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Modeling and Control of a Quad-Rotor Helicopter

Permanent Link: http://ufdc.ufl.edu/UFE0044145/00001

Material Information

Title: Modeling and Control of a Quad-Rotor Helicopter
Physical Description: 1 online resource (63 p.)
Language: english
Creator: Oh, Sang Min
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: controller -- pd -- pid -- quadrotor
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The quad-rotor helicopter is a UAV (Un-manned Air Vehicle) and has four rotors. To get efficient stabilization status, the author applied PD (Proportional, Derivative) and PID (Proportional, Integral and Derivative) controllers. First of all, trying to satisfy stabilization with each controller for the ideal case, which means all motors have same thrusts and speeds. Then the real motor values were applied to see the success of the controller’s performance. When the real values are applied, each input affects other inputs. In this reason the output result was fluctuated. To obtain accurate result, the author took advantage of mixing procedure. The mixing methods will be explained in chapter 5. Lastly, the author tried to show the robustness to actuator variation. It means that the author showed several graphs which are based on the different real values. Through this project, the author accomplished modeling and controlling of a quad-rotor helicopter and the helicopter can be stabilized.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sang Min Oh.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044145:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044145/00001

Material Information

Title: Modeling and Control of a Quad-Rotor Helicopter
Physical Description: 1 online resource (63 p.)
Language: english
Creator: Oh, Sang Min
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: controller -- pd -- pid -- quadrotor
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The quad-rotor helicopter is a UAV (Un-manned Air Vehicle) and has four rotors. To get efficient stabilization status, the author applied PD (Proportional, Derivative) and PID (Proportional, Integral and Derivative) controllers. First of all, trying to satisfy stabilization with each controller for the ideal case, which means all motors have same thrusts and speeds. Then the real motor values were applied to see the success of the controller’s performance. When the real values are applied, each input affects other inputs. In this reason the output result was fluctuated. To obtain accurate result, the author took advantage of mixing procedure. The mixing methods will be explained in chapter 5. Lastly, the author tried to show the robustness to actuator variation. It means that the author showed several graphs which are based on the different real values. Through this project, the author accomplished modeling and controlling of a quad-rotor helicopter and the helicopter can be stabilized.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sang Min Oh.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044145:00001


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1 MODELING AND CONTROL OF A QUAD ROTOR HELI C OPTER By SANG MIN OH A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012

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2 2012 Sang Min Oh

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3 To all of my thankful professors, family, friends and the Republic of Korea ( ROK ) Army

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4 ACKNOWLEDGMENTS I would like to thank my advisor Professor Rick Lind and Dr. Mohamed for providing the material needed to do this project as well as for his advising I also t hank to Professor Lou Cattafesta for teaching the various data measurement and analysis methods which are needed to complete this project Acknowledgement also goes to the Republic of Korea Army for giving me a chance to study abroad and financial support throughout the duration of the master s degree. I cannot omit the thankful expression to my family and friends for the support and the sincere pray for me all the time The y made me cheer up during this thesis L ast but not least, I really want to express thank s to Preethi who is my lab mate. She is a hard working and outstanding student, and her advices and ideas led me to finish this thesis quickly and clearly.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 TABLE OF CONTENTS ................................ ................................ ................................ .. 5 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURE S ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 1.1 Motivation ................................ ................................ ................................ ......... 12 1.2 Problem Statement ................................ ................................ ........................... 14 1.3 Contribution ................................ ................................ ................................ ...... 14 2 QUAD ROTOR HELICOPTER SPECIFICATIONS ................................ ................. 15 2.1 Model Picture ................................ ................................ ................................ .... 15 2.2 Specifications ................................ ................................ ................................ .... 15 2.3 Basic Instruments ................................ ................................ ............................. 15 3 EQUA TIONS OF MODEL ................................ ................................ ....................... 18 3.1 Basic Movements ................................ ................................ .............................. 18 3.1.1 Hovering ................................ ................................ ................................ .. 18 3.1.2 Vertical Movement ................................ ................................ ................... 19 3.1. 3 Roll Movement ................................ ................................ ........................ 19 3.1.4 Pitch Movement ................................ ................................ ....................... 19 3.1.5 Yaw Movement ................................ ................................ ........................ 20 3 .2 Three Different Types o f Mo ments ................................ ................................ .... 20 3.3 Equations o f Motion ................................ ................................ .......................... 20 3.4 Moments of Inertia ................................ ................................ ............................ 21 4 EXPERIMENTAL MODELING ................................ ................................ ................ 27 4.1 Thrust Test ................................ ................................ ................................ ........ 27 4.2 R otor Sp eed Test ................................ ................................ .............................. 27 5 SYSTEM CONTROL ................................ ................................ ............................... 29 5.1 State Space Model ................................ ................................ ............................ 29 5.2 Li nearization o f the System ................................ ................................ ............... 30

