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LINEAR UNKNOWN INPUT OBSERVER FOR

QUADCOPTERS

by

Krishneel Kumar

A thesis submitted in fulfillment of the

requirements for the degree of

Master of Science in Electrical and Electronics Engineering

Copyright © 2018 by Krishneel Kumar

School of Engineering and Physics

Faculty of Science, Technology and Environment

The University of the South Pacific

November, 2018

i

Acknowledgement It is a pleasure to thank those who made this thesis possible.

Firstly my supervisors Prof. Maurizio Cirrincione, Dr. Adriano Fagiolini and Mr.

Sheikh Izzal Azid for their endless guidance and support from the initial to the final

stages which enabled me to develop an understanding of the subject and successfully

.

Moreover, my parents Anil and Jainul Kumar, my brother Praneel Kumar and sister

Maureen Kumar for their encouragement and support towards finishing of the thesis

and being my pillar of strength.

Finally, a heartfelt thanks goes out to my friends and colleagues Hiye Mudaliar, Farhad

Ashab, Dhirendran Kumar, Paayal Kumar, Reginald Nath, Siwan Narayan and Shanil

Nand for their continuous help and support throughout the journey for the completion

of the project.

ii

Abstract A quadcopter is an Unmanned Aerial Vehicle (UAV) which changes its position and

orientation by four rotating blades. Due to its size and manoeuvrability, the quadcopter

has several uses. Some of which include natural disaster management, weather

monitoring, forest fire detection, emergency search and rescue, security, surveillance,

agriculture and academic teaching such as control engineering. Moreover, in the recent

decades, the use of quadcopters in various fields have vastly increased. This is due to

the significant advancements in technologies such as microcontrollers, motors and

sensors. Regardless of these uses and benefits of the quadcopter, a major cause of

concern for current researchers is to obtain stability of quadcopters subjected to

external disturbances (unknown inputs) to the system. An Unknown Input Observer

(UIO) is an observer which estimates the state variables of a system with the utility of

mathematical models by calculating the difference between measured and estimated

states in order to obtain residuals. The advantage of using a UIO over typical

Luenberger type observers is that it does not require any prior knowledge of the

unknown input in order to carry out the estimation. This thesis presents the design and

implementation of a linear Unknown Input Observer for quadcopters. The unknown

input or the external disturbance considered for this thesis is the effect of wind acting

on the body of the quadcopter. Both software and hardware tests were carried out in

order to test the robustness of the UIO on the quadcopter. The software

Matlab/Simulink and Gazebo were utilized for the software simulation of the UIO.

Two test cases were chosen to test the designed UIO in Matlab/Simulink. In the first

test case, step winds were applied to the quadcopter model whereas in the second test

case, military grade wind gusts were applied to the model. A further 3 test cases were

used to test the UIO in Gazebo whereby in two of the test cases, wind disturbances

were applied only to the x and y axis while the third case had wind disturbances applied

to the x, y and z axis. For the simulation of the UIO in Gazebo, ROS toolbox in the

Simulink library was utilized. This was used to establish communication between

Matlab/Simulink and Gazebo. From the results obtained from the software and

hardware tests, it could be said that the UIO was able to successfully estimate the wind

disturbance acting on the body of the quadcopter as the quadcopter was able to reach

its desired position regardless of the external disturbances through compensation of

such disturbances.

iii

Abbreviations 2D Two Dimensional

3D Three Dimensional

6DOF Six Degree of Freedom

APM ArduPilot Mega

DC Direct Current

DO Disturbance Observer

DWT Discrete Wavelet Transforms

ESC Electronic Speed Controller

FDIR Fault Detection, Isolation and Recovery

GPS Global Positioning System

IMU Inertial Measurement Unit

LiPo Lithium-ion Polymer

LQR Linear Quadratic Regulator

LTI Linear Time Invariant

MIMO Multiple Input Multiple Output

OS Operating System

PD Proportional Derivative

PI Proportional Integral

PID Proportional Integral Derivative

PWM Pulse Width Modulation

ROS Robot Operating System

SVD Single Value Decomposition

UAV Unmanned Aerial Vehicle

UIO Unknown Input Observer

iv

UNIX Uniplexed Information and Computing Service

Wi-Fi Wireless Fidelity

v

Table of Contents

Abstract ........................................................................................................................ ii

Abbreviations .............................................................................................................. iii

List of Figures ........................................................................................................... viii

List of Tables ................................................................................................................ x

Chapter 1 ...................................................................................................................... 1

Introduction .............................................................................................................. 1

1.1 Objectives of the Thesis ................................................................................ 2

1.2 Thesis Outline ................................................................................................ 3

Chapter 2 ...................................................................................................................... 5

Literature Review ..................................................................................................... 5

2.1 Unmanned Aerial Vehicles (UAVs) .............................................................. 5

2.2 Quadcopter Definition and its Purpose .......................................................... 6

2.3 History and Evolution of Quadcopters .......................................................... 6

2.4 Quadcopter Model and Operation ................................................................. 8

2.5 Methods Implemented to obtain Quadcopter Control and Stability .............. 9

2.6 Unknown Input Observers and its Applications/State of the art ................. 13

Chapter 3 .................................................................................................................... 19

Quadcopter Mathematical Modelling ..................................................................... 19

3.1 The 12 Quadcopter States ............................................................................ 20

3.2 Coordinate system of Quadcopters .............................................................. 20

3.3 Quadcopter Moments Equations ................................................................. 22

3.4 Dynamic Nonlinear Model of the Quadcopter ............................................ 23

3.5 Attitude Controller ....................................................................................... 23

3.6 Position Controller ....................................................................................... 30

3.7 Simulink Quadcopter Model with Attitude and Position Controllers ......... 34

3.8 Chapter Summary ........................................................................................ 35

vi

Chapter 4 .................................................................................................................... 36

Linear Unknown Input Observer for Quadcopters ................................................. 36

4.1 Unknown Input Observer Theory ................................................................ 36

4.2 Need for UIO for Quadcopters .................................................................... 36

4.3 UIO Design Process .................................................................................... 37

4.4 UIO Design for Quadcopter Model ............................................................. 43

4.5 UIO Implementation in Matlab/Simulink ................................................... 44

4.6 Chapter Summary ........................................................................................ 46

Chapter 5 .................................................................................................................... 47

Gazebo Simulation of Erle-copter with UIO .......................................................... 47

5.1 Launching the Erle-copter model in Gazebo ............................................... 48

5.2 Interfacing Matlab/Simulink with Gazebo via ROS ................................... 49

5.3 Simulation of the Erle-copter ...................................................................... 50

5.4 Chapter Summary ........................................................................................ 52

Chapter 6 .................................................................................................................... 53

Hardware Implementation ...................................................................................... 53

6.1 Components of the Erle-copter .................................................................... 53

6.2 Connection of the Erle-copter to an Ubuntu Machine ................................. 56

6.3 Erle-copter Setup and Calibration ............................................................... 57

6.3.1 Erle-copter Parameters ......................................................................... 58

6.4 Automatic Standalone Node Generation ..................................................... 58

6.4.1 Running the Generated ROS node ....................................................... 63

6.5 Experimental Setup ..................................................................................... 64

6.5.1 Wind Measurement .............................................................................. 64

6.5.2 Measured Wind Data Analysis ............................................................. 64

6.5.3 UIO Hardware Test .............................................................................. 65

6.6 Chapter Summary ........................................................................................ 66

vii

Chapter 7 .................................................................................................................... 67

Results - Software and Hardware ........................................................................... 67

7.1 Software Results .......................................................................................... 67

7.1.1 Matlab/Simulink ................................................................................... 67

7.1.2 Gazebo .................................................................................................. 75

7.2 Hardware Results ......................................................................................... 81

7.3 Chapter Summary ........................................................................................ 84

Chapter 8 .................................................................................................................... 86

Conclusion .................................................................................................................. 86

Reference .................................................................................................................... 88

Appendix .................................................................................................................... 94

Appendix A: Design of the UIO for the Derived Quadcopter Model .................... 94

viii

List of Figures Figure 2-1: UAV Classifications.................................................................................. 5

Figure 2- .6

Figure 2-3: Plus Configured Quadcopter ..................................................................... 6

Figure 2- 7

Figure 2-5: Georges de Bothezat ................................................................................. 7

Figure 2- .7

Figure 2-7: Curtiss-Wright ........................................................................................... 7

Figure 2-

Figure 2-9: Movement of the Quad copter ................................................................... 9

Figure 3-1: Inertial and Body Fixed Frame of Quadcopter ........................................ 20

Figure 3-2: Direction of rotation of the rotors of quadcopter with reference frames . 22

Figure 3-3: Linear Quadcopter Model ....................................................................... 34

Figure 3-4: Graph of Quadcopter Path ....................................................................... 35

Figure 4-1: Quadcopter Model with Wind Disturbance............................................. 36

Figure 4-2: Graph of Quadcopter Path Subjected to Wind Disturbance .................... 37

Figure 4-3: UIO Design.............................................................................................. 43

Figure 4-4: UIO Implementation in Matlab/Simulink ............................................... 44

Figure 4-5: Graph of Estimated and Actual Wind Force ........................................... 45

Figure 5-1: Erle-copter spawned in Gazebo Simulator .............................................. 48

Figure 5-2: Simulink Block Diagram for Gazebo Simulation ................................... 50

Figure 5-3: Segment of Simulink for Subscribing Position and Attitude data........... 51

Figure 5-4: ROS Computation Graph ........................................................................ 52

Figure 6-1: Erle-copter ............................................................................................... 53

Figure 6-2: Erle Brain 3 ............................................................................................. 53

Figure 6-3: Electronic Speed Controller (ESC) ......................................................... 54

Figure 6-4: Brushless DC Motors .............................................................................. 54

Figure 6-5: Propellers of the Erle-copter.................................................................... 55

Figure 6-6: Power Module ......................................................................................... 55

Figure 6-7: The Global Positioning System (GPS) + Compass ................................. 56

Figure 6-8: LiPo Battery ............................................................................................ 56

Figure 6-9: Hardware Board Setting .......................................................................... 59

Figure 6-10: Build Options ........................................................................................ 60

ix

Figure 6-11: Connection to ROS Device Dialog Box ................................................ 60

Figure 6-12: ROS Device Connection Test ............................................................... 61

Figure 6-13: Output of the rosdevice Command ........................................................ 61

Figure 6-14: Network Connection Settings in Simulink ............................................ 62

Figure 6-15: rosdevice Command Properties after Node Generation ........................ 63

Figure 6-16: Experimental Setup ............................................................................... 64

Figure 6-17: Simulink Model for Hardware Testing ................................................. 65


Figure 7-2: 3D Plot of Quadcopter Path..................................................................... 68

Figure 7-3: Graph of Wind Estimation Error ............................................................. 69


Figure 7-5: 3D Plot of Path of Quadcopter ................................................................ 70



Figure 7-8: 3D Plot of Path of Quadcopter ................................................................ 72


Figure 7-10: Graph of Estimated and Actual Wind Force ......................................... 74

Figure 7-11: 3D Plot of Path of Quadcopter .............................................................. 74

Figure 7-12: Graph of Wind Estimation Error ........................................................... 75










Figure 7-22: UIO Estimation for Wind Force (x-axis) .............................................. 82

Figure 7-23: UIO Estimation for Wind Force (y-axis) .............................................. 82

Figure 7-24: 3D Plot of Erle-copter Path ................................................................... 83


Figure 7-26: 2D Plot of Erle-copter Path ................................................................... 84

Figure 7-27: Erle-copter Attitude ............................................................................... 84

x

List of Tables Table 3-1: 12 States of the Quadcopter with its Description ..................................... 20

Table 5-1: ROS Blocks used and their Purpose ......................................................... 49

Table 6-1: Erle-copter Parameters.............................................................................. 58

1

Chapter 1 Introduction In the past few decades, the use of quadcopters has vastly increased in all fields. This

is due to significant advancements in technologies such as microcontrollers, motors

and sensors and also because of reduction of prices of these electronic devices [1, 2].

Some of the uses of quadcopters can be used for natural disaster management, weather

monitoring, forest fire detection, traffic control, cargo transport, emergency search and

rescue, communication relaying, security, surveillance, agriculture and academic

teaching such as control engineering [3, 4]. Advantages of quadcopters include high

maneuverability, a relatively cheap price and a simple construction [5]. Regardless of

these uses and advantages of quadcopters, a major challenge for researchers today is

obtaining the stability of these quadcopters in flight when subjected to external forces

unknown to the system.

In any control system, there exist inputs to the system which are not known as they are

not measurable or are acting as an external disturbance on the system. These factors

cause the system to not function optimally and hence a control mechanism needs to be

designed to cater for such unknown inputs. For the purpose of this thesis, the unknown

input or the external disturbance acting on the quadcopter will be classified as wind.

