DYNAMIC MODELING AND CONTROL OF AN ELECTROMECHANICAL CONTROL ACTUATION SYSTEM

DYNAMIC MODELING AND CONTROL OF AN ELECTROMECHANICAL CONTROL ACTUATION SYSTEM

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF

MIDDLE EAST TECHNICAL UNIVERSITY

BY

ÜMİT YERLİKAYA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

MECHANICAL ENGINEERING

SEPTEMBER 2016

Approval of the thesis:

DYNAMIC MODELING AND CONTROL OF AN ELECTROMECHANICAL CONTROL ACTUATION SYSTEM

submitted by ÜMİT YERLİKAYA in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by,

Prof. Dr. Gülbin Dural Ünver

Dean, Graduate School of Natural and Applied Sciences __________________

Prof. Dr. Tuna Balkan

Head of Department, Mechanical Engineering __________________

Prof. Dr. Tuna Balkan

Supervisor, Mechanical Engineering Dept., METU __________________

Examining Committee Members Prof. Dr. Metin Akkök

Mechanical Engineering Dept., METU __________________

Prof. Dr. Tuna Balkan

Mechanical Engineering Dept., METU __________________

Assoc. Prof. Dr. İlhan Konukseven

Mechanical Engineering Dept., METU __________________

Assoc. Prof. Dr. Yiğit Yazıcıoğlu

Mechanical Engineering Dept., METU __________________

Assoc. Prof. Dr. S. Çağlar Başlamışlı

Mechanical Engineering Dept., Hacettepe University __________________

Date: 09.09.2016

iv

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last Name: Ümit Yerlikaya

Signature :

v ABSTRACT

DYNAMIC MODELING AND CONTROL OF AN ELECTROMECHANICAL CONTROL ACTUATION SYSTEM

Yerlikaya, Ümit

M. S., Department of Mechanical Engineering Supervisor: Prof. Dr. Tuna Balkan

September 2016, 131 pages

Electromechanical simulators, actuators are widely used in miscellaneous applications in engineering such as aircrafts, missiles, etc. These actuators have momentary overdrive capability, long-term storability and low quiescent power/low maintenance characteristics. Thus, electromechanical actuators are applicable option for any system in aerospace industry, instead of using hydraulic actuators. In the same way, they can be used in control actuation section of missiles to deflect flight control surfaces. Mostly used alternatives of control actuation system (CAS) are electromechanical, electrohydraulic and electrohydrostatic CASs. In this thesis, electromechanical control actuation systems that are composed of brushless direct current motor, ball screw and lever mechanism are studied. In this type of control actuation system, there are both nonlinearity and asymmetry which are caused by lever mechanism itself, saturation limits, Coulomb friction, backlash and initial mounting position of lever mechanism. In order to design controller and optimize controller parameters, all equations of motion are derived and so the detailed nonlinear and linear mathematical models of this system are obtained. The servo

vi

drive amplifier of motor is used in current mode. Between position and current loops, inner velocity loop is used to provide extra damping to the system and avoid unnecessary oscillations. Therefore, three control loops are used. By using linear model of electromechanical CAS, according to performance requirements, it is decided that PI and P-controller are sufficient for position and velocity control, respectively. The limitations that are imposed to controllers which have integral gain cause a residual error, so the controllers tend to overshoot target value in order to eliminate it. In order to solve this problem, an anti-windup method is applied. Then, the unknown controller parameters and anti-windup coefficients are found according to the performance requirements by using MATLAB Response Optimization Tools on the nonlinear model. During the optimization, the nonlinear relations and limitations on controller outputs are considered. A prototype of electromechanical CAS with ball screw and lever mechanism is manufactured. All unknown parameters such as dimensions, masses, inertias of components, viscous and Coulomb frictions and backlash of the system are identified. Identified Coulomb friction values are used for friction compensation in real-time application. Real-time tests are performed with optimized controller parameters and anti-windup coefficient by using xPC Target (MATLAB-Simulink). Finally, the nonlinear model of electromechanical control actuation system is verified by making real-time tests on the manufactured prototype with and without external load.

Keywords: Control Actuation System, Electromechanical Actuators, Fin, Control of Brushless DC Motor, Response Optimization, PID, xPC Target

vii ÖZ

ELEKTROMEKANİK KONTROL TAHRİK SİSTEMİNİN DİNAMİK MODELLENMESİ VE KONTROLÜ

Yerlikaya, Ümit

Yüksek Lisans, Makine Mühendisliği Bölümü Tez Yöneticisi: Prof. Dr. Tuna Balkan

Eylül 2016, 131 sayfa

Elektromekanik eyleyiciler uçak ve füzelerde olmak üzere birçok uygulamada sıklıkla kullanılmaktadır. Bu eyleyiciler anlık aşırı sürme kapasitesine, uzun vadeli depolanabilme özelliğine, pasif durumda düşük güç tüketebilme ve az bakım gerektirme karakteristiğine sahiptirler. Bu yüzden elektromekanik eyleyiciler havacılıktaki çoğu sistem için uygulanabilir bir seçenektir. Aynı şekilde bunlar füzelerin kontrol bölümlerinde uçuş kontrol yüzeylerini döndürmek için kullanılabilmektedir. En çok kullanılan kontrol tahrik sistemleri elektromekanik, elektrohidrolik ve elektrohidrostatik alternatifleridir. Bu tez kapsamında, fırçasız doğru akım motoru, bilyeli vida ve kaldıraç mekanizmasından oluşan bir elektromekanik kontrol tahrik sistemi ele alınmıştır. Bu tip kontrol tahrik sistemlerinde, kaldıraç mekanizmasının kendisinden, limitlerden, Coulomb sürtünmelerinden, boşluklardan ve kaldıracın ilk montaj konumlanmasından kaynaklanan bazı doğrusal olmayan durumlar ve simetri bozuklukları mevcuttur.

