FinalYearReport_Angel_Francos-2_PDF-2

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Design of an intelligent fuzzy logic controller for a
quadrotor helicopter
by
Angel J Francos
School of Marine Science and Engineering
Faculty of Science and Technology
University of Plymouth
Honours project submitted in partial fulfilment of the
requirements for the degree of
BEng (Hons) in Mechanical Engineering
Projects co-ordinators:
Dr Sanjay Sharma, Room 104, Reynolds Building
Dr Asiya Khan, Room 009, Reynolds Building
May 2016

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Abstract
In this project a black-box model for the attitude pitch dynamics of a quadrotor
helicopter is obtained. And two fuzzy logic controllers are design using different
number of rules and tuning methods. Then the fuzzy logic controller with the best
performance is compared against a proportional integral derivative controller. The
results shows a better performance was achieve by the fuzzy logic controller
containing a greater number of rules. And the fuzzy logic controller has a better
performance than the proportional integral derivative controller.

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Acknowledgements
I am grateful to: Mrs Maria Aurora Cagigas, Mr Angel Francos, Mrs Ana Laura
Francos, Mr Angel R Francos, Mrs Mari Carmen Cagigas, Dr Rafel Orozco, Mr
Rafael Orozco, Mr Fermin Orozco and Miss Dulce Orozco for all their support and
for being my one and only source of motivation.
I would like to thank: Dr Asiya Khan, Dr Sanjay Sharma and Dr Jian Wan for their
guidance throughout this project and for introducing me to a very interesting topic
that made work, an enjoyable action.
I would also like to thank: Mr Avinash Sithale, Mr Alesh Obornik and Mr Matthew H
Morris for their companionship through this project.

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1. Table of Contents
2. Nomenclature....................................................................................................6
3. List of Figures....................................................................................................9
4. List of tables....................................................................................................11
5. Introduction .....................................................................................................12
5.1. Aim...........................................................................................................13
5.2. Objectives ................................................................................................13
6. Literature review..............................................................................................13
6.1. Modelling the dynamics of the system......................................................13
6.1.1. Mathematical models or white box models........................................13
6.1.2. Black-Box Models .............................................................................13
6.1.3. Grey Box Models...............................................................................14
6.2. Proportional Integral Derivative Controllers ..............................................14
6.3. Fuzzy Logic systems and Fuzzy logic ......................................................15
6.4. Fuzzy logic controllers..............................................................................15
6.4.1. The Mamdani fuzzy model ................................................................16
6.4.2. Takagi-Sugeno-Kang fuzzy model.....................................................16
6.4.3. Adaptive fuzzy PID controller ............................................................16
6.4.4. Signed distance method....................................................................16
6.4.5. Membership functions .......................................................................16
6.4.6. Type-2-fuzzy logic controllers............................................................16
6.4.7. Adaptive fuzzy logic controllers .........................................................17
7. Project specification ........................................................................................17
8. Modelling the quadrotor dynamics...................................................................17
8.1. Quadrotor dynamics.................................................................................18
8.2. ARX-Model...............................................................................................19
8.2.1. Determining the model orders of the ARX model...............................20
8.2.2. Estimating the ARX model parameters..............................................21
8.2.3. Validating the ARX model .................................................................21
9. PID Controller..................................................................................................23
10. Fuzzy Logic Controller .................................................................................25
10.1. Fuzzification..........................................................................................25
10.2. Defuzzification ......................................................................................25
10.3. Inference Mechanism ...........................................................................25
10.4. Rule-base .............................................................................................25
10.5. Design of the first fuzzy logic controller.................................................26

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10.5.1. Rules used for the first controller .......................................................26
10.5.2. Fuzzy sets parameters used for the first controller.............................27
10.6. Design of the second fuzzy logic controller............................................28
10.7. Tuning the FLC controller......................................................................30
11. Results & Discussion ...................................................................................31
11.1. FLC comparison results ........................................................................31
11.2. FLC and PID controller comparison of results .......................................32
11.2.1. Settling time comparison ...................................................................34
11.2.2. Rise time comparison ........................................................................34
11.2.3. Steady state error comparison...........................................................34
11.2.4. Mean square error comparison..........................................................35
11.2.5. Equivalent energy..............................................................................35
11.2.6. Critical analysis..................................................................................36
12. Conclusion and recommendations ...............................................................37
13. References...................................................................................................38
14. Appendix A: Code used to arrange the data.................................................44
15. Appendix B: Data analysis ...........................................................................46
16. Appendix C: Mathematical model of the quadrotor.......................................49
17. Appendix D: Interim report ...........................................................................53

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2. Nomenclature
AI Artificial Intelligence
ANN Artifical neural networks
APD AR Parrot Drone 2.0
AR Autoregressive model
ARMA Autoregressive model with moving avarage
ARX Autoregressive model with exogenous input
FLC Fuzzy Logic Controller
Hz Hertz
IV Instrumental variable method
LSM Least square method
MATLAB MATLAB Sotware
MISO Multiple-Input Single Output
MLE Maximun Likehood method
MSE Mean square error
na Number of poles
nb Number of zeros
nk Delay
PID Proportional Integral Derivative
PWM Pulse width Modulation
Quadrotor Quadrotor-Helicopter
SISO Single-Input Single-Output
TSK Takagi-Sugeno-Kang
UAV Unmanned aerial vehicle
VTOL Vertical take off and landing
Ω𝑖 Angular velocity of corresponding motor
𝐾 𝑑 Derivative gain

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𝐾𝑖 Integral gain
𝐾 𝑝 Proportional gain
𝑇⃗ Total thrust vector
𝑇𝑖 Thrust in corresponding frame
𝑍 𝑛 Data set containing inputs and outputs
𝑍 𝑡−1 Past observations
𝑘 𝑑 Friction constant
𝑛 𝑎 Number of poles
𝑛 𝑏 Number of zeros plus 1
𝑛 𝑘 Number of samples before input affect the output
𝑥̈ Inertial frame acceleration
𝜏𝑖 Torques in corresponding axis
𝐼 Inertial matrix quadrotor
𝑏 Drag coefficient
𝑏 Motor constant
𝑐 Output
𝑒(𝑡) White noise disturbance value
𝑒𝑟𝑟𝑜𝑟 𝑒𝑟𝑟𝑜𝑟 = 𝑦(𝑡) − 𝑟(𝑡)
𝑔 Gravity
𝑙 Distance from the centre of a quadcopter to propeller centre
𝑚 Mass of the quadrotor
𝑞 Forward shift operator
𝑟(𝑡) Reference signal
𝑡 Time
𝑢(𝑡) Input signal
𝑢(𝑡) Input
𝑦(𝑡) Output signal

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𝑦(𝑡) Prediction for the next output
𝜃 Dimensional vector
𝜔 Angular velocity vector inertial frame

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3. List of Figures
Figure 1……………………………… Representation of a closed loop control
system
Figure 2……………………………… Solid works drawing of a symmetrical
quadrotor with the same rotor blade
configuration as the Parrot AR Drone 2.0
Figure 3……………………………… Block-scheme of the global control system
of an UAV
Figure 4……………………………… Graph of the fit of the multi-input ARX model
when comparing to the validation data,
Rotational angle in degrees vs Time. The
red line is the ARX model and the black line
is the recorded output
Figure 5……………………………… Cross correlation of residuals between input
one and output
Figure 6……………………………… Cross correlation of residuals between input
two and output
Figure 7……………………………… Autocorrelation of residuals for the output
Figure 8……………………………… Simulink structure for the PID controller
Figure 9……………………………… Input signal to the black-box model
Figure 10……………………………. Illustration of a FLC
Figure 11……………………………. Simulink structure used to design the FLC
Figure 12……………………………. Fuzzy sets used for input variable one
“Error”
Figure 13……………………………. Fuzzy sets used for input variable two
“Derivative Error”
Figure 14……………………………. Fuzzy sets used for the output variable one
(Error)
Figure 15……………………………. Fuzzy sets used for input variable one
Figure 16……………………………. Fuzzy sets used for input variable two
Figure 17……………………………. Fuzzy sets used for output variable one
Figure 18……………………………. Sketches showing the states of the system
Figure 19……………………………. Step response graph of the FLC’s the pink
line is the first fuzzy logic controller and the

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blue line is the second fuzzy logic controller
Figure 20……………………………. step response graph of the FLC and PID
controllers to a step input, blue colour is
used to indicate the FLC and red colour is
used indicate the PID controller
Figure 21……………………………. Angular Velocity in degrees second vs Time
in seconds
Figure 22……………………………. Output of the FLC
Figure 23……………………………. Output of the PID controller
Figure 24……………………………. Critical evaluation sketch
Figure 25……………………………. Code used to arrange the data obtained
from the experiment
Figure 26……………………………. PWM signal to motor one vs Sample
number
Figure 27……………………………. PWM signal to three vs Sample number
Figure 28……………………………. Rotation in the y-axis vs Number of samples
Figure 29……………………………. Linear velocity in the x-axis vs Number of
samples
Figure 30……………………………. Showing the pictorial representation of the
inertial and body frame coordinates