The wind causes the quadcopter to become unstable in its operation and hence unable

to reach its desired location. The significance of this thesis is to present a method of

estimating external disturbances (i.e. wind) acting on the body of the quadcopter and

then compensating them allowing the quadcopter to precisely reach its desired

location.

In order to estimate the wind acting on the quadcopter, an Unknown Input Observer

(UIO) is proposed in the thesis. The UIO will be utilized to reconstruct the wind gusts

which are unknown to the system that affects the behavior of the rotors in order to

obtain stable operations of the quadcopter in its path. The UIO approach estimates the

state variables of the system with the utility of mathematical models and finally

reconstructs the wind by calculating the difference between measured and estimated

states in order to obtain residuals [6, 7].

2

1.1 Objectives of the Thesis

In order to achieve the main aim of the thesis which is to design a linear UIO for

quadcopters, various objectives have been laid out. These objectives include:

1. Deriving the linearized model of the quadcopter.

For this objective, a linearized model of the quadcopter has to be derived. From the

nonlinear equations, a linearized model of the quadcopter will be obtained by assuming

hovering conditions. A cascaded position and attitude controller will be developed

consisting of a PD control.

2. Designing of the Unknown Input Observer for the linearized model of the

quadcopter.

In order to achieve this objective, the UIO will be designed taking into consideration

the model of the quadcopter. It will be ensured that the necessary conditions for the

existence of the UIO are satisfied then the design process for the development of the

UIO will be followed.

3. Software implementation of the UIO on the quadcopter model using

Matlab/Simulink.

For this objective, the quadcopter mathematical linear model will be implemented in

Matlab/Simulink. Then the designed UIO will be implemented in the design scheme

of the quadcopter model in Matlab/Simulink. The output from the model will be fed

into the UIO together with the known inputs (4 rotor speeds) by using which the UIO

outputs the estimated disturbances (wind) applied to the system, thereby allowing the

system to compensate for these disturbances.

4. Testing of the UIO using Gazebo software.

For this, the UIO will be tested with actual parameters of the quadcopter (Erle-copter)

to visualize how the quadcopter would perform in the actual hardware test by utilizing

the available Erle-copter model provided in Gazebo. To achieve this, Matlab/Simulink

has to be interfaced with Gazebo. For this purpose, the Robot Operating System

toolbox in Simulink will be utilized. Publish ROS blocks will be used to send data to

the Erle-copter model in Gazebo through Simulink while the Subscribe ROS blocks

will be used to get data from the Erle-copter model in Gazebo.

3

5. Hardware implementation of the UIO on the quadcopter via ROS and Simulink.

In this part, the UIO will be tested on a real quadcopter (Erle-Copter). Connection to

the quadcopter will be established through its internal Wi-Fi module and ROS will be

used with Matlab/Simulink to build and upload the program inside the brain of the

quadcopter. The quadcopter will then be tested on windy days and the ability of the

UIO implemented to estimate the wind acting on its body will be assessed.

1.2 Thesis Outline

This thesis is divided into 8 chapters. The content of each of these chapters are as

discussed below:

Chapter 2: In this chapter, a review of the past and present work relating to the thesis

is presented. The readers are provided with the background knowledge of the subject

of the thesis. The chapter concludes with indicating what the thesis aims to achieve

by the proposed work.

Chapter 3: In this chapter, a linearized mathematical model of the quadcopter, more

specifically the linear attitude controller and linear position controller is derived. This

mathematical model is then implemented using the software Matlab/Simulink.

Chapter 4: The linear Unknown Input Observer is designed in this chapter for the

linear model of the quadcopter developed in the previous chapter. All necessary and

sufficient conditions required for the existence of the UIO are presented. The UIO is

then implemented with the quadcopter model in Matlab/Simulink.

Chapter 5: In this chapter, the designed UIO for the quadcopter is tested in Gazebo

software where the actual parameters of the real quadcopter (Erle-copter) is used. This

chapter presents as to how Simulink is interfaced to Gazebo via Robot Operating

System (ROS) toolbox. The entire process of simulating the Erle-copter in Gazebo

software is presented in this chapter.

Chapter 6: In this chapter, steps of hardware implementation of the Erle-copter with

the designed UIO are discussed. The hardware setup of the Erle-copter are presented

and the tasks carried out to achieve hardware implementation of the UIO have been

outlined and discussed.

Chapter 7: The software results from Matlab/Simulink and Gazebo as well as the

hardware results of the quadcopter with the UIO subjected to external disturbances

4

(wind) are presented in this chapter and important conclusions about the behavior of

the designed system are made.

Chapter 8: This is the concluding chapter where discussions are made about the results

and recommendations for future work are proposed.

5

Chapter 2 Literature Review

2.1 Unmanned Aerial Vehicles (UAVs)

An Unmanned Aerial Vehicle (UAV) is a powered aircraft that produces either

computer or remote controlled motion utilizing aerodynamic forces and operates

without human carriers [8]. In the past few decades, the popularity of UAVs have

vastly increased due to the improvement in the technological advancements and the

reduction of prices of electronic devices such as microcontrollers [8]. UAVs have

significantly contributed to the recent technological innovations with their capacity to

replace manned aerial vehicles thus, diminishing numerous aerial operational costs [9].

Introduction of the first successful UAV flight was perceived in 1920s whereby a

sophisticated device was built. With developments in electronics engineering, research

introduced devices of increasing power, low weight and decreased complexity [8].

UAVs can move in any direction and are capable of low speed flights. Specified these

characteristics, UAVs play crucial roles in rescue and search missions, meteorology,

medical emergencies and other suited applications in control engineering [10]. UAVs

can be classified according to the categories shown below:

Figure 2-1: UAV Classifications [11]

6

2.2 Quadcopter Definition and its Purpose

A quadcopter is a flying object which changes its attitude and altitude by four rotating

blades [5]. The four rotors of the quadcopter are directed upwards and are placed in a

square formation, each, an equal distance away from the center of mass of the

quadcopter [2, 12]. There are two major configurations of the quadcopter which are

the plus (+) and cross (x) configurations. In the x configuration quadcopter, the two

rotors spinning in the same direction are tilted at an angle of 45 degrees from the x and

y axis, whereas the same spinning motors in the plus configuration are aligned with

the x and y axis. According to [11, 13], an x configured quadcopter is considered to

have more stability when compared to the + configuration which is more suited for

acrobatic maneuvers.

Figure 2-2: Cross Configured Quadcopter [11] Figure 2-3: Plus Configured Quadcopter [11]

In the past, quadcopters were primarily used for military applications [3] where they

were used for dangerous and hostile conditions which could pose threat to the lives of

humans if carried out by them. However, the recent decrease in the price as well as

size of semi-conductor logic and due to significant advancements in technologies such

as microcontrollers, motors and sensors, the application of quadcopters in several other

fields have been achieved such as natural disaster management, weather monitoring,

forest fire detection, traffic control, cargo transport, emergency search and rescue,

communication relaying, security, surveillance, agriculture and academic teaching

such as control engineering [1-5].

2.3 History and Evolution of Quadcopters

The French Scientist Charles Richet built in the 20th century a small unpiloted

helicopter [10]. While his attempt to accomplish was unsuccessful, it inspired one of

his students Jacques Bréguet, together with his brother Louis Bréguet, to the

construction of an aerial unmanned vehicle, the first quadrotor.

7

The quadrotor was built in 1907 and the first unmanned flight was registered by the

Gyroplane No.1 (Figure ). Each propeller of the quadrotor was 8.1 meters in

diameter and weighed 578 kg which was powered by a 50 horse power (37.3kW)

internal combustion engine [14]. This was however a tethered flight. The main reason

for the helicopter not to fly freely was due to the lack of proper control and stability.

More attempts were made after the flight of the Gyroplane No. 1, but these attempts

were more focused on manned quadrotors for example the Flying Octopus in Figure

2-5 by Georges de Bothezat in 1922 and the Oemichen No. 2 in Figure 2-6 by Etienne

Oemichen also in 1922. Apart from these attempts to manned quadrotors, other

attempts had also been made such as the Convertawings Model A by Oemichen and

de Bothezat in 1922, the Curtiss X-19 by Curtiss-Wright Corporation in 1963 which is

shown in Figure 2-7 below, the Bell X-22A in 1966 by Bell Aircraft Corporation and

the fly vehicles of the Moller Company [14].

Figure 2-4: Bréguet Richet Gyroplane No. 1 [14] Figure 2-5: Georges de Bothezat [14]

Figure 2-6: Oehmichen No.2 [14] Figure 2-7: Curtiss-Wright [14]

A quadcopter has a Six Degree of Freedom (6DOF), i.e. it is able to move in both the

translational and rotational directions (translational x, y and z and rotational roll, pitch

and yaw) [2, 11] and [12]. The position and altitude of quadcopters change in space

and its motion is controlled by varying the speed of the individual four rotors. Since

the number of system inputs (4) are less than the number of outputs, this causes the

8

quadcopters to be severely under actuated and dynamically unstable with four input

forces for each rotor, four propulsors and four controllable variables.

A quadcopter can be defined as a helicopter with 4 rotors. It is a dynamic system

which is under-actuated as it has four input forces where one force acts on each of the

rotors and has a six Degree of Freedom (6DOF) [9].

In order to obtain a stable flight, various high accuracy sensors with fast robust control

algorithms are combined [2]. Currently, quadcopters are being intensively developed

in three major technological advances: military, transportation and entertainment [10].

2.4 Quadcopter Model and Operation

The role of each rotor of the quadcopter is to generate a certain amount of thrust and

torque around the quadcop also to generate a drag force

which will be produced in the quadcopters opposite direction of flight [10]. The

propellers of the quadcopters are always divided into two pairs which are the pusher

blades and the puller blades which operate in contra-rotation. Hence, if all of the four

propellers work with the same angular velocity then the resulting net torque is zero and

the quadcopter remains still around its center of gravity [10].

In the 6DOF, the motion of the quadcopter is controlled by individually varying the

speed of each of the four rotors, which in turns manipulates the lift and rotational

forces. In order to define the orientation of the quadcopter around its center of mass,

three dynamic parameters are essentially defined which are the roll, pitch and yaw

angles [10]. Referring to (Figure 2-8), in order to obtain the changes in the pitch angle

the speed of the propellers 1 and 3 are varied which results in the forward and backward

movement of the quadcopter. Likewise, the change in roll angle is produced by

varying the speed of the propellers 2 and 4 and thus will achieve lateral movement of

the quadcopter. Moreover, the change in the yaw angle is achieved by creating a

mismatch of the thrust generated between the counter rotating pairs of blades.

Therefore, a quadcopter can be maneuvered in a desired direction by controlling the

roll, pitch and yaw angles of the quadcopter [10].

9

Figure 2-8: Configuration of a Quad copter [10] Figure 2-9: Movement of the Quad copter [15]

Figure 2-8 demonstrates the direction in which the roll, pitch and yaw angles act on,

causing the actual movement of the quadcopter. The maneuverability of the

quadcopter can be achieved through the variation shown above in Figure 2-9 which

shows the rotor speed variations needed in order to achieve movements in each desired

direction.

2.5 Methods Implemented to obtain Quadcopter Control and Stability

Although quad rotors have witnessed massive evolution, its control system

are highly non-linear due to coupled translational and rotational motion, very little

friction and additional aerodynamic effects generating an intriguing control obstacle

[2]. Thus, a sophisticated control becomes mandatory. To conquer this, continuous

research is being directed towards the introduction of upgraded robust controllers and

modelling techniques to produce comprehensive and precise representations of real life

unmanned aerial vehicles [10].

Due to the need of quadcopters to maintain a stabilized autonomous system, several

linear and non-linear control methods such as PD, PI, PID, LQR and non-

have been implemented. Karwoski (2011) in a study of Quadrocopter Control Design

and Flight Operation compared the PD controller with PI and PID controllers in

obtaining the stability of the quadcopter and meeting system specifications. Only the

phi (roll) axis was used for the purpose of testing the controllers [16]. The PI

controller, was ineffective in controlling of the quadcopter as the integral component

of the controller was unable to counter the oscillations in the system. Moreover, the

study further showed that when using a PID controller, the overshoot of the system

was reduced slightly but on the other hand, increased the steady state error. This

10

however was not evident in the PD controller. Due to this fact, the PID controller was

deemed unnecessary for the system. Similarly, Gibiansky (2012) in his paper on

Quadcopter Dynamics, Simulation, and Control also compared the use of PD and PID

controllers in the application of quadcopters. The PD controller was used in the

quadcopter model to drive the angular velocities and angles to zero [17]. However,

from simulations, it was noted that the angles were not completely driven to zero

together with the steady state error. This was demonstrated as a common error

associated with the use of PD controllers which could be slightly improved by the use

of PID controllers [17]. The advantages of the PD controller lies in its simplicity as

well as the ease of implementation, but falls short in controlling of mechanical systems

especially in the presence of noise and disturbances and often results in producing

higher steady state error [17]. Hence a PID controller was used to compensate for the

shortcomings of the PD controller. As is known, the PID controller includes an extra

term incorporated which is the integral term, hence, contrary to [16], this enables the

controller to reduce the steady state error as it is able to track the path of the quadcopter

and help stabilize it with a much smaller steady state error when compared to the PD

controller. Likewise, [18] compares the robustness of a classical PID with a cascaded

PID control. The merits of a PID control have been emphasized by several authors

such as in [19-21]. PIDs have been widely used in order to achieve stability for the

quadcopters and has been quite effective as well. According to [18], the main

drawback of PIDs is that when the disturbance causes a large error in the system, the

steady state response is compromised hence providing a less robust control system.