Kontrolcü tasarımı ve kontrolcü parametrelerinin en iyilenmesi için tüm hareket denklemleri türetilmiş, sistemin ayrıntılı doğrusal ve doğrusal olmayan matematiksel

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modelleri elde edilmiştir. Servomotor sürücüsü akım modunda kullanılmaktadır.

Sisteme fazladan sönüm katmak ve oluşabilecek gereksiz salınımları engellemek için konum ve akım döngüsü arasına eklenen hız iç döngüsü ile üç kontrol döngüsü kullanılmaktadır. Doğrusal model kullanılarak, başarım gereksinimlerine göre konum ve hız kontrolcüsü olarak sırasıyla PI ve P kontrolcülerin kullanılmasının yeterli olduğuna karar verilmiştir. İntegral kazancı olan kontrolcü çıkışlarına uygulanan sınırlar hataların artmasına sebep olmaktadır. Gerçekleşen değerin referans komutuna varmasına rağmen, bu hatalar hemen giderilememektedir. Bu durumu ortadan kaldırmak için “anti-windup” yöntemi uygulanmıştır. Sonrasında, belli olmayan kontrolcü parametreleri ve “anti-windup” katsayısı, doğrusal olmayan model kullanılarak performans gereksinimine göre MATLAB Response Optimization Tools (cevap en iyileme araçları) yardımıyla bulunmuştur. En iyileme sırasında, doğrusal olmayan ilişkiler ve kontrolcü çıkışlarına uygulanan sınırlamalar göz önünde bulundurulmuştur. Kaldıraç mekanizmalı ve bilyeli vidalı elektromekanik kontrol tahrik sisteminin prototipi üretilmiştir. Bileşenlerin boyutları, kütleleri, atalet momentleri ve sistemin viskoz/Coulomb sürtünmeleri ve boşluk gibi bilinmeyen parametreler belirlenmiştir. Belirlenen Coulomb sürtünme değerleri gerçek zamanlı uygulamada sürtünme telafisi olarak kullanılmıştır. Gerçek zamanlı testler, en iyilenen kontrolcü parametreleri ve “anti-windup” katsayısı kullanılarak xPC Target (MATLAB-Simulink) ortamında yapılmıştır. Son olarak, üretilen prototip üzerinde harici yük olmadan ve harici yük altında gerçek zamanlı testler yapılarak, doğrusal olmayan benzetim modeli doğrulanmıştır.

Anahtar Kelimeler: Kontrol Tahrik Sistemi, Elektromekanik Eyleyici, Fin, Fırçasız Doğru Akım Motor Kontrolü, Cevap Eniyileme, PID, xPC Target

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ACKNOWLEDGEMENTS

First of all I would like to express my sincere appreciation to my thesis supervisor Prof. Dr. Tuna BALKAN for his guidance throughout my thesis study.

Then I would like to thank TÜBİTAK for scholarship and financial supports which made this project possible. I hope their endless support to science and scientists continue to enrich our scientific heritage.

I wish to express my gratitude to Prof. Dr. Bülent E. PLATİN and Ahmet Can AFATSUN for their guidance throughout my thesis study.

I am in debt of gratitude to Aslı AKGÖZ BİNGÖL in ROKETSAN Inc. for her friendly support and advices.

I also owe thanks to my colleagues in my unit in ROKETSAN Inc. who helped me in every subject throughout the years of my study.

Finally I wish to express my sincere thanks to my father Ahmet YERLİKAYA, my mother Nigar YERLİKAYA, my sister Ayfer YERLİKAYA and lastly I dedicate this thesis work to my lovely son, Onur YERLİKAYA.

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TABLE OF CONTENTS

ABSTRACT ... V ÖZ ... VII ACKNOWLEDGEMENTS ... IX TABLE OF CONTENTS ... X LIST OF TABLES ... XIII LIST OF FIGURES ... XIV LIST OF ABBREVIATIONS ... XVIII LIST OF SYMBOLS ... XIX

