Modelling and Design of a
Quadc...
ii
Declaration
I, Ahiante Stephen Oriasotie bearing matriculation number: 1487/2013 in the second
semester of the final ye...
iii
DEDICATION
This work is dedicated to God, Almighty & Patricia Ahiante-Isemede.
iv
ACKNOWLEGMENTS
God, Almighty:
For keeping me through the years.
Technology Development for Poverty Alleviation Initiati...
v
CONTENTS
Certification and Approval….………………………......……….………..………………..… i
Declaration ………………………………………………………………………………….......
vi
2.5. Rigid Bodies………………………………………………………………… 16
2.6. Review of Related Work…………………………………………..…………19
2.7. Summary of Revie...
vii
3.4. Review of Methodology…………………………………………………….. 52
4. PRESENTATION OF RESULTS …………......................................
viii
LIST OF FIGURES
Figure 1.2.1. The four basic aerodynamic forces.
Figure 1.2.2. The various motions of the quadcopter ...
ix
Figure. 3.2.4.b. The Schematic representation of the one motor test.
Figure. 3.2.4.c The Circuit diagram of the system
...
x
LIST OF ACRONYMS
AES Advanced Encryption Standard
.apk Android Application Package
AT Attention
BR/EDR Bluetooth Basic R...
xi
Abstract
Findings from recent times have shown that the development and usage of drones
(UAVs) such as Quadcopters has ...
1
CHAPTER ONE
INTRODUCTION
1.1 Background of the Work
Over the years, flying aircrafts have evolved, from kites, to hot ai...
2
development of the vehicle was slow because of the difficulty and complexity of
controlling each motor independently.
1....
3
mounted on the same shaft), quadrotor (having four rotors), or multi rotor UAV.
Examples include: Tricopters, quadcopter...
4
same level of control can be achieved by adding two more rotors. The rotation of all
four rotors is set-up such that the...
5
If thrust is greater than drag, the aircraft will increase in speed. Thrust must equal weight
and overcome drag (Kibly, ...
6
to the critical angle, the greater the amount of lift developed, and the greater the induced
drag.
2. Parasitic (parasit...
7
iv. As speed increases, the amount of parasite drag increases. If the speed is
doubled, four times as much drag is produ...
8
Figure 1.2.1. The four basic aerodynamic forces.
In straight and level, un-accelerated flight, the total amount of thrus...
9
The yaw motion of the quadcopter is the motion of the aircraft that determines where it
would be facing in space. Increa...
10
1.3 Purpose of the Work
The sole purpose of the work is to introduce quadcopters and model the quadcopter
system as a r...
11
CHAPTER TWO
REVIEW OF RELATED LITERATURE
This chapter aims to give the general concept of this work, review Bluetooth
t...
12
Bluetooth SIG adopted the code name as a tribute to the tenth-century Viking king
Harald Blatand who peacefully united ...
13
Bluetooth devices operate in the Industrial, Scientific and Medical (ISM) from 2.4 to
2.485 GHz. The Industrial, Scient...
14
Figure 2.2.b. Piconets and Scatternet
Two master devices cannot pair with each other. This is why Bluetooth devices suc...
15
microcontroller is programmed using mainly the Arduino Integrated Development
Environment, a software that is cross-pla...
16
2.5 Rigid Bodies
A rigid body is a physical system of particles that does not deform. It is a system of mass
points sub...
17
In the mechanics of rigid bodies, the centre of mass is considered an important variable,
this is because, the centre o...
18
There is also a motion of a rigid body that is known as the simple harmonic motion; it
is a motion that repeats itself ...
19
Rigid bodies are not really realistic systems because of the characteristic of not being
able to be deformed. Resistant...
20
The work was also extended to the control of up to eight home appliances; this was
achieved by the use of a 12V relay.
...
21
Figure 2.6.b. The Circuit Diagram of the System (Mathieu, 2008).
2.7 Summary of Review of Related Literature
Bluetooth ...
22
CHAPTER THREE
METHODOLOGY
This chapter entails the modelling of the quadcopter system as a rigid body, and the
fabricat...
23
In Fig. 3.1.2, diagram A represents the inertial frame of the system and diagram B
represents the body frame of the sys...
24
Furthermore,
𝑑ℵ
𝑑𝑡
𝑇
= (
𝑑𝜑
𝑑𝑡
𝑑𝜃
𝑑𝑡
𝑑∅
𝑑𝑡
)
𝑇
(3.8)
ℵ̇ 𝑇
= (𝜑̇ 𝜃̇ ∅̇ )
𝑇
(3.9)
ℵ̇ = (
𝜑̇
𝜃̇
∅̇
) (3.10)
But recall tha...
25
= (
∅̇ − 𝜑̇ sin 𝜃
𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅
𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅
)
Let Ω represent the angular velocity.
Ω = (
∅̇ ...
26
3.1.5 The Equations of Motion
The Lagrangian is defined as,
𝐿( 𝑞, 𝑞̇) = 𝑇 − 𝑈 (3.14)
Where; 𝑇 𝑖𝑠 The kinetic energy and...
27
𝐼Ω =
(
(
𝐼𝑥𝑥 0 0
0 𝐼 𝑦𝑦 0
0 0 𝐼𝑧𝑧
) (
∅̇ − 𝜑̇ sin 𝜃
𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅
𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅
)
)
(3.21)
= (
...
28
The quadcopter vehicle is acted upon by external forces and various aerodynamic
effects hence, the model for the quadco...
29
Where; 𝑱 is the Jacobian matrix from the angular velocity, equation (3.13) to the time
derivative of the angular coordi...
30
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕𝑧̇
) −
𝜕𝐿
𝜕𝑧
= 𝐹 𝑧 (3.35)
For the ℵ coordinates,
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕ℵ̇
) −
𝜕𝐿
𝜕ℵ
= 𝜏̃ (3.36)
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕𝜑̇
) −
𝜕𝐿
𝜕𝜑...
31
and equation (3.25) becomes,
𝐿 =
1
2
(𝑚( 𝑥̇2
+ 𝑦̇2
+ 𝑧̇2) + (∅̇ 2
+ 𝜑̇ 2
+ 𝜃̇2
)) − 𝑚𝑔𝑧 (3.40)
Evaluating equation (3.3...
32
𝑚𝑧̈ = 𝐹𝑧
Substituting 𝐹𝑧 from equation (3.31),
𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 (3.43)
Evaluating equation (3.37) with equatio...
33
𝑚𝑥̈ = 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃)
𝑚𝑦̈ = 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅)
𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔
𝜑̈ = 𝜏̃ 𝜑
𝜃...
34
3.2 Design of the Quadcopter
This section covers the design process of the quadcopter system. In designing the
quadcopt...
35
3.2.1 The Components
This section of this work introduces the various components utilized in the design of
the quadcopt...
36
Figure 3.2.1.1.b. The Inner Windings and casing of a Coreless DC motor
The Motors have a girth size of 0.029m, a radius...
37
Figure. 3.2.1.1.c. The Pusher Propellers
Figure. 3.2.1.1.d. The Puller Propellers.
The electric motors are made to rota...
38
The acceleration at the tip of the propellers (the centripetal acceleration) is estimated to
be about 28281𝑚𝑠−2
, 44460...
39
3.2.1.3 The Bluetooth Module
The Bluetooth device used in this work is a HC-05 module. The HC-05 module is an
easy to u...
40
3.2.1.4 The Power Source
The main power source in this work is a 3.7V rechargeable 25C GSP 90540 Lithium
Polymer (LiPo)...
41
Figure. 3.2.1.4.a. The Lithium Polymer Battery Used.
The use of a load of 0.75A would drain the battery given by Fig. 3...
42
3.2.1.5 Other Components
Other components utilized in this work include; a small single pole double throw
(SPDT) switch...
43
3.2.2 The Airframe
The airframe is the body of any quadcopter or similar vehicle; housing all the
components of the sys...
44
Figure 3.2.2.b. The Fabricated airframe for the system
The airframe fabricated is very light weighing only about 1.2gra...
45
direction all running at the same average speed will cause the torque on each rotor to
cancel out; pushing all air mole...
46
is another computer required in this work. The mobile phone is required for the
testing and usage of the mobile applica...
47
 Preparing the Arduino Pro Mini
1. Header Pins and Jumper wires are soldered unto the Arduino Pro Mini
2. The Arduino ...
48
12.The command, “AT+PSWD=9090” was then typed into the serial monitor.
The response was, “OK”; indicating that the pass...
49
the MOSFET and the cathode of the diode connected to the positive terminal
of the LiPo battery. The source code (𝒔𝒆𝒆 𝒂𝒑...
50
Figure. 3.2.4.b. The Schematic representation of the one motor test.
 Building the whole system
1. The airframe was fa...
51
Figure. 3.2.4.c The Circuit diagram of the system
3.3 Interfacing with the Quadcopter
In controlling the quadcopter sys...
52
have to navigate to settings; then to Bluetooth and then scan and pair with a new device.
The PhysicsI Bluetooth will b...
53
CHAPTER FOUR
PRESENTATION OF RESULTS
This chapter entails the presentation of the results obtained in this work. The ch...
54
𝜑̈, 𝜃̈, 𝑎𝑛𝑑 ∅̈ are the angular accelerations in the yaw, pitch and roll directions
respectively
𝜏̃ 𝜑, 𝜏̃ 𝜃, 𝑎𝑛𝑑 𝜏̃∅ are...
55
Figure. 4.1.b. The Quadcopter
Figure. 4.1.c. The LiPo Battery Charger provided in this work
56
4.2 Limitations of the Work
This work serves the purpose of modelling and constructing a basic quadcopter system.
The l...
57
CHAPTER FIVE
CONCLUSION, DISCUSSION &
RECOMMENDATIONS
In this chapter, the discussion of the results, the summary and c...
58
Question Three: Why does this work utilize a battery that is capable of starting a
class C fire?
The LiPo battery was u...
59
3. In latter works, accelerometers, gyroscopes and other sensors should be
incorporated in the construction of quadcopt...
60
References
Anyakoha. M.W. (2010). New School Physics for Senior Secondary Schools
(3rd edition). [paperback copy]. P. 1...
61
Hintenaus P. (2015). Engineering Embedded Systems; Physics, Programs, Circuits.
[pdf version]. Springer International P...
62
Schneider B. (2015). A Guide to LiPo Batteries. [pdf version]. Retrieved from:
https://rogershobbycenter.com/lipoguide/...
A- 1 -
Appendices
Appendix A:
The Inertia Matrix
Moments of inertia describe the resistance of a body to rotation about an...
A- 2 -
Appendix B:
The Rotational Matrix
The two dimensional rotational matrix is defined as:
𝑹∅ = (
cos ∅ −sin ∅
sin ∅ co...
A- 3 -
The rotation about 𝑎2, the pitch axis is;
𝑹 𝜃 = (
cos 𝜃 0 − sin 𝜃
0 1 0
sin 𝜃 0 cos ∅
)
The rotation about 𝑎1, the ...
A- 4 -
Appendix D:
The Centripetal acceleration (the acceleration at the tip of the propellers)
The propellers have a radi...
A- 5 -
Where;
𝜔 = the angular acceleration of the propeller
𝑟 = the radius of the propeller
For 9600 RPM,
Converting RPM t...
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
Dynamics, Modelling & Design of a Quadcopter
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Dynamics, Modelling & Design of a Quadcopter

  1. 1. DYNAMICS, MODELLING AND DESIGN OF A QUADCOPTER BY AHIANTE STEPHEN ORIASOTIE, MAT NO: 1487/2013 DEPARTMENT OF PHYSICS, COLLEGE OF SCIENCE, FEDERAL UNIVERSITY OF PETROLEUM RESOURCES, EFFURUN, DELTA STATE, NIGERIA OCTOBER, 2017
  2. 2. DYNAMICS, MODELLING AND DESIGN OF A QUADCOPTER BY NAME: AHIANTE STEPHEN ORIASOTIE MAT NO: 1587/2013 LEVEL: 400L 1487/2013 SUBMITTED TO THE DEPARTMENT OF PHYSICS, IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF BACHELOR DEGREE IN SCIENCE (B.SC.) IN PHYSICS FEDERAL UNIVERSITY OF PETROLEUM RESOURCES, EFFURUN, DELTA STATE, NIGERIA SUPERVISED BY: DR. AGBALAGBA O.E OCTOBER, 2017
  3. 3. i Certification and Approval This is to certify that this project work entitled “Dynamics, Modelling and Design of a Quadcopter” is a work carried out by Ahiante Stephen Oriasotie with matriculation number: 1487/2013 in partial fulfilment of the requirements for the award of the degree of Bachelor of Science in Physics from the Federal University of Petroleum Resources, Effurun during the year 2016/2017. The work is approved as it satisfies the academic requirements in respect of project work prescribed for the Bachelor of Science degree; we the undersigned attest to this. Signed: Dr. E.O Agbalagba ………………………………………………. Project Supervisor Date: .…………………………………………….... Dr.(Mrs). E.O. Osafile …………………………………………….. Head of Department Date: …………….………………………………… (Physics) External Examiner Date:
  4. 4. ii Declaration I, Ahiante Stephen Oriasotie bearing matriculation number: 1487/2013 in the second semester of the final year in Bachelor of Science at the Federal University of Petroleum Resources, Effurun hereby declare that this project work with title: Dynamics, Modelling and Design of a Quadcopter which is being submitted by me in the partial fulfilment for the award of the degree of Bachelor of Science in Physics from the Federal University of Petroleum Resources, Effurun is an original record from me carried in the 2016/2017 session, under the supervision of Dr. E.O Agbalagba, Department of Physics, Federal University of Petroleum Resources, Effurun. I hereby further undertake that the matter embodied in this dissertation has not been previously submitted for the award of any other degree or diploma by me to any other Institution or University. Ahiante Stephen Oriasotie, 1487/2013, ………………………………………… …………………………………………
  5. 5. iii DEDICATION This work is dedicated to God, Almighty & Patricia Ahiante-Isemede.