Therefore, a cascaded PID has been implemented for the stability of the quadcopter by

lowering the system sensitivity to external disturbances.

Moreover, in [12], a PD controller is implemented in order to achieve stability for the

quadcopter. The PD controller was chosen due to its simplicity and ease of

implementation. However, the accelerations in the x and y directions are not

considered by the PD control method hence giving need for the construction of another

control method that would provide access to all of the positions and angles of the

quadcopter. Therefore, a heuristic method is utilized for the trajectory control of the

quadcopter that enabled the quadcopter to move from the current position to the desired

position by controlling the quadcopter rotor velocities. According to [22], another

method to take into account the possible deviations in the angles, was to integrate a PD

11

controller into the heuristic method. The integrated PD controller was proposed to

reduce the fluctuations in quadcopter behavior under external forces. This proposed

method performed well in the example cases, but performance was dependent on the

parameter values and so if the parameter values were small, the controller would not

respond quickly enough to follow the planned trajectory. A PD control method for the

stabilization of the quadcopter has also been proposed in [23]

observer for fault detection and isolation tasks. From the test results, it was seen that

the accelerometer and inclinometer faults could be detected as well as isolated.

However, disturbance rejection strategy was still under investigation. In addition, [3]

shows that a quadcopter with payload is able to compensate for external disturbances

with a use of a PID controller. According to [3], one of the most popular ways of

controlling a quadcopter is through the use of PID controllers. In order to represent an

external force such as wind acting on the quadcopter, a disturbance was introduced.

From the test results the paper concluded that with the utilization of the PID controller,

the quadcopter was able to compensate the applied disturbances correctly.

Furthermore, [24] proposes the control of a quadcopter with the implementation of a

PID control together with the 1 adaptive control. A PID would be sufficient enough

to control a quadcopter however, in the presence of disturbances and uncertainties, this

control is highly compromised [24]. In order to counter the problem of these

disturbances and uncertainties, the 1 adaptive control is implemented due to its fast

and robust adaptation for the desired performance in the presence of disturbances and

uncertainties. The advantage of 1 adaptive control is that it can be used in a system

with varying parameters and uncertainties [24]. When compared with the PID control,

the 1 adaptive control has a faster adaptation to the present disturbances and

uncertainties due to its high adaptation gain.

Similarly, Balas (2007) in his MSc thesis compared a PID controller, LQR controller

and H infinity controller for the control part of the quadrotor. The PID controller was

implemented on the linearized model for the control of the attitude around the hover

situation [25]. Both software simulation and validation of the simulation on the real

system were carried out and showed efficient results in terms of the steady state error.

However, this situation was only possible if the quadcopter was in hover position and

if strong disturbance was applied on the system, the quadcopter with the PID controller

would not be able to reach the hover position on its own. In addition, two forms of

12

LQR controllers, namely classic and state dependent LQR controllers, were also

discussed by Balas in his thesis. These controllers were used to obtain stable

operations of the quadcopter at hover position. A disadvantage for the classic LQR

controller is that it does not produce satisfactory results at higher thrust levels. Finally,

the H-infinity controller was studied in the thesis for the stabilization of the quadcopter.

The major role of an H-infinity controller was to produce robust stability as well as

good control of the trajectory of the quadcopter [25]. The H-infinity controller

possesses a great advantage of being divided into two loops which are the inner loop,

responsible for the stabilization of roll, pitch and yaw angles as well as the vertical

speed, and the outer loop, responsible for the longitudinal and lateral speed, the height

and the yaw angle of the quadcopter. This configuration enables the controller to

provide satisfactory results when the quadcopter is subjected to various inputs and

disturbances. Domingues (2009) utilized a linear quadratic regulator for the optical

control of the system. The linear quadratic regulator had been utilized since it is able

to control Multiple Input Multiple Output (MIMO) systems since in quadcopters, there

are inputs in terms of motor speed of the four rotors as well as outputs in the form of

sensor readings for which using a classical controller would be quite difficult to

implement [10]. However, after running the tests on the system, it was observed that

when all 12 states of the quadcopter were readable, the linear quadratic regulator was

able to fully control the system but when only 6 of the 12 states of the quadcopter were

readable then the linear quadratic regulator was only able to control the 6 available

states while the other 6 visibly drifted with time [10].

Furthermore, [4] proposes a sliding mode control approach in order to control a

quadcopter in the presence of external disturbances as well as quadcopter actuator

faults. This control method had been specif

insensitivity to the errors of the model, uncertainties in the parameters and external

disturbances. The Sliding Mode control structure has the advantage of rejecting

disturbance in fault free conditions as well as aids in recovery of some control

performances in the situation of a fault. Main disadvantage of this method is that it is

unable to control the system in the presence of a fault. Similarly, for the estimation of

external disturbances acting on the body, [26] presents a nonlinear disturbance

observer where Lyapunov stability theory was used to analyze the stability of the

disturbance observer. From the results, it was concluded that the backstepping

controller with the disturbance observer was able to track the position of the quadcopter

13

despite the presence of external disturbances acting on the body. According to [26], it

should be noted that the control strategies mentioned in [27-32] do not provide

robustness against external disturbances and uncertainties in the value of the

parameters as they assume to have zero external forces acting on the quadcopter. In

[10], a method of using Kalman filter for the control of the quadcopter was proposed.

However, it was observed that when using the Kalman filter, only six (angular speeds

and Euler angles) out of the twelve states of the quadcopter were available for control.

Moreover, it was also found out that none of the system states were reachable.

Following these findings, two LQR controllers were developed, one for an ideal system

where all of the twelve states were available for control and the second one where only

6 states were available from the Kalman filter.

2.6 Unknown Input Observers and its Applications/State of the art

The present study focuses on the use of Unknown Input Observers (UIO) for obtaining

stability in the control of quadcopters being subject to external/unknown disturbance

(wind). The UIO will be utilized to reconstruct the wind gusts which are unknown to

the system that affects the behavior of the rotors in order to obtain stable operations of

the quadcopter in its path. The UIO approach estimates the state variables of the

system with the utility of mathematical models by calculating the difference between

measured and estimated states in order to obtain residuals [6, 7]. Research describes

unknown input observers as linear observers that are entirely independent of the

immeasurable disturbances and principally estimate the error of the state decoupled

from these disturbances (unknown input) [33]. According to [6], an observer can be

classified as an unknown input observer if the error of its state estimation

asymptotically reaches zero regardless of the unknown input present in the system.

Assuming perfect knowledge of the model it exactly decouples the effects of external

disturbances from the state estimation.

A first observer was proposed in early 1960s by Luenberger and further developed.

Afterwards, early literature had focused on observers for purely continuous-time linear

time-invariant systems after which the theory had been modified and developed into

various discrete, stochastic and time-varying systems [10]. According to [34], a

Luenberger type observer cannot be directly designed for systems with unknown

inputs since it requires knowledge of the input for estimation. Linear system with

unknown input is divided into two classes; the full and the reduced order approaches

14

which can be further classified into geometric, algebraic, and generalized inverse

approaches. Full-order designs are simpler and estimate all the states, whereas reduced

order approach are more complex and tend to avoid redundancy caused due to

reconstruction of states [57].

According to [35-39], the problem of designing an observer that is subjected to inputs

which are unknown has been the pole of attraction for many researchers. Since there

are cases which often occur whereby some of the system states are not accessible due

to the unmeasurable disturbances, this problem tends to be of considerable interest

from both the theoretical and practical viewpoint. Since a conventional observer

cannot be utilised in systems subjected to disturbances, this problem gains a greater

significance. According to [34], a Luenberger type observer cannot be directly

designed for systems with unknown inputs. Hence, the reason why several different

approaches to an observer design for unknown inputs have been studied by many

researchers.

As stated in [7], observers add complexity to the system and require computational

resources. Moreover, when the parameters of the plant in operation changes

substantially, the observers become less robust when compared to physical sensors.

However, skillful application of an observer can provide substantial benefits in

performance and is the case in many research papers. Examples of impressive

applications of nonlinear observers to the control of electric machinery can be found

in [40-42]. Two main types of observers are full-order observers and reduced order

observers [43].

Due to the advantage of the reduced order observers being able to avoid redundancy

in the reconstruction of states and saving computational time, many researchers have

based their research on the reduced order observer type. For instance, [37, 39 and 44]

proposes the design of a reduced order observer for the observation of unknown inputs.

In papers [39, 44], the reduced order observer is simplified to a standard observer

through some simple substitutions and coordinate transformations and thus making it

possible to be solved through the famous Luenberger theory. As a result, the design

of the observer is further simplified. Moreover, [37] also presents the design of a

reduced order observer and their approach consists of characterizing the observer with

a delay enabling in the ease of establishment of the necessary conditions of the

existence of the unknown input observer. The results of these papers indicate that the

15

designed observers were successfully able to observe the unknown inputs present in

the system. On the other hand, [38] compares the full order observer with the reduced

order and states that even though the reduced order observer has the advantage of a

faster computational time, it may restrict the rate of convergence of some of the state

estimates. Therefore, due to the simplicity in its design, many researchers have also

preferred utilizing the full order observers for systems with unknown inputs. In [45],

a full order observer is proposed. The full order observer is reduced to a standard

observer in order to simplify the design process by switching from unknown inputs to

inputs which are known. This is similarly seen in [46] where a full order observer is

reduced to standard observer where unknown inputs are converted to inputs which are

known and hence simplifying the overall design.

Unknown input observers (UIOs) have been extensively used in induction motor

applications and it has been investigated by several researchers. Observers with non-

linear decoupling can be used to detect faults in the machines via residual generation,

non-linear approaches can be used for fault diagnosis of the motor. In 2006, Chung-

Wei et al. detected a servomechanism fault by developing a detection procedure

established on the PI controller and discrete wavelet transforms (DWT) method. UIOs

have been useful in fields of studying fault detection, isolation and recovery (FDIR)

systems such as in cases of aircraft elevator redundancy control systems. New

approaches have also introduced UIOs in sensor fault estimation techniques. Spurgeon

in 2008 presented uniqueness in sliding mode observers for its ability to generate

sliding motion between measured and observer outputs [33]. Similarly, Guezmil et al.

in 2015 compared the performance of a High Order Sliding Mode and Unknown Input

Observer which was embedded in the integral sliding mode control scheme for the

drive of an induction machine [47]. The overall stability of the system was achieved

through the theory of Lyapunov. The purpose of the two controllers were to estimate

the machine states. The controllers were compared against each other to determine

how robust each controller was in both high and low speeds under machine resistance

and disturbances in terms of load. The comparisons showed that the High Order

Sliding Mode controller was more robust in achieving better speed and flux tracking

when there were variations in the induction machine parameters [47].

Moreover, UIOs have been constantly associated with fault detection and isolation in

induction motor actuators. UIO is used to compute residuals attained from the

16

difference between the actual and estimated outputs which tend to eliminate the effects

emerging from unknown disturbances on the system. Gaeid et al. deduced a fault

detection and isolation of a multiple input multiple output based system resulting in

the design of several fault banks. These banks generated residuals which are sensitive

to only a specific fault [48]. Actuator faults associated with UIO fault detection was

also studied by Ermeydan et al. which utilized a derived linear model from a nonlinear

model using Jacobian matrix to detect and isolate actuator faults. The technique of

generalized observer scheme based UIO was used. Observer based state estimation is

applied in order to find and isolate the faults in the system so that state estimation error

could be decoupled when there are unknown inputs in the form of disturbances. For

stabilization and control, linear quadratic tracking and linear quadratic regulators are

implemented to compensate for the resulting steady- state errors [6]. Gaddouna et al.

developed a fault detection based UIO for a linear time invariant system subjected to

only random unknown inputs. The method designed utilized a mathematical algorithm

comprising of matrix inverses [49].

In addition, unknown input observers are also used in models associated with control

designs for improving efficiency in motion systems. Hsu et al. (2001) developed an

UIO constructed procedure to decouple the influence of external disturbances which

in real life would impeccably enhance the performance of motion systems. Riccati

equation is applied to diminish the estimation error via the design of an ideal UIO [50].