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4. List of tables
Table 1………………………………….... List of commonly used Black-Box
models
Table 2………………………………….... List of commonly used System
Identification methods
Table 3………………………………….... Parameters used to record the data
during the test-flight
Table 4………………………………….... Range of data used to estimate and
validate the ARX model
Table 5………………………………….... Table of the multi-input ARX model
orders as well as the options used to
estimate the model
Table 6………………………………….... List of parameters for the tuned PID
controller
Table 7………………………………….... Rules used for the first FLC
Table 8………………………………….... Fuzzy sets parameters for input variable
one “Error”
Table 9………………………………….... Fuzzy sets parameters for input variable
two “Derivative of Error”
Table 10………………………………….. Fuzzy sets parameters for output one
Table 11………………………………….. Rules used for the second FLC
controller
Table 12………………………………….. Fuzzy sets parameters for input variable
one “Error”
Table 13………………………………….. Fuzzy sets parameters for input variable
two “Derivative of error”
Table 14………………………………….. Fuzzy set parameters for output variable
one “Output”
Table 15………………………………….. Performance measurement values for
the FLC’s
Table 16………………………………….. Performance measurement for FLC and
PID controller
Table 17………………………………….. Data range used for estimation of the
ARX model
Table 18………………………………….. Data range used to validate the ARX
model

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5. Introduction
Unmanned aerial vehicles (UAVs) have gained a lot of popularity in the past years.
UAVs can fly autonomously or remotely controlled therefore they are becoming
increasingly popular in military and civil applications. (Ali and Gueaieb, 2010)
A particular UAV has gained significant attention; that is the quadrotor helicopter or
quadrotor, because of its vertical take-off and landing (VTOL) capabilities, excellent
manoeuvrability and the available payload capacity. The quadrotor helicopter was
the first helicopter to complete the first-kilometre long helicopter flight. (Sharma,
2012)(Pallet, 1979) (Ali and Gueaieb, 2010) (Coza and Macnab, 2006) (Sikorsky,
1935)
Controlling a quadrotor is a highly complex task because; a quadrotor has open loop
unstable and highly non-linear dynamics the system is under-actuated, the spinning
rotor blades produce a torque that has to be counteracted and there are complex
aerodynamics and gyroscopic effects in the rotor blades. But technological
advances in; Computational power, sensor measurement and micro electrical
systems have endorsed a new helicopter; the electronically actuated and controlled
quadrotor. In this quadrotor the angular velocity of the rotor blades is varied to
control it, therefore, there is no need to use complex mechanical systems to control
the helicopter; this makes this type of helicopters low-cost and easy to Manufacture.
Furthermore, these types of helicopters can have more sophisticated and intelligent
control algorithms. (Coza et al., 2011) (Ryll, Bulthoff, and Giordano, 2012) (Sharma,
2012) (Pallet, 1979)
Using UAV’s to overcome logistic problems is an application that is becoming
increasingly popular. And multi-billion dollar companies like Google Inc. and
Amazon.com, Inc. are a few of the many companies in the pursuit of this dream.
(Soundarajan and Agrawal, 2016) (Santos, Lopez, and Morata, 2010) (Sharma,
2012) (Piskorski et al., 2012)
The use of UAV’s in logistic can help to tackle a major societal problem; that is the
problem faced by individuals living in remote areas that are difficult or too expensive
to access and they are in need of medicine. UAV medicine delivery is a feasible way
of solving this problem, as Raffaello D’Andrea suggested in his paper “Can drones
deliver?” (D’Andrea, 2014); D’Andrea also suggested that research in autonomous
vehicle coordination is needed to make the UAV medicine delivery system a feasible
concept. Therefore the use of an intelligent controller could be a viable method to
increase the autonomous coordination of UAV’s like the quadrotor helicopter.
(Editors et al., 2014) (Andrea, 2016)(D’Andrea, 2014)
A Fuzzy Logic Controller (FLC) attempts to mimic the decision making ability of
humans hence this intelligent controller offers an alternative to traditional control
methods used to control quadrotors like Proportional Integral Derivative (PID)
controllers. (Hema, Scientific, and Publishing, 2013) (Mendel, 1995)

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5.1. Aim
The aim of this project is to compare the performance of a FLC compared to a PID
controller for a quadrotor as to increase the research in autonomous vehicle
coordination.
5.2. Objectives
1. The pitch dynamics of the quadrotor obtained using a black-box model
2. A PID controller is design
3. A FLC is design
4. The controllers are compared and analysed in a simulation
6. Literature review
This literature review is divide into four sections; section 6.1, 6.2, 6.3 and 6.4 which
correspond to modelling the dynamics of a system, PID controllers, Artificial
Intelligence and Fuzzy Logic Controllers respectively.
6.1. Modelling the dynamics of the system
White-box, black box and grey box models can be used to obtain the pitch dynamics
of the quadrotor, as explained below:
6.1.1. Mathematical models or white box models
The forces that affect the controllability of the system can be derived using a set of
equations that describe the system dynamics, known as white box modelling. This
approach is usually used in well understood models. Appendix C shows the
derivation of the equations used to derive a white box model. (Chao et al., 2010)
(Ahmad, Chipperfield, and Tokhi, 2002)(Wu et al., 2014) (Babuška and Verbruggen,
1996)
6.1.2. Black-Box Models
The use of input-output data with the combination of System Identification
techniques is used to get a model of the system dynamics, known as black-box
modelling. This approach is generally used in poorly understood models. Table 1
shows some common model structures used to obtain a black-box model. (Ahmad,
Chipperfield, and Tokhi, 2002) (Babuška and Verbruggen, 1996) (Ljung, 2007)
Table 1: List of commonly used Black-Box models. (Ljung, 2007) (Forssell and Ljung,
1999) (BILLINGS and VOON, 1987)
Model type Model structure
Linear models Autoregressive model with exogenous
input model (ARX)
Autoregressive model with moving
average model (ARMA)
Box-Jenkins model
Autoregressive model (AR)
Non Linear model Artificial Neural Networks (ANN)

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However, System Identification methods have to be used to fit the model to the data
set. Table 2 shows a list of some commonly used system identification methods.
(Ahmad, Chipperfield, and Tokhi, 2002)
Table 2: List of commonly used System Identification methods. (Chao et al., 2010)
(Ahmad, Chipperfield, and Tokhi, 2002) (Ljung, 2007)(Wu et al., 2014)
System Identification methods
Least square method (LSM)
Instrumental variable method (IV)
Maximum Likelihood method (MLE)
Prediction error method (PEM)
The black-box model used in this project is obtained using a similar methodology as
(Kugelberg, 2016); in which he used a closed-loop configuration to acquire observed
data of the system dynamics consequently the least square method is used to fit the
data to the ARX model ignoring the correlation between the input and the
disturbances present when the data was recorded.
(Ahmad, Chipperfield, and Tokhi, 2002) used an open loop approach when
recording the data, this was done to eliminate the undesirable correlation between
the input and the disturbances present when the data was logged, and a rig
configuration was used to overcome the open-loop instability of helicopters.
6.1.3. Grey Box Models
This approach is a combination of black box modelling and white box modelling
where prior information about the system is known but there are certain unknown
parameters that can be determined from experimental data; this is known as grey
box modelling or hybrid-modelling. This approach is usually used in partly known
models. (Kugelberg, 2016) (Ljung, 2007) (Ahmad, Chipperfield, and Tokhi, 2002)
(Babuška and Verbruggen, 1996)
6.2. Proportional Integral Derivative Controllers
PID controllers are linear regulators. Type A, Type B and Type C PID structures are
some commonly used structures as shown in Equations (1), (2) and (3) respectively.
These controllers use a proportional constant, a derivative constant and an integral
constant to control the system, the value of these constants can be modified in order
to achieve a desire performance. (Hema, Scientific, and Publishing, 2013) (Stojković,
Rodić, and Stevanović, 2012)
Type A
𝑢(𝑡) = 𝐾 𝑝 𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡)𝑑𝑡
𝑡
0
+ 𝐾 𝑑 𝑑𝑒(𝑡)/𝑑𝑡 (1)
Type B

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𝑡
0
− 𝐾 𝑑 𝑑𝑦(𝑡)/𝑑𝑡 (2)
Type C`
𝑢(𝑡) = −𝐾 𝑝 𝑦(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡)𝑑𝑡
𝑡
0
− 𝐾 𝑑 𝑑𝑦(𝑡)/𝑑𝑡 (3)
The PID controller chosen in this project is a Type-A PID controller as to make the
comparison between controllers clearer. (Hema, Scientific, and Publishing, 2013)
Further study could be done by comparing a Type-C PID controller with a FLC, as
Type-C controller tends to give smoother control. The use of a Type-B PID
controlled is not considered in this project because the advantages of this controller
are seen when the structure is implemented in hardware and because none of the
controllers designed here are going to be implemented in hardware this controller
structure is not taken into account. (Hema, Scientific, and Publishing, 2013)
6.3. Fuzzy Logic systems and Fuzzy logic
A Fuzzy logic controller derives from Fuzzy logic systems and fuzzy set theory, in
which a Fuzzy logic system is a well-known methodology in artificial intelligence.
(Mendel, 1995) (Zadeh, 1973) (Fukuda and Arakawa, 1998)
Lotfi A. Zadeh proposed fuzzy logic in 1965 where he describes a “methodological
framework which is tolerant of imprecision and partial truths” (Zadeh, 1973).
And Artificial Intelligence System (AI) can be though as the process of mimicking an
intelligent system by using human knowledge. In which (Fukuda and Arakawa, 1998)
defines an intelligent system as “a system that has the ability to adapt to changing
environments”. And perhaps Alan Turing is the father of artificial intelligence.
(Mendel, 1995) (Zadeh, 1973) (Fukuda and Arakawa, 1998)
If the functions of an intelligent system are understood, can these functions be
replicated through the use of machinery? Similar questions to this where asked by
Alan Turing as he was trying to find a way to imitate an adult humans mind. “In
considering the functions of the mind or the brain we find certain operations which
we can explain in purely mechanical terms” (TURING, 1950). However, he argued
that the nervous system cannot be mimicked through a discrete time machine.
(TURING, 1950)
6.4. Fuzzy logic controllers
The introduction to the Fuzzy logic controller (FLC) starts with understanding of the
fuzzy inference mechanism. The inference mechanism is the process of going from
input fuzzy sets to output fuzzy sets or singleton function with use of knowledge in
the form of rules.
Two famous inference systems are used in the FLC literature and that is the
Mamdani fuzzy model and the Takagi-Sugeno-Kang fuzzy model as explained
below. (Babuška and Verbruggen, 1996)(Jang, 1993)