  6. 6. iv ACKNOWLEGMENTS God, Almighty: For keeping me through the years. Technology Development for Poverty Alleviation Initiative (TD4PAI), NGO: For availing to me to the opportunity to work and learn. I may never have possessed the knowledge I now do if not for my experience with you. Dr. Agbalagba E.O, Environmental Physicist: For providing the proper guidance on how to go about this work. Dr. Nenuwe Nelson, Theoretical Physicist: For proper grounding in Classical and Quantum Mechanics. Massachusetts Institute of Technology, www.edx.org: For providing a wonderful Massive Open Online Course (MOOC) in analytical mechanics. University of Pennsylvania, www.coursera.org: For providing a Massive Open Online Course (MOOC) in the building of autonomous robots particularly, quadcopters. Lecturers of the Physics Department, FUPRE: For Knowledge imparted. Mr. Adeyeye Segun, Colleague: For the support given especially at the last minute. The Isemede Family; Sir Patrick, Mac-Henry, Ejemen, & Monica: For all the love and support shown. Ahiante Stephen Oriasotie.
  7. 7. v CONTENTS Certification and Approval….………………………......……….………..………………..… i Declaration …………………………………………………………………………………......ii Dedication ……...………………...………………….……………………….…………….… iii Acknowledgements ……………………..…………………………………….……….……… iv Contents…………………………………………………………………..…………..…………. v List of Figures ……………...…………………………………………………….…………. viii List of Acronyms ……………………..…………………………………………….………….. x Abstract …………...……………………………………………………………..…...…….…. xi Chapter 1. INTRODUCTION.................................................................................................. 1 1.1. Background of the Work................................................................................... 1 1.2. Quadcopters ………………….....…………………......…………………...… 3 1.2.1. Principles of Flight………………………………………..…………….4 1.2.2. Flight Manoeuvres of the Quadcopter ……………………………..….. 8 1.3. Purpose of the Work........................................................................................ 10 1.4. Relevance of the Work…………………........................................................ 10 2. REVIEW OF RELATED LITERARURE……………..…………………...... 11 2.1. Conceptual Framework ………………………….......................................... 11 2.2. Bluetooth Technology ……….……............................................................... 11 2.3. The Arduino Platform …………………………………………...…………. 14 2.4. Massachusetts Institute of Technology Mobile Application Inventor…....... 15
  8. 8. vi 2.5. Rigid Bodies………………………………………………………………… 16 2.6. Review of Related Work…………………………………………..…………19 2.7. Summary of Review of Related Literature ……………………………...….. 21 3. DESIGN & MODELLING OF THE QUADCOPTER.…….……….………. 22 3.1. Modelling the Quadcopter …………………..............……………………… 22 3.1.1. The Quadcopter as a Rigid Body ……………………......................... 22 3.1.2. Frames of Reference ……………………………...............………….. 22 3.1.3. Time Derivative of the System……………………………………….. 23 3.1.4. Body Orientation …………………………………………...………... 25 3.1.4.1. Euler Angles ………………………....……...…………………. 25 3.1.5. The Equations of Motion....................................................................... 26 3.2. Design of the Quadcopter…………………......…………………………….. 34 3.2.1. The Components……….…………………………………………….. 35 3.2.1.1. The Motors and Propellers ….………………………………….. 35 3.2.1.2. The Microcontroller Unit…..…...………………………………. 38 3.2.1.3. The Bluetooth Module…….……………………………………. 39 3.2.1.4. The Power Source.……..……………………………………….. 40 3.2.1.5. Other Components ..………...…………………………………. 42 3.2.2. The Airframe ……………...…………………………………………. 43 3.2.3. Mounting the Electric Motors ……..…………….…………………... 44 3.2.4. The Requirements and Procedures Carried Out ……………………... 45 3.3. Interfacing with the quadcopter …………………………………………….. 51
  9. 9. vii 3.4. Review of Methodology…………………………………………………….. 52 4. PRESENTATION OF RESULTS …………..................................................... 53 4.1. The Quadcopter …………………………………………………………….. 53 4.2. Limitations of this work ……………………………………………………. 56 4.3. Summary of the Results ……………………………………………………. 56 5. CONCLUSION, DISCUSSION & RECOMMENDATIONS......................... 57 5.1. Discussion of Results……………………………………………………….. 57 5.2. Conclusion…………………………………………………………………... 58 5.3. Recommendations…………………………………………………………... 58 5.4. Suggestions for Further Studies…………………………………………….. 59 REFERENCES APPENDICES
  10. 10. viii LIST OF FIGURES Figure 1.2.1. The four basic aerodynamic forces. Figure 1.2.2. The various motions of the quadcopter with reference to the RPM on each individual rotor. Figure 2.2.a. Industrial, Scientific and Medical Data Band. Figure 2.2.b. Piconets and Scatternet Figure 2.6.a. The Block Diagram of the system (Mathieu, 2008). Figure 2.6.b. The Circuit Diagram of the System (Mathieu, 2008). Figure. 3.1.2. Body Axes of the quadcopter system Figure 3.2. The Block Diagram of the system Figure 3.2.1.1.a. A coreless DC motor utilized in this work Figure 3.2.1.1.b. The Inner Windings and casing of a Coreless DC motor Figure. 3.2.1.1.c. The Pusher Propellers Figure. 3.2.1.1.d. The Puller Propellers Figure. 3.2.1.1.e. The Propellers and Motors Figure. 3.2.1.2. The Arduino Pro Mini Figure 3.2.1.3. The HC-05 Bluetooth module used Figure. 3.2.1.4.a. The Lithium Polymer Battery Used. Figure 3.2.1.4.b. The Lithium Polymer Charger Module Figure. 3.2.2.a. The design of the airframe Figure 3.2.2.b. The Fabricated airframe for the system Figure. 3.2.3. The Mounted Electric Motors. Figure 3.2.4.a. The test of the Application using one electric motor.
  11. 11. ix Figure. 3.2.4.b. The Schematic representation of the one motor test. Figure. 3.2.4.c The Circuit diagram of the system Figure 4.1.a. A section of the android application. Figure. 4.1.b. The Quadcopter Figure. 4.1.c. The LiPo Battery Charger Provided in this work
  12. 12. x LIST OF ACRONYMS AES Advanced Encryption Standard .apk Android Application Package AT Attention BR/EDR Bluetooth Basic Rate/ Enhanced Data Rate BLE Bluetooth Low Energy CM Centre of Mass DOF Degree of Freedom EMF Electromotive Force IC Integrated Circuit ISM Industrial, Scientific and Medical LiPo Lithium Polymer M2M Machine-to-Machine MCU Microcontroller Unit PWM Pulse Width Modulation RC Remote Controlled RXD Receive SIG Special Interest Group SMD Surface Mount Device TXD Transmit UAV Unmanned Aerial Vehicle USB Universal Serial Bus
  13. 13. xi Abstract Findings from recent times have shown that the development and usage of drones (UAVs) such as Quadcopters has and will become a ubiquitous tool in the hands of developed and even developing countries such as Nigeria. The purpose of this work was to assume the quadcopter system to be a rigid body; obtain its equations of motion using the Lagrange’s Formulations as well as construct a small quadcopter model capable of hovering and landing, and controlled by means of Bluetooth Technology via an Android Mobile Application that was designed using the Massachusetts Institute of technology application inventor. It was recommended in this work that further studies should be carried out on the various navigational patterns of Quadcopter system as well as the navigational and flight manoeuvres of other UAV systems. The work also recommends extensive local based studies on the applications of semiconductor physics in robotics. Keywords: Quadcopter, robotics, semiconductor physics, Lagrange’ Formulations.
  14. 14. 1 CHAPTER ONE INTRODUCTION 1.1 Background of the Work Over the years, flying aircrafts have evolved, from kites, to hot air balloons, to planes, space shuttles, to drones. All of these aerial objects all exploit fundamental laws of physics for manoeuvring the air. This chapter aims to introduce a fast rising popular aerial vehicle known as the quadcopter; and its flight manoeuvring techniques. This work is based on the design and modelling of quadcopters. A quadcopter is an unmanned aerial vehicle. An unmanned aerial vehicle was first manufactured by Lawrence and Sperry (USA) in the year 1916. They called it the Aviation Torpedo show and they were able to fly it for a distance of 30 miles. It was reported that Lawrence and Sperry of the American Radio Relay League, performed the first public demonstration of remote-controlled flights. In the summer and fall of 1937, they designed and built sailplanes with a 13-foot wingspan, completing over 100 successful radio-controlled flights in Hartford, Connecticut. During this era, Hull set the pace for homebrew radio apparatus design. He increased transmitter efficiency by receiver for model aircraft. In the summer of 1958, Jack Kilby—a new employee at Texas Instruments and young inventor at the time— revolutionized the electronics industry with the introduction of his integrated circuit (IC); this gave rise of microchips. The advent of microchips (microcontrollers and microprocessors) has allowed for the development of computers as well as drones. In contemporary times, the development of quadcopters as well as many other robots and devices have been proved feasible because of the decreasing cost of microprocessors and microcontrollers, and the development of sensing and actuating technologies have made the electronic control (and even autonomous control) possible for military, commercial and even hobbyist purposes. This is unlike the past wherein the
  15. 15. 2 development of the vehicle was slow because of the difficulty and complexity of controlling each motor independently. 1.1.1 Unmanned Aerial Vehicles (UAVs) Drones are robots that are either autonomous in movement (but still remotely controlled) or completely remotely controlled. Drones are differentiated by the terrain in which they traverse (Baichtal, 2016). There are:  Rovers Which are remote controlled (RC) vehicles (that usually feature knobby or tank-like tires) to navigate the earthbound terrains using sensors such ultrasonic, Radio Frequency Identification (RFID), etc to detect obstructions.  Remotely operated vehicles These are underwater drones that are usually connected to a boat for the transfer of data; since radio waves are hindered by water.  Unmanned Aerial Vehicles Unmanned Aerial Vehicles are drones that navigate in the air by means of onboard computers and not human pilots. UAVs can be classified into four main categories based on their aerodynamic configuration: i. Fixed-Wing UAVs These are UAVs that have immobile wings that are fixed. They require a runway to take off and can fly for a long time at high cruising speeds. Examples include the predator, the general atomics MQ-9 reaper, etc. ii. Rotary Wing UAVs These are UAVs in which their propellers rotate. They can hover and fly with high manoeuvrability. They can also take off and land vertically. Rotary wind UAVs can be a single rotor (having only one rotor), co-axial (having two rotors
  16. 16. 3 mounted on the same shaft), quadrotor (having four rotors), or multi rotor UAV. Examples include: Tricopters, quadcopters, hexacopters, octocopters, y6, x6, etc. iii. Flapping Wing UAVs This class of UAV is still under development. They have a small wing and extremely low payload and endurance. iv. Blimps Blimps may look like balloons or airships; they ensure lifting by their helium- filled body. They are very light and have a large size. They can fly for a long time and at low speeds. The quadcopter is a rotary wing Unmanned Aerial Vehicle. 1.2 Quadcopters Quadcopters; also known as quadrotors or quadrotor helicopters are helicopters with four equally spaced, independently controlled rotors. A helicopter is a flying vehicle that utilizes rapid spinning rotors to push air downwards (thus producing a thrust force) to keep the helicopter aloft. Conventional helicopters typically possess one main rotor. This rotor is used for both up and down motions as well as longitudinal and lateral motions through the use of mechanical manipulation of the collective and cyclic pitch of the rotor blades using a swash plate (to change the angle of attack on it) with accompanying linkages and joints. Quadcopters unlike helicopters have no requirement for a tail rotor or any cyclic pitch controls. This is because, they utilize independently controlled speeds to increase or decrease the thrust and torque generated by each of the four rotors. This is advantageous in that, the vehicle becomes mechanically less complex and the rotors can easily be modelled. All four rotors of the quadcopter are directed upwards and usually arranged at the corners of a square body with equal distance from the centre of mass of the vehicle. With four rotors, the need for a swash plate mechanism is pressing. There is the need for a swash plate mechanism to allow the aircraft more degrees of freedom although the
  17. 17. 4 same level of control can be achieved by adding two more rotors. The rotation of all four rotors is set-up such that there are two oppositely rotating pairs; each member of a pair located across the other such that they are able to counteract each other at a lesser or greater degree depending on their inputs. Quadcopters are controlled by adjusting the angular velocity of the rotors. The main body of the vehicle houses the power source (battery), the sensors employed, the microcontroller (or microprocessor) employed and the control hardware. Due to the simple structure of quadcopters; as a flying robot, they have been used as a typical design for unmanned aerial vehicles (UAV, drones). As an unmanned aerial vehicle, they can be applied in agriculture (for precision farming), construction (3D mapping, inspection, etc.), Photography, search and rescue (first responders), Archaeology (3D information, etc.), surveillance, etc. 1.2.1 Principles of Flight There are four basic aerodynamic forces that act on an aircraft. They are considered basic because they act upon the aircraft in all manoeuvres (Burke, 2005). These forces are: Thrust, Lift, Drag, and Gravity (Weight). These four forces are opposing forces completely independent of one another and do not only affect aircrafts but also cars, trucks as well as every other vehicle. The four forces are explained in turn:  Thrust Thrust is a forward force that pushes an aircraft through the air. It is produced by the aircraft’s propulsion system or engine; in the case of quadcopters, it is produced by the rotors. The direction of thrust dictates the direction in which the aircraft will move. To every action, there is equal and opposite reaction (Newton, 1687); this is the principle on which thrust works hence, the engines, and propulsion system on the aircraft propel it forward by moving large quantities of air backwards. The direction of the thrust force is referred to as thrust line.
