Dynamics, Modelling & Design of a Quadcopter

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

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

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:

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,
…………………………………………
…………………………………………

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 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.

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

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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

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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

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.

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

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

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.

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

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

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

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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

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/ .

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.

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.

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.

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).

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).

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.

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

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

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
)

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
) (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.

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 ,

27
𝐼Ω =
(
(
𝐼𝑥𝑥 0 0
0 𝐼 𝑦𝑦 0
0 0 𝐼𝑧𝑧
) (
∅̇ − 𝜑̇ sin 𝜃
)
)
(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.

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)

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)

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
)

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)
𝜕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)
𝜕𝐿
𝜕𝑧̇
= 𝑚𝑧̇ ;
𝜕𝐿
𝜕𝑧
= 0
𝑑
𝑑𝑡
( 𝑚𝑧̇) − 0 = 𝐹𝑧
𝑚 (
𝑑
𝑑𝑡
( 𝑧̇)) = 𝐹𝑧

32
𝑚𝑧̈ = 𝐹𝑧
Substituting 𝐹𝑧 from equation (3.31),
𝑚𝑧̈ = 𝜌(cos 𝜃 cos ∅) − 𝑚𝑔 (3.43)
𝜕𝐿
𝜕𝜑̇
= 𝜑̇ ;
𝜕𝐿
𝜕𝜑
= 0
𝑑
𝑑𝑡
( 𝜑̇ ) − 0 = 𝜏̃ 𝜑
𝜑̈ = 𝜏̃ 𝜑 (3.44)
𝜕𝐿
𝜕𝜃̇ = 𝜃̇ ;
𝜕𝐿
𝜕𝜃
= 0
𝑑
𝑑𝑡
(𝜃̇) − 0 = 𝜏̃ 𝜃
𝜃̈ = 𝜏̃ 𝜃 (3.45)
𝜕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.

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.

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.

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

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.

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.

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

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.

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.

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

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.

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.

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

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

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:

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”.

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

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.

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.

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

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.

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

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.

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 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.

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.

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.

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.

60
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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
)

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 ∅
)

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.

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

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,
= (
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

A- 6 -
For 23000 RPM,
= (
23000 𝑟𝑒𝑣
𝑚𝑖𝑛
) (
2𝜋 𝑟𝑎𝑑
1 𝑟𝑒𝑣
) (
1 𝑚𝑖𝑛
60 𝑠𝑒𝑐𝑠
)
= 2408.554368 𝑟𝑎𝑑 𝑠⁄
≈ 2410 𝑟𝑎𝑑 𝑠⁄
Then,
𝑎 𝑟𝑎𝑑 = (2410)2
∗ (0.028)
= 162626.8 𝑚𝑠−2
𝑎 𝑟𝑎𝑑 ≅ 162630𝑚𝑠−2
𝑎 𝑟𝑎𝑑 ≈ 1.62630 ∗ 105
𝑚𝑠−2
∴ The acceleration at the tip of the propellers (the centripetal acceleration) is:
28281𝑚𝑠−2
, 44460𝑚𝑠−2
, and 162630𝑚𝑠−2
at 9600RPM, 12000RPM and 23000RPM
respectively.
Appendix E:
Potential Fire Case due to the usage of a LiPo Battery
When a LiPo battery becomes unstable; it can burst into flames. The battery becomes
unstable when the it overused (in the case of the battery used in this work; when a load
of 18.75A or greater is placed on it). This over usage causes excess atoms of oxygen
and lithium to build up at the cathode and anode of the of the battery; this build up results
to the formation of Lithium Oxide (Li2O). This build-up of Li2O causes the internal
resistance of the battery to increase. As the internal resistance of the battery increases,
the operating temperature of the battery increases as well. As the oxygen and the Li2O
continues to build up and the operating temperature of the battery continues to increase,
the battery begins to swell until it bursts open from the pressure thus triggering a reaction
between the oxygen, the air, heat and the lithium of the battery and thus causing a very
dangerous fire. The same process occurs when the battery is punched or poorly handled.

A- 7 -
The same process occurs when the battery starts reaching the end of its life span. The
typical lifespan of a LiPo battery is 300-400 charge/discharge cycles. The lifespan of
the battery used in this work is 500 charge/discharge cycles.
The fire caused by a LiPo battery is a chemical fire i.e. a Class C fire; which is very
dangerous. Whenever charging the battery in this work, the user must be around in case
something happens.
Appendix F:
Back EMF
Electric Motors have coils turning in a magnetic field. As the coils cut the magnetic
flux, EMF is induced; this induced EMF is known as back EMF and it acts in opposition
to the applied voltage that cause the electric motor(s) to spin. The Back EMF usually
becomes noticeable when the electric motor(s) stops spinning.
When the electric motors(s) stops spinning, the back EMF produced is dangerous to the
rest of the circuit to which the electric motor(s) are(is) connected. Hence, there is the
need to protect the rest of the circuitry. This “protection” is usually done by employing
what is known as flyback diode. The cathode of the diode is connected to the positive
terminal of the power source and the anode of the diode is connected to terminal chosen
to be the negative terminal of the electric motor(s).
Flyback diodes are also known as snubber diode, commutating diode, catch diode, clamp
diode or suppression diode. The diode used as the flyback diode in this work is the
1N4001 diode.