Liu et al. (2002) proposed a sophisticated scheme to simultaneously approximate

unknown states as well as disturbances of a linear time invariant system employing an

output feedback observer. Researchers applied Luenberger-like observer to estimate

states and inverse dynamics to estimate disturbance signals. The structured design

apparently can be applied to a diverse range of disturbances since the strategy does not

incorporate any disturbance model for estimation [51]. Pillosu et al. (2012) linearized

a nonlinear system and combined a conventional linear UIO with a non-linear

Disturbance Observer (DO) constructed on the sliding mode method to design an

algorithm. The designed scheme efficiently estimated the time varying infiltration and

discharge parameters of an open channel hydraulic arrangement [52].

Other work on the UIO includes in [34] where a novel approach for the design of the

UIO was presented. According to [34], a UIO has a derivation of system inputs which

is not available from the measurement directly. Hence, a Block function differential

17

operations method for the design of the UIO is proposed which would enable the

designed UIO to not need the derivation of the system outputs for the estimation of the

unknown inputs. Moreover, in [53], an output feedback controller is proposed that

would simultaneously estimate both the unknown states and the disturbance of linear

time invariant (LTI) systems. For the estimation of the states, a Luenberger-like

observer is used while an inverse-dynamics motivated algorithm is used for the

disturbance estimation. A major advantage highlighted for this method was that the

proposed method could be applied to any system with different disturbances since no

disturbance model is required for the estimation. In [54], the method of Single Value

Decomposition (SVD) to design a Luenberger observer that could be used in a system

with unknown inputs is presented. The advantage of using the SVD method is that it

provides a practical way of implementing the observer design. In addition, [55]

proposes a method of an algebraic design of UIO that does not satisfy the observer

matching condition for systems which are linear and subjected to unknown input.

According to [55, 56], the decoupling principle of the UIO where the observer is

designed to be insensitive to unknown inputs has been one of the most successful

observer techniques. The method however has a drawback which is that the dimension

of the observer increases considerably.

To sum up, Quadcopters as discussed above have various applications and advantages

but a major cause of concern is the ability to obtain control over the quadcopter in its

trajectory due to unexpected obstacles and external noises. In order to achieve this

desired control and stability of the quadcopters, several researches have been carried

out proposing different approaches to obtain this control and stability. One possible

approach which could be used on the quadcopters for this control and stability is the

use of an Unknown Input Observer. This technique has been previously used in

induction machines for detection of faults and hence this study would like to further

extend this technique by implementing it on quadcopters.

This thesis presents a Linear Unknown Input Observer for quadcopters. The unknown

input observer will be used to reconstruct the wind gusts (external

disturbance/unknown inputs) in order to estimate the unknown input acting on its body

and thus compensate for these external disturbances through some control strategy.

The UIO will be designed for a linear model of the quadcopter. Both software

simulations using Matlab/Simulink and Gazebo and hardware tests will be carried out

18

as part of the project in order to validate the performance of the UIO with the linear

quadcopter model subjected to unknown inputs (wind).

19

Chapter 3 Quadcopter Mathematical Modelling A quadcopter is a flying object which changes its attitude and altitude by four rotating

blades [5]. The four rotors of the quadcopter are directed upwards and are placed in a

square formation, each, an equal distance away from the center of mass of the

quadcopter [2, 12]. This chapter presents the derivation of the mathematical model of

the quadcopter. More specifically, the linear position controller and the linear attitude

controller from the nonlinear model of the quadcopter will be derived. The purpose of

deriving the mathematical model of the quadcopter is to describe the behavior and the

movement of the quadcopter with respect to input and external disturbance values [58].

This enables the simulation of the behavior of the quadcopter in various conditions.

Nomenclature

- Length of the quadcopter

F Force

M Moment

- Inertia in the x-axis

- Inertia in the y-axis

- Inertia in the z-axis

- Force constant

- Moment constant

- Motor speeds

- Hovering speed of Quadcopter

Desired x coordinate

Desired y coordinate

Desired z coordinate

Desired rotation in the z axis

20

3.1 The 12 Quadcopter States Table 3-1: 12 States of the Quadcopter with its Description

Name Description

Describes the linear position of the quadcopter in the x direction

Describes the linear position of the quadcopter in the y direction

Describes the linear position of the quadcopter in the z direction

Describes the angular position (orientation) of the quadcopter in the x

direction

Describes the angular position (orientation) of the quadcopter in the y

direction

Describes the angular position (orientation) of the quadcopter in the z

direction

Describes the linear velocity of the quadcopter in the x direction

Describes the linear velocity of the quadcopter in the y direction

Describes the linear velocity of the quadcopter in the z direction

Describes the angular velocity of the quadcopter in the x direction

Describes the angular velocity of the quadcopter in the y direction

Describes the angular velocity of the quadcopter in the z direction

3.2 Coordinate system of Quadcopters

In order to derive the mathematical model of the 6DOF quadcopter, we must define

two reference frames.

Inertial frame:

Body frame:

Body Frame

Roll

Pitch

Yaw

z

y

x

Inertial Frame

21

The inertial reference frame is defined by the ground in which the gravity points in the

negative direction of the z axis while the orientation of the quadcopter is defined by

the body frame. In this reference frame, the rotor axes are pointing in the positive z

direction while the arms of the quadcopter in the x and y direction [2].

The linear position and the linear velocity of the quadcopter are defined in the inertial

frame as and , respectively, while the body frame is used to

define the angular position and the angular velocity of the quadcopter as

and , respectively.

In order to align the axes of the inertial frame to those of the body frame, we choose

the proper representation of the orientation of the quadcopter. For this thesis, the ZXY

Euler angles are chosen. This means that in order to align the inertial reference frame

with the body frame, we must first apply a rotation of the yaw angle then a rotation

of the roll angle , and then a final rotation of the pitch angle . Hence, the complete

rotation matrix of the quadcopter can be obtained as follows. The elementary rotations

are

Therefore, the total rotation is

CosSin

0 0 1 0 Sin Cos Sin

22

(3.1)

Equation (3.1) above represents the rotation matrix of the quadcopter used in the thesis.

3.3 Quadcopter Moments Equations

In this section, the equation for the quadcopter is derived. According to the

direction of rotation of the rotors with respect to the appropriate axes, the equations of

moments are derived. Figure 3-2 shows the configuration of the quadcopter with the

direction of rotation of each of the rotors as well as the direction of each of the three

axes.

Figure 3-2: Direction of rotation of the rotors of quadcopter with reference frames

From Figure 3-2, it can be seen that rotors 2 and 4 rotate in the positive direction of

the z axis of the body frame while rotors 1 and 3 rotate in the negative direction of the

axis. Moreover, since the produced moment on the quadcopter is in the opposite

direction to the rotation of the propellers, therefore, moments and will act in

the negative direction of the z axis of the body frame of the quadcopter while moments

and will act in the positive direction of the z axis. Furthermore, since the rotors

of the quadcopter are separated from the center of mass with a distance of , therefore

the rotors labelled 2 and 4 will produce moments along the y axis of the body frame in

23

the positive and negative directions respectively. Meanwhile rotors labelled 1 and 3

will produce moments along the x axis of the body frame in the negative and positive

directions respectively. Finally, the moments equation of the quadcopter can be

represented as described in equation (3.2) below:

(3.2)

3.4 Dynamic Nonlinear Model of the Quadcopter

The dynamic nonlinear model of the quadcopter is as follows:

(3.3)

(3.4)

(3.5)

These nonlinear equations of the quadcopter are utilized in deriving the linear attitude

and position controllers discussed in the next two sections of the chapter.

3.5 Attitude Controller

This section of the chapter will discuss the derivation of a linear attitude controller for

the quadcopter. In order to design an attitude controller from the equations discussed

above, we use the dynamic equations (3.4) and (3.5):

24

(3.6)

Dividing both sides of equation (3.6) above with and rearranging the equation,

we obtain:

(3.7)

Moreover, since the quadcopter is assumed to be in hovering condition (i.e. in

equilibrium state), the angles roll and pitch are almost equal to zero i.e. . Due

to this condition and referring to equation (3.4), replacing and angles with 0,

and . Therefore, equation (3.7) can be represented as:

(3.8)

From the above dynamic system, we can obtain an attitude controller by linearising the

system around the hovering condition. It is assumed that at the hovering condition,

(where m is the mass of the quadcopter and g is the gravitational

force). Moreover, the following identities have been used for the linearisation of the

system in equation (3.8); and .

Solving for :

25

Since hovering condition is assumed, Therefore the term is

cancelled out. Hence,

Making use of the identity , we obtain:

and

Hence, by replacing this identity into the above equation, can be rewritten as:

Expanding the above equation we obtain:

Since terms can be cancelled from the above equation:

Morover, it is known that therefore:

and

4 4 o and

Hence, the equation for simplifies to:

26

In addition, since is defined as , we obtain through substitution:

Finally, simplifies to:

Solving for :

Since hovering condition is assumed, Therefore the term is


Utilising the identity , we obtain:

3 o 3 and

Hence, through the substitution of this identity in the abobe equation, can be

rewritten as:


Cancelling out the terms from the above equation due to the opposite signs:

27

Moreover, since , therefore:

3 3 o and

and

Hence, the above equation simplifies to:

Moreover, since is defined as , by substitution we obtain:

Finally, simplifies to:

F3 1

yy

mgkI

Solving for :

Since hovering condition is assumed, Therefore the term yy xx

zz

I II

is


Utilising the identity , we get:

, , and 4 o 4

Hence, through substitution can be rewritten as:

2 2 2 2M1 o 2 o 3 o 4 o

zz

k ((I

28


Cancelling out the terms due to opposite signs from the above equation, we

obtain:

Moreover, since , therefore:

and

and 2 ow w

and

and

Therefore, the above equation simplifies to:

Furthermore, since is defined as and we obtain:

Finally, by combining all the derived equations (i.e. , and ), the linearised

model can now be represented as equation (3.9) below:

29

It should be noted that the objective in deriving an attitude controller is to force the

orientation of the quadcopter according to a well-behaving and known dynamics.

Therefore, the derived model is compared with a reference model shown below in

equation (3.10):

(3.10)

The constants in the above equation can be obtained by choosing the location of the

desired eigenvalues appropriately. Moreover, from the above equation, and

where is the commanded roll and is the commanded pitch angles.

Comparing equations (3.9) and (3.10) and equating the equations together, we obtain

the following:

(3.11)

Where,

30

Furthermore, we need to determine the variation laws of the rotor speeds of the

quadcopter. Hence, in the derivation of a linear attitude controller, we introduce a

new quantity and define it as:

Hence, variations for the speed of the rotors of all can be described as:

(3.12)

Then, making the subject of the formula in equation (3.12),

we obtain the following set of equations:

(3.13)

Finally, after putting everything together and expressing all quantities in the original

coordinate frames, the final linear feedback attitude controller can be represented as:

(3.14)

3.6 Position Controller

This section of the chapter will discuss the derivation of a linear position controller for

the quadcopter. Its objective will be to control the position of the center of mass of the

31

quadcopter. The center of mass of the quadcopter is controlled by the total thrust

generated by the propellers as well as the values of the commanded roll and the

commanded pitch. The dynamic equation that describes the position of the quadcopter

described in equation (3.3) is represented as:

In the above equation, the variable f is defined as:

Similar to the attitude controller, a linear feedback position controller will be designed

around the hovering condition. The parameters taken into consideration are

and . Moreover, the variables of the position error are defined as

, , and . Finally, the variables for input

variation are defined as , and .

Solving for :

In order to obtain the linearized position controller, hovering condition was assumed.

Hence

Note: and , likewise, (this is achieved

due to the assumption of hovering condition)

Hence, through substitution of the above in the equation we obtain:

With the use of the variables of the position error defined above and through

simplifications, the following equations are obtained for :

32

Finally, can be represented as:

Solving for :

y Sin

With the use of the variables of the position error defined above and through

simplifications, the following equations are obtained for :

Finally, can be represented as:

c d c dy

Solving for :

Since (due to assumption of hovering condition):

Then, according to the input variation variable , can be represented as:

33

Hence, the linearized model describing the position of the quadcopter can finally be

represented as:

(3.15)

In addition, we can also express the term in terms of the speed of the 4 rotors:

(3.16)

The next step was to compare the derived linearized model with a reference model in

equation (3.17) in order to stabilize the quadcopter in its center of mass. Hence:

(3.17)

Therefore, equating equation (3.17) with the derived linearized model in (3.15):

(3.18)

Hence, the final linearized position controller of the quadcopter can be represented as:

(3.19)

Where:

34

3.7 Simulink Quadcopter Model with Attitude and Position Controllers

Figure 3-3: Linear Quadcopter Model

Figure 3-3 shows the implementation of the linear quadcopter model in Simulink. The

linear attitude and position controllers which have been derived in this chapter have

been utilized in the implementation of the quadcopter model. The reference or desired

coordinates ( are entered by the user from which the appropriate

motor speeds are generated and fed into the linear quadcopter model represented as the

state space block in figure 3-3 above. Finally, the linear quadrotor model outputs the

12 states of the quadcopter as the output of the model. Presented below is a test case

conducted for the linear quadcopter model shown in figure 3-3. The coordinates

entered as reference values were:

35

Figure 3-4: Graph of Quadcopter Path

Figure 3-4 shows the path of the quadcopter for the test case explained above. From

the graph above, it can be concluded that the quadcopter has successfully managed to

reach the desired coordinates which were entered at the start of the program.