  18. 18. 5 If thrust is greater than drag, the aircraft will increase in speed. Thrust must equal weight and overcome drag (Kibly, 2016).  Lift Lift is an aerodynamic force that acts upwards against gravity and makes it possible for the aircraft to rise in air. It is a mechanical force generated by a solid body being in contact with a fluid as well as a difference in velocity between the solid body and the fluid it’s in contact with hence, no fluid, no lift; no motion, no lift (National Aeronautics and Space Administration, NASA Physical Science-aeronautics, 2015). In the case of winged air-craft, lift comes from air moving across an airfoil shape of a wing or propeller. The air moving above the airfoil is moving faster; therefore, it has lower pressure. Slower-moving air below the wing has higher pressure. This is governed by Bernoulli’s principle which states that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow (Young et al, 2008). The amount of lift generated by the wing is dependent on several factors which include: air density, angle of attack, speed of the wing through the air, wing area, and planform of the wing. To hover, lift must equal weight; to climb, lift must be greater than weight. Lift acts perpendicularly.  Drag Drag is the force of resistance to the motion of a vehicle’s body as it moves through a fluid. Drag acts in the opposite direction of thrust. Drag also acts parallel to and in the same direction as the relative wind. Every part of an aircraft that is exposed to air while the aircraft is in motion produces some resistance and contributes to the total drag (Burke, 2005). Total drag is classified into two basic types: 1. Induced drag Induced drag is a by-product of lift created by the higher pressure air below the wing traveling around the side of the wing to the lower pressure area. Induced drag increases in direct proportion to increases in the angle of attack. The greater the angle of attack up
  19. 19. 6 to the critical angle, the greater the amount of lift developed, and the greater the induced drag. 2. Parasitic (parasite) drag Parasitic drag is the resistance of the air produced by any part of the aircraft that does not produced lift. Parasitic drag can be classified into: a. Interference drag This type of drag is caused by two different airflows meeting and resisting each other. This is commonly seen where the wing is attached to the fuselage of an aircraft, otherwise known as the root. b. Form drag This type of drag is caused by the design of an aircraft. While the body of an aircraft may be extremely smooth and aerodynamic, the many objects attached to it, such as radio, sensors, antennas, etc. are not. These objects create drag. c. Skin-friction drag This is type of drag caused by air passing over the aircraft’s surfaces. It increases considerably if the surface of the aircraft is rough and dirty. This can be reduced by polishing or smoothing the surface exposed to the air. The total drag of an aircraft determines the amount of thrust required at a given airspeed. Several factors affect parasite drag. When each factor is considered independently, it must be assumed that other factors remain constant. These factors are: i. The more streamlined an object is, the less the parasite drag. ii. The denser the air moving past the airplane, the greater the parasite drag. iii. The larger the size of the object in the airstream, the greater the parasite drag.
  20. 20. 7 iv. As speed increases, the amount of parasite drag increases. If the speed is doubled, four times as much drag is produced.  Gravity (Weight) The earth attracts every object existing in the earth’s gravitational field. This attraction is called gravitational attraction and its effect is to change the velocity of objects under its influence, i.e. to accelerate such objects. The acceleration of objects due to the earth’s gravitational attraction is called acceleration due to gravity; it is represented by the symbol 𝑔 whose average value is about 9.81𝑚𝑠−2 . Weight is the force or pull with which the earth attracts a body towards the centre of the earth (Anyakoha, 2007); it is a response of mass to the pull of gravity. Gravity is the force that pulls down on objects and gives them weight. Mathematically, 𝑊 = 𝑚𝑔 (1.1) Where; 𝑊 is weight in Newton (𝑁), 𝑚 is mass in Kilograms (𝑘𝑔) and 𝑔 is acceleration due to gravity (𝑚𝑠−2 ) in equation (1.1). Although the weight is distributed throughout the aircraft, its centre of gravity has the most effect on the aircraft’s ability to fly. Centre of gravity is the point at the entire weight of a body appears to be concentrated (Anyakoha, 2007). The exact location of the centre of gravity is important during flight, because of its effect on airplane stability and performance. For anything to fly, or even hover, it must somehow continuously balance or overcome the force of gravity. In designing an aircraft, it is required to keep the weight to the minimum; this is because, the lighter the aircraft, the less power it consumes for flight and the more payload it can carry.
  21. 21. 8 Figure 1.2.1. The four basic aerodynamic forces. In straight and level, un-accelerated flight, the total amount of thrust is equal to the total drag, while the total amount of lift is equal to the total weight (National Aeronautics and Space Administration, NASA Physical Science-aeronautics, 2015). 1.2.2 Flight Manoeuvres of the Quadcopter In order, to move all different directions, left, right, vertically, etc., the speed of each rotor is adjusted. In defining an aircraft’s orientation, three angles have defined namely; roll, pitch and yaw angles. These angles are important because, the forces used to control a quadcopter causes the aircraft to roll, pitch or yaw. In hovering, the RPM (Revolution per minute; which gives the speed) on all four rotors must be equivalent and all torques must be in equilibrium. To climb (ascend vertically) the RPM on all four rotors is increased simultaneously. If the RPM on all four rotors is reduced simultaneously, the quadcopter would descend vertically. The pitch motion of the quadcopter is the motion of the aircraft that determines whether it flies forward or backward. In order to pitch forward, the RPM on the rear rotors is increased and the RPM on the front motors is reduced. To pitch backwards (pitch aft), the RPM on the front rotors is increased and the RPM on the rear rotors is reduced. The roll motion of the quadcopter is obtained when the RPM on the rotors on one side of the aircraft is increased and the RPM on the rotors on the other side is reduced. When the RPM on the rotors on the left hand side of the aircraft is increased and the RPM on the rotors on the right hand side reduced, the quadcopter would roll to the right. When the RPM on the rotors on the right hand side is increased and the RPM on the left hand side of the aircraft is reduced, the air vehicle would roll to the left.
  22. 22. 9 The yaw motion of the quadcopter is the motion of the aircraft that determines where it would be facing in space. Increasing or decreasing the RPM on opposing rotors increases or decreases torque in that particular pair’s direction; this results in the yaw motion. Hence, in order to yaw left, the RPM on the right rear rotor and the left front rotor would be increased while that of the left rear rotor and the right front rotor would be decreased. To yaw right, the RPM on the left rear rotor and the right front rotor would be increased while that of the right rear rotor and the left front rotor would be decreased. In order to manage all the control inputs for these various motions, quadcopters employ a flight control system (an on-board computer). The various possible flight patterns is shown in figure 1.1.2. Figure 1.2.2. The various motions of the quadcopter with reference to the RPM on each individual rotor.
  23. 23. 10 1.3 Purpose of the Work The sole purpose of the work is to introduce quadcopters and model the quadcopter system as a rigid body whose motion is described using the Lagrangian physical technique. It is also aim of this work to design a quadcopter prototype model capable of hovering when the right command is sent through a specialized developed android application communicating through Bluetooth. 1.4 Relevance of the Work This work is highly relevant because, it is imperative that the physics behind existing, experimental, and emerging technologies are exposed, revisited, and improved upon. Furthermore, this work is relevant because, Quadcopters pose a fundamentally difficult and interesting physical problem: its control. In controlling quadcopters, the major challenge is that the vehicle possesses six degrees of freedom (three translational and three rotational) and it only has four independent control input (the rotor speeds); this means that the vehicle is under-actuated. In order to achieve six degrees of freedom, the translational and rotational movements are coupled. The resulting dynamics of coupling both the translational and rotational motions of the vehicle is highly non-linear, especially when aerodynamics effects such as gravity, drag, etc. are accounted for. The work is also highly relevant because more research has to be carried out in the area of drones and unmanned aerial vehicles for the collective benefit of the country, Nigeria.
  24. 24. 11 CHAPTER TWO REVIEW OF RELATED LITERATURE This chapter aims to give the general concept of this work, review Bluetooth technologies, as well as the Arduino Platform, shed more light on the purpose of this work and highlight how the knowledge offered by the preceding works will be incorporated into this project work. The review is presented under the following sections:  Conceptual Framework  Bluetooth Technology  The Arduino Platform  Massachusetts Institute of Technology Mobile Application Inventor  Review of Related Work  Summary of Review of Related Literature 2.1 Conceptual Framework The concept of this work is to describe the motion of the quadcopter system using the Lagrangian Formalism, design a quadcopter basic airframe, and construct a quadcopter that is capable of hovering and communicating via Bluetooth. The approach to this is by first establishing the equations of motion after which the design of the quadcopter system will be made after which, the rotors of the model will be connected to the Arduino Pro Mini which in turn will be connected to a Bluetooth module to serve as a communication interface. 2.2 Bluetooth Technology Bluetooth is a wireless technology for exchanging data over short distances using short wavelength of ultra-high frequency radio waves. Bluetooth wireless technology was developed in 1994 at Ericsson in Sweden. Bluetooth technology is now managed by the Bluetooth special interest group (SIG). The
  25. 25. 12 Bluetooth SIG adopted the code name as a tribute to the tenth-century Viking king Harald Blatand who peacefully united Denmark and Norway. Harald liked to eat blueberries, which gave his teeth the coloration that lead to the nickname "Bluetooth." There are basically two “flavours” of Bluetooth technology; Basic Rate/Enhanced Data Rate (BR/EDR) and Low Energy (LE). (Bluetooth SIG, 2017). Bluetooth BR/EDR is a Bluetooth technology for the connecting one device to another (point-to-point connection). It is capable of continuous data streaming and as such is utilized in audio streaming devices such as wireless headsets, speakers, etc. Bluetooth Low energy is aimed for short burst connections and as such, can be utilized in point-to-point connections, broadcast connections (which involves the connection of one to many devices), and mesh connections (which involves the connection of many devices to one another). Hence, Bluetooth Low energy can be utilized in smart watches, way finding beacons, asset tracking, wireless sensor networks, as it is aimed at very low power applications (peripheral devices that operate on batteries) and sacrifices range (50m instead of 100m) and data throughput (0.27 Mbps instead of 0.7-2.1 Mbps). Bluetooth has constant been evolving since its advent, the first versions; Bluetooth 1.0, 1.1 and 1.2 is characterized by poor compatibility, noise and data transmission speed up to 721kbps. Versions 2.0 and 2.1, released in 2004 and 2007 respectively featured the enhanced data rate technology, has a theoretical data transmission speed of about 3mbps and improved device compatibility including backward compatibility with version 1.0. The version 3.0 was released in 2009; it pioneered the use of high speed (HS) data broadcasting, with a disadvantage of increased power consumption. Bluetooth 3.0 is supported by almost all smart phones, and has a maximum data transfer range of 10m. Bluetooth versions 4.0 and 4.1 specifications introduced in 2010 and 2013 respectively fixed the demerit of increased power consumption due to high speed data transmission, and featured a connectivity time of 5ms, permission AES-encryption, coordinated data packets, transmission range of 100m and a maximum data transmission speed of 25mbps. Bluetooth version 5.0 is a new version of Bluetooth specification introduced in 2017 featuring data safety, lower energy consumption and data transfer within the maximum range of 240m.
  26. 26. 13 Bluetooth devices operate in the Industrial, Scientific and Medical (ISM) from 2.4 to 2.485 GHz. The Industrial, Scientific and Medical band is shown in figure x. In order to minimize and prevent eavesdropping and interference from other devices that use the ISM band, Bluetooth devices utilize a technique known as frequency hopping for data transmission. With frequency hopping, the data is divided into small pieces called packets. The transmitter and receiver exchange a data packet at one frequency, and then they hop to another frequency to exchange another packet. They repeat this process until all the data is transmitted. Bluetooth devices randomly hop between frequencies up to 1600 times per second. Figure 2.2.a. Industrial, Scientific and Medical Data Band. When two Bluetooth devices locate each, they connect (pair); the device that initiates the connection is known as the master and the device that connects is called the slave device. When a master connects with up to seven slaves, a piconet is formed. Each piconet has a different frequency-hopping pattern to differentiate its signals from the signals of other piconets. When two piconets are linked by a bluetooth device, a scatternet is formed.
  27. 27. 14 Figure 2.2.b. Piconets and Scatternet Two master devices cannot pair with each other. This is why Bluetooth devices such as Bluetooth Printers, Audio Players, etc. are usually embed with a slave Bluetooth device. However, there are Bluetooth devices that can act as slave devices and as master devices. For communication to take place through Bluetooth, there must be a master and a slave; and the password must match between the two or more devices. Bluetooth technology is a relatively safe data & voice transmission medium because it offers 128 encryptions, frequency hoping and pairing authentication. There are various forms in which Bluetooth devices exists; they can exist as PC cards, USB devices, mini PCI, ultraport, modules, etc. The form in of Bluetooth device utilized in this work is the Serial to Bluetooth module, HC-05 module; a master/slave module. 2.3 The Arduino Platform The Arduino Platform is an open-source hardware prototyping platform. The Arduino Platform offers a variety of development boards. A development board offered by the Arduino platform is usually based around a particular microcontroller; this
  28. 28. 15 microcontroller is programmed using mainly the Arduino Integrated Development Environment, a software that is cross-platform. The Arduino board utilized in this project work is the Arduino Pro Mini. It is a small sized development board measuring only about 33mm x 18mm x 2mm. The Arduino Pro Mini is a microcontroller board based on the ATmega328 microcontroller. The Board has 14 digital input/output pins (of which 6 can be used as PWM outputs), 6 analogy inputs, an on-board resonator, a reset button, and holes for mounting pin headers. 2.4 Massachusetts Institute of Technology Application Inventor The Massachusetts Institute of Technology application inventor (MIT app inventor) is an open source web based application for building android applications (apps). It a visual programming tool in which its users are able to perform programming tasks without actually entering any computer code (Walter et al, 2015). The application was initially built by Google in 2010 and later open-sourced and given to the Massachusetts Institute of Technology (MIT) Centre for mobile learning in 2012. It has since then been maintained by the Massachusetts Institute of Technology; receiving sponsorship from Google, Motorola, Samsung, Verizon Foundation, Motorola Mobility Foundation, Ford Motor company, Centro Superior para la Ensenanza Virtual (CSEV), and other sponsors. The aim of the MIT app inventor is to direct people away from technology consumption to technology creation. The app inventor is very useful for making mobile apps and it is suitable for people with little or no programming experience. (Penta, 2016). The Massachusetts Institute of Technology Application Inventor can be located on http://ai2.appinventor.mit.edu/ .