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6.4.1. The Mamdani fuzzy model
The Mamdani fuzzy model; was proposed by Mamdani and Assilian in 1974. This is
the first example of fuzzy logic control where they design a controller for an
industrial steam engine. (Ali Raza and Gueaieb, 2009)(Mamdani and Assilian, 1975)
(Zadeh, 1988)
6.4.2. Takagi-Sugeno-Kang fuzzy model
The Takagi-Sugeno-Kang (TSK) fuzzy model; was proposed by Takagi and Sugeno
in 1985. (Ali Raza and Gueaieb, 2009) (Babuška and Verbruggen, 1996)
(Syed, Raza, and Gueaieb, 2012) paper compares the Mamdani and TSK fuzzy
inference systems by implementing both inference systems to a Fuzzy Logic flight
controller and found that the Mamdani inference system converges to the desire
state faster than the TSK inference system and that FLC’s are very good at handling
external disturbances.
Both (Syed, Raza, and Gueaieb, 2012) and (Ali Raza and Gueaieb, 2009) results
concur that the TSK inference system uses less computational power when
compared to the Mamdani inference system.
6.4.3. Adaptive fuzzy PID controller
FLC can also be used to aid controllers. (Gautam and H, 2013b) designed Fuzzy
PID controller in which the gains of the PID controller are the outputs of the Fuzzy
Logic Controller the results showed an increase in the PID controller performance.
6.4.4. Signed distance method
(Ishaque et al., 2011) Designed a Single Input Single Output (SISO) FLC for an
underwater vehicle using the signed distance method. The rule table is a one-
dimensional table, therefore; it uses less computational power. However, the author
believes that the main advantage of using the signed distance method is the
simplification of the tuning process of the rules and membership functions.
6.4.5. Membership functions
(Sasaki et al., 1998) Used Gaussian membership functions to build a FLC for a
helicopter and he found that these functions are very good at stabilizing it. The
author considers that using a Gaussian membership functions can make the tuning
process harder than it already is. Therefore, triangular functions are utilized in the
FLC designed in this project.
6.4.6. Type-2-fuzzy logic controllers
(Raju and Pillai, 2016) and (Sabahi, Ghaemi, and Badamchizadeh, 2016) used a
type-2-fuzzy logic controller in which the membership functions of this controller
consist of a combination of a two membership functions; a primary and a secondary
function.
The type-2 membership functions allow the modelling of uncertainty in the system.
However the model used in this project is a simulated model hence sensors are not
used to feed-back the state of the system therefore the uncertainty of the state of
the system is minimal and this type of controller is not considered in this project.
(Raju and Pillai, 2016) (Sabahi, Ghaemi, and Badamchizadeh, 2016)

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However, more research in type-2-fuzzy logic controllers could help to reduce the
price of the UAV medicine delivery system making it more economically feasible this
is because cheaper sensors can be used to construct UAV’s if type-2-fuzzy logic
controllers are used, assuming cheap sensors produce more measurement
uncertainty. (Sepúlveda et al., 2007) (Sabahi, Ghaemi, and Badamchizadeh, 2016)
(Raju and Pillai, 2016)
6.4.7. Adaptive fuzzy logic controllers
A still controversial topic in the FLC literature relates on how to tune a FLC, and
authors like (Kaur and Kaur, 2012) have leap frog the tuning process by using
Adaptive Fuzzy Logic controllers to improve the performance and (Khan et al., 2008)
used genetic algorithms to optimize the membership functions and if the FLC has
scaling gains this can be tuned as well. However there is still a gap in the Fuzzy
Controller literature; that is the lack of understanding on how to tune a FLC to
achieve certain performance parameters.
7. Project specification
The type of control design for the FLC and PID controller in this project was chosen
to be a closed loop control, because helicopters are unstable in open-loop control.
A closed loop system uses feedback to get the error of the system which is the
difference between the desired state and the current state. As shown in Fig. 1.
(Passino and Yurkovich, 1998)
Figure 1: Representation of a closed loop control system. (Passino and Yurkovich,
1998)
To obtain the closed loop controllers; a model that describes the pitch dynamics is
obtained, consequently a PID controller and FLC were designed, and the controllers
performance relative to each other is analysed using performance measurements as
shown in sections 8, 9, 10 and 11 respectively. (Passino and Yurkovich, 1998)
8. Modelling the quadrotor dynamics
In this section a black-box model is going to be obtained for the pitch attitude
dynamics, the data obtained to estimate the model was obtained from a test-flight
using the AR Parrot Drone 2.0 (APD).
The black-box model used in this project is a Linear ARX model; consequently the
least square error is going to be used to fit the model to the observed data.

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8.1. Quadrotor dynamics
The pitch of the quadcopter can be controlled by increasing the speed of the rotor
blade three by the same amount that the speed of rotor blade one is decreased, as
shown in Equation (4). Figure 2 shows the Parrot AR Drone 2.0 rotor configuration.
(Coza and Macnab, 2006) (Piskorski et al., 2012)
𝑈 𝜃 = [
Ω1 − 𝛥𝑢
Ω2
Ω3 + 𝛥𝑢
Ω4
]
(4)
Figure 2: Solid work drawing of a symmetrical quadrotor with the same
configuration as the Parrot AR Drone 2.0
The pitch of the quadrotor controls the rotation in the y-axis consequently this
motion is coupled with the translational motion of the quadcopter in the x-axis.
A position controller can be used to control the translational motion of the quadrotor.
The inputs to the position controller are the desire translational motions and the
outputs are the reference angles needed to achieve this translational motion, as
shown in Fig. 3. (Coza et al., 2011) (Stojković, Rodić, and Stevanović, 2012)
(Razinkova et al., 2014)

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Figure 3: Block-scheme of the global control system of an UAV. (Stojković, Rodić,
and Stevanović, 2012)
In this project only the y-axis rotation in the body frame of the quadrotor is going to
be controlled, this is also known as attitude control. (Stojković, Rodić, and
Stevanović, 2012)
8.2. ARX-Model
The structure of the ARX model is described below.
An ARX model is represented by a linear differential equation as shown in Equation
(5). This model is a discrete time system that represents the relationship between
the input and output at a given time, given a disturbance is present. (Ljung, 2007)
(Kugelberg, 2016)
The ARX model structure uses regression to estimate the output. Essentially the
model contains old values of the variable to be described; hence the model structure
shown in Equation (5) is called an Auto-Regressive model with exogenous input.
(Ljung, 2007) (Kugelberg, 2016)
A more generalised representation of the model is shown in Equations (6), (7) and
(8). (Kugelberg, 2016)
𝐴(𝑞)𝑦(𝑡) = 𝐵(𝑞)𝑢(𝑡) + 𝑒(𝑡) (6)
𝐴(𝑞) = 1 + 𝑎1 𝑞−1
+ ⋯ + 𝑎 𝑛𝑎 𝑞−𝑛𝑎 (7)
𝐵(𝑞) = 𝑏1 𝑞−1
+ ⋯ + 𝑏 𝑛𝑏 𝑞−𝑛𝑏 (8)
The ARX model used in this project has a Multi-Input-Single-Output (MISO).
Therefore it has two inputs and one output. (Coza et al., 2011)
The inputs to the ARX model are: the Pulse width Modulation (PWM) signal to motor
one and the PWM signal to motor three. And it is assumed that the PWM input
signal to the motors is proportional to the angular velocity of the rotor blades. (Coza
et al., 2011)
The output of the ARX model is the rotational angle of the quadrotor in the y-axis
measured in degrees.
The observed data of the pitch dynamics was obtained by connecting the ADP to a
laptop via WIFI. A manually controlled test flight was conducted to obtain the data.
The test-flight lasted a total of 162.27 seconds and it consisted of 32453 samples.
The test flight and data parameters used to record the data are shown in Table 3.
𝑦(𝑡) + 𝑎1 𝑦(𝑡 − 1) + ⋯ + 𝑎 𝑛𝑎 𝑦(𝑡 − 𝑛 𝑎)
= 𝑏1 𝑢(𝑡 − 𝑛 𝑘) + ⋯ + 𝑏 𝑛𝑏 𝑢(𝑡 − 𝑛 𝑏 𝑛 𝑘 + 1) + 𝑒(𝑡)
(5)