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6 5. 3 PID ( Proportional, Integral and Derivative ) Control Technique ......................... 32 5. 4 Simulation Results ................................ ................................ ............................ 32 5.4.1 PD (Proportional and Derivative) Controller for the Ideal Case ............... 32 5.4.2 PD Controller without Mixing for the Real Case ................................ ...... 33 5.4. 3 PD Controller with Mixing for the Real Case ................................ ........... 33 5.4. 4 PID Controller for the Ideal Case ................................ ............................. 33 5.4. 5 PID Controller without Mixing for the Real Case ................................ ..... 34 5.4.6 PID Controller with Mixing for the Real Case ................................ .......... 34 5.4.7 Robustness to Actuator Variation ................................ ............................ 34 6 CONCLUSION ................................ ................................ ................................ ........ 61 LIST OF REFERENCES ................................ ................................ ............................... 62 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 63

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7 LIST OF TABLES Table page 2 1 Weights ................................ ................................ ................................ .............. 17 2 2 Lengths ................................ ................................ ................................ ............... 17 2 3 Basic instru ments ................................ ................................ ............................... 17 3 1 Rolling moments ................................ ................................ ................................ 26 3 2 Pitching moments ................................ ................................ ............................... 26 3 3 Yawing moments ................................ ................................ ................................ 26 4 1 Thrusts (Experimental Results) ................................ ................................ .......... 28 4 2 Rotor speed (Experimental Results) ................................ ................................ ... 28 5 1 Definition of symbols ................................ ................................ .......................... 58 5 2 Control gains (PD Controller case) ................................ ................................ ..... 59 5 3 Control gains (PID Controller ca se) ................................ ................................ .... 60

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8 LIST OF FIGURES Figure page 2 1 Quad rotor helicopter model picture ................................ ................................ ... 16 3 1 Quad rotor Helicopter Schematic (Hovering) ................................ ...................... 22 3 2 Vertical movement ................................ ................................ .............................. 22 3 3 Ro ll movement ................................ ................................ ................................ .... 23 3 4 Pitch movement ................................ ................................ ................................ .. 23 3 5 Yaw movement ................................ ................................ ................................ ... 24 3 6 Moment o f inertia for solid cylinder ................................ ................................ ..... 24 3 7 Moment of inertia for the solid cuboid ................................ ................................ 25 5 1 PID controller (Block diagram) ................................ ................................ ............ 35 5 2 Simulink block diagram (PD controller : ideal case) ................................ ............ 35 5 3 Output roll angle (input : roll 30 degree, PD controller) ................................ ....... 36 5 4 Output pitch angle (input : roll 30 degree, PD controller) ................................ .... 36 5 5 Output yaw angle (input : roll 30 degree, PD controller) ................................ ..... 36 5 6 Output roll angle (input : pitch 30 degree, PD controller) ................................ .... 37 5 7 Output pitch angle (input : pitch 30 degree, PD controller) ................................ 37 5 8 Output yaw angle (input : pi tch 30 degree, PD controller) ................................ .. 37 5 9 Output roll angle (input : yaw 30 degree, PD controller) ................................ ..... 38 5 10 Output pitch angle (input : yaw 30 degree, PD controller) ................................ .. 38 5 11 Output yaw angle (input : yaw 30 degree, PD controller) ................................ ... 38 5 12 Simulink block diagram (PD controller : real case without mixing ) ..................... 39 5 1 3 Simulink block diagram (PD controller : real case with mixing ) .......................... 39 5 1 4 Output roll angle (input : roll 30 degree, PD controller without mixing ) .............. 40 5 1 5 Output pitch angle (input : roll 30 degree, PD controller without mixing ) ........... 40

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9 5 1 6 Output yaw angle (input : roll 30 degree, PD controller without mixing ) ............ 40 5 1 7 Output roll angle (input : pitch 30 degree, PD controller without mixing ) ........... 41 5 1 8 Output pitch angle (input : pitch 30 degree, PD controller without mixing ) ........ 41 5 1 9 Output ya w angle (input : pitch 30 degree, PD controller without mixing ) .......... 41 5 20 Output roll angle (input : yaw 30 degree, PD controller without mixing ) ............ 42 5 21 Output pitch angle (input : yaw 30 degree, PD controller without mixing ) .......... 42 5 22 Output yaw angle (input : yaw 30 degree, PD controller without mixing ) ........... 42 5 2 3 Output roll angle (input : roll 30 degree, PD controller, real case) ...................... 43 5 2 4 Output pitch angle (input : roll 30 degree, PD controller, real case) ................... 43 5 2 5 Output yaw angle (input : roll 30 degree, PD controller, real case) ..................... 43 5 2 6 Output roll angle (input : pitch 30 degree, PD controller, real case) ................... 44 5 2 7 Output pitch angle (input : pitch 30 degree, PD controller, real case) ................. 44 5 2 8 Output yaw angle (input : pitch 30 degree, PD contr oller, real case) .................. 44 5 2 9 Output roll angle (input : yaw 30 degree, PD controller, real case) ..................... 45 5 3 0 Output pitch angle (input : yaw 30 degree, PD controller, real case) .................. 45 5 3 1 Output yaw angle (input : yaw 30 degree, PD controller, real case) ................... 45 5 3 2 Simulink block diagram (PID controller : ideal case) ................................ ........... 46 5 33 Simulink block diagram (PID controller : real case without mixing ) .................... 46 5 3 4 Simulink block diagram (PID controller : real case with mixing ) ......................... 47 5 35 Output roll angle (input : roll 30 degree, PID controller) ................................ ...... 48 5 36 Output pitch angle (input : roll 30 degree, PID controller) ................................ ... 48 5 37 Output yaw angle (input : roll 30 degree, PID controller) ................................ .... 48 5 38 Output roll angle (input : pitch 30 degree, PID controller) ................................ ... 49 5 39 Output pitch angle (input : pitch 30 degree, PID controller) ................................ 49 5 40 Output yaw angle (input : pitch 30 degree, PID controller) ................................ 49