  29. 29. 16 2.5 Rigid Bodies A rigid body is a physical system of particles that does not deform. It is a system of mass points subject to the holonomic constraints that the distance between all pairs of points remain constant throughout the motion (Goldstein, 1950). This means that in a rigid body, assuming that A and B are any of its particles, the distance between A and B will remain the same without changes assuming elasticity and breakage are the limits. Invariably, if A and B are any two points on a rigid body, the distance between them is always constant. Hence, it can be concluded that rigid bodies are extended bodies that are a collection of N points so that the distance between them is fixed. i.e. |𝑟𝑖 − 𝑟𝑗 | = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (2.1) Where: 𝑟𝑖 is the position of an 𝑖 𝑡ℎ particle, 𝑟𝑗 is the position of a 𝑗 𝑡ℎ particle and 𝑖 = 𝑗 = 1,2,3,4, … . 𝑁 No real body is absolutely rigid but there exist cases where a body can be regarded as rigid. Like any other physical system, forces act on rigid bodies. The forces that act on a rigid body can be separated into two broad categories namely: External forces and Internal forces. External forces are forces that are exerted on rigid bodies. These forces are the only forces that can make a rigid body exhibit some form of motion; translational, rotational or some cases, both. Internal forces are forces that act within the rigid body. These forces are responsible for holding together the particles forming the rigid body. A rigid body has six coordinates required to describe its position i.e. three translational and three rotational about a point. Hence, a rigid body has six degrees of freedom. These coordinates are most important in the mechanics (kinematics and dynamics) of the rigid body.
  30. 30. 17 In the mechanics of rigid bodies, the centre of mass is considered an important variable, this is because, the centre of mass is the point at which the total mass of the system appears to be concentrated. For small rigid objects, the centre of mass coincides with the centre of gravity (the centre of gravity is the point at which the entire weight of the rigid body appears to be concentrated). Thus, in computing the point where the centre of mass is placed, 𝑚𝑟⃗𝐶𝑀 = ∑ 𝑚𝑖 𝑟⃗𝑖 (2.2) or 𝑚𝑟⃗𝐶𝑀 = ∫ 𝑟⃗𝑑𝑚 (2.3) Where: 𝑚 is the total mass of the rigid body, 𝑟⃗ is the position vector of the system 𝑟⃗𝐶𝑀 is centre of mass position vector, 𝑚𝑖 is the mass of the 𝑖 𝑡ℎ particle and 𝑟𝑖 is the position vector of the 𝑖 𝑡ℎ particle In describing the motion of a rigid body without considering the forces involved, the motion can be divided into five broad categories: translational motion, rotational motion, general plane motion, motion about a fixed point, general motion. Translational motion is any motion in which any straight line inside the rigid body keeps the same direction during the movement. All the particles forming the body move along parallel paths. If these paths are straight lines, the motion is said to be a rectilinear translation; if the paths are curved lines, the motion is a curvilinear motion. In rotational motion, the particles forming the rigid body move in parallel planes along circles centred on the same fixed axis. If this axis, called the axis of rotation intersects the rigid body, the particles located on the axis have zero velocity and zero acceleration. Plane motion is that motion in which all the particles of the body move in parallel planes. Any motion that is neither rotational or translational is referred to as general plane motion. Motion about a fixed point is the motion of a rigid body attached to a fixed point. Any motion of a rigid body that does not fall into any of the category previously mentioned is known as general motion of the body.
  31. 31. 18 There is also a motion of a rigid body that is known as the simple harmonic motion; it is a motion that repeats itself periodically. The equations of motion in the kinematic description captures the velocity (linear or angular), acceleration, momentum, impulse of the system but not the forces associated with it. In the rectilinear motion for example, considering a particle moving along the plane within a stipulated time, the velocity would be the time derivative of the distance in which case is. 𝑣 = 𝑑𝑥 𝑑𝑡 = 𝑥̇ (2.4) Again, 𝑎 = 𝑑2 𝑥 𝑑𝑡2 = 𝑥̈ (2.5) Where: 𝑣 is the same as 𝑥̇ and is the velocity of the body, and 𝑎 is the same as 𝑥̈ and is the acceleration of the body. The dynamic description of the motion of a rigid body on the other hand captures the force associated with the motion. The orientation of a rigid body as well as any other physical system is the description of how the body or system is placed in space. The orientation tells the rotation required to move the system or rigid body from a reference position to its current position. Orientation is usually represented using Euler angles, Quaternions, rotation matrices, miller indices (as in crystallography), Tait-Bryan angles and angle-axis representation methods. The Tait-Bryan angles and Euler angles method utilize three angles known as yaw, pitch and roll for orientation representation. The orientation of a rigid body is also known as its attitude. The mechanics (statics, kinematics and dynamics) of a rigid body can be evaluated using several physical techniques; commonly used are the Newtonian methods, Lagrangian formalism methods and more robustly, Hamiltonian formalism methods. Each technique has its own drawback.
  32. 32. 19 Rigid bodies are not really realistic systems because of the characteristic of not being able to be deformed. Resistant bodies however, are most feasible physical systems. 2.6 Review of Related Work What has been is what will be, and what has been done is what will be done; there is nothing new under the sun (Ecclesiastes 1: 9, The New Revised Standard Version, Catholic Edition). Prior to the development of this work, several similar work have been done. This section aims to review some of such similar work. The Quadcopter body system can be modeled by representing the system as a solid body with a main thrust acting in a three dimensional (3D) space; subject to three torques: pitch, yaw and roll. Approaching such a system with the Euler-Lagrangian technique, the generalized co-ordinates of the system can be expressed by as (Carillo et al, 2013): 𝑞 = (𝑥, 𝑦, 𝑧, 𝜓, 𝜃, 𝜑) ∈ 𝑅6 (2.6) In the work of (Roldofo et al, 2013), the time derivative of (2.6) was taken and obtained to be (2.7): 𝑞̇ = (𝑥̇, 𝑦̇ , 𝑧̇, 𝜑̇ − 𝜓̇ 𝑠𝑖𝑛 𝜃 , 𝜃̇ 𝑐𝑜𝑠 𝜑 + 𝜓̇ 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜑 , 𝜓̇ 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜑 − 𝜃̇ 𝑠𝑖𝑛 𝜑) The lagrangian is a function of generalized position and generalized, 𝐿(𝑞, 𝑞̇ ) = 𝑇 – 𝑈 (2.8) The equations of motion of the model were then obtained from the Euler-Lagrangian Equations with external generalized forces. 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑞̇ ) − 𝜕𝐿 𝜕𝑞 = [ 𝐹 𝜏 ] (2.9) Where 𝜏 represents the yaw, pitch and roll moments. The equations of motions in the work were then obtained using equations (2.6), (2.7), (2.8), and (2.9). In another work, an HC-05 Bluetooth Module has proved to be feasible for use in the control Light Emitting Diodes through an Android mobile application (Shrestha, 2015).
  33. 33. 20 The work was also extended to the control of up to eight home appliances; this was achieved by the use of a 12V relay. Bluetooth technology has also been proved feasible in the construction of a Bluetooth remote control Audio device (Mathieu S, 2008). In the work, the Bluetooth module employed was the LMX9838 from National Semi-conductors. The Bluetooth module was interfaced with the AT90USB1287, an 8-bit microcontroller that runs at 8MHz, 8KB of Random Access Memory (RAM) and 128KB of Flash Memory. The system utilized the audio codec capability of the LMX9838 for parsing audio data files. The communication between the Bluetooth Module and the Audio Codec was achieved using a Winbond W681310 Pulse-Code Modulation (PCM) bus; since the LMX9838 already supports audio codecs. The system also possesses the capability of connecting to computers via Universal Serial Bus (USB) interface. The block diagram of the system is given by figure 2.6.a. Figure 2.6.a. The Block Diagram of the system (Mathieu, 2008).
  34. 34. 21 Figure 2.6.b. The Circuit Diagram of the System (Mathieu, 2008). 2.7 Summary of Review of Related Literature Bluetooth Technology has revolutionized wireless communication between devices with its ubiquitous and simple characteristics. (Abhishek et al, 2016). This revolution has made it possible for normal everyday devices to be able to send data to a designated device or devices once the device(s) is(are) in range of communication. The communication range and data transfer speed of Bluetooth has improved with recent versions of the technology and will continue to improve in the future. From the review, from the works of (Shrestha, 2015) and (Mathieu, 2008), it can be seen that with the advent of a simple microcontroller programming platform such as the Arduino platform, interfacing with everyday devices via Bluetooth has become very much feasible and applicable. From the section, rigid bodies are also highlighted upon. These bodies (rigid bodies) are in real life, not really existent as pointed out in the review but it is necessary to make the assumption that a system does not deform for proper modelling to be carried out on that system.
  35. 35. 22 CHAPTER THREE METHODOLOGY This chapter entails the modelling of the quadcopter system as a rigid body, and the fabrication of a quadcopter vehicle using various components; whilst giving in detail the processes carried out. 3.1 Modelling the Quadcopter This chapter aims to give the frame of reference in which the quadcopter system operates while modelling the system as a rigid body. 3.1.1 The Quadcopter as a Rigid Body The quadcopter system is being modelled as a rigid body with six degrees of freedom (DOF); three translational coordinates and three rotational coordinates. This is so because, the quadcopter is capable of translating and rotating due to the forces caused by its electric motors and propellers. As a rigid body, the motion of the quadcopter system can be described using various physical techniques. The quadcopter system will be modelled using the Lagrangian technique in the subsequent sections. 3.1.2 Frames of Reference The Frames of reference of the quadcopter is shown in figure 3.1.2 Figure. 3.1.2. Body Axes of the quadcopter system
  36. 36. 23 In Fig. 3.1.2, diagram A represents the inertial frame of the system and diagram B represents the body frame of the system. The body axes centre is assumed to be in the same position as the centre of gravity while the x and y axes are in the position of the arms of the quadcopter. f1, f2, f3, f4 in diagram B represent the forces created by the four motors; w1, w2, w3, w4, also in diagram B represent the angular velocities of the four motors. TM1, TM2, TM3, TM4 represent the torques of the four motors. In the inertial frame, the linear position of the quadcopter is given by: 𝜉 𝑇 = ( 𝑥 𝑦 𝑧 )T (3.1) The angular position of the quadcopter is defined by the inertial frame with three Euler angles ℵ. The pitch angle, 𝜃 determines the rotation of the quadcopter around the y-axis. The rotation around the x-axis is given by the roll angle, , ∅ and the angle, 𝜑 which is known as the yaw angle gives the rotation of the vehicle around the z-axis. ℵ 𝑇 = ( 𝜑 𝜃 ∅ )T (3.2) The position vector, 𝑞⃗ is the generalized coordinate that describes the position of the quadcopter in space. It can be expressed as: 𝑞⃗ 𝑇 = ( 𝑥 𝑦 𝑧 𝜑 𝜃 ∅ )T (3.3) 𝑞⃗ 𝑇 = ( 𝜉 ℵ )T (3.4) 3.1.3 Time Derivative of the System From the inertial frame, 𝑑𝜉 𝑇 𝑑𝑡 = ( 𝑑𝑥 𝑑𝑡 𝑑𝑦 𝑑𝑡 𝑑𝑧 𝑑𝑡 ) 𝑇 (3.5) 𝜉̇ 𝑇 = ( 𝑥̇ 𝑦̇ 𝑧̇) 𝑇 (3.6) Hence, 𝜉̇ = ( 𝑥̇ 𝑦̇ 𝑧̇ ) (3.7) Equations (3.6) and (3.7) give the linear velocity of the quadcopter
  37. 37. 24 Furthermore, 𝑑ℵ 𝑑𝑡 𝑇 = ( 𝑑𝜑 𝑑𝑡 𝑑𝜃 𝑑𝑡 𝑑∅ 𝑑𝑡 ) 𝑇 (3.8) ℵ̇ 𝑇 = (𝜑̇ 𝜃̇ ∅̇ ) 𝑇 (3.9) ℵ̇ = ( 𝜑̇ 𝜃̇ ∅̇ ) (3.10) But recall that the quadcopter rotates about these angles, 𝜑, 𝜃, 𝑎𝑛𝑑 ∅ hence, equations (3.9) and (3.10) do not give the angular velocities but rather the time derivative of these angles. In other to obtain the angular velocity, the time derivatives must be multiplied by: 𝜔 = ( − sin 𝜃 0 1 cos 𝜃 sin ∅ cos ∅ 0 cos 𝜃 cos ∅ − sin ∅ 0 ) (3.11) Where; 𝜔 is the rotational matrix according to the Euler angle resolution Multiplying equation (3.11) by equation (3.10), ( − sin 𝜃 0 1 cos 𝜃 sin ∅ cos ∅ 0 cos 𝜃 cos ∅ − sin ∅ 0 ) ( 𝜑̇ 𝜃̇ ∅̇ ) (3.12) The multiplication in equation (3.12) is only possible because, the number of columns on the left hand side is equal to the number of rows on the left hand side. = ( −𝜑̇ sin 𝜃 + 0 ∗ 𝜃̇ + ∅̇ 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ + ∅̇ ∗ 0 𝜑̇ cos 𝜃 cos ∅ + (− sin ∅ 𝜃̇) + ∅̇ ∗ 0 ) = ( −𝜑̇ sin 𝜃 + 0 + ∅̇ 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ + 0 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅ + 0 )
  38. 38. 25 = ( ∅̇ − 𝜑̇ sin 𝜃 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅ ) Let Ω represent the angular velocity. Ω = ( ∅̇ − 𝜑̇ sin 𝜃 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅ ) (3.13) Equation (3.13) gives the angular velocity. 3.1.4 Body Orientation The orientation of any rigid body; rods, tables, or even a quadcopter can be represented in more than one way with the most common methods being the Euler’s angles method and the Quaternion method. The Euler angle method involves the use of three angles to describe orientation. The quaternion method is more complicated and involves the use of four parameters to describe orientation. 3.1.4.1 Euler Angles The Euler angles are used to define the orientation of the body frame axes with respect to the inertial frame axes. The Euler angles are: 𝜑, 𝜃, 𝑎𝑛𝑑 ∅. They are called yaw, pitch and roll respectively although this is not universal. The main disadvantage of using Euler angles is that singularities occur in the change of coordinates matrices that is at.