20
Table 3: Observed data and test flight parameters
Frequency (sampling rate) 200(Hz)
Number of Samples 32453
Time length of data 162.27 (seconds)
Configuration of quadcopter Close loop configuration
Noise to the quadcopter None
From the test-flight, a text document containing different arrays of information was
obtained. This text file was imported into MATLAB and the code shown in Appendix
A was used to organize the data into a series of arrays.
To estimate and validate the ARX model parameters two different data sets are
used, the first data set is used for the estimation of the parameters and the second
data is used to validate the ARX model. (Ahmad, Chipperfield, and Tokhi, 2002)
(Ljung, 2007)
Because the test flight was controlled manually, only certain ranges of the data
contain the correct excitation signal that exposes the relevant properties of the pitch
attitude dynamics. Therefore the data was plotted and analysed to find these ranges
as shown in Appendix B, Table 4 shows the ranges of data used for estimation and
validation of the ARX model. (Ahmad, Chipperfield, and Tokhi, 2002)
Table 4: Range of data used to estimate and validate the ARX model
Range of samples used to estimate the
ARX model
[25710:26020]
Range of samples used to validate the
ARX model
[27690:27940]
8.2.1. Determining the model orders of the ARX model
In order to estimate the parameters of the model structure, the number of zeros,
poles and the delay in the system has to be input manually. The “order selection”
function in MATALB can be used to select the most suitable model structure. This
function allowed the author to select the model orders with the lowest residuals.
However, trial and error was also used to estimate the most suitable model orders.
Table 5 contains the model orders used to estimate the ARX-structure. (Ljung, 1998)
Table 5: Table of the multi-input ARX model orders as well as the options used to
estimate the model
Number of poles (na) 14
Number of zeros (nb) [1 1]
Delay (nk) [3 5]
Initial state (option) Auto
Focus (option) Prediction
Covariance (option) Estimate
Sample time 0.005 (seconds)

21
8.2.2. Estimating the ARX model parameters
The ARX parameters are estimated using the Least-Square-error Method; This
method works by rearranging Equation (5) to give a prediction of the next output
based on previous observations, as shown in Equation (9). Therefore, the
parameters in Equation (11) can be adjusted, in order to fit the predicted output of
the model as close as possible to the recorded output. (Ahmad, Chipperfield, and
Tokhi, 2002) (Chao et al., 2010) (Ljung, 2007)
After using the least square error to fit the model to the estimation data, the following
structure was obtained as shown in equation (10), equations (11), (12) and (13). T
he model has a fit of 99.71% to the estimation data.
𝐴(𝑞)𝑦(𝑡) = 𝐵(𝑞)𝑢(𝑡) + 𝑒(𝑡) (10)
𝐴(𝑞) = 1 − 1.656 𝑧^ − 1 + 0.08202 𝑧^ − 2 + 0.8692 𝑧^ − 3
+ 0.1443 𝑧^ − 4 − 0.8325 𝑧^ − 5 + 0.4325 𝑧^ − 6
− 0.3216 𝑧^ − 7 + 0.7183 𝑧^ − 8 − 0.5296 𝑧^ − 9
− 0.3743 𝑧^ − 10 + 0.5855 𝑧^ − 11 + 0.1981 𝑧^ − 12
− 0.4307 𝑧^ − 13 + 0.1162 𝑧^ − 14
(11)
𝐵1(𝑞) = −2.531𝑒 − 05 𝑧^ − 3 (12)
𝐵2(𝑞) = 0.0001759 𝑧^ − 5 (13)
The coefficients of the equation 11, 12 and 13 alone have no significant meaning but
the pitch dynamics are dependent upon them (Ahmad, Chipperfield, and Tokhi,
2002)
8.2.3. Validating the ARX model
This model was then validated using the second data set to check if the model is
describing the pitch dynamics and not the disturbances present when the data was
collected. (Kugelberg, 2016)
𝑦(𝑡) = −𝑎1 𝑦(𝑡 − 1) − ⋯ − 𝑎 𝑛𝑎 𝑦(𝑡 − 𝑛 𝑎) + 𝑏1 𝑢(𝑡 − 𝑛 𝑘) + ⋯ + 𝑏 𝑛𝑏 𝑢(𝑡 − 𝑛 𝑏 𝑛 𝑘 + 1)
+ 𝑒(𝑡)
(9)
𝑦(𝑡) = 𝜃𝜑(𝑡) (10)
𝜃 = [ 𝑎1, ⋯ , 𝑎 𝑛𝑎 𝑏1, ⋯ 𝑏 𝑛𝑏] 𝑇 (11)
𝜑(𝑡) = [−𝑦(𝑡 − 1) ⋯ − 𝑦( 𝑡 − 𝑛 𝑎) 𝑢(𝑡 − 1) ⋯ 𝑢(𝑡 − 𝑛 𝑏)]
𝑇 (12)

22
The ARX model shows a fit of 75.49 % to the validation data as shown in Fig. 4.
Figure 4: Graph of the fit of the multi-input ARX model when comparing to the
validation data, Rotational angle in degrees vs Time. The red line is the ARX model
and the black line is the recorded output
To further validate the ARX model residual analysis was used. The residuals are the
prediction errors of the model. The residual confidence interval was set to 99%.
Figure 5, shows the cross correlation between the first input and the output, Fig. 6,
shows the cross correlation between the second input and the output and Fig. 7
shows the auto-correlation between the residuals and the output.
Figure 5 and Fig. 6, shows that the cross correlation between the inputs and the
output lies between the 99% confidence region therefore the model description of
how the output is generated from given inputs is 99% accurate and this model is
going to be used to design and test the controllers. However from Fig. 7 it can be
seen that autocorrelation of residuals does not lie within 99% confidence interval this
could be because a close loop configuration was used to obtain the observed
data.(Tang and Mullholland, 1987)(Westwick and Perreault, 2011)(Kugelberg, 2016)

23
Figure 5: Cross correlation of residuals between input one and output
Figure 6: Cross correlation of residuals between input two and output
Figure 7: Autocorrelation of residuals for the output
9. PID Controller
PID controllers are linear regulators, and is estimated that 90% of the controllers
used today are PID controllers. (Passino and Yurkovich, 1998) (Stojković, Rodić,
and Stevanović, 2012)
A Type-A PID controller works by obtaining the derivative of the error, the integral of
the error and the error this values are then multiplied by a gain constant whose

24
value can change, to tune the controller this values are then added and the result is
the input signal to the model as shown in Equation 14. (Hema, Scientific, and
Publishing, 2013)
Type-A PID controller
𝑡
0
+ 𝐾 𝑑 𝑑𝑒(𝑡)/𝑑𝑡 (14)
The structure used to make the pitch PID controller is shown in Fig. 8.
Constant and summing blocks were used to simulate the matrix used to
couple the pitch, roll, yaw and height controllers in a complete attitude
quadrotor controller, therefore the input signal to motor one and three is equal
and inversely proportional, as shown in Fig. 9. (Santos, Lopez and Morata,
2010) (Coza et al., 2011)
Figure 8: Simulink structure for the PID controller
Figure 9: Input signal to the black-box model
The PID controller was tuned using the PID Tuning tool in MATLAB. The PID was
tuned to get the smallest settling time; the following parameters where obtained as
shown in Table 6. One limitation of tuning the controller by just taking the settling
time into consideration is that the controller has a big overshoot.

25
Table 6: List of parameters for the tuned PID controller
Parameter Value
Proportional gain 19.6312672482784
Integral gain 47.5371083298902
Derivative gain 1.66097140544201
Filter coefficient 48.1589094901108
10. Fuzzy Logic Controller
In this section two fuzzy logic controllers are design; the first fuzzy logic controller
uses nine rules and the second one uses 25 rules. The reason to design two FLC is
to show how the performance of the FLC is dependent on the number of rules used.
(Ishaque et al., 2011)
A fuzzy logic controller has four components as illustrated in Fig. 10 and the
components are explained below.
10.1. Fuzzification
Fuzzification is the process of mapping an input in the universe of discourse to a
fuzzy set in which the fuzzy set has a degree of membership to the given input.
(Bellman, Kalaba, and Zadeh, 1966)(Mendel, 1995)
10.2. Defuzzification
Defuzzification is the inverse operation of the fuzzification process, and this is the
process of going from output fuzzy sets to the universe of discourse.(Mendel, 1995)
10.3. Inference Mechanism
For the Mamdani inference system, the inference mechanism is the process of
going from input fuzzy set to output fuzzy sets with the use of the rules. (Babuška
and Verbruggen, 1996) (Mendel, 1995)
10.4. Rule-base
This part of the controller holds the knowledge on how to control the system in set of
“If- Then “rules. (Fukuda and Arakawa, 1998) (Mendel, 1995)

26
Fig 10: Illustration of a FLC
The Simulink structure shown in Fig. 11 was used to design the FLC. The FLC’s
inference system used to design both controllers is the Mamdani inference system
and the centroid method is for the defuzzification process in both controllers.
Figure 11: Simulink structure used to design the FLC
The type of FLC designed in this project is PD type controllers therefore; the FLC
has two inputs which are the error and the derivative of the error which corresponds
to the angle and the angular velocity of the quadrotor in the y-axis and a single
output which correspond to PWM signal to the motors. (Ishaque et al., 2011)
10.5. Design of the first fuzzy logic controller
10.5.1. Rules used for the first controller
The rules used for this controller are shown in Table 7 and they are shown below:
1. If error is Negative and the change in error is Negative then output is
Negative
2. If error is Negative and the change in error is then Zero then output is
Negative
3. If error is Negative and the change in error is then Positive then output is
Zero
4. If error is Zero and the change in error is then Negative then output is
Negative
5. If error is Zero and the change in error is then Zero then output is Zero
6. If error is Zero and the change in error is then Positive then output is Positive
7. If error is Positive and the change in error is then Negative then output is
Zero
8. If error is Positive and the change in error is then Zero then output is Positive
9. If error is Positive and the change in error is then Positive then output is
Positive