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10 5 41 Output roll angle (input : yaw 30 degree, PID controller) ................................ .... 50 5 42 Output pitch angle (input : yaw 30 degree, PID controller) ................................ 50 5 43 Output yaw angle (input : yaw 30 degree, PID controller) ................................ .. 50 5 44 Output roll angle (input : roll 30 degree, PID controller without mixing ) ............. 51 5 45 Output pitch angle (input : roll 30 degree, PID controller without mixing ) .......... 51 5 46 Output yaw angle (input : roll 30 degree, PID controller without mixing ) ........... 51 5 47 Output roll angle (input : pitch 30 degree, PID controller without mixing ) .......... 52 5 48 Output pitch angle (input : pitch 30 degree, PID controller without mixing ) ....... 52 5 49 Output yaw angle (input : pitch 30 degree, PID controller without mixing ) ......... 52 5 50 Output roll angle (input : yaw 30 degree, PID controller without mixing ) ........... 53 5 51 Output pitch angle (input : yaw 30 degree, PID controller without mixing ) ......... 53 5 52 Output yaw angle (input : yaw 30 degree, PID controller without mixing ) .......... 53 5 5 3 Output roll angle (input : roll 30 degree, PID controller, real case) ..................... 54 5 5 4 Output pitch angle (input : roll 30 degree, PID controller, real case) .................. 54 5 5 5 Output yaw angle (input : roll 30 degree, PID controller, real case) .................... 54 5 5 6 Output roll angle (input : pitch 30 degree, PID controller, real case) .................. 55 5 5 7 Output pitch angle (input : pitch 30 degree, PID controller, real case) ................ 55 5 5 8 Output yaw angle (input : pitch 30 degree, PID controller, real case) ................. 55 5 5 9 Output roll angle (input : yaw 30 degree, PID controller, real case) .................... 56 5 6 0 Output pitch angle (input : yaw 30 degree, PID controller, real case) ................. 56 5 6 1 Output yaw angle (input : yaw 30 degree, PID controller, real case) .................. 56 5 62 Robustness to actuator variation (input : pitch, output : roll) ............................... 57 5 63 Robustness to actuator variation (input : pitch, output : pitch) ............................ 57 5 64 Robustness to actuator variation (input : yaw, output : yaw) .............................. 57

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MODELING AND CONTROL OF A QUAD ROTOR HELIIOPTER By Sang Min Oh A ugust 201 2 Chair : Rick Lind Major: Aerospace Engineering The quad rotor helicopter is a UAV (Un manned Air Vehicle) and has four rotors. To get efficient stabilization status, data measurement and analysis phase and simulation and control phase are needed. During data measurement and analysis phase, people are struggling with inaccurate data due to the external disturbance or the battery issue The author tried to get the moment of inertia of the quad rotor helic opter as accurate as possible based on the experimental data. During simulation and control phase the author applied PD (P roportional, D erivative ) controller. First of all, trying to satisfy stabilization for the ideal case, which means all motors have sa me thrust s and speed s Then the real motor values were applied to see the success of the controller s performance.

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12 CHAPTER 1 I NTRODUCTION 1.1 Motivation Recently, there are many researches about helicopters, UAV and MAV (Micro Air Vehicle). Q uad rotor helicopters which are inexpensive to prepare, are of less mechanical complexity, better safety and higher payload. Especially this research is related to the real helicopters, so it is great topic to the author as a current army aviation officer. Therefore the author decided to study and research for VTOL (Vertical Take Off and Landing) aircraft field with a quad rotor helicopter Quad rotor helicopter can be utilized such as surveillance, transporting or researching geography, and so on. And it is a good research project topic, because there is not a unique way to define control the quad rotor helicopter, then anybody can try to approach different method s to find control a quad rotor helicopter Fundamentally the helicopter has six degrees of fre edom and in case of the quad rotor model, it has four motors. In order to complete this project, all courses such as control theory, robot geometry and dynamics, those are required to graduate master s degree program. During preparing and researching this topic, the author could review and understand the relationship among all courses and utilized them to complete this research. First of all, the author measured and searched all specifications from th e real quad rotor helicopter model. Then the author built a system model which is based on its dynamics and state space form. In order to satisfy stabilization and proper control, the author applied PD (Proportional, Derivative) controller with an ideal ca se and the measured values.