  39. 39. 26 3.1.5 The Equations of Motion The Lagrangian is defined as, 𝐿( 𝑞, 𝑞̇) = 𝑇 − 𝑈 (3.14) Where; 𝑇 𝑖𝑠 The kinetic energy and 𝑈 𝑖𝑠 The Potential energy Since the quadcopter translates as well as rotates, its kinetic energy is the sum of the translational kinetic energy and the rotational kinetic energy. 𝐿( 𝑞, 𝑞̇) = 𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + 𝑇𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 − 𝑈 (3.15) Now, the translational kinetic energy is given by; 𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 1 2 𝑚𝜉̇ 𝑇 𝜉̇ (3.16) = 1 2 𝑚 ((𝑥̇ 𝑦̇ 𝑧̇) ( 𝑥̇ 𝑦̇ 𝑧̇ )) = 1 2 𝑚( 𝑥̇ ∗ 𝑥̇ + 𝑦̇ ∗ 𝑦̇ + 𝑧̇ ∗ 𝑧̇) (3.17) 𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 1 2 𝑚(𝑥̇2 + 𝑦̇2 + 𝑧̇2 ) (3.18) The rotational energy is given by; 𝑇𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 1 2 Ω 𝑇 𝐼Ω (3.19) Where; 𝐼 𝑖𝑠 the matrix of inertia (𝑠𝑒𝑒 𝑎𝑝𝑝𝑒𝑛𝑑𝑖𝑥) The quadcopter vehicle is assumed to have a symmetric structure with its four arms aligned with the body x and y axes. Thus, the inertia matrix is the diagonal matrix 𝐼 for which: 𝐼𝑥𝑥 = 𝐼 𝑦𝑦 = 𝐼𝑧𝑧 𝐼 = ( 𝐼𝑥𝑥 0 0 0 𝐼 𝑦𝑦 0 0 0 𝐼𝑧𝑧 ) (3.20) Solving for 𝐼Ω ,
  40. 40. 27 𝐼Ω = ( ( 𝐼𝑥𝑥 0 0 0 𝐼 𝑦𝑦 0 0 0 𝐼𝑧𝑧 ) ( ∅̇ − 𝜑̇ sin 𝜃 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅ ) ) (3.21) = ( 𝐼𝑥𝑥(∅̇ − 𝜑̇ sin 𝜃) + 0 + 0 0 + 𝐼 𝑦𝑦(𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅) + 0 0 + 0 + 𝐼𝑧𝑧(𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅) ) (3.22) 𝐼Ω = ( 𝐼𝑥𝑥(∅̇ − 𝜑̇ sin 𝜃) 𝐼 𝑦𝑦(𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅) 𝐼𝑧𝑧(𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅) ) (3.23) Solving now for the rotational translational energy, 1 2 Ω 𝑇 𝐼Ω. Ω 𝑇 = (∅̇ − 𝜑̇ sin 𝜃 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅) Ω 𝑇 𝐼Ω = (∅̇ − 𝜑̇ sin 𝜃 𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos ∅ 𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅) ( 𝐼𝑥𝑥(∅̇ − 𝜑̇ sin 𝜃) 𝐼 𝑦𝑦(𝜑̇ cos 𝜃 sin ∅ + 𝜃̇ cos∅) 𝐼𝑧𝑧(𝜑̇ cos 𝜃 cos ∅ − 𝜃̇ sin ∅) ) Hence, 1 2 Ω 𝑇 𝐼Ω which will be known now as equation (3.24) is the rotational kinetic energy and is equal to: 1 2 (𝐼 𝑥𝑥(∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 𝑠𝑖𝑛2 𝜃) + 𝐼 𝑦𝑦(𝜑̇2 𝑐𝑜𝑠2 𝜃𝑠𝑖𝑛2 ∅ + 2𝜑̇ cos 𝜃 sin∅ 𝜃̇ cos∅ + 𝜃̇2 𝑐𝑜𝑠2 ∅) + 𝐼𝑧𝑧(𝜑̇2 𝑐𝑜𝑠2 𝜃𝑐𝑜𝑠2 ∅ − 2𝜑̇ cos 𝜃 cos∅ 𝜃̇sin∅ + 𝜃̇2 𝑠𝑖𝑛2 ∅)) (3.24) The potential energy 𝑈 is equal to 𝑚𝑔𝑧 where; 𝑚 is the mass of the quadcopter, 𝑔 is the acceleration due to gravity and 𝑧 is the altitude climbed through the quadcopter. Hence, 𝐿 = 1 2 𝑚(𝑥̇2 + 𝑦̇2 + 𝑧̇2)+ 1 2 (𝐼 𝑥𝑥(∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 𝑠𝑖𝑛2 𝜃) + 𝐼 𝑦𝑦(𝜑̇ 2 𝑐𝑜𝑠2 𝜃𝑠𝑖𝑛2 ∅ + 2𝜑̇ cos 𝜃 sin∅ 𝜃̇ cos∅ + 𝜃̇2 𝑐𝑜𝑠2 ∅) + 𝐼𝑧𝑧(𝜑̇2 𝑐𝑜𝑠2 𝜃𝑐𝑜𝑠2 ∅ − 2𝜑̇ cos 𝜃 cos∅ 𝜃̇sin ∅ + 𝜃̇2 𝑠𝑖𝑛2 ∅)) − 𝑚𝑔𝑧 (3.25) Equation (3.25) is the Lagrangian equation for the quadcopter vehicle.
  41. 41. 28 The quadcopter vehicle is acted upon by external forces and various aerodynamic effects hence, the model for the quadcopter dynamics is obtained from the Lagrangian equation with external generalized forces. 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑞̇ ) − 𝜕𝐿 𝜕𝑞 = [ 𝐹𝜉 𝜏 ] (3.26) Where; 𝐹𝜉 is the translational force applied to the quadcopter due to the main thrust obtained by resolving 𝐹𝜉 = 𝑹𝐹̂ ; 𝑹 is the rotational matrix (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑩) and 𝜏 is the generalized torque. Also, 𝐹̂ = ( 0 0 𝜌 ) (3.27) Where; 𝜌 is the main thrust directed from the bottom of the quadcopter 𝜌 = ∑ 𝑓𝑖 4 𝑖=1 (3.28) For 𝑖 = 1, 2,3 𝑎𝑛𝑑 4 ( 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟 𝑟𝑜𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑞𝑢𝑎𝑑𝑐𝑜𝑝𝑡𝑒𝑟) and is the force 𝑓𝑖 produced by each rotor. And, 𝜏 = ( 𝜏 𝜑 𝜏 𝜃 𝜏∅ ) ≜ [ ∑ 𝜏𝑀𝑖 4 𝑖=1 ( 𝑓2 − 𝑓4) 𝑙 ( 𝑓3 − 𝑓1) 𝑙 ] (3.29) Where; 𝜏𝑀𝑖 is the moments produced by each rotor and is the distance between the rotors and the centre of gravity of the quadcopter vehicle. Equation (3.29) gives the generalized torque The generalized torque is directly related to the generalized moment, 𝜏̃. 𝜏̃ = ( 𝜏̃ 𝜑 𝜏̃ 𝜃 𝜏̃∅ ) = 𝑱−1 𝑱ℵ̈ (3.30)
  42. 42. 29 Where; 𝑱 is the Jacobian matrix from the angular velocity, equation (3.13) to the time derivative of the angular coordinates (3.10) and is obtained by resolving 𝜔 𝑇 𝐼𝜔, 𝑱−1 is the inverse of the Jacobian matrix, 𝑱 and ℵ̈ is the second order time derivative of the angular coordinates Evaluating 𝐹𝜉 = 𝑹𝐹̂, 𝑹𝐹̂ = ( cos 𝜃 cos 𝜑 cos 𝜑 sin 𝜃 sin ∅ − cos ∅ sin 𝜑 sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃 cos 𝜃 sin 𝜑 cos ∅ cos 𝜑 + sin 𝜃 sin ∅ sin 𝜑 cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅ − sin 𝜃 cos 𝜃 sin ∅ cos 𝜃 cos ∅ ) ( 0 0 𝜌 ) = ( 0 + 0 + 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃) 0 + 0 + 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅) 0 + 0 + 𝜌(cos 𝜃 cos ∅) ) 𝐹𝜉 = ( 𝐹𝑥 𝐹𝑦 𝐹𝑧 ) = ( 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃) 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅) 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 ) (3.31) The value of 𝐹𝑧 is like so because, gravity acts in the direction of the z axis. The quadcopter’s position is described by two sets of coordinates; the translational, 𝜉 and the angular, ℵ, hence the Lagrangian equation can be divided into the Lagrangian equation for translational motion and the Lagrangian equation for rotational motion. Hence, for 𝜉; 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝜉̇ ) − 𝜕𝐿 𝜕𝜉 = 𝐹 𝜉 (3.32) 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑥̇ ) − 𝜕𝐿 𝜕𝑥 = 𝐹 𝑥 (3.33) 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑦̇ ) − 𝜕𝐿 𝜕𝑦 = 𝐹 𝑦 (3.34)
  43. 43. 30 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑧̇ ) − 𝜕𝐿 𝜕𝑧 = 𝐹 𝑧 (3.35) For the ℵ coordinates, 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕ℵ̇ ) − 𝜕𝐿 𝜕ℵ = 𝜏̃ (3.36) 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝜑̇ ) − 𝜕𝐿 𝜕𝜑 = 𝜏̃ 𝜑 (3.37) 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝜃̇ ) − 𝜕𝐿 𝜕𝜃 = 𝜏̃ 𝜃 (3.38) 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕∅̇ ) − 𝜕𝐿 𝜕∅ = 𝜏̃∅ (3.39) Recall equation (3.24), 1 2 (𝐼 𝑥𝑥(∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 𝑠𝑖𝑛2 𝜃) + 𝐼 𝑦𝑦(𝜑̇2 𝑐𝑜𝑠2 𝜃𝑠𝑖𝑛2 ∅ + 2𝜑̇ cos 𝜃 sin ∅ 𝜃̇ cos∅ + 𝜃̇2 𝑐𝑜𝑠2 ∅) + 𝐼𝑧𝑧(𝜑̇2 𝑐𝑜𝑠2 𝜃𝑐𝑜𝑠2 ∅ − 2𝜑̇ cos 𝜃 cos∅𝜃̇ sin ∅ + 𝜃̇2 𝑠𝑖𝑛2 ∅)) For a symmetrical body with unit inertia, 𝐼𝑥𝑥 = 𝐼 𝑦𝑦 = 𝐼𝑧𝑧 = 1 hence, 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇ 2 𝑠𝑖𝑛2 𝜃 + 𝜑̇2 𝑐𝑜𝑠2 𝜃𝑠𝑖𝑛2 ∅ + 2𝜑̇ cos 𝜃 sin ∅ 𝜃̇ cos ∅ + 𝜃̇2 𝑐𝑜𝑠2 ∅ + 𝜑̇2 𝑐𝑜𝑠2 𝜃𝑐𝑜𝑠2 ∅ − 2𝜑̇ cos 𝜃 cos∅ sin∅ + 𝜃̇2 𝑠𝑖𝑛2 ∅) 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇ 2 𝑠𝑖𝑛2 𝜃 + 𝜑̇2 𝑐𝑜𝑠2 𝜃𝑠𝑖𝑛2 ∅ + 𝜃̇2 𝑐𝑜𝑠2 ∅ + 𝜑̇2 𝑐𝑜𝑠2 𝜃𝑐𝑜𝑠2 ∅ + 𝜃̇2 𝑠𝑖𝑛2 ∅) 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇ 2 𝑠𝑖𝑛2 𝜃 + 𝜑̇2 𝑐𝑜𝑠2 𝜃(𝑠𝑖𝑛2 ∅ + 𝑐𝑜𝑠2 ∅) + 𝜃̇2 (𝑠𝑖𝑛2 ∅ + 𝑐𝑜𝑠2 ∅)) = 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 𝑠𝑖𝑛2 𝜃 + 𝜑̇2 𝑐𝑜𝑠2 𝜃 + 𝜃̇2) = 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 (𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃) + 𝜃̇2) = 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin 𝜃 + 𝜑̇2 + 𝜃̇2) As the quadcopter rotates at 𝜃 = 360° , = 1 2 (∅̇ 2 − 2∅̇ 𝜑̇ sin(360) + 𝜑̇2 + 𝜃̇2) = 1 2 (∅̇ 2 − 0 + 𝜑̇2 + 𝜃̇2) Hence, 𝑇𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 1 2 (∅̇2 + 𝜑̇2 + 𝜃̇2 )
  44. 44. 31 and equation (3.25) becomes, 𝐿 = 1 2 (𝑚( 𝑥̇2 + 𝑦̇2 + 𝑧̇2) + (∅̇ 2 + 𝜑̇ 2 + 𝜃̇2 )) − 𝑚𝑔𝑧 (3.40) Evaluating equation (3.33) with equation (3.40), 𝜕𝐿 𝜕𝑥̇ = 𝑚𝑥̇ ; 𝜕𝐿 𝜕𝑥 = 0 𝑑 𝑑𝑡 ( 𝑚𝑥̇) − 0 = 𝐹𝑥 𝑚 ( 𝑑 𝑑𝑡 ( 𝑥̇)) = 𝐹𝑥 𝑚𝑥̈ = 𝐹𝑥 Recall equation (3.31) 𝑚𝑥̈ = 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃) (3.41) Evaluating equation (3.34) with equation (3.40), 𝜕𝐿 𝜕𝑥̇ = 𝑚𝑦̇ ; 𝜕𝐿 𝜕𝑦 = 0 𝑑 𝑑𝑡 ( 𝑚𝑦̇) − 0 = 𝐹𝑦 𝑚 ( 𝑑 𝑑𝑡 ( 𝑦̇)) = 𝐹𝑦 𝑚𝑦̈ = 𝐹𝑦 And from equation (3.31), 𝑚𝑦̈ = 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅) (3.42) Evaluating equation (3.35) with equation (3.40), 𝜕𝐿 𝜕𝑧̇ = 𝑚𝑧̇ ; 𝜕𝐿 𝜕𝑧 = 0 𝑑 𝑑𝑡 ( 𝑚𝑧̇) − 0 = 𝐹𝑧 𝑚 ( 𝑑 𝑑𝑡 ( 𝑧̇)) = 𝐹𝑧
  45. 45. 32 𝑚𝑧̈ = 𝐹𝑧 Substituting 𝐹𝑧 from equation (3.31), 𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 (3.43) Evaluating equation (3.37) with equation (3.40), 𝜕𝐿 𝜕𝜑̇ = 𝜑̇ ; 𝜕𝐿 𝜕𝜑 = 0 𝑑 𝑑𝑡 ( 𝜑̇ ) − 0 = 𝜏̃ 𝜑 𝜑̈ = 𝜏̃ 𝜑 (3.44) Evaluating equation (3.38) with equation (3.40), 𝜕𝐿 𝜕𝜃̇ = 𝜃̇ ; 𝜕𝐿 𝜕𝜃 = 0 𝑑 𝑑𝑡 (𝜃̇) − 0 = 𝜏̃ 𝜃 𝜃̈ = 𝜏̃ 𝜃 (3.45) Evaluating equation (3.39) with equation (3.40), 𝜕𝐿 𝜕∅̇ = ∅̇ ; 𝜕𝐿 𝜕∅ = 0 𝑑 𝑑𝑡 (∅̇ ) − 0 = 𝜏̃∅ ∅̈ = 𝜏̃∅ (3.46) Equations (3.41), (3.42), (3.43), (3.44), (3.45), (3.46) give the equations of motions of the quadcopter system.