CHAPTER 1 ... 1

1. INTRODUCTION ... 1

1.1 Literature Survey ... 1

1.1.1 Electromechanical Control Actuation Systems (EM-CASs) ... 2

1.1.1.1 EM-CAS with Screw and Lever Mechanism ... 3

1.1.1.2 EM-CAS with Clutch Actuator ... 6

1.1.1.3 EM-CAS with Worm Gear ... 7

1.1.2 Electrohydrostatic Control Actuation Systems (EHS-CASs) ... 8

1.1.3 Electrohydraulic Control Actuation Systems (EH-CASs) ... 9

1.2 Objective and Scope of Thesis ... 12

1.3 Thesis Outline ... 13

CHAPTER 2 ... 17

2 DYNAMIC MODELING OF EM-CAS ... 17

2.1 Mechanical System ... 17

2.1.1 Kinematic Relations ... 18

2.1.2 Equations of Motion ... 20

2.2 Electrical System ... 28

2.3 Block Diagram of EM-CAS ... 29

CHAPTER 3 ... 33

3 IDENTIFICATION OF THE SYSTEM PARAMETERS ... 33

3.1 System Performance Criteria ... 34

3.2 The Components of EM-CAS ... 35

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3.2.1 BLDC Motor ... 35

3.2.2 Servo Drive Amplifier... 36

3.2.3 Real-time Target Machine (RTTM) ... 40

3.2.4 Incremental Encoder ... 40

3.2.5 Fin ... 41

3.2.6 Fin Shaft ... 42

3.2.7 Fork ... 43

3.2.8 Screw and Nut ... 44

3.3 The Friction Characterization of EM-CAS ... 45

3.3.1 Friction Measurement Test Setup ... 45

3.3.2 Viscous Friction ... 46

3.3.2.1 Ambient Temperature, 24ºC ... 47

3.3.2.2 Ambient Temperature, -6ºC ... 50

3.3.2.3 The Effect of Different Ambient Temperatures on Viscous Friction ... 52

3.3.2.4 Curve Fitting to Viscous Friction Graph ... 55

3.3.3 Coulomb Friction ... 57

3.3.3.1 Ambient Temperature, 24ºC ... 57

3.3.3.2 Ambient Temperature, -6ºC ... 59

3.3.3.3 The Effect of Different Ambient Temperatures on Coulomb Friction ... 60

3.3.3.4 Curve Fitting to Coulomb Friction Graph ... 60

3.4 Measuring Backlash in EM-CAS ... 64

3.5 Loading Test Setup ... 66

CHAPTER 4 ... 69

4 LINEAR SYSTEM MODELING AND CONTROLLER DESIGN ... 69

4.1 Linear System Modeling ... 70

4.1.1 Using P-Controller in Position Control ... 73

4.1.2 Using PI-Controller in Position Control ... 76

4.2 Controller Design ... 78

4.2.1 Optimizing of Controller Parameters ... 79

CHAPTER 5 ... 83

5 SIMULATION AND TEST RESULTS ... 83

5.1 Nonlinear Simulink Model of EM-CAS ... 83

5.1.1 Controllers ... 84

5.1.2 Motor & Drive ... 84

5.1.3 Screw and Nut ... 85

5.1.4 Fin ... 86

5.2 Comparison of Linear and Nonlinear Models ... 87

5.3 Real-time Test Model in xPC Target ... 89

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5.3.1 Homing ... 91

5.3.2 Setup Files ... 92

5.3.3 Incremental Encoder ... 93

5.3.4 Friction Compensation ... 94

5.3.5 Loading Model ... 94

5.4 Evaluation and Comparison of Test and Simulation Results ... 95

5.4.1 Unloaded Tests ... 95

5.4.1.1 Step Test ... 96

5.4.1.2 Modified Square Wave Test ... 98

5.4.1.3 Bandwidth (Chirp) Test ... 100

5.4.2 Loaded Tests ... 103

5.4.2.1 The Comparison of Simulation and Real-Time Test Results ... 104

CHAPTER 6 ... 109

6 DISCUSSION OF THE RESULTS AND CONCLUSIONS ... 109

6.1 Summary and Discussion ... 109

6.2 Conclusions and Future Work ... 113

REFERENCES ... 115

APPENDIX A ... 117

APPENDIX B ... 129

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LIST OF TABLES

TABLES

Table 1: Parameters to Calculate Frequencies ... 32

Table 2: Performance Requirements of EM-CAS ... 34

Table 3: Motor Parameters ... 35

Table 4: Determining Units of Current Loop of Ba-mobil ... 38

Table 5: Specifications of Servo Drive Amplifier ... 39

Table 6: Required IO Modules ... 40

Table 7: Specifications of the Fin ... 42

Table 8: Specifications of the Fin Shaft ... 43

Table 9: Specifications of the Fork ... 44

Table 10: Specifications of the Screw and Nut ... 44

Table 11: Components of FMTS [18] ... 46

Table 12: Parameters of FMTSM [18] ... 46

Table 13: Effects of Ambient Temperatures on the Friction ... 53

Table 14: Friction Parameters of EM-CAS @ 24ºC ... 57

Table 15: Coefficients of Polynomials ... 61

Table 16: Specifications of the Components of LTS [21] ... 68

Table 17: Requirements of Position Controller ... 69

Table 18: Parameters to Be Optimized ... 79

Table 19: Optimized Parameters ... 81

Table 20: Realized Performance Values of EM-CAS... 103

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LIST OF FIGURES

FIGURES

Figure 1.1: Types of Control Actuation Systems (CASs) [1] ... 2

Figure 1.2: Control Actuation System (CAS) [5], [6] ... 3

Figure 1.3: EM-CAS with Screw and Lever Mechanism [5], [6] ... 4

Figure 1.4: Ristanović's EM-CAS Simulation Model [5], [6], [7] ... 5

Figure 1.5: Özkan's EM-CAS Simulation Model [9], [10] ... 5

Figure 1.6: Habibi's EM-CAS Simulation Model [8] ... 6

Figure 1.7: EM-CAS with Clutch Actuator [3], [4] ... 7

Figure 1.8: EM-CAS with Worm Gear ... 7

Figure 1.9: EHS Top Level System Schematic [1] ... 8

Figure 1.10: An Example of EHS [25] ... 9

Figure 1.11: Typical Valve Controlled Hydraulic Circuit [2] ... 10

Figure 1.12: Schematic Diagram of the Servo Mechanism ... 11

Figure 2.1: Assembly State of EM-CAS ... 18

Figure 2.2: Motion State of EM-CAS ... 19

Figure 2.3: Free Body Diagram of Motor and Screw Pairs ... 21

Figure 2.4: Free Body Diagram of Screw and Nut Pairs ... 21

Figure 2.5: Free Body Diagram of Aero Fin and Fork Pairs ... 22

Figure 2.6: Representation of Earth’s Fixed and Missile’s Local Axes ... 25

Figure 2.7: Nonlinear Block Diagram of EM-CAS ... 29

Figure 2.8: Details of Motor Shaft and Screw ... 31

Figure 3.1: Electromechanical Control Actuation System ... 34

Figure 3.2: Torque vs Speed Graph of Motor ... 36

Figure 3.3: Servo Drive Amplifier [17] ... 37

Figure 3.4: Motor Actual Current /Reference Analog Voltage Input, ... 37

Figure 3.5: Modeling of Motor Drive’s Current Loop ... 38

Figure 3.6: Result of Current Loop (Test and Simulation) ... 39

Figure 3.7: Speedgoat^® Real-time Target Machine [19] ... 40

Figure 3.8: Baumer Hollow Shaft Incremental Encoder [22] ... 41

Figure 3.9: Initial Position of Fin w.r.t. Earth’s Fixed Frame ... 42

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Figure 3.10: Initial Position of Fin Shaft w.r.t. Earth’s Fixed Axis ... 42