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13 The main purpose of this thesis is showing stabilization and control through MATLAB and SIMULINK. The author will match with a real model in the future. There are many conference papers, thesis about quad rotor helicopters. Th ose projects try to set up their model and simulation for effective controlling based on different quad rotor models. A nonlinear dynamic MIMO (Multi Input Multi Output) model of a 6 DOF (Degrees Of Freedom) quad rotor helicopter is derived based on Newton Euler formalism. [1] Derivation about q uad rotor helicopter d ynamics and c ontrol [2] P resent the mechanical design, dynamic al modeling, sensing, and control ling of an indoor quad rotor helicopter [3] Different types of controller ( Lyapunov PID LQ, back stepping and sliding mode concepts are used) are implemented for validating through various flight experiments [4] P resent the results of two nonlinear control techniques ( back stepping and sliding mode ) applied to a q uad rotor helicopter [5] D escribes the control approach (Integral Back stepping) and the scheme for full control of quad rotor helicopters (attitude, altitude and position). [6] The dynamic system modeling and the control algorithm evaluation for a quad rotor helicopter [7] A probl em of attitude stabilization and robust regulation of an indoor quad rotor helicopter ( The paper shows the design of continuous time controller based on Dynamic Contraction Method ) [8] A dynamic model of such a vehicle using bond graphs [9] A quad rotor he licopter using custom built chassis and avionics with off the shelf motors and batteries, to be a highly reliable experimental platform ( A linear SISO controller was designed to regulate flyer attitude ) [10]

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14 1.2 Problem Statement What are basic movements of the quad rotor helicopter? What is the equation of motion for the quad rotor helicopter? How to control the quad rotor helicopter properly and accurately when the different inputs / thrust (roll, pitch and yaw) are given? How to develop the controller w hen the ideal cases (pure movements) are implemented and when the measured values are applied? 1.3 Contribution Setting up the computer model which is based on the real quad rotor helicopter model Applying PD (Proportional, Derivative) and PID (Proportiona l, Integral and Derivative) controllers for satisfying stabilization and controlling the model properly and accurately Verifying the PD and PID controllers by applying the ideal cases and the measured values

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15 CHAPTER 2 Q UAD ROTOR H ELICOPTER S PECIFICATIONS 2.1 Model Picture Figure 2 1 shows the real quad rotor helicopter which was used in this thesis. This quad rotor helicopter model belongs to Geo matics lab. There are four motors, and each motor has a propeller. 2.2 Specifications Table 2 1 and Table 2 2 show the specifications of the model and the motor. A nd the motor model was Turnigy SK 4250 650. Compared to other general quad rotor helicopter, this model is big and it is not easy to con trol without controller. 2.3 Basic Instruments Table 2 3 shows the instruments that were needed during this research. During using digital tachometer and thrust meter, the motor from the quad rotor helicopter must be linked to the solid body. Then the mete r equipment could measure the maximum and the minimum value and it showed on the screen.

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16 Figure 2 1 Quad rotor helicopter model picture

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17 Table 2 1 Weights Motor Battery Frame (Whole body) Hub Unit (Kg) 0.288 0.356 0.701 0.293 Table 2 2 Lengths From the beginning of the arm (mm) To the center of the motor To the end of the arm 415 440 From the center of the frame (mm) 480 505 Table 2 3 Basic i nstruments Name (Model) Full scale (Range) Accuracy Electronic balance (Kern 440 45N) 1 Kg 0.2 g measuring tape (Assist 3M) 3 m 0.001 m Digital tachometer (DT 2234C+) 2.5 ~ 99,999 RPM 0.05% + 1 digit Digital multi meter (CEN TECH 98025) 200mV/2000mV 20/200/1000V (@200mV) 0.5% 1D (@2000mV 200V) 1% 2D (@1000V) 1% 2D Thrust meter (Tahmazo) 5 ~ 9,995 g 5 g