3.8 Chapter Summary

The goal of this chapter was to derive a mathematical model of the quadcopter. More

specifically, a linear attitude controller and a linear position controller for the

quadcopter model were derived. Hovering conditions were assumed for the derivation

of the two controllers. The position and attitude controllers were derived in order to

control the translational position and the attitude of the quadcopter respectively.

36

Chapter 4 Linear Unknown Input Observer for Quadcopters

4.1 Unknown Input Observer Theory

An Unknown Input Observer is an observer used to estimate the unknown input or

disturbance acting on a system. A major advantage of the UIO over other observers

is that it does not require knowledge of the unknown input in order for the estimation

unlike the simple Luenberger observers. The UIO approach estimates the state

variables of the system with the utility of mathematical models by calculating the

difference between measured and estimated states in order to obtain residuals [6, 7].

As discussed previously, dynamic systems can often be modelled as a system having

unknown inputs and various methods have been presented in the literature for the

design of observers for these systems. This chapter will discuss the need for an

unknown input observer for quadcopters. Moreover, the design procedure for the

construction of an unknown input observer to be used for with quadcopters will be

discussed. The unknown input/disturbance taken into consideration for this thesis is

the influence of wind acting on the body of the quadcopter.

4.2 Need for UIO for Quadcopters

For the quadcopter model implemented in chapter 3 of the thesis, a step disturbance

(representing the effect of wind) is applied to the quadcopter model, which is unknown

to the system, and is tested if the quadcopter will still be able to reach its target position

regardless of the applied disturbance. This is seen in Figure 4-1.

Figure 4-1: Quadcopter Model with Wind Disturbance

A step disturbance of magnitude 3.5 m/s and 2.5 m/s were applied to the system in the

x and y axis respectively. In addition, the target position for the quadcopter was set as

37

3m, 4m and 5m in the x, y and z axis respectively. The results obtained are discussed

below.

Figure 4-2: Graph of Quadcopter Path Subjected to Wind Disturbance

Figure 4-2 shows the graph of the path of the quadcopter and the reference position

which was the target position of the quadcopter. Since wind disturbance was applied

to the x and y axis of the system, it can be seen that the target positions for the x and y

axis which were 3 and 4 meters respectively has not been reached by the quadcopter

but the quadcopter has ended up drifting away from its target position.

Hence, from this test, it can be concluded that the need for an Unknown Input Observer

is necessary for the quadcopter model when it is subjected to external disturbances

which are unknown to the system in order to allow for the compensation of the

disturbances based on the estimations and enabling the quadcopter to reach its target

position. The advantage of using an UIO for the wind estimation instead of sensors is

that the UIO provides a cheaper alternative of implementation. Moreover, the UIO

caters for delay in its estimation unlike the sensors used for measuring of the

disturbance.

4.3 UIO Design Process

Firstly, a discrete time linear model of a system is considered:

(4.1)

38

With being the state vector, the input vector and being the

output vector. Matrices A, B, C and D are the system matrices. It should be noted that

for the development of the UIO, only the unknown inputs are considered in equation

(4.1) above. It is also assumed that the matrix is a full rank matrix. This

assumption can be achieved by making appropriate transformations.

Furthermore, the response of the above system over a time step of is given as:

(4.2)

In equation (4.2), is the observability matrix, is the invertability matrix and L is

the system delay. The form of the observability matrix and the invertability matrix are

represented below in equations (4.3) and (4.4) respectively:

(4.3)

(4.4)

Therefore, from the definition of the observability and invertability matrices, equation

(4.2) can be rewritten as:

(4.5)

Moreover, if we consider an observer of the form:

(4.6)

39

The observer in equation (4.6) can only be considered as an unknown input observer

with a delay L if, as regardless of the (input) sequence.

For the observer defined in equation (4.6), the observer matrices E and F are

determined by examining the error in the estimation. The estimation error is derived

as follows:

(4.7)

Replacing in the above equation with equation (4.6) we obtain:

(4.8)

In order to simplify this equation, we add extra terms while at the same time

ensuring that it does not affect the original equation in any way. After adding the extra

terms, we get an equation of the estimation error of the following form:

(4.9)

After grouping the like terms together and factorizing equation (4.9), we obtain the

following error function in equation (4.10):

(4.10)

Now, replacing in the above equation with the equation (4.2), we obtain:

(4.11)

Further simplifying equation (4.11) above, gives us the final error estimation

function:

(4.12)

Therefore, if the error of the above error function is to be driven to zero, the matrix E

must be a stable matrix and matrix F must satisfy the following condition in equation

(4.13):

40

(4.13)

(Must be stable) (4.14)

It should be noted that a matrix F will only exist, if and only if the following condition

is satisfied. i.e.

(4.15)

Where m is the number of columns of the disturbance matrix.

For the design of this type of observer, the value of L can be obtained by starting from

zero and moving up until the above condition is satisfied. It should be noted that, the

maximum value for L for which the observer would exist is L = n. If the value of L

increases and becomes greater than n, then asymptotic estimation of the states is not

possible.

Furthermore, from equation (4.13), it can be said that matrix F will lie in the left null

space of the last columns of represented as . From this, we let matrix

be the left null space of the matrix . Hence, the matrix

(4.16)

becomes a matrix whose rows form the basis of the left null space of where W

is an invertible matrix.

In order to find the invertible matrix W, we use the following matrix calculations:

(4.17)

41

From equation (4.17) the bottom half of matrix W is obtained by taking the basis of

the left null space of the product of the matrices while the top

half of matrix W is obtained by taking the left inverse of the product of the above

mentioned matrices. i.e.

(4.18)

Note: The product of matrices is referred to as Matrix 1.

Once matrix W is obtained, matrix N can be calculated using equation (4.16) above.

After obtaining the matrix N, it becomes possible to form the matrix . This matrix

can be calculated as follows:

(4.19)

In the matrix defined in equation (4.19), the last m rows is equal to the matrix Then,

the final equation of the observer matrix E becomes:

(4.20)

Observer matrix F is then calculated as:

(4.21)

Since it was stated that for the error estimation to reach zero, matrix E must be stable,

therefore the eigenvalues for

42

must be stable i.e. eigenvalues .

If the eigenvalues for the above matrix are stable, then the value of matrix is set to

zero. However if the eigenvalues of the above equation are unstable, then in order to

place the eigenvalues at the desired locations, matrix can be found by using the

place command in Matlab. For example, where p is the

vector specifying the desired locations of the eigenvalues.

Once both the observer matrices E and F are successfully obtained, satsifying their

conditions of existence, the final observer can be formulated as:

Finally, in order to find the estimate of the unknown inputs we firstly rewrite the

equation of the original discrete model as in equation (4.24):

For the reconstruction of the input, a matrix G needs to be found satisfying the

condition in equation (4.25):

(4.25)

From equation (4.25), it can be seen that matrix G can be obtained by calculating the

inverse of the matrix .BD

Moreover, once the convergence of the estimate to

is reached, in equation (4.24) can be replaced with and by left

multiplying matrix G, the estimated input can be obtained as:

(4.26)

43

4.4 UIO Design for Quadcopter Model

Based on the steps mentioned in the above section of the chapter, the UIO for the

derived quadcopter model has been designed. The calculations and the steps utilised

to achieve this has been attached in appendix A of the thesis.

Figure 4-3 shows the construction of the Unknown Input Observer in Simulink

designed for the quadcopter model discussed in this chapter. The observer matrices E

and F and the matrix G used for the reconstruction of the input used in the figure below

have all been calculated and attached in appendix A as mentioned above. Note: Matrix

in Figure 4-3 is for the known inputs (i.e. four motor speeds).

Figure 4-3: UIO Design

44

4.5

UIO

Impl

emen

tatio

n in

Mat

lab/

Sim

ulin

k

Figu

re 4

-4: U

IO Im

plem

enta

tion

in M

atla

b/Si

mul

ink

45

In Figure 4-4, the designed UIO in the chapter has been implemented in the Simulink

model with the linear quadcopter. In order to represent the existence of wind

disturbance on the model, a step disturbance or a discrete wind gust model can be

applied which has been implemented in the Simulink block labelled Wind in Figure 4-

4. As it can be seen form Figure 4-4, the inputs to the UIO block are the linear and

angular positions of the quadcopter and its four motor speeds. Based on these input

values, the UIO is then able to estimate the wind disturbance that has been applied to

the system as well as estimate the states of the quadcopter.

Presented below is a test case conducted for the UIO implementation with the linear

quadcopter model shown in Figure 4-4. The wind disturbance values applied were a

step disturbance of:

Figure 4-5: Graph of Estimated and Actual Wind Force

Figure 4-5 shows the output from the UIO implemented with the linear quadcopter

model in Simulink. A step disturbance was applied to each of the axis representing the

effect of wind on the quadcopter and from the results obtained it can be stated that the

UIO has successfully estimated the applied disturbances.

46

4.6 Chapter Summary

The goal of this chapter was to design an Unknown Input Observer for quadcopters.

An Unknown Input Observer is an observer used to estimate the unknown input or

disturbance acting on a system. A major advantage of the UIO over other observers

is that it does not require knowledge of the unknown input in order for the estimation

unlike the simple Luenberger observers. Firstly, in this chapter, the design procedure

for an Unknown Input Observer was discussed. Then, the steps discussed for designing

the Unknown Input Observer was utilized in designing the UIO for the quadcopter

model. The UIO designed for the quadcopter was then implemented with the model

of the quadcopter in Simulink and a test case was used to demonstrate the functionality

of the designed UIO.

47

Chapter 5 Gazebo Simulation of Erle-copter with UIO In this chapter, the designed UIO for the quadcopter is tested in Gazebo software where

the actual parameters of the real quadcopter (Erle-copter) is used. This chapter

presents as to how Simulink is interfaced to Gazebo via Robot Operating System

(ROS) toolbox. The entire process of simulating the Erle-copter in Gazebo software

is presented in this chapter.

Robot Operating System (ROS) is a meta-operating system which is used for robots.

The services ROS provides are similar to any operating system. These services may

include abstraction of hardware, control of low level devices, ability to pass messages

between processes and the management of packages. Moreover, it also provides users

with the resource of libraries and tools which aid in building, writing and running code

across multiple computers. With the use of ROS, users are provided with a variety of

communication styles such as synchronous RPC-style communication over services,

asynchronous streaming of data over topics and storage of data on a parameter server

[59]. The Robot Operating System currently works only on systems based on UNIX

platforms as primary ROS software testing has been carried out on Ubuntu and Mac

OS systems [59].

Gazebo is a robotics simulator, which can be utilized for creating applications for real

robots in a virtual environment. This software can be used to simulate robots with

actual world parameters in order to visualize how the robot would behave in the actual

hardware tests in the real environment. Gazebo-simulated hardware is designed to

reflect the behavior of its equivalent in reality. Because of this, the client software

uses an interface identical to the real robot [60]. This would thus save time and money

in carrying out tests directly on hardware without the knowledge of how it would

actually behave in the physical world scenario. Advantages of using Gazebo software

are firstly, it enables the user to simulate various types of position sensors such as laser

scanning, sonar and Global Positioning System (GPS). Secondly, in its library it

contains robots which are commonly used and finally, it enables realistic simulation of

rigid body physics [60].

48

5.1 Launching the Erle-copter model in Gazebo

Since the Erle-copter would be utilized for the hardware testing of the UIO, the Erle-

copter model which is already available in Gazebo will be utilized. Firstly a workspace

is created and initialized, and inside the workspace, the Erle-copter model and

resources are downloaded. The steps listed in the Erle robotics documentation page

for the configuration of the environment were followed to achieve this. Once the

workspace has been created and initialized with Erle-copter model and resources

downloaded, the Erle-copter can then be launched. In order to launch the Erle-copter

in Gazebo, the following commands are entered into the terminals in an Ubuntu

machine:

In Terminal 1:

In Terminal 2:

This enables the Erle-copter model to be launched in Gazebo. Figure 5-1 shows the

Erle-copter model spawned in Gazebo simulator.