Figure 3.11: Initial Position of Fork w.r.t. Earth’s Fixed Axis ... 43

Figure 3.12: Initial Position of Screw and Nut w.r.t. Earth’s Fixed Axis ... 44

Figure 3.13: Friction Measurement Test Setup (FMTS) ... 45

Figure 3.14: Changes of FMTSM Speed and Current @ 24ºC, with 50 rpm Speed Command ... 47

Figure 3.15: Changes of FMTSM Speed and Current @ 24ºC, with 500 rpm Speed Command ... 48

Figure 3.16: Changes of FMTSM Speed and Current @ 24ºC, with 1000 rpm Speed Command ... 48

Figure 3.17: Changes of FMTSM Speed and Current @ 24ºC, with 1500 rpm Speed Command ... 49

Figure 3.18: Changes of FMTSM Speed and Current @ 24ºC, with 2000 rpm Speed Command ... 49

Figure 3.19: Changes of FMTSM Speed and Current @ -6ºC, with 50 rpm Speed Command ... 50

Figure 3.20: Changes of FMTSM Speed and Current @ -6ºC, with 500 rpm Speed Command ... 50

Figure 3.21: Changes of FMTSM Speed and Current @ -6ºC, with 1000 rpm Speed Command ... 51

Figure 3.22: Changes of FMTSM Speed and Current @ -6ºC, with 1500 rpm Speed Command ... 51

Figure 3.23: Changes of FMTSM Speed and Current @ -6ºC, with 2000 rpm Speed Command ... 52

Figure 3.24: Friction Torque of whole EM-CAS vs FMTSM Speed @ 24ºC ... 54

Figure 3.25: Friction Torque of whole EM-CAS vs FMTSM Speed @ -6ºC ... 54

Figure 3.26: Effect of Ambient Temperatures on Friction Torque of whole EM-CAS vs Motor Speed ... 55

Figure 3.27: Find the Viscous Friction Torque Coefficients of EM-CAS ... 56

Figure 3.28: Changes of Current, Speed and Position of FMTSM @ 24ºC Ambient Temp., 25 rpm Speed Command... 58

Figure 3.29: Coulomb Friction Torque of EM-CAS @Motor Axis vs Fin Position @ 24ºC ... 58

Figure 3.30: Changes of Current, Speed and Position of FMTSM @ -6ºC Ambient Temp., 25 rpm Speed Command... 59

Figure 3.31: Coulomb Friction Torque of EM-CAS @Motor Axis vs Fin Position @ -6ºC ... 59

Figure 3.32: Effect of Ambient Temperatures on Coulomb Friction Torque of whole EM-CAS vs Fin Position ... 60

Figure 3.33: Curve Fitting to Coulomb Friction Torque for Positive Direction ... 62

Figure 3.34: Curve Fitting to Coulomb Friction Torque for Negative Direction ... 62

Figure 3.35: Combination of Curve Fittings for Both Directions ... 63

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Figure 3.36: Position of Encoders to Measure Backlash ... 64

Figure 3.37: Direct and Indirect Measurements of Fin Angle and Change of Backlash ... 65

Figure 3.38: Backlash of EM-CAS ... 66

Figure 3.39: Design of Loading Test Setup (LTS) [21] ... 67

Figure 3.40: Photo of EM-CAS Mounted Loading Test Setup ... 68

Figure 4.1: Linear Block Diagram of EM-CAS ... 70

Figure 4.2: Using “Anti-windup” in Position Controller ... 79

Figure 4.3: Entering Controller Requirements into Simulink Block [20] ... 80

Figure 4.4: Optimization of Design Parameters by Using Simulink Response Optimization Tool ... 81

Figure 5.1: Nonlinear Simulink Model of EM-CAS ... 83

Figure 5.2: Details of "Controllers" Sub-block ... 84

Figure 5.3: Details of "Motor &Drive" Sub-block ... 84

Figure 5.4: Details of "Screw" Sub-block ... 85

Figure 5.5: Details of "Nut" Sub-block ... 85

Figure 5.6: Details of "Fin" Block ... 86

Figure 5.7: Details of "Nonlinear Kinematic Relation" Block ... 87

Figure 5.8: Comparison of Linear and Nonlinear Model Responses for 1º Step Command ... 88

Figure 5.9: Comparison of Linear and Nonlinear Model Responses for 10º and 20º Step Command ... 89

Figure 5.10: Test Prototype of EM-CAS ... 90

Figure 5.11: Real-time Test Model in Simulink-xPC Target ... 90

Figure 5.12: Homing Operation Steps ... 91

Figure 5.13: Setup Files of Controller's Modules ... 92

Figure 5.14: Settings of the Modules ... 93

Figure 5.15: Details of "Incremental Encoder" Block ... 93

Figure 5.16: Details of "Friction Compensation" Block ... 94

Figure 5.17: Details of "Loading Model" Block ... 95

Figure 5.18: Unloaded Step Test ... 96

Figure 5.19: Difference between Fin Angles of Simulation and Test Results (Step Test) ... 97

Figure 5.20: Fin Angular Speed (Unloaded Step Test) ... 97

Figure 5.21: Fin Angle Command for Modified Square Wave Test (MSWT) ... 98

Figure 5.22: Results of MSWT (Fin Angle, Angle Difference, Angular Speeds) .... 99

Figure 5.23: Bandwidth (Chirp) Test ... 100

Figure 5.24: Bandwidth (Chirp) Test (Zoomed) ... 101

Figure 5.25: Difference between Fin Angles of Simulation and Test Results (Bandwidth Test) ... 101

Figure 5.26: Bode Plot of Bandwidth Test ... 102

Figure 5.27: Fin Angle and Command for Loaded Tests ... 104

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Figure 5.28: Real-time Applied Load on Fin Axis ... 105 Figure 5.29: Results of Real-time Loaded Tests ... 106 Figure 5.30: Results of Real-time Loaded Tests (Zoomed) ... 107