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18 CHAPTER 3 E QUATIONS OF M ODEL 3.1 Basic Movements The quad rotor helicopter can be defined as a VTOL UAV whose take off or lift is powered by four motors and each motor has its own rotors. According to Figure 3 1 rotor number 1 and 3 (or front and rear rotor) rotates CCW (Counter Clock Wise), and rotor n umber 2 and 4 (or right and left rotor) rotates CW (Clock Wise). Because of this arrangement, aerodynamic torque is canceled by each other. Initially the helicopter is on the ground before taking off. There are five status of movement condition for the qu ad rotor helicopter after taking off. Those are hovering, vertical (or thrust) roll ing pitch ing and yaw ing movement s Each movement condition is characterized by different motor speed s With different colors, shapes and length s each movement condition can be described simply and clearly. T he f ollowing indicators will be used generally for any of the different cases. Black lines (and circles) Frame and four rotors Green lines Body fixed frame Blue lines (straight and its length) Velocity Blue lines (curve) Rotation direction Red line (straight or curve) Whole body moving direction 3.1.1 Hovering Figure 3 1 shows the quad rotor helicopt er under hovering condition. In this case, the four rotors have the same speed so generate the same thrust. The total thrust from all four motors equals the weight, so the helicopter hovers at an altitude. T he helicopter maintains its equilibrium and bala nce, so it does not move in any direct ion unless acted upon by other forces.

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19 Clearly, in hovering movement condition, there is no red arrow because the quad rotor helicopter does not move anywhere (up, down, roll, pitch or yaw) but just maintain s its current position. 3.1.2 Vertical M ovement When increasing or decreasing all rotor speed s by the same amount, the quad rotor helicopter will move upward or downward along the z axis. Figure 3 2 shows this vertical movement. 3.1.3 Roll M ovement In the c ase of roll movement, front and rear rotors have the same speed, but left and right rotors have different amount s of speed. If the left rotor increases (decreases) its speed, the right rotor decreases (increases) its speed simultaneously. Therefore the qua d rotor helicopter rotates (or rolls) left and right. Figure 3 3 shows a simple diagram of roll movement as distinguish ed by the length of each straight arrow. Clearly front and rear arrows have the same length but the left and right arrows have different length s 3.1.4 Pitch M ovement P itch movement is characterized by left and right rotors hav ing same speed but front and rear rotors hav ing different speed. If the front rotor increases (decreases) its speed, the rear rotor decreases (increases) its speed simultaneously. Therefore the quad rotor helicopter rotates (or pitche s) front and rear. The quad rotor helicopter cannot dist inguish four directions, so it must be marked which one is front, rear, right and left. Then roll movement and pitch movement can be characterized by its initial setting

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20 Figure 3 4 shows a simple diagram of pitch movement. Clearly right and left arrows h ave same length but front and rear arrows have different length s If the observer or the experimenter confuses the direction of each motor, pitching and rolling movements must be flipped over. So, the direction of each motor has to be decided firmly and cl early. 3.1.5 Yaw M ovement In case of yaw movement, the front rear rotors have the same speed and the left right rotor s have same speed but each pair of rotors have different amount s of speed. If the front rear rotor pair decreases (increases) its speed, then the left right rotor pair increases (decreases) its speed simultaneously. Therefore the quad rotor helicopter rotates CCW (CW) along z axis. 3 .2 Three Different Types o f Mo ments According to the general equations of motion for the quad rotor helicopte r [4], quad rotor helicopter movements are caused by the series of moments and forces. T able 3 1 through Table 3 3 shows the three different types of moments of a quad rotor helicopter. 3.3 Equations o f Motion C ombined moments consists equations of motion for six degrees of freedom (6 DOF) dynamics of a quad rotor helicopter. [4]

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21 (3 1) 3. 4 Moments of Inertia In order to calculate the moments of inertia of the quad rotor helicopter for this thesis, two types of basic formula are needed. Those are the moment of inertia for the solid cylinder shape (for motor), and the moment of inertia for the solid cuboid shape (for central hub). The quad rotor helicopter for this thesis is approximated with four solid c ylinder shape motors and solid cuboid shape central hub. of the solid cuboid (central hub) can be calculated with of the solid cylinder (motor) because they share same z axis. L ikewise can be compared to because they share same x axis. L astly, can be calculated with because they share same y axis. Thanks to above equations, the moments of inertia for the quad rotor helicopter for this thesis are, (3 2)

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22 Figure 3 1 Quad rotor Helicopter Schematic (Hovering) Figure 3 2 Vertical movement X B Z B Y B RIGHT FRONT REAR LEFT FRONT RIGHT REAR LEFT

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23 Figure 3 3 Roll movement Figure 3 4 Pitch movement FRONT RIGHT REAR LEFT FRONT RIGHT REAR LEFT

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24 Figure 3 5 Yaw movement Figure 3 6 Moment of inertia for solid cylinder FRONT RIGHT REAR LEFT z x y r h

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25 Figure 3 7 Moment of inertia for the solid cuboid h w d

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26 Table 3 1. Rolling m oments Effect Equations Body gyro effect Rolling moment due to forward flight Propeller gyro effect Hub moment due to sideward flight Roll actuators action Table 3 2. Pitching m oments Effect Equations Body gyro effect Pitching moment due to forward flight Propeller gyro effect Hub moment due to sideward flight Pitch actuators action Table 3 3. Yawing m oments Effect Equations Body gyro effect Yawing moment due to forward flight Propeller gyro effect Hub moment due to sideward flight Yaw actuators action

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27 CHAPTER 4 E XPERIMENTAL M ODELING 4.1 Thrust Test In order to do the thrust test, the author attached the thrust meter to a solid frame and linked each motor from the quad rotor helicopter. T hen the half throttle was given to the motor and the thrust meter measured the thrust at the status. T he unit is and Table 4 1 shows the results of the thrust test. 4.2 R otor Speed Test In case of the motor speed test, digital tachometer was located in front of the motor. I t measured the highest rpm while the rotors rotate. E ach motor s rotor s have almost the same speeds, if the same amount of throttle is given.