27
Table 7: Rules used for the first FLC
𝑒𝑟𝑟𝑜𝑟/𝛥𝑒𝑟𝑟𝑜𝑟 N Z P
N N N Z
Z N Z P
P Z P P
The two FLC controllers designed in this project contain the rules used by (Passino
and Yurkovich, 1998), in which he uses these rules to control an inverted pendulum.
10.5.2. Fuzzy sets parameters used for the first controller
The first input to the FLC is the error in degrees; the range given to the error is: [-1
3]. Consequently three membership functions where used as shown in Table 8 and
Fig. 12. It is important to understand one major limitation of this range; if the
reference input is bigger than the range of the error input variable, the reference
input would not belong to any Fuzzy set.
Table 8: Fuzzy sets parameters for input variable one “Error”
Linguistic variable for the
fuzzy set
Range of values Shape
Negative (N) [-1 -1 0] Triangular
Zero (Z) [-1 0 1] Triangular
Positive (P) [0 1 3 3] Trapezoidal
Figure 12: Fuzzy sets used for input variable one “Error”
The second input to the FLC is the change in error in degrees per second; the range
given to the change in error was from: [-6 6]. Three membership functions where
used as shown in Table 9 and Figure 13.
Table 9: Fuzzy sets parameters for input variable two “Derivative of Error”
fuzzy set
Negative (N) [-6 -6 -3 0] Trapezoidal

28
Figure 13: Fuzzy sets used for input variable two “Derivative Error”
The output to the FLC is the input signal to the plant and the range given to the input
signal is: [-10 30]. Consequently three membership functions where used as shown
in Table 10 and Fig. 14.
Table 10: Fuzzy sets parameters for output one “Output”
Figure 14: Fuzzy sets used for the output variable one (Error)
The performance measurements and time response for the first Fuzzy logic
controller is shown in section 11.
10.6. Design of the second fuzzy logic controller
Table 11, contains the rules used to design this controller furthermore a more
intuitive tuning method was used as shown in section 10.7 this was done to increase
the performance of the FLC.
Linguistic variable for
the fuzzy set

29
Table 11: Rules used for the second FLC controller
𝑒𝑟𝑟𝑜𝑟/𝛥𝑒𝑟𝑟𝑜𝑟 NB NS Z PS PB
NB NB NB NB NS Z
NS NB NB NS Z PS
Z NB NS Z PS PB
PS NS Z PS PB PB
PB Z PS PB PB PB
The new parameters for input one, input two and the output are shown in Table 12,
Fig.15, Table 13, Fig. 16, Table 14 and Fig. 17 respectively.
Table 12: Fuzzy sets parameters for input variable one “Error”
fuzzy set
Negative Big (NB) [-3 -3 -0.3 -0.1] Trapezoidal
Negative (N) [-0.2 -0.05 0] Triangular
Zero (Z) [-0.06 0 0.06] Triangular
Positive (P) [0 0.05 0.2] Triangular
Positive Big (PB) [0.1 0.3 3 3] Trapezoidal
Figure 15: fuzzy sets used for input variable one
Table 13: Fuzzy sets parameters for input variable two “Derivative of error”
fuzzy set
Negative Big (NB) [-6 -6 -5 -3] Trapezoidal
Zero (Z) [-0.95 0 0.95] Triangular
Positive (P) [0 2 4] Triangular
Positive Big (PB) [3 5 6 6] Trapezoidal

30
Figure 16: Fuzzy sets used for input variable two
Table 14: Fuzzy set parameters for output variable one “Output”
fuzzy set
Negative Big (NB) [-35 -35 -15] Triangular
Positive (P) [0 22 30] Triangular
Positive Big (PB) [15 35 35] Triangular
Figure 17: Fuzzy sets used for output variable one
The performance measurements and the time response of this controller are shown
in section 11.
10.7. Tuning the FLC controller
This FLC was tuned using a combination of trial and error and the following thinking
process:
To achieve the certain performance desired by the designer, the fuzzy set ranges
used to model the state of the system are changed as to fire the rules that designer
thinks are more appropriate at that given range.

31
Graphs of the error, the derivative of the error and the output are plotted against
time and are used with the combination of sketches like the one shown in Fig 18, to
attempt to predict which rules need to be fired at the certain ranges to and
consequently changing the range and shapes of the fuzzy sets in order to force fire
the rules that the designer wants to fire in order to achieve the desire performance
and deep understanding of the system is needed if this approach is to be used.
Figure 18: Sketches showing the states of the system
11. Results & Discussion
11.1. FLC comparison results
Table 15, and Fig. 19 shows the performance parameters and time response to a
step input with a final value of one for both of the FLC designed in this project.
It can be seen that second FLC controller that uses 25 rules outperforms the first
FLC that only uses 9 rules in almost every single performance parameter. This
shows that the performance of the controller is dependent on the number of rules
and the tuning process.

32
Table 15: Performance measurement values for the FLC’s
Performance measurement FLC 1 (9 Rules) FLC 2 (25
Rules)
Steady-state-error (degrees) 0.132 0.014
Percentage Overshoot (%) None None
Rise time to 1% (seconds) Inconclusive 0.31
Settling time to 2%(seconds) Inconclusive 0.55
Equivalent energy 53.74 62.88
Mean Square Error (MSE) 0.1005 0.0389
Figure 19: Step response graph of the FLC’s the pink line is the first fuzzy logic
controller and the blue line is the second fuzzy logic controller
11.2. FLC and PID controller comparison of results
Is important to understand that model used to run the simulations is a linear model
and no disturbances where applied during the simulation. Therefore the controllers
were controlling a highly linear system and if these controllers are to be compared in
a real environment via hardware implementation, and disturbances like wind gust
are present this perturbations will make the system less linear and the performance
of the FLC controller is expected to increase when compared to PID controller
because a FLC is a nonlinear controller. (Leigh, 1992) (Stojković, Rodić, and
Stevanović, 2012)

33
To compare the performance of the Second FLC to the PID controller; a step input
with a final value of one was applied to both controller the simulation lasted five
seconds. The results are shown in Table 16, and in Fig. 20.
Table 16: Performance measurement for FLC and PID controller
Performance measurement PID FLC
Steady-state-error (degrees) 0 0.014
Percentage Overshoot (%) 12 None
Rise time to 2% (seconds) 0.14 0.31
Settling time to 2%(seconds) 0.98 0.55
Equivalent energy 51.64 62.88
Mean Square Error (MSE) 0.0228 0.0389

34
Figure 20: step response graph of the FLC and PID controllers to a step input, blue
colour is used to indicate the FLC and red colour is used indicate the PID controller
11.2.1. Settling time comparison
The FLC reaches a steady state value within 2% of the final value 78% faster than
the PID controller and this shows that the set of rules used to stabilize an inverted
pendulum can be implemented to control the pitch rotation of a quadrotor very
effectively.
11.2.2. Rise time comparison
Figure 20 and Table 16 shows that the PID controller is 121% faster than the FLC at
reaching the reference angle; this is because the PID Controller manages to achieve
an angular velocity of 10 degrees/ second whereas the FLC only achieves a velocity
of 4 degrees/ second as shown in Fig. 21. Therefore the PID controller achieves a
faster angular velocity hence the faster rise time.
Figure 21: Angular Velocity in degrees second vs Time in seconds
11.2.3. Steady state error comparison
From Table 16, it can be seen that PID outperforms the FLC in terms accuracy to
reach the reference angle, this is because the PID controller has a zero steady state
error and the FLC has a steady state error of 0.014. However the difference in the
steady state errors can be considered insignificant if they would be to be
implemented in a quadrotor attitude controller.

35
11.2.4. Mean square error comparison
From Table 16, it is seen that Mean Square Error (MSE) is smaller for the PID
controller, however the MSE measurement was not used to evaluate the
performance of the controllers because: The PID controller has an overshoot and
undershoot which reduces the value of the MSE even though the overshoot and
undershoot are undesirable characteristics in a controller. Furthermore, the FLC
does not achieve steady state error hence the MSE measurement obtained for the
FLC controller is greatly influenced by the length of the simulation and the steady
state error of the controller.
11.2.5. Equivalent energy
From Table 16 it can be seen that the equivalent energy measurement obtained
from the PID controller is smaller than the one obtained from the FLC, hence the
PID controller is more energy efficient this can be understood by observing the
output of the FLC and PID controller shown in Fig. 22 and Fig. 23 respectively.
Figure 22: Output of the FLC

36
Figure 23: Output of the PID controller
11.2.6. Critical analysis
To further analyse the performance of the controllers relative to each other, a
scenario is created to simulate the trajectory that a quadrotor might have to follow in
the near future. As shown in Fig. 24.
In this scenario the quadrotor is controlled autonomously by a position controller as
shown in Fig. 3, the quadrotor has to achieve the fastest linear velocity in the x-axis
but without touching the imaginary or physical boundaries.
Equation 15 can be used to by the position controller, to know the biggest angle ∝
the quadrotor can have if the length of the quadrotor 2 ∗ 𝐿 is bigger than the height
of the imaginary boundary 2 ∗ ℎ2.
∝ = sin−1
(
ℎ2
𝐿
)
(15)
If a position controller is used to control the FLC and PID controller designed in this
project, the FLC would get to 2% of the reference value 75% faster than the PID
controller furthermore the PID has an overshoot of 12% and this would cause the
quadrotor to overshoot the boundary hence failing the objective of the simulation.