  46. 46. 33 𝑚𝑥̈ = 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃) 𝑚𝑦̈ = 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅) 𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 𝜑̈ = 𝜏̃ 𝜑 𝜃̈ = 𝜏̃ 𝜃 ∅̈ = 𝜏̃∅ Where; 𝑚 is the mass of the quadcopter 𝑥̈, 𝑦̈, 𝑎𝑛𝑑 𝑧̈ are accelerations in the 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 directions respectively 𝜌 is the main thrust from the bottom of the quadcopter 𝜑, 𝜃, 𝑎𝑛𝑑 ∅ are the yaw, pitch and roll angles respectively 𝜑̈, 𝜃̈, 𝑎𝑛𝑑 ∅̈ are the angular accelerations in the yaw, pitch and roll directions respectively 𝜏̃ 𝜑, 𝜏̃ 𝜃, 𝑎𝑛𝑑 𝜏̃∅ are the yaw moment, pitch moment and roll moment respectively. In this section, the operational frames of the reference of the quadcopter vehicle has been highlighted, the equations of motion of the quadcopter has also being obtained using Lagrangian techniques owing to the assumption that a quadcopter is a physical rigid body.
  47. 47. 34 3.2 Design of the Quadcopter This section covers the design process of the quadcopter system. In designing the quadcopter, the adequate choice of components was made; this choice was made whilst keeping the entire weight of the system in mind. If the weight of the system is heavier than the thrust being generated by the electric motors, lift would not be achieved. The block diagram given in figure 3.2 shows the block diagram of the quadcopter system. Figure 3.2. The Block Diagram of the system From the Bluetooth diagram, a user of the proposed system will connect to the Bluetooth module, the data sent from the user will be evaluated by the Microcontroller receiving the data from the Bluetooth module. If the data received is the expected data, the motors will come off or come on as specified by the data.
  48. 48. 35 3.2.1 The Components This section of this work introduces the various components utilized in the design of the quadcopter. The components are presented as:  The Motors and Propellers  The Microcontroller Unit  The Bluetooth Module  Other Components 3.2.1.1 The Motors and Propellers The motors utilized in this work are coreless Direct Current (DC) motors. The motors are small and compact in design and have low electrical noise. The motors have high acceleration rates up to 48000RPM; and can be controlled seamlessly. Figure 3.2.1.1.a. A coreless DC motor utilized in this work
  49. 49. 36 Figure 3.2.1.1.b. The Inner Windings and casing of a Coreless DC motor The Motors have a girth size of 0.029m, a radius of 0.004m and a diameter of 0.008m. The visible part of the motor shaft measures about 0.00525m. The Propellers are propulsion devices that create relative motion by forcing the fluid in the environment backwards. The propellers are driven by the engines. The electric motors used in this work serve as the driving engine for the propellers. There are basically two types of quadcopter propellers namely: Pusher propellers and Puller (tractor) propellers. The pusher propellers are designed to spin in a clockwise manner. The pusher propeller can be identified when an observer places the pusher propeller on a surface vertically, it will be noticed that the thicker upward part of the upper half of the propeller faces the right. The Pusher propellers utilized in this work are labeled “B” underneath. The puller propellers on the other hand are typically designed to rotate counter- clockwise. They are the normal type of propellers used by tractor air planes. When placed in a vertical manner and looked upon by an observer, it is noticed that the thicker part of the the upper half of the propeller is on the left. The Puller propellers utilized in this work are labeled “A” underneath.
  50. 50. 37 Figure. 3.2.1.1.c. The Pusher Propellers Figure. 3.2.1.1.d. The Puller Propellers. The electric motors are made to rotate in accordance with the propeller specification. This is possible because electric motors can be made to rotate in a clockwise or counter- clockwise manner. The propellers have a diameter of 0.056cm and a radius of 0.028cm.
  51. 51. 38 The acceleration at the tip of the propellers (the centripetal acceleration) is estimated to be about 28281𝑚𝑠−2 , 44460𝑚𝑠−2 , and 162630𝑚𝑠−2 at 9600RPM, 12000RPM and 23000RPM respectively (𝑺𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙; 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝑫) Figure. 3.2.1.1.e. The Propellers and Motors 3.2.1.2 The Microcontroller Unit The microcontroller unit of the quadcopter system in this work is the ATmega328p provided by the Arduino pro mini board. Figure. 3.2.1.2. The Arduino Pro Mini
  52. 52. 39 3.2.1.3 The Bluetooth Module The Bluetooth device used in this work is a HC-05 module. The HC-05 module is an easy to use Bluetooth SPP (Serial Port Protocol) module, designed for transparent wireless serial connection setup. The HC-05 Bluetooth Serial port Bluetooth module is a fully qualified Bluetooth V2.0+EDR (Enhanced Data Rate) 3Mbps Modulation device with complete 2.4GHz radio transceiver and baseband. It uses CSR Bluecore 04-External single chip Bluetooth system with CMOS technology and with AFH (Adaptive Frequency Hopping Feature). Figure 3.2.1.3. The HC-05 Bluetooth module used The HC-05 Bluetooth Module measures 0.0127m X 0.027m. It has six pins for connecting to it; a VCC pin, a GND pin, a receive (RXD) pin, a transmit (TXD) pin, an enable pin and a state pin. The HC-05 Bluetooth module communicates using Attention (AT) commands. AT commands are commands used for communicating and programming machine-to- machine (M2M) modems.
  53. 53. 40 3.2.1.4 The Power Source The main power source in this work is a 3.7V rechargeable 25C GSP 90540 Lithium Polymer (LiPo) battery with a capacity of 750mAh. Lithium is an alkali metal capable of reacting with water and combustion. The Lithium Polymer battery uses a polymer as its electrolyte. The Lithium Polymer battery provides power for the Microcontroller Unit, the Bluetooth module, the electric motor drivers as well as the electric motors in this work. The use of a Lithium Polymer battery instead of a Nickel-Metal Hydride (NiMH) or a Nickel-Cadmium (NiCd) battery in this work is because of the following reasons: 1. Lithium Polymer (LiPo) are far lighter in terms of weight compared to Nickel- Cadmium batteries, Nickel-Metal Hydride batteries or any other battery currently available. 2. Lithium Polymer batteries generally hold more power than other batteries i.e. Lithium Polymer batteries have greater Capacities compared to other batteries. 3. Lithium Polymer batteries generally offer more Discharge rates compared to other batteries. For these reasons, a Lithium Polymer battery is the preferred choice for a power source for the construction of a quadcopter. But, even with these reasons, the Lithium Polymer battery is disadvantageous because Lithium Polymer batteries generally have shorter life span compared to other options of power source in addition to the fact that Lithium Polymer batteries are capable of exploding and catching fire.
  54. 54. 41 Figure. 3.2.1.4.a. The Lithium Polymer Battery Used. The use of a load of 0.75A would drain the battery given by Fig. 3.2.1.4 completely in one hour. In a likewise manner, the battery given by Fig. 3.2.1.4 would be completely charged in about an hour if it were charged at 0.75A. Furthermore, the battery given by Fig. 3.2.1.4. can by all calculations (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑪) sustain a maximum load of 18.75A. Any load higher than 18.75A would result in the battery becoming unstable and cause the battery to potentially burst into flames (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑬). In order to charge the Lithium Polymer battery dedicated to this work, a charger was supplied. The charger module utilized in this work is based on the TP4056 Integrated Circuit (IC) Surface Mount Device (SMD) chip. Figure 3.2.1.4.b. The Lithium Polymer Charger Module
  55. 55. 42 3.2.1.5 Other Components Other components utilized in this work include; a small single pole double throw (SPDT) switch, four N-Channel Metallic-Oxide Semiconductor Field Effect Transistors (MOSFETs); specifically, IRFZ34N transistors of TO-220 package, a DC-DC converter (~3V – 5V), Arduino Uno development board and jumper wires. The function of the switch was to cut off or allow the flow of current in the circuitry of the quadcopter system whenever necessary. The MOSFETs were utilized in this work as the motor drivers. The MOSFETs were used in place of the TB6612FNG motor driver. The IRFZ34N was used because of the following reasons:  The IRFZ34N has a gate-to-source voltage (VGS) of ±20V.  The IRFZ34N supports fast switching; hence, it is possible to employ Pulse Width Modulation (PWM).  The IRFZ34N can supply current of up to 29A and the proposed quadcopter system does not require more than about 3A.  The IRFZ34N is reasonably light in weight. An alternate to the IRFZ34N is the IRFZ44N MOSFET. The jumper wires used were employed for making connections in the circuitry of the quadcopter system. The Arduino Uno development board was used in testing components utilized in this work. The board was also used for programming the ATmega328 of the Arduino Pro Mini.
  56. 56. 43 3.2.2 The Airframe The airframe is the body of any quadcopter or similar vehicle; housing all the components of the system. Figure. 3.2.2.a. The design of the airframe The airframe utilized in this work was fabricated from thin plastic material. The plastic was cut to form a square of 0.047m x 0.047m. Rectangular sheets of thin plastic was also cut. These sheets serve as the arm of the central square part of the airframe connected to it by means of glue.