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28 Table 4 1 Thrusts (Experimental Results) Motor No. 1 st result 2 nd result 3 rd result 4 th result mean S tandard D eviation 1 1.275 1.100 1.165 1.145 1.171 3 0.0743 2 1.325 1.060 1.125 1.160 1.16 75 0.1129 3 1.310 1.265 1.235 1.265 1.26 88 0.0309 4 1.435 1.295 1.375 1.280 1.346 3 0.0724 Table 4 2 Rotor speed (Experimental Results) 1 st result 2 nd result 3 rd result 4 th result mean S tandard D eviation Unit (rpm) 2807 2727 2676 2579 2697 95.5070

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29 CHAPTER 5 S YSTEM C ONTROL 5.1 State Space Model In order to get a system control, the quad rotor helicopter dynamics model (3 1) can be expressed by state space form as, (5 1) F rom (3 1) and (5 1), we obtain : (5 2)

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30 W here (5 3) 5.2 Linearization o f the System S tate space model (5 2) is not a linear system, so in order to apply PID control technique, the system must be linearized. Equilibrium points are, and (5 4) T herefore, linearized state space model of the system is, (5 5)

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31 W here, (5 6) (5 7) (5 8) (5 9)

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32 5. 3 PID ( Proportional, Integral and Derivative ) Control Technique There are many kinds of controlling methods to satisfy stabilization or control properly for the quad rotor helicopter. According to other papers and thesis, Lyapunov / optimal control theories and PID (Proportional, Integral and Derivative) / LQR ( Linear quadratic regulator ) / back stepping sliding mode techniques were used for controlling the quad rotor helicopter. [3 6] I n this thesis, the author only focused on PID control technique B ecause it is the most general control m ethod, and its performance is satisfied with the expectation for this thesis. Additionally PD controller was introduced for comparing two different types of controllers. Figure 5 1 shows a block diagram of a PID controller. Basically, P (Proportional term) controls the system based on the current error proportionally. I (Integral term) contributes reducing the error in steady state. D (Derivative term) prevents the rapid change of the controller s output. It contributes decreasing the overshoot and improvin g the stability of the system. For this thesis, PD and PID controllers were applied. 5. 4 Simulation Results 5.4.1 PD (Proportional and Derivative) C ontroller for the I deal C ase The input of the PD controller is the error which is combined the expected ang le and the output angle. Kr, Kp and Ky are the proportional control gains for rolling, pitching and yawing errors. Kdr, Kdp and Kdy are the derivative control gains for rolling, pitching and yawing errors after implementing derivative controller. The output of rolling for the rolling input is same as the output of pitching for the pitching input and the output of pitching for the rolling input is same as the output of

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33 rolling for the pitching input. Yawing movement does not occur in response to the rolling and pitching commands. Also, rolling and pitching movements are not created in response to a yawing command. 5.4.2 PD Controller without Mixing for the Real Case T he purpose of this chapter is that the author tried to show the bad result of the P D controller without mixing for the real case. I t is the reason why the mixing process is needed to control the system for the real case properly. I n this case, the real actuator was implemented between PD controller and the plant model. U 1 through u4 mean the real calculated input thrusts. According to the results, output of the roll and pitch angle is oscillate d, so we can define this result as a failure of controlling. 5.4. 3 PD C ontroller with Mixing for the R eal C ase I n this chapter, the real measured and calculated thrust values are applied for the system. Thanks to math process, the whole system are not affected by the real thrust values. Therefore, this model can be controlled by the expected movements such as the pure rolling, pitching and yawing be cause the mixing process was added. U1 is the summation of all four motor s thrust (speed). U2 is the combination of difference between 4 th motor and 2 nd motor. U3 is the combination of difference between 3 rd motor and 1 st motor. Lastly, U4 is the combinat ion of difference between the summation of 2 nd motor and 4 th motor, and the summation of 1 st motor and 3 rd motor. 5.4. 4 PID C ontroller for the I deal C ase Figure 5 3 2 shows Simulink block diagram for PID controller. Each control gains affect all output values Therefore tuning process is required.