37
Figure 24: Critical evaluation sketch
12. Conclusion and recommendations
In conclusion the project shows that the ARX model can model the quadrotor pitch
dynamics effectively and it has a fit of 75.49% to validation data without any cross
correlation between the input and the output. It was also shown that the
performance of the FLC is greatly affected by the number of rules used and the how
well the controlled is tuned. The results show that the FLC is a more effective
attitude controller with the slight draw back that is less energy efficient.
The author would like to recommend the use of an open-loop configuration to obtain
the pitch dynamics as this can improve the accuracy of the black-box model.
Furthermore this model can be used to compare the performance of the PID
controller compared to a FLC, as to validate the results obtained in this project.
The author would like to recommend using quadrotor with a microcontroller that is
compatible with the code generation function in MATLAB like the use of an Arduino
Microcontroller like the “Arducopter” consequently to compare the performance of
the PID against the FLC in a real world environment.

38
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44
14. Appendix A: Code used to arrange the data
The data was collected in a text file; this text file consisted of a single array of strings
and figures. This data was imported into MATLAB using the “import data from file”
option then the code shown, in Fig. 25 was used to read and arrange the single
array data into an organized multiple array structure. The code consist of the
following functions: “fopen” “~feof” “fscanf” “fprintf” “fclose”, it has a
single loop that ends when all the data has been read, and a number is defined as
“%f” representing a figure and a linguistic variable is defined as a “%s” representing
a string.
%%reading and opening files
fid = fopen('dataset2_praises.txt','r');
file_out1= fopen('out.dat_1','w');
file_outix= fopen('out.dat_rix','w');
file_outiy= fopen('out.dat_riy','w');
file_outiz= fopen('out.dat_riz','w');
file_outlvy= fopen('out.dat_lvy','w');
file_outlvx= fopen('out.dat_lvx','w');
file_outtime= fopen('out.dat_time','w');
%scanning data
%%loop data
%%
while ~feof(fid) %loping data with the feof()command
header= fscanf(fid, '%s', 1);
seq= fscanf(fid, '%f', 1);
stamp= fscanf(fid, '%s', 1);
secs= fscanf(fid, '%f', 1);
nsecs= fscanf(fid, '%f', 1);
frame_id= fscanf(fid, '%s', 1);
battery= fscanf(fid, '%f', 1);
state= fscanf(fid, '%f', 1);
magx= fscanf(fid, '%f', 1);
magy= fscanf(fid, '%f', 1);
magz= fscanf(fid, '%f', 1);
pressure= fscanf(fid, '%f', 1);
temp= fscanf(fid, '%f', 1);
ws= fscanf(fid, '%f', 1);
wa= fscanf(fid, '%f', 1);
wc= fscanf(fid, '%f', 1);
rotx= fscanf(fid, '%f', 1);
roty= fscanf(fid, '%f', 1);
rotz= fscanf(fid, '%f', 1);
altd= fscanf(fid, '%f', 1);
vx= fscanf(fid, '%f', 1);
vy= fscanf(fid, '%f', 1);
vz= fscanf(fid, '%f', 1);
ax= fscanf(fid, '%f', 1);
ay= fscanf(fid, '%f', 1);
az= fscanf(fid, '%f', 1);
motor1= fscanf(fid,'%f',1);

45
tagsc= fscanf(fid, '%s', 1);
tagst= fscanf(fid, '%s', 1);
tagsx= fscanf(fid, '%s', 1);
tagsy= fscanf(fid, '%s', 1);
tagswi= fscanf(fid, '%s', 1);
tagshe= fscanf(fid, '%s', 1);
tagso= fscanf(fid, '%s', 1);
tagsd= fscanf(fid, '%s', 1);
tm= fscanf(fid, '%f', 1);
blank= fscanf(fid, '%s', 1);
fprintf(file_out1, '%fn',motor1);
fprintf(file_outix, '%fn',rotx); %f/n used to print an array
file
fprintf(file_outiy, '%fn',roty);
fprintf(file_outiz, '%fn',rotz);
fprintf(file_outlvy, '%fn',vy);
fprintf(file_outlvx, '%fn',vx);
fprintf(file_outtime, '%fn',tm);
end
%closing the files ;)
fclose(fid);
fclose(file_out1);
fclose(file_out2);
fclose(file_out3);
fclose(file_out4);
fclose(file_outix);
fclose(file_outiy);
fclose(file_outiz);
fclose(file_outlvy);
fclose(file_outlvx);
fclose(file_outtime);
Figure 25: Code used to arrange the data obtained from the experiment

46
15. Appendix B: Data analysis
Once the data was organized in different arrays, the data was analysed in order to
find the sample range were the pitch input was applied to it.
The values that described the pitch dynamics are dependent on the: PWM signal to
motor one and three, the rotational speed in the y-axis and the linear velocity in the
x-axis as shown in Figures 26, 27, 28 and 29 respectively.
Figure 26: PWM signal to motor one vs Sample number
Figure 27: PWM signal to three vs Sample number

47
Figure 28: Rotation in the y-axis vs Number of samples
Figure 29: Linear velocity in the x-axis vs Number of samples
Range selection
Because the data was obtained by flying the APD manually only certain ranges of
the data contain pitch inputs, and is only in this ranges that the data describes the
pitch dynamics of the APD.
Therefore looking at the graphs shown above it was stablish that the pitch inputs
were applied somewhere in between sample 24000 and sample 37000 because the

48
linear speed in x-direction and the rotational angle in y-direction increases in
between those samples. Therefore the following data sets were used for validation
and estimation.
Data range used for estimation of the model structure
The range of samples used to estimate the numerical values in the model was
chosen to be samples [25710:26020], this range was chosen as it where the pitch
input affects the plant the most, Fig. 1.5 contains further information about the
estimation range.
Table 17: Data range used for estimation of the ARX model
Number of Samples 310 (samples)
Sampling range [25710:26020]
Time 1.55 (seconds)
Validation data range
The range of samples used to estimate the numerical values in the model was
chosen to be samples [27690:27940]; Fig. 1.6 contains further information about the
validation range.
Table 18: Data range used to validate the ARX model
Number of Samples 250 (samples)
Sampling range [27690:27940]
Time 1.25 (seconds)

49
16. Appendix C: Mathematical model of the quadrotor
To derive a mathematical model of a Quadrotor, the following literature was used:
(A.Abdelkhalek, .El-Demerdash, and .Dief, 2015) (Phang et al.,6 2012)(Gibiansky,
2012)(Li and Li, 2011)(Sinha, 2012)(abeywardena et al., 2009) (Phang et al.,6 2012)
(Santos, Lopez and Morata, 2010)
Mathematical model using a white box technique
To derive the mathematical model of the system, Euler equations are used even
though other papers used the quaternion differential equations, to prevent losing a
degree of freedom when the quadrotor has a pitch angle of 90 degrees also the
following assumptions where made:
 The quadcopter consist of two uniform rods crossed at the origin with the
mass of the motor at each end of the rod
 Quadrotor and its components are rigid bodies
 The origin of inertial coordinate system is in the same position with the
geometric centre and centroid of the quadrotor
 Aerodynamics forces and gyroscopic effects are neglected in this study
Coordinate system
The coordinate system can be acquired through it states. A quadrotor has 12 states
and six degrees of freedom; these consist on the orientation and the position of the
quadrotor. (Santos, Lopez and Morata, 2010)
Two coordinate frames will be used to derive the dynamic behaviour of the system:
Inertial frame
Consist of six states: the position of the centre of gravity of the quadcopter and
respectively the linear velocity components. This frame is time-invariant.
The position and the velocity of the quadrotor are defined as x = (x, y, z)T
,
ẋ = ( ẋ ẏ ż)T
respectively.
Body frame
Consist of six states: the roll, yaw, pitch and the angular velocities around each axis
of the body of frame.
The rotations and the angular velocities of the quadrotor in the body frame are
defined as 𝜃 = (∅, 𝜃, Ψ) 𝑇
, 𝜃̇ = (∅̇ 𝜃̇ Ψ̇ ) 𝑇
respectively.
The equations of motion are formulated in the body frame, the orientation of this
body of frame coordinate system B(Oxyz), has the origin at the centroid of the
quadrotor and the z-axis is pointing upwards as described in Fig. 30.