  57. 57. 44 Figure 3.2.2.b. The Fabricated airframe for the system The airframe fabricated is very light weighing only about 1.2grams; when weighed on a beam balance. 3.2.3 Mounting the Electric Motors The electric motors were mounted to the airframe; this was done by making holes in each arm of the fabricated airframe. Each of the four motors were then placed in each arm of the airframe and glued to it. The motors spinning in the clockwise direction (and bearing the pusher propellers) were placed in alternate adjacent positions. Likewise, for the motors spinning on counter clockwise direction (and bearing the puller propellers). It is essential that the electric motors and propellers are position in this manner. This is because, each rotor (electric motor and propeller) produces a torque that has an effect on the airframe and tends to rotate the airframe in the direction of that rotor. If all the rotors of the quadcopter rotate in the same direction, the overall torque would make the whole system move in haphazardly in the direction of the rotors. But placing the clockwise rotors in the adjacent direction and the counter-clockwise rotors in adjacent
  58. 58. 45 direction all running at the same average speed will cause the torque on each rotor to cancel out; pushing all air molecules downward thus allowing the quadcopter system to hover. Figure. 3.2.3. The Mounted Electric Motors. 3.2.4 The Requirements & Procedures Carried Out This section entails gives the requirements necessary to design (construct) the quadcopter system and proceeds to spell the procedures carried out in building the quadcopter vehicle in this work. The requirements are foremost given in turn: 1. Computers A computer (PC or workstation) with at least 2GB or Random Access Memory (RAM) and processor clock speed of at least 1.41GHz speed is required for carrying out this work. The computer used in this work is a Hewlett Packard (HP) Pavilion laptop with 2.4GHz of processor speed running Linux Mint 17.3 (Rosa) Operating System. A mobile phone running an android operating system
  59. 59. 46 is another computer required in this work. The mobile phone is required for the testing and usage of the mobile application consequently developed in latter sections of this work. The mobile phone used in this work is a blackberry z10. Although, not running an android operating system is capable of the emulation of android applications. 2. The Arduino Integrated Development Environment The Arduino Integrated Development Environment (IDE) is a software provided by the Arduino platform for programming the microcontrollers on which each Arduino board is based upon. The Arduino IDE is installed on the computer used and used to program the ATmega328 microcontroller of the Arduino Pro Mini. 3. Fritzing Software The Fritzing software is a hardware simulation and schematic diagram representation software. The Fritzing software is installed on the computer utilized in this work and all the schematic diagrams drawn for this work was done using the Fritzing software. 4. The Components The components have been given by section 3.2.1 of this work. 5. C/C++, Arduino Programming Knowledge of C/C++ or Arduino or both is highly essential for the design of the quadcopter system if it is to be built upon the Arduino platform. The procedures for building the system was carried out in phases. The phases are given below:  Preparing the Arduino Pro Mini  Preparing the Bluetooth Module  Designing the Mobile Application  Testing the Application with a motor  Building the whole system The phases are elaborated upon in turn:
  60. 60. 47  Preparing the Arduino Pro Mini 1. Header Pins and Jumper wires are soldered unto the Arduino Pro Mini 2. The Arduino Pro Mini was connected to the Arduino Uno 3. The Arduino Uno was used to program the Arduino Pro Mini to make an LED blink and fade using PWM; this is to make sure the Arduino Pro Mini is capable of driving the electric motors using the transistors.  Preparing the Bluetooth Module 1. The HC-05 Bluetooth module was connected to the Arduino Uno. 2. The transmit pin (TXD) of Bluetooth module was connected to pin 10 of the Arduino Uno. 3. The Receive (RXD) pin of the Bluetooth module was connected to pin 11 of the Arduino Uno. 4. Pins 10 and 11 were declared in the Arduino IDE as RXD and TXD pins using the software Serial library; while writing a communication code for the Bluetooth module (see appendix). The code was uploaded to the Arduino board. 5. The P1011 Pin of the HC-05 Bluetooth module was connected to HIGH (5V of the Arduino) and the button of the module pressed down before powering up the module; this was to put the Bluetooth module in the state for communicating via AT commands. 6. The Bluetooth module was powered up. 7. Once powered up, the module blinked its red led at a steady one second rate; indicating it was in the right state for receiving AT commands. 8. The serial monitor of the Arduino IDE was opened. 9. The command, “AT” was sent via the serial monitor; the response was “OK”; indicating the Bluetooth module is connected and functional. 10.The command, “AT+NAME=PhysicsI”, the response was, “OK”; indicating the name of the Bluetooth Module had been successfully changed to PhysicsI. 11.The command, “AT+PSWD?” was typed in, the response was, “1234”; indicating that the password to the Bluetooth module is “1234”.
  61. 61. 48 12.The command, “AT+PSWD=9090” was then typed into the serial monitor. The response was, “OK”; indicating that the password to the Bluetooth module has been successfully changed to “9090”. 13.The command, “AT+ROLE” was then typed into the serial monitor, the response was, “1”; indicating that the Bluetooth module was in Master modem. 14.The command, “AT+ROLE = 0” was then typed into the serial monitor, the response was, “OK”; indicating that the role of the Bluetooth module had been changed to Slave mode. 15.The command, “AT+ADDR” was then typed into the serial monitor and the response was, “98d3:31:fd04ef”; indicating that the physical address of the of the Bluetooth module is: “98:D3:31:FD:04:EF”. 16.The details of the Bluetooth Module were noted down.  Designing the Mobile Application 1. The MIT App inventor site was opened. 2. A new project was opened and named, “PhysicsI”. 3. The layout of the application was done. 4. The programming blocks of the application was then developed (see appendix for the pseudocode of the application). 5. The application was then built into an apk file and installed on a test mobile phone.  Testing the Application with a motor 1. The gate of an IRFZ34N MOSFET was connected to a PWM pin (pin 9) of the Arduino Uno through a 100 ohm resistor. 2. The source of the MOSFET was connected to the GND of the 3.7V LiPo Battery. 3. A terminal of the electric motor was connected to the drain of the MOSFET. 4. The GND pin of the Arduino Uno was connected to the GND of the LiPo Battery. 5. A 1N4001 diode was employed as a fly back diode to prevent back EMF (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑭). The anode of the diode was connected to the drain of
  62. 62. 49 the MOSFET and the cathode of the diode connected to the positive terminal of the LiPo battery. The source code (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑮) was uploaded to the MCU. 6. Whenever the “hover” button in the application was clicked, the electric motor ran; the motor also stopped whenever the “land” button was clicked. Figure 3.2.4.a. gives the physical connections carried out. In the Figure 3.2.4.a., a LiPo battery of 1A capacity was used and the Arduino Pro Mini was used even though the actual connections were done using Arduino Uno. Figure 3.2.4.a. The test of the Application using one electric motor.
  63. 63. 50 Figure. 3.2.4.b. The Schematic representation of the one motor test.  Building the whole system 1. The airframe was fabricated as elaborated upon in section 3.2.2. 2. The electric motors were mounted as given in section 3.2.3. 3. Similar connections were done for three other electric motors as in testing only one electric motor. The gate of the three other motors were connected to PWM enabled pins of the Arduino Pro Mini. 4. A small Vero board was used for connections. 5. The Bluetooth module was connected to the Arduino Pro Mini. 6. The switch was connected to the terminals of the battery. 7. The source code (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑯) was uploaded to the Arduino Pro Mini. 8. All the components were attached to the airframe by means of glue via a glue gun.
  64. 64. 51 Figure. 3.2.4.c The Circuit diagram of the system 3.3 Interfacing with the Quadcopter In controlling the quadcopter system, an android mobile application was developed using the Massachusetts Institute of Technology (MIT) app inventor. The application is meant to connect an android mobile device to the quadcopter. Once connected, the two buttons responsible for the hovering and landing of the quadcopter are visible. The application was uploaded to a Google drive and can be downloaded from: https://drive.google.com/open?id=0B5lAV8ql0xu4QjU0dHpUUjVpRTA. The name of the application is: PhysicsI. The two buttons responsible for the hovering and landing of the quadcopter vehicle are labelled Hover and Land in the mobile application. The characters; “a” and “b” were mapped in the source code of the application to the Hover and Land buttons respectively. In order to use the application, once, the quadcopter is turned on by means of the switch, the Bluetooth of the android phone being used should be turned on. The user would
  65. 65. 52 have to navigate to settings; then to Bluetooth and then scan and pair with a new device. The PhysicsI Bluetooth will be available in the list of available devices, once clicked and prompted for a password, the user will have to enter the password; 9090. Once pairing is complete, the user can open the PhysicsI application and then click on the “select Bluetooth device” button. The PhysicsI Bluetooth device together with its address will be displayed in a list, the user will then have to select the PhysicsI device. Whenever the Hover Button of the Application is clicked from an android phone connected to the quadcopter, the android phone sends the character “a” to the quadcopter system. Once the character, “a” is received by the Atmega328 microcontroller unit (MCU) through the HC-05 Bluetooth module, all the rotors of the quadcopter system is turned on through the motor drivers interfaced to the ATmega328 MCU. In a similar manner, when the Land Button is clicked, the character, “b” is sent to the quadcopter system, once received by the ATmega 328 MCU, the rotors are made to stop spinning gradually by means of pulse width modulation (PWM) until, it finally stops spinning. Whenever the Disconnect button is clicked on, the application will disconnect the Bluetooth communication from the quadcopter system. 3.4 Review of Methodology In this chapter, the quadcopter system has been modelled as a rigid and the Lagrangian formalism has been used to obtain the equations of motion for a quadcopter system. The chapter also covers the processes carried on in designing, as well as programming a quadcopter system.
  66. 66. 53 CHAPTER FOUR PRESENTATION OF RESULTS This chapter entails the presentation of the results obtained in this work. The chapter is presented under the following sections:  The Quadcopter  Limitations of the work  Summary of the Results 4.1 The Quadcopter In this work, the quadcopter system was introduced as a model for UAVs, the quadcopter system was also model as a rigid body whose dynamics was given in the work using the Lagrangian technique; after which the equations of motion were obtained to be: 𝑚𝑥̈ = 𝜌(sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃) 𝑚𝑦̈ = 𝜌(cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅) 𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 𝜑̈ = 𝜏̃ 𝜑 𝜃̈ = 𝜏̃ 𝜃 ∅̈ = 𝜏̃∅ Where; 𝑚 is the mass of the quadcopter 𝑥̈, 𝑦̈, 𝑎𝑛𝑑 𝑧̈ are accelerations in the 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 directions respectively 𝜌 is the main thrust from the bottom of the quadcopter 𝜑, 𝜃, 𝑎𝑛𝑑 ∅ are the yaw, pitch and roll angles respectively
  67. 67. 54 𝜑̈, 𝜃̈, 𝑎𝑛𝑑 ∅̈ are the angular accelerations in the yaw, pitch and roll directions respectively 𝜏̃ 𝜑, 𝜏̃ 𝜃, 𝑎𝑛𝑑 𝜏̃∅ are the yaw, pitch and roll moments respectively The work then proceeded to the construction of a model. The model constructed is only capable of hovering in mid-air and landing. The control of the hovering and landing process of the constructed model is made possible through an android mobile application that employs Bluetooth communication technology. Figure 4.1.a. A section of the android application. A TP4056 charger module enclosed a small circular box is provided for charging the LiPo battery of the constructed quadcopter model. The charger provided uses a Micro-B USB connector for charging. It takes about one hour, thirty minutes for the LiPo battery to complete charging if depleted. When plugged without the battery connected, the charger displays a green light and when the battery is connected to the charger, the charger displays red light when charging and displays green light when fully charged. The LiPo battery used in this work must never be discharged below 3.0V.
  68. 68. 55 Figure. 4.1.b. The Quadcopter Figure. 4.1.c. The LiPo Battery Charger provided in this work
  69. 69. 56 4.2 Limitations of the Work This work serves the purpose of modelling and constructing a basic quadcopter system. The limitations of this work are therefore given in this section. The limitations are given below: 1. The quadcopter constructed in this work is only capable of hovering and landing. 2. The use of Euler angles in modelling the quadcopter system as a rigid is disadvantageous because, singularities tend to occur in changing coordinate matrices. 3. The quadcopter constructed therein this work cannot be controlled 100metres away from it; hence, it cannot hover beyond 100meters. 4. The quadcopter loses balance when landing. 5. The quadcopter system uses a LiPo battery; a potentially dangerous power source capable of causing a class C fire. 6. The Quadcopter can only hover for about seven minutes and has a charging time of about one hour, thirty minutes. 7. The components utilized are not readily available in the country. 4.3 Summary of the Results The Equations of the motion of a quadcopter system was successfully obtained using the Lagrangian technique. The properties of the constructed quadcopter model as well as its charger is given by Table 4.1. Table 4.1. Properties of the quadcopter and its Charger System Part Flight Time (𝑠𝑒𝑐𝑠) Charge Time (𝑠𝑒𝑐𝑠) Maximum Voltage Drawn(𝑉) Maximum Current Delivered(𝐴) Maximum Current Drawn(𝐴) Size (𝑚2 ) Weight (𝑘𝑔) Quadcopter 420 5400 5 - 5.35 0.92 0.07 8 Charger - 5400 - 0.75 - 0.0011 Battery 420 5400 - 18.75 - 0.088 0.02 the total weight of the quadcopter system is 0.098kg and the system can hover for 7minutes.
  70. 70. 57 CHAPTER FIVE CONCLUSION, DISCUSSION & RECOMMENDATIONS In this chapter, the discussion of the results, the summary and conclusion of the work as well as recommendations have been presented in this chapter. 5.1 Discussion of the Results This work differs from that of (Luukkonen, 2011), (Balas, 2007), and (Nabil, 2014) as their works focused more on the modelling and control of the quadcopter system. The work is also different from that of (Carillo et al 2013) as their work is primary on the navigation and control of a quadcopter system. This work is primarily on the modelling of a quadcopter system and the construction of a basic model. In this section, the following inquiries are made: Question One: Why is the Quadcopter system in this work only capable of and Landing? The quadcopter is in work is only capable of hovering and landing because, the quadcopter constructed in this work is intended to be a basic model to illustrate the working principle of a quadcopter system. Furthermore, the quadcopter in this work is capable of only hovering and landing because, the quadcopter here does not employ the use of accelerometers, gyroscopes, tilt sensor and other sensors required to balance the quadcopter while performing all various flight manoeuvre patterns of as shown in Figure 1.2.2. Question Two: The Quadcopter system in this work has a flight time of seven minutes and a charging time of over one hour; why such high discrepancy? The total power consumption of the components of the quadcopter in this work is far higher than the capacity of the battery. Also, the battery charges at a slow rate because of its “tricky” sensitive chemical nature.