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34 Likewise previous results, the output of rolling for th e rolling input is same as the output of pitching for the pitching input and the output of pitching for the rolling input is same as the output of rolling for the pitching input. Yawing movements does not occur when the rolling and pitching commends, and r olling and pitching movements does not create when the yawing commend. 5.4. 5 PID C ontroller without Mixing for the Real Case T he author intentionally put this chapter due to explain the reason why the tuning or mixing process is needed for the real case. L ike the previous results of PD controller without mixing for the real case, all output angles were oscillate d, so the PID controller cannot perform well without mixing. 5.4.6 PID Controller with Mixing for the Real Case Similar to the result of Simulink b lock diagram for the real case of PD controller e ach control gains affect all output values, so tuning process is required. Thanks to same process, the whole system can be controlled by the expected input value not because of the real thrust values. 5.4. 7 Robustness to Actuator Variation PID controller with mixing for the real case performed almost same even if the different real measured thrusts were given. Figure 5 62 through Figure 5 64 show the results of the robustness to actuator variation.

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35 Figure 5 1 PID controller (Block diagram) Figure 5 2 Simulink block diagram (PD controller : ideal case) P lant / Process I D P

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36 Figure 5 3 Output roll angle (input : roll 30 degree, PD controller) Figure 5 4 Output pitch angle (input : roll 30 degree, PD controller) Figure 5 5 Output yaw angle (input : roll 30 degree, PD controller)

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37 Figure 5 6 Output roll angle (input : pitch 30 degree, PD controller) Figure 5 7 Output pitch angle (input : pitch 30 degree, PD controller) Figure 5 8 Output yaw angle (input : pitch 30 degree, PD controller)

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38 Figure 5 9 Output roll angle (input : yaw 30 degree, PD controller) Figure 5 10 Output pitch angle (input : yaw 30 degree, PD controller) Figure 5 11 Output yaw angle (input : yaw 30 degree, PD controller)

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39 Figure 5 12 Simulink block diagram (PD controller : real case without mixing ) Figure 5 1 3 Simulink block diagram (PD controller : real case with mixing )

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40 Figure 5 1 4 Output roll angle (input : roll 30 degree, PD controller without mixing ) Figure 5 1 5 Output pitch angle (input : roll 30 degree, PD controller without mixing ) Figure 5 1 6 Output yaw angle (input : roll 30 degree, PD controller without mixing )

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41 Figure 5 1 7 Output roll angle (input : pitch 30 degree, PD controller without mixing ) Figure 5 1 8 Output pitch angle (input : pitch 30 degree, PD controller without mixing ) Figure 5 1 9 Output yaw angle (input : pitch 30 degree, PD controller without mixing )

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42 Figure 5 20 Output roll angle (input : yaw 30 degree, PD controller without mixing ) Figure 5 21 Output pitch angle (input : yaw 30 degree, PD controller without mixing ) Figure 5 22 Output yaw angle (input : yaw 30 degree, PD controller without mixing )

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43 Figure 5 2 3 Output roll angle (input : roll 30 degree, PD controller, real case) Figure 5 2 4 Output pitch angle (input : roll 30 degree, PD controller, real case) Figure 5 2 5 Output yaw angle (input : roll 30 degree, PD controller, real case)

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44 Figure 5 2 6 Output roll angle (input : pitch 30 degree, PD controller, real case) Figure 5 2 7 Output pitch angle (input : pitch 30 degree, PD controller, real case) Figure 5 2 8 Output yaw angle (input : pitch 30 degree, PD controller, real case)

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45 Figure 5 2 9 Output roll angle (input : yaw 30 degree, PD controller, real case) Figure 5 3 0 Output pitch angle (input : yaw 30 degree, PD controller, real case) Figure 5 3 1 Output yaw angle (input : yaw 30 degree, PD controller, real case)

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46 Figure 5 3 2 Simulink block diagram (PID controller : ideal case) Figure 5 33 Simulink block diagram (PID controller : real case without mixing )

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47 Figure 5 3 4 Simulink block diagram (PID controller : real case with mixing )

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48 Figure 5 35 Output roll angle (input : roll 30 degree, PID controller) Figure 5 36 Output pitch angle (input : roll 30 degree, PID controller) Figure 5 37 Output yaw angle (input : roll 30 degree, PID controller)

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49 Figure 5 38 Output roll angle (input : pitch 30 degree, PID controller) Figure 5 39 Output pitch angle (input : pitch 30 degree, PID controller) Figure 5 40 Output yaw angle (input : pitch 30 degree, PID controller)

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50 Figure 5 41 Output roll angle (input : yaw 30 degree, PID controller) Figure 5 42 Output pitch angle (input : yaw 30 degree, PID controller) Figure 5 43 Output yaw angle (input : yaw 30 degree, PID controller)

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51 Figure 5 4 4 Output roll angle (input : roll 30 degree, PID controller without mixing ) Figure 5 4 5 Output pitch angle (input : roll 30 degree, PID controller without mixing ) Figure 5 4 6 Output yaw angle (input : roll 30 degree, PID controller without mixing )