50
Figure 30: Showing the pictorial representation of the inertial and body frame
coordinates
Transition matrix
The transition matrix 𝑅 in equation (16) is used to convert the vector components in
the body frame to the inertial frame and vice-versa, where equation (17), (18) and
(19) represents the x-axis, y-axis and z-axis projections of the body frame into the
inertial frame respectively
𝑅 = 𝑅 𝑥 𝑅 𝑦 𝑅 𝑧 = (
cos𝛹cos∅ 𝑐𝑜𝑠𝛹𝑠𝑖𝑛𝜃𝑠𝑖𝑛∅ 𝑐𝑜𝑠𝛹𝑠𝑖𝑛𝜃𝑐𝑜𝑠∅ + 𝑠𝑖𝑛𝛹𝑠𝑖𝑛∅
𝑠𝑖𝑛𝛹𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝛹𝑠𝑖𝑛𝜃𝑠𝑖𝑛∅ 𝑠𝑖𝑛𝛹𝑠𝑖𝑛𝜃𝑐𝑜𝑠∅ − 𝑠𝑖𝑛∅𝑐𝑜𝑠𝛹
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃𝑠𝑖𝑛∅ 𝑐𝑜𝑠𝜃𝑐𝑜𝑠∅
) (16)
𝑅 𝑥 = (
1 0 0
0 𝑐𝑜𝑠∅ −𝑠𝑖𝑛∅
0 𝑠𝑖𝑛∅ 𝑐𝑜𝑠∅
) (17)
𝑅 𝑦 = (
𝑐𝑜𝑠𝜃 0 𝑠𝑖𝑛𝜃
0 1 0
−𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃
) (18)
𝑅 𝑧 = (
𝑐𝑜𝑠𝛹 −𝑠𝑖𝑛𝛹 0
𝑠𝑖𝑛𝛹 𝑐𝑜𝑠𝛹 0
0 ∅ 1
) (19)
Forces and thrust
From the literature it was establish that the thrust in a single propeller is proportional
to torque. And the thrust is proportional to the square angular velocity therefore the
thrust in the body frame is shown in equation (20).
𝑇𝐵 = 𝑘 (
0
0
∑ Ω𝑖
2
) (20)
To model the torque generated in z-axis it was found that torque is given by the sum
of all torques in the propeller as shown in equation (21), to model the torque in the x-
axis it was found that the torque can be modelled by multiplying the torques

51
generated in the x-axis by the distance of the centre of the quadcopter to the
propeller as shown in equation (22) similarly the same can be done to get the torque
generated in the y-axis as shown in equation (23) therefore the torques in the body
frame are given by equation (24).
𝜏Ψ = 𝑏(Ω1
2
− Ω2
2
+ Ω3
2
− Ω4
2
) (21)
𝜏∅ = 𝑙𝑏(Ω4
2
− Ω2
2
) (22)
𝜏 𝜃 = 𝑙𝑏(Ω1
2
− Ω3
2
) (23)
𝜏 𝐵 = (
𝑙𝑏(Ω4
2
− Ω2
2
)
𝑙𝑏(Ω1
2
− Ω3
2
)
𝑏(Ω1
2
− Ω2
2
+ Ω3
2
− Ω4
2
)
) (24)
To model friction in the inertial frame complex aerodynamic effects are ignored and
the shape of the quadcopter is assumed to be symmetrical therefore a friction
constant is used and multiplied by the linear velocity as shown in equation (25)
𝐹 = (
−𝑘 𝑑 𝑥̇
−𝑘 𝑑 𝑦̇
−𝑘 𝑑 𝑧̇
) (25)
Equations of motion
Using Newton’s second law the linear motion of the quadcopter can be modelled by
taking into account the thrust, friction and gravity and converting the corresponding
body frames vectors to the inertial frame as shown in equation (26)
𝑚𝑥̈ = [
0
0
−𝑚𝑔
] + 𝑅𝑇𝐵 + 𝐹 (26)
Euler’s equations shown in equation (27) can be rearranged and used to model the
rotation of the quadcopter in the body frame as shown in equation, where equation
(29) represents the inertial matrix. From these equation the rotation of the quadrotor
in the body frame are obtained as shown in equation (30)
𝐼 𝜔̇ + 𝜔(𝐼𝜔) = 𝜏 (27)
𝜔̇ = 𝜔(𝐼𝜔) (28)
𝐼 = (
𝐼 𝑥𝑥 0 0
0 𝐼 𝑦𝑦 0
0 0 𝐼𝑧𝑧
) (29)

52
From these the body frame rotations can be obtained.
𝜔̇ =[
𝜏∅ 𝐼 𝑥𝑥
−1
𝜏 𝜃 𝐼 𝑦𝑦
−1
𝜏Ψ 𝐼𝑧𝑧
−1
] −
[
𝐼 𝑦𝑦−𝐼 𝑧𝑧
𝐼 𝑥𝑥
𝜔 𝑦 𝜔 𝑧
𝐼 𝑧𝑧−𝐼 𝑥𝑥
𝐼 𝑦𝑦
𝜔 𝑥 𝜔 𝑧
𝐼 𝑥𝑥−𝐼 𝑦𝑦
𝐼 𝑧𝑧
𝜔 𝑥 𝜔 𝑦 ]
(30)

53
17. Appendix D: Interim report
Design of a linear PID fuzzy logic control for a quadrotor helicopter
Introduction
The UAV type quadrotor helicopter has; low dimensions, good manoeuvrability,
simple mechanics and available payload capacity (Olivares et al., 2015),
furthermore a hybrid configuration which consist of a fixed wing combined with a
quadrotor configuration would enable this rotorcraft to have the advantages of fixed
wing aircraft vertical flight and the advantages of helicopter vertical take-off and
landing (Flores and Lozano, 2013), therefore the use of quadrotors in logistics is a
very controversial topic, the multi-billion dollars company named amazon.com, Inc.
has proposed the use of a UAV quadrotor configuration to handle their complex
logistics(Amazon, no date)
Aims
To design a Liner PID Fuzzy logic controller for an unmanned quadrotor helicopter
consequently to test the control in a Test-bed using Simulink and in a real
environment using the AR parrot 2.0.
Objectives
The objective of this project is to design a Linear PID Fuzzy logic controller for an
unmanned quadrotor helicopter. To reach this objective:
 The non-linear dynamics of the system are derived.
 A suitable linearization method has to be used for the transformation of the
non-linear dynamics.
 A model of the fuzzy logic controller has to be design in Simulink.
 A test bed for the quadrotor dynamics has to be design using Simulink.
Helicopter history
The control dynamics of helicopters are more challenging than that of airplanes
dynamics, and it took longer for the helicopters to be a mass produced vehicle when
compared to airplanes even though the construction of the first quadrotor dates back
to 1907, later on 1924 George De Bothezat design a quadrotor that achieved the
first kilometre long helicopter flight (Ali and Gueaieb, 2010) however the first mass
produced helicopter was the Sikorsky R-4 in 1942 where a tail rotor was used to
counter act the torque generated by blades, (sikorsky, 1935) this rotorcraft
configuration was the first type to be mass produced mainly because of the
lightweight design of the Sikorsky R-4 but many other rotorcraft configurations have
been implemented in helicopters the most common types are: Tandem, side-by-side,
coaxial, synchropter, quadrotor, et. (Pallet, 1979)
Quadrotor helicopter as UAV
Unmanned aerial vehicles (UAV) can fly autonomously or remotely controlled and
are becoming increasingly popular in civil and military applications that are too risky
humans to conduct. (Ali and Gueaieb, 2010) furthermore recent advances in

54
computational power and in the fields of microelectromechanical system and
sensors (Ryll, Bulthoff, and Giordano, 2012) have enabled the UAV to have more
complex, robust and intelligent controllers.
Quadrotor helicopter background
The quadrotor helicopter or quadrotor, has a four rotor configuration and it does not
requires complex mechanical control linkages (Sharma, 2012) therefore the use of
cyclic, collective pitch or tail rotors is not required when compared to previous
rotorcraft configurations. (Coza and Macnab, 2006) Instead the speed of the blades
is varied to achieve control and the directional flight of the controller is achieved by
tilting the vehicle towards the intended flight direction. (Abeywardena et al., 2009)
The quadrotor will have two pairs of rotors rotating clockwise and the other two
rotating anti-clockwise to counteract the moments similarly varying these speeds will
determine the roll, yaw and spin. (Sharma 2012)
Roll
The roll is the movement of the quadrotor sideways.
Fig. 1(Syed, Raza, and Gueaieb, 2012)
“Increasing the speed of the left motor by the same amount that the speed of the
right motor is decreased will keep the total thrust provided by the four rotors
approximately the same” and roll rate is determined (Coza and Macnab, 2006)
Pitch
The pitch is either the movement of the quadrotor either forward or backwards.

55
Fig. 2 (Syed, Raza, and Gueaieb, 2012)
Increasing the speed of front motor by the same amount that the speed of the right
motor is decreased will keep the total thrust approximately the same and the pitch
rate is determined. (Coza and Macnab, 2006)
Yaw
The yaw is the rotation of the head of the quadrotor.
The yaw can be obtained by can be obtained by increasing the speed of the front
and rear rotor together and decreasing the speed of the lateral rotors therefore the
thrust is keep constant. (Sharma 2012)
Thrust
The thrust or height is the movement of the quadrotor in the horizontal plane.