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LIST OF ABBREVIATIONS

AC : Alternating Current BLDC : Brushless Direct Current CAS : Control Actuation System CFD : Computational Fluid Dynamic

CM : Center of Mass

DC : Direct Current

DOF : Degree of Freedom

EH : Electrohydraulic

EHA : Electrohydraulic Actuator EHS : Electrohydrostatic

EH-CAS : Electrohydraulic Control Actuation System EHS-CAS : Electrohydrostatic Control Actuation System EM-CAS : Electromechanical Control Actuation System EMA : Electromechanical Actuator

EMF : Electromotive Force

FA : Fin Angle

FBD : Free Body Diagram

FMTS : Friction Measurement Test Setup FMTSM : Friction Measurement Test Setup Motor IAF : Indirect Angle of Fin

IO : Input / Output

ISF : Indirect Speed of Fin LTS : Loading Test Setup LTSM : Loading Test Setup Motor

MA : Motor Angle

MSWT : Modified Square Wave Test PPR : Pulses Per Revolution PWM : Pulse Width Modulation RPM : Revolution Per Minute RTTM : Real-time Target Machine TF : Transfer Function

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LIST OF SYMBOLS

: Circular Cross Sectional Area of the Beam [m²]

: Bandwidth [Hz]

: Coulomb Friction on Fin Axis [Nm]

: Coulomb Friction on Motor Axis (CCW) [Nm]

: Coulomb Friction on Motor Axis (CW) [Nm]

: Average Coulomb Friction on Motor Axis [Nm]

: Equivalent Inertia of EM-CAS [kg.m²]

: Equivalent Damping Coefficient of EM-CAS [Nm.rad/s]

: Moment Arm [m]

: Equivalent Disturbances of EM-CAS [Nm]

: Diameter of the Beam [m]

: Elastic Modulus of the Beam [Pa]

: Resolution of the LTS's Encoder [PPR]

: Force on Nut by Motor [N]

: Equivalent Coulomb Friction of EM-CAS [Nm]

: Load on Nut by Thrust of Missile [N]

⃗ : Load Vector by Thrust of Missile on the Fin [N]

⃗ : Load Vector by Thrust of Missile on the Fork [N]

⃗ : Load Vector by Thrust of Missile on the Fin Shaft [N]

⃗ : Load Vector by Thrust of Missile on the Nut [N]

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: Load on Nut by Gravity [N]

⃗ : Load Vector by Gravity on the Fin [N]

⃗ : Load Vector by Gravity on the Fork [N]

⃗ : Load Vector by Gravity on the Fin Shaft [N]

⃗ : Load Vector by Gravity on the Nut[N]

⃗ : Gravity Vector [m/s²]

: Shear Modulus of the Beam [Pa]

: Controller of Current Loop of Drive [V/A]

: TF of Motor Electrical Part [A/V]

: TF of EM-CAS [rad/ Nm s]

: Speed Controller of EM-CAS [V.s/rad]

: Position Controller of EM-CAS [1/s]

: TF of Integral Operation [s]

: Current of EM-CAS's Motor [A]

: Second Moment of Inertia of the Beam [m⁴]

: The Inertia of Screw [kg.m²]

: Equivalent Inertia on the Motor Axis [kg.m²] : Inertia of Fin Around Hinge Axis [kg.m²]

:

:

Total Inertia at Fin Axis [kg.m²]

: Actual Current of FMTSM [A]

: Inertia of Fork Around Hinge Axis [kg.m²]

: Inertia of Fin Shaft Around Hinge Axis [kg.m²] : Inertia of EM-CAS's Motor [kg.m²]

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: Total Inertia at Motor Axis [kg.m²]

: LTSM Actual Current/Analog Reference Signal[A/V]

: Back EMF Constant of EM-CAS's Motor [V.s/rad]

: Equivalent Stiffness of the Beam in Tension [N/m]

: Equivalent Stiffness of the Beam in Torsion [Nm/rad]

: Integral Gain of Current Loop in Servo Drive [V/A.s]

: Proportional Gain of Current Loop in Servo Drive [V/A]

: Integral Gain of Position Loop EM-CAS [rad/rad.s²]

: Proportional Gain of Position Loop of EM-CAS [rad/rad.s]

: Proportional Gain of Speed Loop of EM-CAS [V.s/rad]

: Torque Constant of EM-CAS's Motor [Nm/A]

: Torque Constant of FMTSM [Nm/A]

: Torque Constant of LTSM[Nm/A]

: EM-CAS's Motor Actual Current/Analog Reference Signal [A/V]

: EM-CAS's Motor Current Offset [A]

: LTSM Motor Actual Current/Analog Reference Signal [A/V]

: Forward Gain of EM-CAS in Illustration of TFs [Nm/A]

: Feedback Gain of EM-CAS in Illustration of TFs [V.s/rad]

: Inductance of EM-CAS's Motor [H]

: Length of the Beam [m]

: Equivalent Mass on the Ball Screw [kg]

: Mass of Fin [kg]

: Mass of Fork [kg]

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: Mass of Fin Shaft [kg]

: Mass of Nut [kg]

: LTS Transmission Ratio

: Pole Number of EM-CAS's Motor : Pitch Rotational Matrix

: Resistance of EM-CAS's Motor [Ohm]

: Roll Rotational Matrix

⃗ : Position Vector of Reference Point at Hinge Axis [m]

⃗ : Position Vector of CM of Fin [m]

⃗ : Position Vector of CM of Fork ,while is zero [m]

⃗ : Position Vector of CM of Fin Shaft [m]

: Approximate Transport Delay in EM-CAS's Servo Drive [ms]

: Aerodynamic Load Applied Fin Axis [Nm]