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52 Figure 5 4 7 Output roll angle (input : pitch 30 degree, PID controller without mixing ) Figure 5 4 8 Output pitch angle (input : pitch 30 degree, PID controller without mixing ) Figure 5 4 9 Output yaw angle (input : pitch 30 degree, PID controller without mixing )

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53 Figure 5 50 Output roll angle (input : yaw 30 degree, PID controller without mixing ) Figure 5 51 Output pitch angle (input : yaw 30 degree, PID controller without mixing ) Figure 5 52 Output yaw angle (input : yaw 30 degree, PID controller without mixing )

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54 Figure 5 5 3 Output roll angle (input : roll 30 degree, PID controller, real case) Figure 5 5 4 Output pitch angle (input : roll 30 degree, PID controller, real case) Figure 5 5 5 Output yaw angle (input : roll 30 degree, PID controller, real case)

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55 Figure 5 5 6 Output roll angle (input : pitch 30 degree, PID controller, real case) Figure 5 5 7 Output pitch angle (input : pitch 30 degree, PID controller, real case) Figure 5 5 8 Output yaw angle (input : pitch 30 degree, PID controller, real case)

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56 Figure 5 5 9 Output roll angle (input : yaw 30 degree, PID controller, real case) Figure 5 6 0 Output pitch angle (input : yaw 30 degree, PID controller, real case) Figure 5 6 1 Output yaw angle (input : yaw 30 degree, PID controller, real case)

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57 Figure 5 6 2 Robustness to actuator variation (input : pitch, output : roll) Figure 5 6 3 Robustness to actuator variation (input : pitch, output : pitch ) Figure 5 6 4 Robustness to actuator variation (input : yaw output : yaw )

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58 Table 5 1. Definition of symbols Symbol s Definition Position vector R Rotation matrix Skew symmetric matrix Roll angle Pitch angle Yaw angle Rotor speed Body inertia Rotor inertia Motor inertia Propeller inertia Torque on airframe body b Thrust factor d Drag factor Lever

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59 Table 5 2 Control g ains (PD Controller case)

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60 T able 5 3 Control g ains (P I D Controller case) I 0.001 0.001 100 0.001 1 1 100 0.001 0.01 0.001 1 1 1 1 1 1 0.01 0.001

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61 CHAPTER 6 CONCLUSION The author got data fundamentally from the real experiment with basic measurement instruments that are announced in experimental setup. Even though the instruments have good accuracy range, there were errors among data. In the result of real experiment, each value was changed with a big difference. Therefore the standard deviation was high. On the other hand, the a uthor tried to generate one thousand random values (normal distributed random data) between the highest and the lowest value of the real experimental result. Through this process, the author could calculate less standard deviation than previous value. Usi ng this simple method, the author does not need to experiment over again and again. With several real data, the author calculates those means and standard deviations. Then with numbers of generating data based on real data s measured range, the author calc ulates those means comparing to real mean. When the author tries to calculate the moment of inertia and thrust factor, these modified values can be used. In case of simulation and controlling phase, the author tried to get high performance for satisfying s tabilization with PD and PID controller. Therefore, the quad rotor helicopter model for this thesis can be controlled by the expected inputs. It implies that the helicopter can operate rolling, pitching and yawing movements properly when the specific comma nds are requested.

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62 LIST OF REFERENCES Zhejiang University SCIENCE A, 2008, pp. 539 545. Rep. 2008. mode techniques applied to an i in Proc. (IEEE) International Conference on Robotics and Automation on Intelligent Robots and Systems, 200 7, pp. 153 158. Control, P. Albertos and A. Sala, Eds. London, U.K.:Springer Verlag, 2002, pp. 3 16. Silesian University of Technology, 2009. Rotor d in Proceedings of the Australasian Conference on Robotics and Automation (Auckland, New Zealand), 2006. http://www.embedded.com 2000.

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63 BIOGRAPHICAL SKETCH Sang Min Oh was born in Seoul, South Korea in 1981. H e graduated from Soong Sil University with his Bachelor of Science degree in 2004. During his junior and senior period in the University, Sang Min participated in Reserve Officer Training Corps (ROTC) program. After he commissioned in 2004, he served in the army as a communication officer. Sang Min changed his branch to army aviation in 2005, and he became a pilot with about 300 flight hours of 500MD and UH 1H helicopters. Therefore, he will contribute for improving and developing h is branch with his knowledge. Thanks to the Korean army, he experienced training, commanding and S 4 (logistics) officer tasks A nd he got advanced military education such as Military English Course (MEC) and Officers Advanced Course (OAC). R ecently, he i s interested in the Peace Keeping Operations (PKO) and its related jobs in the United Nations (UN). He sincerely hopes that he can serve for the world peace. S ang Min is proud of serving for his country and he always tries to do his best for improving the army. H e wants to contribute for enhancing the relationship between the United States and Korea. H e will do his best at any time, any location and any job.