  71. 71. 58 Question Three: Why does this work utilize a battery that is capable of starting a class C fire? The LiPo battery was utilized in this work because of its relatively light weight and high capacity as compared to other batteries. The battery can start the fire due to its very nature. (𝒔𝒆𝒆 𝒂𝒑𝒑𝒆𝒏𝒅𝒊𝒙 𝑬). Furthermore, the battery can only start the fire if not properly handled. Question Four: Why can the Quadcopter constructed in this work not be controlled from a distance farther than 100metres? This is so because, the Bluetooth Module employed in this work works on Bluetooth V2.0 + EDR and has a transmission range of about 100meters. 5.2 Conclusion Based on the results of this work, the following conclusions were made that: Quadcopters can be modelled as rigid bodies (incapable of deformation) and their dynamics can be evaluated using various physical techniques; A simple and small quadcopter model can be constructed and controlled using various forms of communication such as Bluetooth owing to the existence of small sized microcontrollers and even more conveniently, small sized development boards such as the Arduino Pro mini that employ these small sized microcontrollers. 5.3 Recommendations The results, and limitations of this work serve as the basis for proffering the following recommendations: 1. Further Studies should be carried be carried out in UAV systems such as multirotor vehicles like the Quadcopter, Octocopter, Y6, etc. 2. Cutting edge research on applications should be carried out in the branch of solid state physics: semi-conductor physics.
  72. 72. 59 3. In latter works, accelerometers, gyroscopes and other sensors should be incorporated in the construction of quadcopter as well as other UAV model prototypes. 5.4 Suggestions for Further Study The following topics have been suggested for investigation in order to carry out further studies on Quadcopters and Unmanned Aerial vehicles. 1. Applications of semiconductor devices for Unmanned Aerial Vehicles. 2. Modelling the Quadcopter system with Quaternions. 3. Applications of Unmanned Aerial Vehicles. 4. Autonomous Flight of Unmanned Aerial Vehicles. 5. Effects of aerodynamic forces on aerial vehicles. 6. The Dynamics of Propellers. 7. Advanced modelling for Robotics. 8. The Nigerian aerial space; properties, range and laws 9. Applications of Unmanned aerial vehicles in Nigeria. 10.Solar Powered Flying Robots. 11.Vertical Take-off and Landing (VTOL). 12.Hybrid aerial Vehicles.
  73. 73. 60 References Anyakoha. M.W. (2010). New School Physics for Senior Secondary Schools (3rd edition). [paperback copy]. P. 125-160. Africana First Publishers Limited, Enugu. Baichtal J. (2015). Building your own drones: A beginner’s guide to Drones, UAVs and ROVs. [pdf version]. Que Publishing. Balas C. (2007). Modelling and Linear Control of a Quadrotor. [pdf version]. M.Sc Thesis, School of Engineering, Cranfield University Bluetooth Special Interest Group. (2017). What is Bluetooth? Retrieved from: https://www.bluetooth.com/what-is-bluetooth-technology/ Bostrom, A. (2012). Rigid Body Dynamics. [pdf version]. Retrieved from: http://www.am.chalmers.se/~paja/RBD/Handouts/Compendium.pdf Burke R. (2005). Principles of Flight. [pdf version]. Retrieved from: https://www.google.com.ng/url?q=http://archives.sparkflows.io/principles_of_fl ight_for_pilots_aerospace_series.pdf&sa=U&ved=0ahUKEwjn7f284pfXAhVF BMAKHb5JD44QFggUMAM&usg=AOvVaw1K0JeHujwVRKa3so4pPcOW Carillo L.G.R, L.A (n.d.). (2013). Quad rotorcraft control, vision based hovering and navigation. [pdf version]. Retrieved from http://www.springer.com/978-1-4471-4398-7 Chun Y.C. (2011). Quadcopters. (M.Sc. Thesis). Retrieved from: http://www4.ncsu.edu/~kksivara/sfwr4c03/projects/4c03projects/CYChan- Project.pdf Deyst E. (2003). Lagrange’s Equations. Lecture Notes [pdf version]. Massachusetts Institute of Technology. Goldstein H, Poole C & Safko J. L. (2001). Classical Mechanics (3rd edition). [pdf version]. Addison-Wesley. Retrieved from: http://bookzz.org/book/2373839/gfgdhee
  74. 74. 61 Hintenaus P. (2015). Engineering Embedded Systems; Physics, Programs, Circuits. [pdf version]. Springer International Publishing, Switzerland ITEAD. (2017). Serial Port Bluetooth Module (Slave/Master): HC-05 Retrieved from https://www.itead.cc/wiki/Serial_Port_Bluetooth_Module_(Master/Slave)_:_ HC-05 Kilby, T & Kilby, B. (2016). Make: getting started with drones. [pdf version]. Retrieved from http://bookzz.org/book/2610493/fe7cce Luukkonen T. (2011). Modelling and Control of Quadcopter. [pdf version]. Independent Research Project in Applied Mathematics, Aalto University, School of Science Mathieu S. (2008). Bluetooth Remote Control. Semester Project. [pdf version]. Retrieved from: http://www.limpkin.fr/public/BTP_Final_Report.pdf McDermott-Wells P. (2004). Digital Vision (1st edition). Retrieved from https://projetos.inf.ufsc.br/arquivos_projetos/projeto_555/01368913.pdf National Aeronautics and Space Administration. (2011). Four Forces: Principles of Flight k-4 Series Lesson. [pdf version]. Nabil R. (2014). Design and Control of Quadcrotors. [pdf version]. University of Science and Technology, Bangladesh National Aeronautics and Space Administration. (2015). Four Forces: Principles of Flight. [pdf version]. Nenuwe N. (2017). Lagrange’s equations. Lecture Notes. [paperback copy]. Federal University of Petroleum Resources, Effurun Newton I. (1687). Mathematical Principles of Natural Philosophy (1st edition). [pdf version]. Penta J (2016). Getting Started with MIT app Inventor. (1st edition). [pdf version]. Pillai S.O. (2015). Solid State Physics (7th edition). [paperback copy]. P.521-527, 828-862. New Age International Publishers.
  75. 75. 62 Schneider B. (2015). A Guide to LiPo Batteries. [pdf version]. Retrieved from: https://rogershobbycenter.com/lipoguide/ Shrestha R. (2015). Study and Control of Bluetooth Module HC-05 using Arduino Uno. [pdf version]. Department of Physics, Tribhuvan University, Nepal. Teppo L. (2011). Modelling and control of quadcopter. (M.Sc. Thesis). [pdf version]. School of Science. Independent research project in applied mathematics Walter Derek, & Sherman Mark. (2015). Learning MIT App Inventor: A Hands-on Guide to Building Your Own Android Apps (1st edition). [pdf version]. Addison-Wesley. Wingo B. (2016). Rose-Hulman Undergraduate Research Publications Mathematical modelling of quadcopter dynamics. [pdf version].11. Retrieved from http://scholar.rose-hulman.edu/udergrad_research_pubs/11 Uzoagulu, A.E (2011). Practical Guide to Writing Research Project Reports in Tertiary Institutions (2nd edition). [paperback copy]. Cheston LTD. Young H & Freedman R. (2006). University Physics (12th edition). [paperback copy]. P 53 -70, P.205-401. Pearson, Addison Wesley.
  76. 76. A- 1 - Appendices Appendix A: The Inertia Matrix Moments of inertia describe the resistance of a body to rotation about an axis. It is analogous to mass to resists translation. The moments of inertia for a rigid body with respect to the Cartesian coordinates system is defined by: 𝐼 𝒙𝒙 = ∫( 𝑦2 + 𝑧2) 𝑑𝑚 𝐼 𝒚𝒚 = ∫( 𝑥2 + 𝑧2) 𝑑𝑚 𝐼𝒛𝒛 = ∫( 𝑥2 + 𝑦2) 𝑑𝑚 The integration is over the whole body. Similarly, the products of inertia is defined by: 𝐼 𝒙𝒚 = ∫ 𝑥𝑦 𝑑𝑚 𝐼 𝒙𝒛 = ∫ 𝑥𝑧 𝑑𝑚 𝐼 𝒚𝒛 = ∫ 𝑦𝑧 𝑑𝑚 In discussing the rotations of a coordinate system, the inertia matrix (also known as the inertia tensor) is introduced. 𝐼 = ( 𝐼𝑥𝑥 −𝐼𝑥𝑦 −𝐼𝑥𝑧 −𝐼 𝑦𝑥 𝐼 𝑦𝑦 −𝐼 𝑦𝑧 −𝐼𝑥𝑧 −𝐼 𝑦𝑧 𝐼𝑧𝑧 ) The inertia matrix is a real symmetric matrix and thus can be transformed to a diagonal form in another coordinates. In principle, 𝐼 = ( 𝐼𝑥𝑥 0 0 0 𝐼 𝑦𝑦 0 0 0 𝐼𝑧𝑧 )
  77. 77. A- 2 - Appendix B: The Rotational Matrix The two dimensional rotational matrix is defined as: 𝑹∅ = ( cos ∅ −sin ∅ sin ∅ cos ∅ ) the above matrix is the so called rotation matrix, which is nonlinear mapping that rotates any vector from the inertial frame to the body frame and vice versa, the yaw angle ∅. The mapping 𝑹∅ is an orthonormal matrix i.e. 𝑹∅ −1 = 𝑹∅ 𝑇 𝒅𝒆𝒕 𝑹∅ = 1 The rotation about 𝑎3, the roll axis is; 𝑹∅ = ( 1 0 0 0 cos ∅ sin ∅ 0 − sin ∅ cos ∅ )
  78. 78. A- 3 - The rotation about 𝑎2, the pitch axis is; 𝑹 𝜃 = ( cos 𝜃 0 − sin 𝜃 0 1 0 sin 𝜃 0 cos ∅ ) The rotation about 𝑎1, the yaw axis is; 𝑹 𝜑 = ( cos 𝜑 sin 𝜑 0 − sin 𝜑 cos 𝜑 0 0 0 1 ) Hence, the rotation is given by; 𝑹 = 𝑹∅ 𝑹 𝜃 𝑹 𝜑 𝑹 = ( cos 𝜃 cos 𝜑 cos 𝜑 sin 𝜃 sin ∅ − cos∅ sin 𝜑 sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃 cos 𝜃 sin 𝜑 cos ∅ cos 𝜑 + sin 𝜃 sin ∅ sin 𝜑 cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅ − sin 𝜃 cos 𝜃 sin ∅ cos 𝜃 cos ∅ ) Where 𝑹 is the full rotation and 𝑹 is also an orthonormal matrix i.e. 𝑹−1 = 𝑹 𝑇 𝒅𝒆𝒕 𝑹 = 1 Appendix C: The Capacity of the Lithium Polymer Battery The Lithium Polymer battery utilized is rated 25C. This 25C rating is the discharge rating of the battery where C = Capacity of the Battery in Amps (A). The battery has a Capacity of 0.75A (750mAh). Hence, the discharge rating is: 25 ∗ 𝐶 = 25 ∗ 0.75 (𝐴) = 18.75𝐴 ∴ 18.75A is the maximum load the battery can sustain.
  79. 79. A- 4 - Appendix D: The Centripetal acceleration (the acceleration at the tip of the propellers) The propellers have a radius of 2.8cm. From the datasheet of the coreless DC motor, it can be obtained that the motors can spin up to 48000 times per minute. For the obtaining the centripetal acceleration of the propeller (s) in this work, the cases: of 9600, 12000 and 23000 RPM are used. These cases are used because of the unavailability of a tachometer, and making the assumption that the propeller (s) have a forward airspeed of 25𝑚𝑠−1 with the speed of the tips of the propellers through the air not exceeding 270𝑚𝑠−1 . If the propellers move too close to the speed of sound, tremendous amount of noise will be made. The centripetal acceleration can be obtained by evaluating, 𝑎 𝑟𝑎𝑑 = 𝜔2 𝑟
  80. 80. A- 5 - Where; 𝜔 = the angular acceleration of the propeller 𝑟 = the radius of the propeller For 9600 RPM, Converting RPM to rad/sec……… = ( 9600 𝑟𝑒𝑣 𝑚𝑖𝑛 ) ( 2𝜋 𝑟𝑎𝑑 1 𝑟𝑒𝑣 ) ( 1 𝑚𝑖𝑛 60 𝑠𝑒𝑐𝑠 ) = 1005.309649 𝑟𝑎𝑑 𝑠⁄ ≈ 1005 𝑟𝑎𝑑 𝑠⁄ Then, 𝑎 𝑟𝑎𝑑 = (1005)2 ∗ (0.028) = 28280.7 𝑚𝑠−2 𝑎 𝑟𝑎𝑑 ≅ 28281𝑚𝑠−2 𝑎 𝑟𝑎𝑑 ≈ 2.8281 ∗ 104 𝑚𝑠−2 For 12000 RPM, Converting RPM to rad/sec……… = ( 12000 𝑟𝑒𝑣 𝑚𝑖𝑛 ) ( 2𝜋 𝑟𝑎𝑑 1 𝑟𝑒𝑣 ) ( 1 𝑚𝑖𝑛 60 𝑠𝑒𝑐𝑠 ) = 1256.637061 𝑟𝑎𝑑 𝑠⁄ ≈ 1260 𝑟𝑎𝑑 𝑠⁄ Then, 𝑎 𝑟𝑎𝑑 = (1260)2 ∗ (0.028) = 44452.8 𝑚𝑠−2 𝑎 𝑟𝑎𝑑 ≅ 44460 𝑚𝑠−2 𝑎 𝑟𝑎𝑑 ≈ 4.4460 ∗ 104 𝑚𝑠−2
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The Dynamics of a quadcopter using the Lagrangian Technique and the design of a small model

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