Figure 5-1: Erle-copter spawned in Gazebo Simulator

Once the Erle-copter model has been launched in Gazebo, the next step is to load its

parameters. The parameters of the copter are loaded using the following command:

source ~/simulation/ros_catkin_ws/devel/setup.bash cd ~/simulation/ardupilot/ArduCopter ../Tools/autotest/sim_vehicle.sh -j 4 -f Gazebo

source ~/simulation/ros_catkin_ws/devel/setup.bash roslaunch ardupilot_sitl_gazebo_plugin erlecopter_spawn.launch

49

In Terminal 1

After carrying out the above mentioned steps, one should be able to launch the Erle-

copter model in Gazebo. In the next section of this chapter, the interfacing of

Matlab/Simulink with Gazebo via ROS is explained.

5.2 Interfacing Matlab/Simulink with Gazebo via ROS

In order to simulate the Erle-copter, Matlab/Simulink must be interfaced with Gazebo.

This is achieved through the ROS toolbox in the Simulink library. Two main types of

functions, specifically Publish and Subscribe functions are used in order for

Matlab/Simulink and Gazebo to communicate with each other. This communication

is achieved through choosing appropriate topics and its corresponding message types

of the ROS blocks in the ROS toolbox. The publish ROS block is used to send data

from Matlab/Simulink to Gazebo while the subscribe ROS block is used to receive

information from Gazebo. For the purpose of simulating the Erle-copter in Gazebo to

test the UIO, position data generated in Matlab/Simulink is published to the Erle-copter

model in Gazebo while the path the Erle-copter takes in Gazebo is obtained by

Matlab/Simulink through the use of the subscribe ROS block. These values subscribed

from Gazebo are then fed into the designed UIO as input together with the speed of the

four rotors. Shown below are the Subscribe and Publish ROS blocks used in the

Simulink model of the quadcopter together with the topic names and their

corresponding message types.

Table 5-1: ROS Blocks used and their Purpose

Type of ROS Block

Purpose Topic Name Message Type

Publish Publish position to Gazebo

/mavros/setpoint_position/local geometry_msgs/PoseStamped

Subscribe Subscribe position and orientation data from gazebo

/erlecopter/ground_truth/odometry nav_msgs/Odometry

Subscribe Subscribe Erle-copter motor speeds.

/erlecopter/motor_speed/motor_number std_msgs/Float32

param load /home/simulation/ardupilot/Tools/Frame_params/Erle-Copter.param

50

5.3

Sim

ulat

ion

of th

e E

rle-

copt

er

Figu

re 5

-2: S

imul

ink

Blo

ck D

iagr

am fo

r Gaz

ebo

Sim

ulat

ion

The A

rdup

ilot 3

.5 co

ntro

ller w

hich

is th

e def

ault

cont

rolle

r in

the G

azeb

o si

mul

ator

has

bee

n ut

ilize

d fo

r the

test

ing

of th

e UIO

in G

azeb

o si

mul

atio

n.

This

pro

ves t

hat t

he U

IO a

ppro

ach

is va

lid fo

r any

con

trolle

r whi

ch is

con

trolli

ng th

e dr

one

in it

s flig

ht.

51

Figure 5-2 represents the Simulink model of the quadcopter with the UIO

implementation for simulation in Gazebo simulator. Publish and subscribe blocks used

in Figure 5-2 are from the ROS toolbox in Simulink.

Once the Erle-copter model has been launched and Gazebo and Matlab/Simulink

interfaced, the simulation of the Erle-copter can be carried out. With the Gazebo

simulator running, the Simulink model of the quadcopter is run. The user first selects

the desired coordinates for the Erle-copter to travel (i.e. . These values

are published to the Gazebo simulator via the Publish ROS block. This prompts the

Erle-copter model in gazebo to move to the desired coordinates entered by the user.

The path which the Erle-copter takes to reach its desired position is extracted to

Matlab/Simulink through the Subscribe ROS block. This includes the linear and the

angular positions of the quadcopter. It should be noted that the orientation in Gazebo

are represented in quaternions i.e. w, x, y and z and hence must be converted to Euler

angles roll, pitch and yaw. This step is carried out in the Gazebo Position and

Orientation block in Figure 5-2. In Figure 5-3, it shows how the linear and angular

positions of the Erle-copter are extracted from gazebo using the Subscribe ROS blocks

from the ROS toolbox in Simulink.

Figure 5-3: Segment of Simulink for Subscribing Position and Attitude data Furthermore, the motor speeds of the Erle-copter when moving to the desired

coordinates are also extracted to Matlab/Simulink through the Subscribe ROS block.

Once all the required data (i.e. linear and angular positions and the motor speeds) have

been extracted from Gazebo to Matlab/Simulink, it is fed into the UIO block as input

52

as shown in Figure 5-2. Finally, the UIO block outputs the estimated wind disturbance

which was applied to the Erle-copter model in the gazebo simulation environment.

Figure 5-4: ROS Computation Graph

Figure 5-4 shows the rqt_graph which provides a Graphical User Interface plugin

useful in visualizing the ROS computational graph. The figure illustrates how the

simulation is executed and how each component is linked to the other. In the mavros

block, the two topics used for publishing and subscribing of data are shown. Moreover,

the tf block in Figure 5-4 is a package that allows the user to keep track of multiple

coordinate frames over some time period. Moreover, it also allows users to transform

points and vectors between any two coordinate frames at any desired point in time.

5.4 Chapter Summary

The goal of this chapter was to provide the steps to simulating the quadcopter model

implemented with the UIO in the Gazebo software. Firstly, the model and resources

of the Erle-copter were download into the system. Steps for doing these are well

documented in the Erle robotics documentation page. The next step was to launch the

Erle-copter in Gazebo. For this, specific commands discussed in the chapter are to be

entered. Entering these commands enabled the launching of the Erle-copter in Gazebo.

Once the Erle-copter model was launched, communication between Matlab/Simulink

and Gazebo had to be obtained. This was done through the aid of the ROS toolbox in

Simulink. Once communication between the two had been achieved, the simulation

of the Erle-copter in Gazebo was carried out and the UIO was tested.

53

Chapter 6 Hardware Implementation In order to validate the results obtained through simulations, hardware implementation

of the designed system is very important. In this chapter, steps of hardware

implementation of the Erle-copter with the designed UIO will be discussed. The

hardware setup of the Erle-copter will be presented and the tasks carried out to achieve

hardware implementation of the UIO will be discussed.

Figure 6-1: Erle-copter

Figure 6-1 shows the quadcopter (Erle-copter) which will be utilized for the hardware

testing of the designed UIO. The Erle-copter was the first Linux based smart drone

that utilized robotics frameworks for example the Robot Operating System (ROS) [61].

The Erle-copter comes equipped with the Erle brain 3 which is the latest edition of the

robot brain from Erle robotics.

6.1 Components of the Erle-copter

1. Erle Brain 3

Figure 6-2: Erle Brain 3 [62]

Figure 6-2 shows the third version of Erle Brain which is called the Erle brain 3. This

is the third and latest installment of artificial robot brains based on Linux systems. The

54

main purpose of the brain is to allow support of ROS for drones and robots. The brain

also allows for Wi-Fi connection as it sets an internal Wi-Fi (802.11n, 2.4 GHz) of the

Erle-copter. The Erle brain 3 comes equipped with a variety of sensors namely

gyroscope, temperature and pressure sensor, gravity sensor and a digital compass [62].

The version of the operating system in the Erle brain 3 is Frambuesa 3.5.

2. Electronic Speed Controller (ESC)

Figure 6-3: Electronic Speed Controller (ESC) [63]

The electronic speed controller illustrated in Figure 6-3 is the device responsible for

controlling the speed as well as the direction of rotation of the rotors of the Erle-copter

[64]. An ESC can be considered as the link between the brain and the rotors of the

Erle-copter.

3. Brushless DC Motors

Figure 6-4: Brushless DC Motors [65]

The Erle-copter contains 4 brushless DC motors where two of the motors rotate in the

clockwise direction while the other two rotate in the anticlockwise direction. The main

function of these 4 motors is to convert the electrical energy into mechanical energy

55

(by inducing rotation) which is transmitted to the four propellers of the Erle-copter

[64].

4. The Propellers

Figure 6-5: Propellers of the Erle-copter [63]

In Figure 6-5, the propellers of the Erle-copter are shown. There are two pairs of

propellers utilized in the copter. The first pair tightens itself when rotated in the

clockwise direction whereas the second pair of propellers tighten itself when rotated

in the anticlockwise direction. The propellers are the components of the Erle-copter

that provide propulsion from the rotary motion provided by the brushless DC motors.

5. The Power Module

Figure 6-6: Power Module [63]

Figure 6-6 represents a power module. The power module is an essential component

of the system since it is used to power up the autopilot and other add on accessories as

well as report the current and voltage values of the battery.

56

6. The Global Positioning System (GPS) + Compass

Figure 6-7: The Global Positioning System (GPS) + Compass [63]

The Global Positioning System (GPS) + Compass is represented in Figure 6-7. This

component of the Erle-copter is used to determine its position and orientation at any

particular time. The GPS, in order to avoid any interference and for a better connection

to the satellite, must be placed in the highest part of the vehicle [64].

7. LiPo Battery (4S or 3S)

Figure 6-8: LiPo Battery [63]

Figure 6-8 illustrates the LiPo battery used by the Erle-copter for its flight. Either a 4s

or a 3s LiPo battery can be utilized. With the battery fully charged, the Erle-copter can

obtain a flight time of up to twenty minutes [66].

6.2 Connection of the Erle-copter to an Ubuntu Machine

The machine (laptop with Ubuntu) with which the Erle-copter calibration and testing

is to be done has to be connected to the Erle-copter. In order to achieve this connection,

the internal Wi-Fi provided by the Erle brain 3 is utilized. The Wi-Fi provided by the

Erle brain is activated once the brain is powered up after connecting the LiPo battery.

Shown below is the Wi-Fi hotspot name and password of the Erle-copter.

57

Note: It must be ensured that the IP address of the host computer is set correctly. The

correct addresses are as follows:

Once connected to the Erle copter through its internal Wi-Fi, the brain of the Erle-

copter can be accessed by entering the following command in the terminal of the

Ubuntu machine.

6.3 Erle-copter Setup and Calibration In order to obtain the optimal performance from the Erle-copter, various settings and

calibrations are carried out. In this section of the chapter, the three important

calibrations which were carried out are discussed.

I. ESC Calibration

ESC calibration is carried out so that the ESCs are aware of the maximum and

minimum Pulse Width Modulation (PWM) values the flight controller will send. The

is chosen where all the ESCs are calibrated

together. Calibrating the ESC is important as it ensures that all motors rotate at the

same speed and that correct power is given to the motors.

II. Compass calibration

Compass calibration is one of the most important calibrations in order to ensure a flight

which is safe as well as controlled. The purpose of the compass calibration is to allow

compensation of the magnetic inclination and deviation. If not corrected, these

inclinations and deviations will cause inconsistencies between the compass and the

GPS which could probably result in losing control of the Erle-copter [67]. Live

calibration for calibrating the compass is chosen in the APM software.

ssh [email protected]

Wi-Fi hotspot Username: erle-robotics-frambuesa

Password: holaerle

ROS_MASTER_URI = http:// 10.0.0.1:11311

ROS_HOSTNAME: = 10.0.0.2

58

III. Accelerometer calibration

Accelerometers can be defined as inertial sensors which can sense mechanical motion.

This mechanical motions can then be converted to an electrical quantity which can be

easily measured. The Erle-copter contains three Inertial Measurement Units (IMU)

and each of these have to be calibrated in order to obtain satisfactory flight from the

Erle-copter. The process of the accelerometer calibration will require placing the

copter in various positions. It should be noted that the level position of the Erle-copter

is the most important position to get right since the Erle-copter controller will consider

this attitude as level while in flight [68].

6.3.1 Erle-copter Parameters Table 6-1 below shows the parameters of the Erle-copter used for both software and

hardware tests.

Table 6-1: Erle-copter Parameters

Parameter Name Value

Mass, m 1.30 kg

Gravitation force, g 9.80

Length, 0.18 m

Inertia, 0.03

Inertia, 0.05

Inertia,

Motor constant, 8.55

Moment constant, 1.60

6.4 Automatic Standalone Node Generation

The Simulink model in Figure 5-2 used for the gazebo simulation is also utilized for

the hardware implementation of the UIO on the Erle-copter. In order to run the

program implemented in Simulink, the program must be built into the hardware (i.e.

the brain of the Erle-copter). In order to accomplish this, the following steps in the

automatic generation of a standalone node are followed:

Step 1: Model Configuration for Code Generation

The Model Configuration Parameters option is chosen by first clicking on the

Simulation option in the menu bar in Simulink. In the Model Configuration

59

Parameters dialog box, Hardware Implementation is chosen and under Hardware

Implementation, the Hardware Board is set to Robot Operating System (ROS). Figure

6-9 shows a graphical representation of step 1.