: Torque of EM-CAS's Motor [Nm]

: Nominal Torque of EM-CAS's Motor [Nm]

: Nominal Torque of LTSM [Nm]

: Rise Time [s]

: Settling Time [s]

: Stall Torque of EM-CAS's Motor [Nm]

: Nominal Torque of LTSM [Nm]

: Coulomb Friction Torque of FMTSM [Nm]

⃗⃗ : Load Vector on Fin Axis by Thrust of Missile (Fin, Fin Shaft and Fork) [Nm]

: Load on Fin Axis by Thrust of Missile (Fin, Fin Shaft and Fork) [Nm]

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: Load on Fin Axis by Gravity (Fin, Fin Shaft and Fork) [Nm]

⃗⃗ : Load Vector on Fin Axis by Gravity (Fin, Fin Shaft and Fork) [Nm]

: Input Torque of Nut [Nm]

: Voltage of EM-CAS's Motor [V]

: Viscous Friction Coefficient on Fin Axis [Nm.rad/s]

: Viscous Friction Coefficient on Motor Axis(CCW) [Nm.rad/s]

: Viscous Friction Coefficient on Motor Axis(CW) [Nm.rad/s]

: Average Viscous Friction Coefficient on Motor Axis [Nm.rad/s]

: Maximum Speed of EM-CAS's Motor [rpm]

: Maximum Speed of LTSM [rpm]

: Nominal Speed of EM-CAS's Motor [rpm]

: Nominal Speed of LTSM [rpm]

: Viscous Friction Coefficient of FMTSM [Nm/ krpm]

: Yaw Rotational Matrix

: Linear Displacement of Nut [m]

̇ : Linear Speed of Nut [m/s]

̈ : Linear Acceleration [m/s²]

: Initial Displacement of Nut Because of Angle [m]

: Overshoot [%]

: Natural Frequency [Hz]

: Natural Frequency of the Beam in Tension [Hz]

: Natural Frequency of the Beam in Torsion [Hz]

⃗ : Acceleration Vector of Missile [m/s²]

xxiv

: Damping Ratio

: Fin Angle [rad]

̇ : Fin Angular Speed [rad/s]

̈ : Fin Angular Acceleration [rad/s²] ̇ : Angular Speed of FMTSM [rpm]

: Initial Assembly Angle of Fork [rad]

: Motor Angle [rad]

̇ : Motor Angular Speed [rad/s]

̈ : Motor Angular Acceleration [rad/s²] : Pitch Angle of Missile [rad]

: Lead of Ball Screw [m/rad]

: Roll Angle of Missile [rad]

: Yaw Angle of Missile [rad]

º : Degree

1 CHAPTER 1

1. INTRODUCTION

1.1 Literature Survey

One of the most important subsystems of guided missiles is the control actuation systems (CASs), in other words fin (wing) actuators. The aim of CASs is to steer the missile towards target according to command signal that comes from the autopilot of the missile [9], [10], [11]. The autopilot defines the fin deflection angle demand during flight and simultaneously sends this demand to the control actuation system.

After receiving the command, aero fins (control surfaces) are deflected to desired angles under the effect of aerodynamic loads acting on these surfaces, as well as viscous and Coulomb friction, backlash and etc. [11]. There are several types of CASs which are widely used in aerospace applications such as Electromechanical Control Actuation Systems (EM-CASs), Electrohydrostatic Control Actuation Systems (EHS-CASs) and Electrohydraulic Control Actuation Systems (EH-CASs) [1], [2], [3], [4], [8]. These systems and their components are shown as simple diagrams in Figure 1.1.

2

Figure 1.1: Types of Control Actuation Systems (CASs) [1]

1.1.1 Electromechanical Control Actuation Systems (EM-CASs)

The use of electromechanical actuation is becoming increasingly popular in aerospace industry because of their momentary overdrive capability, low quiescent power/low maintenance characteristics and long-term storability [5], [6], [7], [9], [12], [13]. After developing servomotors these systems start to become prominent and more used in defense industry. However, modeling of electromechanical actuator is subjected to some uncertainty because of several reasons such as, electrical noises, frictions, backlash, changing of operating point, external disturbances (ex: external loads), parametric variations due to temperature changes, asymmetric behavior and non-modeled dynamics [12], [13], [14].

3

To steer missiles towards to target, CASs are used to deflect fin angle and direct the missiles. Aero fins are illustrated in Figure 1.2. As mentioned before, CASs may be electromechanical or electrohydraulic.

Figure 1.2: Control Actuation System (CAS) [5], [6]

There are some electromechanical CAS types which can be implemented into the missile control section. Mounting servomotor directly to the fin axis is difficult due to volume restriction and torque-speed characteristic of the actuators. Therefore, there should be gearing mechanisms (or transmission mechanisms) both to change rotational axis (about 90º) and to adjust the whole system to desired torque-speed region. Types of EM-CASs are as follows.

1.1.1.1 EM-CAS with Screw and Lever Mechanism

As seen in Figure 1.3, CAS is composed of DC Motor, Planetary gear, Screw shaft, Encoder, Grid fin and Lever mechanism [5], [6], [9]. In this type, ball screws are frequently used instead of acme screw due to their great efficiency which is generally more than %95. To convert linear displacement of the ball screw, nut, lever mechanisms (fork mechanism) are used. On the lever mechanism, there is a longitudinal slot which lets pin of nut to slip through. Moreover, the ratio between

4

motor and fin angle is not constant and it changes with tangent of fin angle.

Therefore, this situation causes system to be nonlinear. In this study, this kind of CASs will be investigated, mathematically modeled and a controller designed for them.

Figure 1.3: EM-CAS with Screw and Lever Mechanism [5], [6]

Modeling of EM-CAS in Literature:

In the literature, dynamic modeling of EM-CAS with Screw and Lever Mechanism has some differences. Ristanović used a brush DC motor in his work and therefore PWM control was chosen as the control strategy [5], [6], [7]. As seen in Figure 1.4, in his model there was no current loop and back EMF voltage was not included.