56
The thrust or height movement can be achieved by increasing the speed of all the
rotors by the same amount therefore keeping the yaw, spin and roll constant.
Hardware description
The Quadrotor helicopter used in this project is the AR Drone 2.0, which has the
following:
 A 1500 mAh rechargeable battery
 Four brushless in-runner motors, that have a maximum power and angular
velocity of 14.5 watts and 28 500 rpm respectively.
 A 3 axis accelerometer
 A 3 axis gyroscope
 A pressure sensor
(Flying toys Ltd, 2014)
Brushless DC motor 2.2
The main advantage of a brushless motor includes optimized torque simplified
system structure and longer life span, compared to brushed motor. A brushless
motor works by using the coil as the stator of the motor and the coil as the
permanent magnet. (H.E.A. and F.N., 2013)
Control Strategies used for quadrotor helicopter
The most commons type of controllers for a quadrotor helicopter is as follows:
 Robust feedback controllers
 Fuzzy controllers
 PDI controllers
 PD controllers
 Back-stepping controllers
 Neural-network adaptive flight control
 Linear quadratic method

57
Table. 5 (Saad Shaikh, 2011) (Coza and Macnab, 2006) ) (Abeywardena et al.,
2009) (Belkheiri et al., 2012)
And the main focus of this project is on fuzzy logic controllers.
Fuzzy logic controllers background
Fuzzy logic controllers are intelligent systems, the first paper on fuzzy logic was on
1965 by Lotfi A. Zadeh (Mendel, 1995) his papers he describes a “methodological
framework which is tolerant of imprecision and partial truths” (Zadeh, 1973) the
behaviour of complex systems can be described without the need of precise
mathematical models therefore the use a fuzzy logic controller for a complex system
like the quadrotor system would offer many advantages.
Fuzzy logic structure
 A fuzzy logic system is a non-linear mapping of an input data vector into a
scalar output. (Mendel, 1995)
 A fuzzy logic control has the following elements: Pre-processing,
Fuzzification, rule table, inference engine, defuzzification, post-processing.
As shown in (fig. 6)
Inference engine
The two most common types of inference engines are the: Mamdani type, and
Takagi-Sugeno-Kang. It can though as system that maps inputs into outputs. (Syed,
Raza, and Gueaieb, 2012)
Mamdani inference system
Output membership functions can be higher order models. (Syed, Raza, and
Gueaieb, 2012)
Takagi-Sugeno-kang;
Output membership functions are either linear or constant. (Syed, Raza, and
Gueaieb, 2012)

58
Mamdani requires more computational power but the controller converges to the
desire states relative faster than Takagi-Sugeno-Kang inference engine. (Syed,
Raza, and Gueaieb, 2012)
Rules
Consists of a collection of if-then rules which can be summarised in a relationship
matrix and then the computer is able to compute a control signal based on the error
and the derivative of the error. (Rai, 2012)
Fuzzification
The fuzzification of the crisp data into a fuzzy set is done via membership function.
(Rai, 2012)
Defuzzification
“Defuzzification is when all the actions that have been activated are combined and
converted into a single non-fuzzy output signal which is the control signal of the
system” (Rai, 2012)
Types of fuzzy logic controllers:
 single input fuzzy logic controller (ishaque et al.,2011),
 Smart self-tuning fuzzy PID controller (Gautam and H, 2013),
 Adaptive fuzzy controller (Coza and Macnab, 2006)
 PDI fuzzy logic (Santos, Lopez, and Morata, 2010) (Sharma,
2012) (Abeywardena et al., 2009)
Table 2
Fuzzy logic system design
A fuzzy PID controller for quadrotor configuration looks like the one shown in figure
10, it has six fuzzy logic controllers, 3 fuzzy logic controllers to control the yaw, roll,
and pitch and 3 fuzzy logic controllers to control the x, y, and z quadrotor position.
(YordanovA and JAIN, 2012)
The controller decisions are based on the desire combination of four motions: yaw,
roll, pitch, and height. The PID controller uses the error, the derivative of the error,
and the integral as inputs. The output is the control action and the corresponding
gain. (Santos, Lopez, and Morata, 2010)
The type of controller being design is a fuzzy PID controller and It uses the error and
the derivative of the error as inputs (Santos, Lopez, and Morata, 2010) (Sharma
2012) (Abeywardena et al., 2009)

59
Fig. 10 (YordanovA and JAIN, 2012)
The aggregation block
Each controller gives the power require by each engine, and there is four controller
inputting an output to the dynamics therefore the aggregation block implements the
coupling of the control actions of the four controllers (Santos, Lopez, and Morata,
2010)
Mathematical model
To derive the mathematical model of the system, Euler equations are used even
though other papers used the quaternion differential equations, to prevent losing a
degree of freedom when the quadrotor has a pitch angle of 90 degrees.
(A.Abdelkhalek, .El-Demerdash, and .Dief, 2015) (Phang et al., 2012) (Li and Li,
2011) (sinha, 2012)(loogbook)
Also the mathematical model derived in here uses the same assumptions,
coordinate system orientation, and method as (A.Abdelkhalek, .El-Demerdash,
and .Dief, 2015) (Li and Li, 2011)
Coordinate system
The coordinate system can be acquired through it states. A quadrotor has 12 states
and six degrees of freedom. (Santos, Lopez and Morata, 2010) These consist on the
orientation or attitude and the position of the quadrotor. (Santos, Lopez and Morata,
2010)
Two main coordinate frames will be used in the derivation of the dynamic behaviour
of the system:

60
Inertial frame; consist of six states: the position of the centre of gravity and
respectively linear velocity components. This frame is time-invariant
Body frame; Consist of six states: the roll, yaw, pitch and the angular velocities
around each axis of the body of frame. The equations of motion are formulated in
the body frame, the orientation of this body of frame coordinate system B(Oxyz), has
the origin at the centroid of the quadrotor and the z-axis is pointing upwards as
described in Figure 11
Fig. 11 (Li and Li, 2011)
The definitions of the Euler angles and the transition matrix from the body frame to
the inertial frame of their corresponding axis are shown below:
Yaw angle; angle between the X-axis and the projection of the body frame in the
inertial frame
(Li and Li, 2011) (1)
Pitch angle; angle between the Z-axis and the projection of the body frame in the
inertial frame
(Li and Li, 2011) (2)
Roll angle; angle between the Y-axis and the projection of the body frame in the
inertial frame
(Li and Li, 2011) (3)

61
The transition matrix R will enable the two coordinates (inertial frame and body of
frame) to be converted to each other.
Therefore the transition matrix R from the body frame or quadrotor coordinate
system B (Oxyz) to the inertial frame or inertial coordinate system E (OXYZ) is
derived:
(Li and Li, 2011)
(4)
To mathematically model the movement of the aircraft Newton’s second law of
motion is used to get the net force and net moment on the quadrotor body.
Assumptions:
Quadrotor and its components are rigid bodies;
The origin of inertial coordinate system is in the same position with the geometric
centre and centroid of the quadrotor
Aerodynamics forces and gyroscopic effects are neglected in this study
1st
equation of motion: translational equations of motion
F=m*a (5)
Where “F” and “a” are the force and acceleration in the body axes respectively.
The force vector “F” in the body, can be derived by: calculating the total thrust vector
in the body axes “ ” and the gravitational force vector in the inertial frame of axis
“ ”
(6) (A.Abdelkhalek, .El-Demerdash,
and .Dief, 2015)
and .Dief, 2015)
Transforming the gravitational force from the inertial frame of axis to the body frame
axis using the rotation transformation:

62
and .Dief, 2015)
Combining the thrust force and gravitational force in the 1st
equation of motion:
(9) (A.Abdelkhalek, .El-
Demerdash, and .Dief, 2015)
This equation can be rearranged to give the following:
(10) (11)
(A.Abdelkhalek, .El-Demerdash, and .Dief, 2015)
In the equation above the acceleration in the body of axis is the subject.
(12)
Where “ ” is the linear velocity in the body axis [u, v, w]
Where “ ” is the resultant angular velocity vector due to the three rotation angles,
yaw, spin, roll as described by equation (13)
(13)
Where [p, q, r] are angular velocities component in the body axes.
Using the equations (10) and (13), equation (14) is derived where “ ” is the vector
acceleration in the inertial frame of reference.

63
(14)
Using Equation (12) the following equation is derived:
(15)
2nd
equation of motion: Rotational equations of motion
and .Dief, 2015)
Where “ ” is the moment vector in the body axes.
Using the angular velocities “ ” the torque “ ”equations can be derived:
(17)(18) (A.Abdelkhalek, .El-Demerdash,
and .Dief, 2015)
and .Dief, 2015)
Equation (19) gives the torque results from drag reactions on blades. From this
equation the moment vector in the body axes can be derived. As the assumption
that the quadrotor is rigid body the inertia tensor becomes a rigid body.

64
Therefore using equation (16),
Since,
The following equation is obtained:
Conclusion and revised project plan
From appendix 1 the project Literature review on Thursday the 26 of
November on the mathematical model was supposed to be finish on Friday

65
the 25 of November and the testing or checking of the mathematical
equations was going to be finish on Friday the 4 of December, however the
complexity of the mathematical model was underestimated which lead to only
having half of the mathematical model completed, as the mathematical model
equations where supposed to be solved using mathlab and then input into
the word document using the word format instead the equations and
diagrams where solved in the logbook and screen shoots of the answers
where use instead as the time was limited to construct the equations.
Therefore the plan proposed on the 7th
of October has changed significantly
one of the changes is that the amount of work that was meant for this part of
the Gantt chart is going to increase because the mathematical modelling was
not done on time the other change is a more detail Gantt chart as the 1st
Gantt chart was very broad, consequently it did not provided good managing
reference of how well or bad the project was going in conclusion a more
detail Gantt chart should be able to break the projects into smaller chunks
which will make the project easier keep track of.
To conclude the aim of this literature is to make a case on the importance
that quadrotor helicopters and artificial intelligence (Fuzzy logic) have in the
present world.

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