Figure 6-9: Hardware Board Setting

Step 2: Configuration of the connection to the ROS device (Erle-copter)

Simulink has the capability of transferring a generated node to a ROS device, build an

executable as well as run the resulting ROS node as long as the system running

Simulink is connected to the ROS device. Therefore, firstly, connection to the Erle-

copter is obtained through its internal Wi-Fi. Once the system is connected to the Erle-

copter, the Build Options field under Hardware Implementation is modified. Three

options for the type of build action are available which are; None , Build and Load

and Build and Run t setting where the code is only

option generates the code as well as transfers it to the external ROS device and builds

an executable ROS node but does not run the node. In t option, the

executable node at the end of the build process. For the purpose of this thesis, the

Figure 6-10 shows the fields available for

modification under the Build Options.

60

Figure 6-10: Build Options

as well as create a Catkin workspace where the generated node will be built. Apart

from these options, the user is also required to enter the address of the ROS device, its

username as well as its password. The information entered for the fields discussed

above are shown in Figure 6-11.

Figure 6-11: Connection to ROS Device Dialog Box

Once all the information required for the connection to the Erle-copter is updated, the

connection of the PC to the ROS device is tested. Figure 6-12 shows the result of the

connection test which was carried out.

61

Figure 6-12: ROS Device Connection Test

Since the catkin workspace entered above does not exist in the Erle-copter brain as

seen in the fifth point of Figure 6-12, the connection test fails. The folder is then

created and initialized by clicking on the fix button. Upon making these changes, the

ROS device connection test finishes successfully.

Step 3: ROS Node Generation in C++

In this step, a standalone ROS node code is generated. This code is then transferred to

the Erle-copter where it is then built and run. In order to perform code generation, a

valid connection to a ROS master is necessary. A ROS master can be started on a ROS

device by the rosdevice command in Matlab.

Entering the above command in the Matlab command window yields the following

output as shown in Figure 6-13:

Figure 6-13: Output of the rosdevice Command

The output of the rosdevice command as seen in Figure 6-13 shows the address of the

Erle-copter, its username, the ROS folder and the Catkin Workspace which were

d = rosdevice

62

entered in step 2 for obtaining connection to the Erle-copter. It should be noted that

currently there are no available nodes as seen in the figure because the node is yet to

be generated.

After starting the ROS master on the Erle-copter, the ROS master is required to be run.

The command to run the ROS master is:

The next step is to connect Matlab with the ROS master which is running on the ROS

device (i.e. the Erle-copter). For this purpose the following command is entered in the

Matlab command window:

Moreover, the same ROS connection settings which are used in Matlab are also

transferred to Simulink. This is done by selecting the ROS Master and Node

Host network addresses as Default. This setting can be found under Tools menu

in Simulink. Figure 6-14 shows the graphical representation of the connection settings

for Simulink.

Figure 6-14: Network Connection Settings in Simulink

Once the connection settings have been accomplished, the C++ code of the model can

be generated and deployed to the Erle-copter. This is done by selecting ode

menu in Simulink and choosing C/C++ code option and finally selecting the Deploy

to Hardware option. After the successful completion of the generation of the code,

the ROS node gets transferred into the Catkin Workspace which was created initially

runCore(d)

rosinit(d.DeviceAddress)

63

in the Erle-copter brain. Once the node is transferred, it builds itself in the workspace

and finally starts running automatically.

Step 4: Verification of the Generated Node on the ROS Device

This is done by entering the rosdevice command in the Matlab Command window.

The output of the command should show the correct address of the Erle-copter, its

username, the ROS folder, the Catkin Workspace and the name of the generated node.

Figure 6-15 shows the output obtained at the end of the steps for the automatic

generation of a standalone node.

Figure 6-15: rosdevice Command Properties after Node Generation

From Figure 6-15 above, it can be seen that the ROS node of the Simulink model was

successfully created on the ROS device (Erle-copter). The name of the generated ROS

node in the brain of the Erle-

6.4.1 Running the Generated ROS node

To run the ROS node which was generated in the above section, we follow the steps

listed below:

a. Power up the brain

b. Connect to the Erle-copter via its internal Wi-Fi

c. Get access to the brain of the Erle-copter via the following command:

d. Once inside the brain, change the directory to the folder where the Catkin

Workspace was created for the generation of the ROS node. The command to

change the directory in the brain is:

ssh [email protected]

cd /home/erle/catkin_ws_copter

64

e. After entering the catkin workspace, we need to source the folder in order to

add the environment variables to our path for ROS to function. This is done

by the following command:

f. Finally, the generated ROS node in the Catkin Workspace can be run using the

rosrun command. The full command to run the node on the Erle-copter is as

follows:

6.5 Experimental Setup

Figure 6-16: Experimental Setup

Figure 6-16 shows the experimental setup for the hardware test of the UIO. In order

to measure the wind speed and wind direction, an anemometer and a wind vane have

been used respectively. In order to save the wind data during the time of flight of the

Erle-copter, a data logger has been utilized.

6.5.1 Wind Measurement

A pulse output A101M anemometer was utilized for the measuring of the wind speed

during the hardware tests. In addition, the wind direction was measured via the

potentiometer wind vane (W200P). The wind speed and direction data were logged to

the PC with the aid of the CR1000 data logger. Logger Net 4.5 software was utilized

to extract data from the data logger.

6.5.2 Measured Wind Data Analysis

Using the wind speed and wind direction data obtained from the data logger, the wind

speed was converted to wind force as the designed UIO was used to estimate the wind

rosrun experimental experimental_node

source devel/setup.bash

65

force acting on the body of the Erle-copter. For this purpose, wind speed was

converted into wind force using the following formula:

Where: Area = 0.0988

Air Density = 1.225

Once the wind speed had been converted to wind force, it was decoupled into wind

force acting in the x axis and the y axis. This was achieved as follows:

Where: ; ;

6.5.3 UIO Hardware Test

While the Erle-copter was commanded a path to move to, the anemometer and the

wind vane measured the wind speed and direction respectively. This data was logged

using the CR1000 data logger. Upon the completion of the Erle- s

position (linear and angular) as well as its motor speeds which were logged using the

APM (Ardupilot Mega) software were utilized in estimating the amount of wind force

acting on the Erle-

from the UIO was compared against the calculated wind forces in the x and y axis

using the data logged during the time of the flight. Based on this comparison, the

validation of using an Unknown Input Observer for quadcopters was obtained. Figure

6-17 below shows the Simulink model used to command the target position for the

Erle-copter to follow.

Figure 6-17: Simulink Model for Hardware Testing

66

Figure 6-17 shows the Simulink model used for the hardware tests. The x, y and z

coordinates entered are published to the Erle-copter through the Publish ROS block.

The topic and the message type used for the publishing of the position data to the Erle-

copter has been discussed in Table 5-1. Once all the required data i.e. the linear and

angular positons of the Erle-copter as well as its motors speeds during the time of the

flight was obtained, the Simulink scheme of Figure 5-2 was used to test the UIO.

6.6 Chapter Summary

The goal of this chapter was to discuss the hardware setup and provide the steps for

the hardware implementation of the Erle-copter with the designed UIO. Firstly, the

hardware of the Erle-copter was discussed. The importance of each of the components

of the copter were outlined. Secondly, the steps required to connect to the Erle-copter

and its brain were presented. Thirdly, to achieve optimal performance from the Erle-

copter, three important calibrations namely ESC calibration, compass calibration and

accelerometer calibration were discussed. Moreover, the steps required to generate a

standalone ROS node from the Simulink model of the quadcopter into the Erle-copter

and running the ROS node from the brain of the Erle-copter was presented. Finally,

the experimental setup for the hardware test was discussed. The results obtained from

the hardware tests are presented in the next chapter of the thesis.

67

Chapter 7 Results - Software and Hardware The software results from Matlab/Simulink and Gazebo as well as the hardware results

of the quadcopter with the UIO subjected to external disturbances (wind) is presented

in this chapter. Different test cases were used to carry out the test of the UIO both on

software and hardware. The results obtained from each of the test cases are discussed

in this chapter.

7.1 Software ResultsIn this section, the results obtained from software simulations using Matlab/Simulink and Gazebo simulator are presented.

7.1.1 Matlab/Simulink

Test Case 1.1: Step Wind

For the first test case, the quadcopter was commanded the following coordinates as

shown below:

Moreover, disturbance was applied to the system which represented the wind

disturbance acting on the body of the quadcopter. The disturbance applied to the

system was a step input and was applied to all 3 axes. The values for the disturbances

were:

Given the above coordinates and disturbance values, the UIO was tested with the

quadcopter model and results were analyzed to determine if the UIO was able to

estimate the disturbances which were applied to the system and if the system was then

able to compensate for these disturbances based on the estimations accordingly.

68


Figure 7-1 shows the graph of the estimated and actual wind force for test case 1. The

solid line in the graph above demonstrates the estimated wind force by the UIO while

the dashed lines represent the actual wind force which was applied to the system. From

the graph it can be said that the UIO was successful in estimating the wind force acting

on the system.

Figure 7-2: 3D Plot of Quadcopter Path

Figure 7-2 shows the 3D plot of the path of the quadcopter for the first test case with a

step disturbance acting on the system. The target position for the quadcopter to reach

was 5m, 6m and 7m in the x, y and z axis respectively as shown by the red marker on

the graph above and it can be clearly stated from the graph that the quadcopter was

able to do that successfully.

69

Figure 7-3: Graph of Wind Estimation Error

Figure 7-3 shows the graph of the error in the estimation of the wind disturbance

applied to the system. It can be seen from the graph that the UIO designed was able

to successfully estimate the applied wind disturbances on each of the axes.

Test Case 1.2: Military Grade Wind Gust

In the second part of the first test case, the quadcopter was again commanded to go to

the coordinates shown below:

However, instead of the step disturbance which was applied in the first part of the test

case, Military grade wind gusts were applied in this part of the test. The values for the

wind gusts in each of the axes were:

The wind gusts were applied at the beginning of the simulation at t = 0 seconds and

took t = 10 seconds to reach its maximum specified magnitude.

70


Figure 7-4 represents the graph of the estimated and the actual wind force for the

second part of the first test case where military grade wind gusts were applied to the

system instead of step disturbances. As seen from the graph above, the UIO was

successful in estimating the wind gusts that were applied to the quadcopter model.

Figure 7-5: 3D Plot of Path of Quadcopter

The 3D plot of the path of the quadcopter is shown above in Figure 7-5. The red

marker on the graph represents the target position of the quadcopter which was 5m,

6m and 7m in the x, y and z axis respectively. Hence it can be stated that the

quadcopter was able to reach its target position after successful compensation of the

applied disturbances.

71


The error in the estimation of the applied disturbance on the quadcopter model is

depicted in Figure 7-6. From the above graph, it can be concluded that the UIO was

able to correctly estimate the wind disturbances acting on the system.

Test Case 2.1: Step Wind

For the second test case, the quadcopter was commanded to move in a circular path.

The parameters used to achieve this task were as follows:

For the first part of the second test case, step disturbance was applied to the system

which would represent the wind disturbance acting on the body of the quadcopter. The

values for the disturbances were:

w = 1

72


Figure 7-7 shows the graph of the estimated and actual wind force for the second test

case. The solid lines in the graph represents the estimated wind forces by the UIO and

it can be seen that these values correspond to the actual values of the wind force

represented by dashed lines in the above graph. Hence, it can be concluded that the

UIO was successful in estimating the effect of wind force acting on the system.


The 3D plot of the circular path taken by the quadcopter is illustrated in Figure 7-8.

As specified in the test condition, the quadcopter was commanded to make a circular

path at a height of 5m and with a radius of 10m. From the above graph, it can be said

that the quadcopter was successful in achieving its target position regardless of the

disturbance which was applied to the system.

73


Figure 7-9 shows the graph of the error in the estimation of the wind disturbance

applied to the system. It can be seen from the graph that the UIO designed was able

to successfully estimate the applied wind disturbances on each of the axes.

Test Case 2.2: Military Grade Wind Gust

Similar to the first test case, the second part of the second test case will command the

quadcopter to move in a circular path while military grade wind gusts are applied to

the system. The magnitude of the wind gusts are as follows:

The wind gusts were applied at the beginning of the simulation at t = 0 seconds and

took t = 10 seconds to reach its maximum specified magnitude.

74


The graph in Figure 7-10 represents the estimated and the actual wind force acting on

the system. The applied disturbances were wind gusts of magnitude 1.5 m/s, 2.5 m/s

and 0 m/s in the x, y and z axis respectively. Moreover, since the solid lines represents

the estimated wind force by the UIO, it can be stated that the designed UIO was

successfully able to estimate the military grade wind gusts which were applied to the

quadcopter model.