Also, whole system parameters, such as inertias, viscous and Coulomb frictions were reduced to the motor shaft with constant gear ratio assumption. Moreover, there were also rate saturation for speed due to absence of current loop and voltage limitation in that model. Kinematic saturations could cause some oscillations or overshoot for position control. Also, a torsional bar was used where the load increases if the angle of fin increases in order to simulate aerodynamic loads.

5

Figure 1.4: Ristanović's EM-CAS Simulation Model [5], [6], [7]

In the Özkan's EM-CAS Simulation Model, a brushless DC motor was used instead of a brush one. There is not much information given about the motor drive. As seen in Figure 1.5, it was assumed that the current loop of motor drive is ideal one;

therefore no back EMF voltage exists like as in Ristanović's model. Also, whole system parameters, such as inertias, viscous and Coulomb frictions are reduced to the motor shaft with constant gear ratio assumption [9], [10]. Another difference from Ristanović's model is that the shaft of the motor was modeled as a flexible body.

Figure 1.5: Özkan's EM-CAS Simulation Model [9], [10]

The different points from the other modeling, in Habibi's EM-CAS model, current, velocity and position loops are constructed and a Backlash model is included as shown in Figure 1.6. Also all components which are used in CAS were assumed as rigid bodies [8]. In Habibi's model similarly whole system parameters, such as inertias, viscous and Coulomb frictions were reduced to the motor shaft with constant gear ratio assumption.

6

Figure 1.6: Habibi's EM-CAS Simulation Model [8]

In this study EM-CAS is modeled similarly like Habibi's model [8]. However, whole system parameters, such as inertias, viscous and Coulomb frictions are not reduced to the motor shaft and constant gear ratio is not assumed. Also viscous and Coulomb friction which can change with respect to fin position is measured and modeled. The backlash in whole CAS will be measured and decided if it can be ignored. After that, control strategy is determined and controller parameters optimized by using MATLAB Response Optimization Tools [20]. At least loading test is performed and all results are compared.

1.1.1.2 EM-CAS with Clutch Actuator

This second type is nearly same with first one with exception of including clutch in the system [3], [4]. During flight of missiles, aerodynamic loads work sometimes against the motor, sometimes together with motor. The case of working against the motor is not a problem but for other situation there occurs some regenerative energy on the motor and motor drive due to requirement of braking. As seen in Figure 1.7 clutch mechanism can be implemented into CAS to ensure that the CAS is not affected by regenerative energy that is caused by external disturbances.

7

Figure 1.7: EM-CAS with Clutch Actuator [3], [4]

1.1.1.3 EM-CAS with Worm Gear

In this alternative worm gear is used as transmission mechanism due to low backlash and linear behavior. The gear ratio between motor and fin angle are constant and it does not change with fin angle which makes the mechanism linear. Moreover, worm gears have self-locking property which protects servomotor actuators from the regenerative energy. However, self-locking means low efficiency where efficiency of worm gear is around %50 and it decreases with increasing gear ratio. Therefore, optimization should be made between locking and efficiency.

Figure 1.8: EM-CAS with Worm Gear

8

1.1.2 Electrohydrostatic Control Actuation Systems (EHS-CASs)

The electrohydrostatic control actuation systems, in other words, pump controlled hydraulic systems are composed of both electric and hydraulic actuation attributes.

As seen in Figure 1.9, these systems consist of servomotor, pump, relief valves, reservoir, hydraulic cylinder, filters, bypass valve and etc. Hydraulic transmissions are more attractive than mechanical transmission which has potential mechanical jamming [1], [2], [8].

Figure 1.9: EHS Top Level System Schematic [1]

In EHS system which is illustrated in Figure 1.10, the number of turns of servomotor is proportional to the position of the hydraulic cylinder. To measure the piston displacement, position feedback transducers are generally used which sends position information back to the flight control computer. Also EHS contains internal fluid reservoir which provides a minimum pressure to make flow of hydraulic fluid easier into the suction port of the pump [1], [2].

9

Figure 1.10: An Example of EHS [25]

EHS has similar advantages and attributes to electromechanical actuation for aerospace applications, but it also has leakage concerns and fluid contamination. [2]

Advantages [2]:

 In this actuation there is no backlash or lost motion and it behaves similarly to a conventional hydraulic system.

 They are more reliable because of their jam proof attitude.

 Return porting and hydraulic supply is self-contained; therefore the requirement of hydraulic maintenance is little or none.

1.1.3 Electrohydraulic Control Actuation Systems (EH-CASs)

There are two types of hydraulic transmissions which are often used in industry due to their high performance quality, one of them is valve controlled (EH) and other is pump controlled (EHS) [2]. The pump controlled systems are already discussed in section 1.1.2. A typical valve controlled circuit is presented in Figure 1.11.

10

Figure 1.11: Typical Valve Controlled Hydraulic Circuit [2]

EH (valve controlled) systems are generally composed of hydraulic cylinder, control valve (servovalve or proportional valve), accumulator, filter, check and relief valves, pump, electric motor, cooler and oil tank. While the hydraulic pump rotates at constant speed (i.e. gives a constant flow rate), EH systems use a hydraulic valve as a control element and this valve directs the oil flow that is generated by a pump to hydraulic actuator (a hydraulic cylinder or motor). If the control element of the EH systems would be servovalve, then the best performance can be obtained. This system usually requires large oil reservoir that is exposed to the atmospheric pressure. The efficiency of the system can reach values under 30% that is the disadvantage of the circuit because of throttle losses at the valve [2]. As seen in Figure 1.12, EH systems (valve controlled) can be used in CAS of aircraft and missiles in order to deflect the control surface at desired angles.