A- 8 -
Appendix G:
The Source for the single motor test
/*this code was written by Stephen Ahiante*/
char data; //declaring a variable; data to hold a character
int motor = 9; //declaring a variable; motor and storing pin 9 in
int i; //declaring a variable; i
void setup() //a function that runs once
{
Serial.begin(38400); //starting serial communication at a baud rate of 38400
pinMode(motor, OUTPUT); //setting pin 9 as an output pin
}
void loop() //a functions that runs repeatedly
{
if (Serial.available() > 0) //checking for any form of serial communication
{
data = Serial.read(); //reading data from serial communication and storing in a
// variable; data
if (data == 'a') //checking if the data received is the character; a
{
Hover(); // calling the Hover function
}
else if (data == 'b') //checking if the data received is the character; b
{
Land(); //calling the land function
}
}
}

A- 9 -
void Hover() //defining a function; Hover
{
if (i == 0) //checking if i is at 0 level
{
for (i = 0; i < 255; i++) //adding a duty level to i till it reaches 255
{
analogWrite(motor, i); //turning the motor on according to the duty level
delay(30); //waiting for 30 milliseconds after each addition
}
}
}
void Land() //defining a function; Land
{
if (i == 255) //checking if i is at the highest level
{
for (i = 255; i > 0; i--) //subtracting a level from i till it reaches 0
{
analogWrite(motor, i); //spinning the motor according to the current level
delay(30); //waiting for 30 milliseconds after each subtraction
}
}
}
Appendix H:
Source Code for the Work
/*this code was written by Stephen Ahiante*/
char data; //declaring a variable; data to hold a character
int motor_1 = 9; //declaring a variable; motor and storing pin 9 in it

A- 10 -
int i; //declaring a variable; i
void setup() //a function that runs once
{
Serial.begin(38400); //starting serial communication at a baud rate of 38400
pinMode(motor_1, OUTPUT); //setting pin 9 as an output pin
}
void loop() //a functions that runs repeatedly
{
if (Serial.available() > 0) //checking for any form of serial communication
{
data = Serial.read(); //reading data from serial communication and storing in a
// variable; data
if (data == 'a') //checking if the data received is the character; a
{
Hover(); // calling the Hover function
}
else if (data == 'b') //checking if the data received is the character; b
{
Land(); //calling the land function
}
}
}
void Hover() //defining a function; Hover
{
if (i == 0) //checking if i is at 0 level

A- 11 -
{
for (i = 0; i < 255; i++) //adding a duty level to i till it reaches 255
{
analogWrite(motor_1, i); //turning the 1st motor on according to the duty level
analogWrite(motor_2, i); //turning the 2nd motor on according to the duty level
analogWrite(motor_3, i); //turning the 3rd motor on according to the duty level
analogWrite(motor_4, i); //turning the 4th motor on according to the duty level
delay(30); //waiting for 30 milliseconds after each addition
}
}
}
void Land() //defining a function; Land
{
if (i == 255) //checking if i is at the highest level
{
for (i = 255; i > 0; i--) //subtracting a level from i till it reaches 0
{
analogWrite(motor_1, i); //turning the 1st motor on according to the duty level
analogWrite(motor_2, i); //turning the 2nd motor on according to the duty level
analogWrite(motor_3, i); //turning the 3rd motor on according to the duty level
analogWrite(motor_4, i); //turning the 4th motor on according to the duty level
delay(30); //waiting for 30 milliseconds after each subtraction
}
}
}

A- 12 -
Appendix I:
Cost Implication of this work
In the construction of the quadcopter in this work, components were purchased, the cost
implication of constructing the quadcopter is given here in a tabular representation:
S/N Component Unit Price (₦) Quantity Amount (₦)
1 Arduino Uno (rev3) 4000 1 4000
2 Arduino Pro Mini (5V) 1500 1 1500
3 Bluetooth HC-05 module 3000 1 3000
4 Coreless DC motors and
propellers
1250 8 10000
5 TB6612FNG motor drivers 1200 2 2400
6 IRFZ34N 150 4 600
7 IRFZ44N 150 4 600
8 Jumper Wires 1500 1 1500
9 Solder core wire 250 2 500
10 Soldering Iron 2000 1 2000
11 Glue Gun + Glue sticks 4560 1 4560
12 LiPo battery 3.7V 2000 2 4000
13 LiPo Charger Module 500 2 1000
14 DC-DC converter 2570 1 2570
15 Resistor Pack 2380 1 2350
16 Heliway Mini Drone 23,500 1 23,500
The total cost of obtaining the components used in this works amounts to ₦64080.

Dynamics, Modelling & Design of a Quadcopter

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Dynamics, Modelling & Design of a Quadcopter