Figure 7-11 represents the circular path taken by the quadcopter subjected to military

grade wind gusts. According to the graph, it can be seen that the quadcopter has been

able to meet the specifications for the second test case i.e. to move in a circular path

with a radius of 10m and at a height of 5m.

75


The error in the estimation of the wind disturbances applied to the system is shown in

Figure 7-12. From the graph it can be said that the UIO was able to estimate the wind

disturbances correctly with just a slight error.

7.1.2 Gazebo

Test Case 1: Hovering

In the first test case for Gazebo simulation, hovering condition was tested. The Erle-

copter was commanded to travel to a height of 10m and then hover in its position. The

coordinates of this test case were:

Moreover, the magnitude of the wind force applied to the Erle-copter model in the

Gazebo simulation environment were as follows:

76


The estimated wind force by the UIO and the actual wind force which were applied to

the system are represented in the graph in Figure 7-13. Wind force of magnitude 1.2,

0.8 and 0 m/s were applied in the x, y and z axis respectively. In the graph above, the

solid lines represents the estimated wind force by the designed UIO and hence it can

be concluded that the UIO has successfully estimated the applied wind force correctly.


Figure 7-14 illustrates the path the quadcopter took to reach its desired target for the

first test case. The red marker in the figure above represents the target position set by

the user. Hence, it can be said that the quadcopter was able to reach its desired

coordinates.

77


Figure 7-15 shows the error in the wind estimation of the UIO in Gazebo software.

According to the graph, it can be said that the UIO implemented in the Gazebo software

was able to correctly estimate the wind disturbances present.

Test Case 2: Movement in the x, y and z axis

In the second test case for Gazebo simulation, the Erle-copter was commanded to travel

to the coordinates presented below:

In this test case, the magnitude of the wind force applied to the Erle-copter model were

as follows:

78


Figure 7-16 represents the graph of the estimated and actual wind force for the second

test case. The dashed lines in the graph above represents the wind force which was

applied to the system while the solid lines represent the UIO estimation of the applied

wind force. Since it can be seen from the graph above that the solid lines match the

dashed lines, therefore, the UIO has been able to estimate the applied wind force

correctly.


Figure 7-17 shows the 3D plot of the path of the quadcopter when moving to its target

position for the second test case. It can be seen from the figure above that the

quadcopter was successful in reaching its target position which is represented by the

red marker in the graph above.

79


Figure 7-18 shows the error in the wind estimation of the UIO implemented in Gazebo

software. According to the graph, it can be said that the UIO had a slightly large

estimation error in the beginning of the simulation but was later able to correctly

estimate the wind disturbances present.

Test Case 3: Movement in the x, y and z axis with vertical wind

In the final test case for Gazebo simulation, the Erle-copter was commanded to travel

to the coordinates presented below:

In this final test case, apart from the wind force being applied to the x and y axis,

vertical wind force acting in the z axis had also been applied. The magnitude of the

wind force applied to the Erle-copter model were as follows:

80


The graph in Figure 7-19 represents the estimated wind force by the UIO (solid lines)

and the actual wind force (dashed lines) applied to the system. Wind force were

applied to all three axes with magnitude of 2.5, 1.5 and 0.5 m/s. It can be seen from

the graph that the estimated wind force by the UIO corresponds to the actual wind

force which were applied to the system. Hence, it can be said that the UIO was

successful in estimating the wind force correctly.


The 3D plot of the quadcopter path for the third test case is represented in Figure 7-20.

The target position for the quadcopter to travel to was set as 7m, 8m and 9m in the x,

y and z axis respectively. This target position is represented by the red marker in the

figure above and it can be said that the quadcopter was successful in reaching its target.

81


Figure 7-21 shows the error in the wind estimation of the UIO implemented in Gazebo

software. According to the graph, it can be said that the UIO had a slightly large

estimation error in the beginning of the simulation but was later able to correctly

estimate the wind disturbances present.

7.2 Hardware Results

Test Case: Hovering

For the hardware test, the Erle-copter was commanded the following coordinates:

Moreover, during the time of the flight, the wind speed and wind direction were

measured and recorded. After converting the wind speed to wind force, the results

obtained from the UIO estimation were compared. As seen in Figure 7-15 and Figure

7-16, the wind force estimation of the UIO in the x and y axis of the Erle-copter has

the same pattern as the calculated wind force, however, has a bias of about 0.7m. This

is due to the fact that the GPS error of the Erle-copter is about ±5m and requires at

least 6 satellites during its flight to obtain better GPS accuracy.

82

Figure 7-22: UIO Estimation for Wind Force (x-axis)

Figure 7-22 shows the UIO estimation of the wind force acting on the body of the Erle-

copter in the x-axis. From the above graph, it can be said that the UIO was successful

in estimating the correct pattern of the wind force however had a bias of about 0.7m in

its estimation. The reason of this as stated previously was due to the high GPS error

(±5m).

Figure 7-23: UIO Estimation for Wind Force (y-axis)

Figure 7-23 represents the UIO estimation for the wind force acting on the y axis of

the body of the Erle-copter. Upon comparing the calculated and estimated wind force

it can be said that the UIO was able to estimate the pattern of the wind force but also

83

had a bias of about 0.7m similar to the wind force estimation in the x-axis. The reason

for this is again due to the high GPS error (±5m).

Figure 7-24: 3D Plot of Erle-copter Path

Figure 7-24 represents the 3D plot of the path of the Erle-copter for the commanded

coordinates for the hardware test. The red marker in the graph above represents the

desired target position of the Erle-copter. From Figure 7-24, it can be said that the

Erle-copter was able to approximately reach its desired target position.


Figure 7-25 shows the wind estimation error of the UIO in the hardware test. As

discussed earlier in the chapter, there is a slight bias in the estimation of the wind

84

disturbance by the UIO due to the GPS inaccuracy. Hence the error in the wind

disturbance estimation is slightly larger when compared to the simulation results.

Furthermore, Figure 7-26 shows the 2D path the Erle-copter followed while Figure 7-

27 illustrates the attitude of the Erle-copter during its flight.

Figure 7-26: 2D Plot of Erle-copter Path

Figure 7-27: Erle-copter Attitude

7.3 Chapter Summary

In this chapter, the software and the hardware results have been discussed. For the

software tests of the UIO, Matlab/Simulink and Gazebo have been utilized. In the

software simulations, it had been seen that the UIO was successful in correctly

estimating the applied wind forces in each of the respective test cases. Moreover, for

85

the hardware test, it had been seen that the pattern of the wind force estimated by the

UIO was similar to the calculated wind force during the time of the flight however,

had a bias of approximately 0.7m in its estimation. This was due to the fact of the high

GPS error (±5m) of the Erle-copter and its need to have at least 6 satellites in its range

for a better GPS accuracy.

86

Chapter 8 Conclusion Since quadcopters are subjected to external disturbances (wind) in its flight, a system

for estimating these unknown disturbances in order for the quadcopter to reach its

desired position through compensation of the disturbances is required. Hence, in this

thesis, a linear Unknown Input Observer for quadcopters had been designed and

implemented. As discussed in the thesis, an UIO had been chosen over the simple

Luenberger type observers as it did not require any prior knowledge of the unknown

input for the estimation process.

Firstly, in the thesis, the linear mathematical model of a quadcopter was derived using

Matlab/Simulink. More specifically, the linear position and attitude controllers were

designed under hovering conditions. The position and attitude controllers were

designed to control the translational position and the attitude of the quadcopter

respectively.

Secondly, an Unknown Input Observer was designed for the derived linear model of

the quadcopter. This was done to enable the quadcopter to reach its target position

despite of external disturbances (which are unknown to the system) acting on it. The

designed UIO was implemented with the linear model of the quadcopter in

Matlab/Simulink. Two test cases were used to test the designed UIO in

Matlab/Simulink. In the first test case, step winds were applied to the quadcopter

model whereas in the second test case, military grade wind gusts were applied to the

quadcopter model.

Thirdly, the functionality of the designed linear UIO was further tested using the

Gazebo simulator. In order to do this, Matlab/Simulink was interfaced with Gazebo

via the ROS toolbox in the Simulink library. The Gazebo software was chosen due to

its ability to reflect the behavior of its equivalent in reality. Moreover, the Ardupilot

controller was used in the Gazebo simulation of the UIO which was the default

controller in the Gazebo software. Three test cases were used to test the UIO in Gazebo

whereby in two of the test cases, wind disturbances were applied only to the x and y

axis while the third case had wind disturbances applied to the x, y and z axis.

87

The results obtained from the simulation tests showed that the UIO was able to

correctly estimate the applied disturbances to the quadcopter model as the wind force

estimates of the UIO were similar to the applied disturbances to the quadcopter model.

Moreover, due to the accurate estimation of the wind disturbances, the quadcopter was

able to reach its desired position through proper compensation of the external

disturbances.

Finally, the designed UIO was tested on the actual quadcopter (Erle-copter). ROS

toolbox was again utilized in establishing the link between the Erle-copter and the

laptop running Matlab/Simulink. In order to get the values of the wind speed and

direction during the time of flight for comparison with the UIO estimations an

anemometer and a wind vane were used. These values were logged to the PC using

the CR1000 data logger. It was seen from the results that the UIO was able to correctly

estimate the pattern of the wind force disturbance acting on the Erle-copter however,

had a bias of about 0.7m in its estimation. This was due to the high inaccuracy of the

GPS module of the Erle-copter (±5m) and its need to require at least a coverage of 6

satellites for better GPS resolution.

Therefore, from the results obtained from the software simulations and the hardware

tests, it can be concluded that the designed linear Unknown Input Observer for

quadcopters was able to estimate the wind disturbances acting on the body of the

quadcopter hence allowing compensation of the disturbance on the system enabling

the quadcopter to reach its desired position.

88

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Appendix Appendix A: Design of the UIO for the Derived Quadcopter Model

A discrete model of the quadcopter is firstly obtained as the form discussed in equation

(4.1):

The system matrices namely A, B, C and D are as follows:

;

Where u[k] is the known input and w[k] is the unknown input

95

The first step was to determine the smallest possible value of L for which the system

was invertible. The condition was strictly followed. For

this case, m = 3.

The smallest value of L found satisfying the above condition was L = 2. The proof of

this is shown below:

Therefore, L was chosen to be equal to 2 for the derived quadcopter model.

In the next step, matrix N had to be calculated. In order to find matrix N, matrix

was obtained. The Matrix was obtained by finding the basis of the left nullspace of

the invertability matrix .

This was found by the Matlab command .

Hence, the matrix found in Matlab is shown below:

Once matrix was obtained, the next step was to find an invertible matrix W before

matrix N could be found. The steps followed to finding matrix W are as follows:

96

From the above equation, matrix was defined as Matrix 1 and matrix

was defined as Matrix 2. Moreover, since p = 6 and L = 2, these matrices

simplified to:

97

Then, a third matrix was defined as Matrix 3. By letting Matrix 1 * Matrix 2 = Matrix

3, we obtained:

Finally, the invertible matrix W could be calculated. The configuration of matrix W

was as follows; the bottom half of the matrix was the basis of the left null space of

Matrix 3 while the top half of the matrix was obtained by finding the left inverse of

Matrix 3.

The left null space of Matrix 3 was found by the Matlab command. The left inverse of Matrix 3 was found by the

98

Matlab command . Hence, matrix W was obtained as:

After obtaining the invertible matrix W, it was then possible to find the matrix N by

the equation below:

Simple matrix multiplications led to matrix N as:

Furthermore, after the calculation of matrix N, matrix 1

2

S

Swas obtained. The method

to finding this matrix is as follows:

99

This yielded matrix as:

From the above matrix, was equal to the last m = 3 rows of matrix .

In the third step, the observer matrices E and F were found for the quadcopter model.

In order to find the observer matrix E, the equation was utilized.

Since matrix E had to be stable, the eigenvalues of should be less than

or equal to zero.

The eigenvalues of E were found using the Matlab command Eigen_Value = eig (E)

100

Since not all of the eigenvalues of E were stable, matrix had to be calculated by

placing the poles at zero using the place command in Matlab. i.e.

Hence, matrix was calculated to be:

Then, the observer matrix E was calculated by which produced

the observer matrix E equal to:

Following this, the observer matrix F was obtained using the equation

.

101

Hence, the final observer for the quadcopter model with a delay of 2 was obtained as:

Where the observer matrices E and F are calculated above.

For the reconstruction of the input, a matrix G needed to be found satisfying the

condition:

From the above condition, the matrix G was then found to be:

Finally, the estimated inputed was obtained as:

linear unknown input observer fordigilib.library.usp.ac.fj/gsdl/collect/usplibr1/index/... ·...

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