11

Figure 1.12: Schematic Diagram of the Servo Mechanism

There are some transmission mechanisms such as four bar mechanism between the Piston Connection and Control Surface. While hydraulic cylinder piston is moving, the control surface starts to rotate.

12 1.2 Objective and Scope of Thesis

The objective of this thesis is to obtain a detailed mathematical model of an electromechanical control actuation system which is composed of a ball screw-lever mechanism and a brushless direct current motor and also to design its controller.

EM-CAS with ball screw and lever mechanism has some nonlinearity such as Coulomb friction, backlash, rate saturations, external loads and etc. Also the gear ratio between input and output shaft of this system is not constant and it changes with respect to their position. This situation adds some extra nonlinearity to the system.

Therefore, the detailed mathematical model of EM-CAS will be nonlinear. In order to decide which controller should be used to satisfy performance criteria, the block diagram which is composed of all transfer functions is utilized after linearizing mathematical model of EM-CAS by making necessary assumptions.

The linear model is only used to decide controller type and check if controller satisfies the performance criteria. Then, in order to determine controller parameters, MATLAB Response Optimization Tools are used on the nonlinear model of EM- CAS which has all nonlinearities including external disturbances.

In order to verify the detailed nonlinear mathematical model, a prototype of EM- CAS with ball screw and lever mechanism is manufactured. After manufacturing, all uncertain parameters such as friction, backlash and conversion coefficients between components, the unknown parameters of servo drive amplifier are found by making real-time tests and embedded into nonlinear model. In this manner, the overall identification of system parameters of the prototype is handled.

Moreover, after obtaining controller parameters by optimizing, the performance test of overall prototype of EM-CAS is conducted for both without external load and under external load whether if the performance criteria is satisfied and the results of simulation and real-time tests are overlapped.

13 1.3 Thesis Outline

In the first chapter an introduction to control actuation systems is made by presenting what types of CASs are used in aerospace industry. Three types of CAS are mostly used which are electromechanical, electrohydraulic and electrohydrostatic CASs.

The advantages and disadvantages of each CAS are given in detail. After developing servomotors electromechanical actuators start to become prominent and more used in defense industry. There are three different electromechanical control actuation systems which transmission mechanisms are different from each other. In the scope of this thesis, EM-CAS with ball screw-lever mechanism is studied. The mathematical models of EM-CAS in literature are investigated and the differences between these models and the model studied in this thesis are presented.

In the second chapter of the study, all equations of motion are derived after determining kinematic relations between positions, velocities and accelerations of input and output shafts. By adding the electrical system to the mechanical system, the nonlinear model of all system is obtained. At the end of second chapter, by using all derived equations, block diagram of EM-CAS is presented.

In the third chapter, the performance requirements of the desired CAS are given.

Accordingly, a prototype of electromechanical control actuation system which is composed of BLDC motor, ball screw and lever mechanism is manufactured in order to satisfy performance criteria and verify developed nonlinear model. Afterwards, all system parameters that are used in nonlinear mathematical model are identified in this chapter. These parameters are necessary dimensions, positions of CM, inertias and masses of components, also viscous, Coulomb frictions, backlash and etc. In order to use Coulomb friction torque of EMCAS as friction compensation, “curve fitting” method is used and a 6^th order polynomial is fitted to the test data. The backlash in whole EM-CAS is measured and it is decided that backlash can be ignored because its value is much smaller than position accuracy limit. At the end of third chapter, loading test setup (LTS) is explained clearly which is used to apply external load to the EM-CAS.

14

The fourth chapter of the study is dedicated to linearized model of electromechanical control actuation system and controller design. By making small angle assumption for fin angle after assuming that the EM-CAS is symmetric with respect to home position, the nonlinear equations of motion which are obtained in the second chapter are linearized. In order to ensure that the EM-CAS is affected by inner dynamics of servo drive amplifier as little as possible, torque mode is decided to use which has minimum loop inside.

In this study, in order to provide some extra damping to the system and prevent the possible oscillations during controlling, the inner velocity loop is created between current and position loops. By combining these three loops with linearized equations, the linear model of EM-CAS is obtained. Therefore, the transfer function between any two variables can be found. According to controller requirements, by using Final Value Theorem (FVT) it is found that PI and P-controller are sufficient for position and velocity control respectively with and without external load. Also “anti-windup”

method is used due to limitation on the controller output which has integral gain.

At the end of fourth chapter, the steps during controller design are presented.

Controller parameters are found by using MATLAB Response Optimization Tools.

Both controller parameters , , and anti-windup coefficient are selected as design parameters and optimized according to the controller’s performance requirements by using Simulink block which is named "Check Step Response Characteristics".

In the fifth chapter, the nonlinear mathematical model of EM-CAS is constructed in Simulink by using all mechanical and electrical equations. Afterwards, linear and nonlinear models are compared which behave similarly for small reference commands (< 3°). However, for large reference command the responses differ from each other as expected and nonlinear model is slower than linear model due to limitations imposed on the controller’s output. Thereafter real-time test software is explained in detail which is constructed in MATLAB/Simulink as well.

At the end of fifth chapter, several real-time tests are performed without and under external loads if prototype satisfies performance criteria. Three unloaded tests are

15

done which are step test, modified square wave test and bandwidth test. Also loaded test is done under applying 100 Nm external load on fin axis. After performing these all performance tests, it is seen that simulation results and real-time test results are very similar and consistent.

The concluding remarks and future works are presented in Chapter 6 as a conclusion.

16

17 CHAPTER 2

2 DYNAMIC MODELING OF EM-CAS

The detailed mathematical model of Electromechanical Control Actuation System (EM-CAS) consists of two parts which are mechanical and electrical systems.

Mechanical modeling is created by using the equations of motion after determining the kinematic relationship between mechanisms. Besides, electrical model consist of the model of brushless motor, inner loops of motor drive and real-time controller which are used in EM-CAS. After combining of all equations, nonlinear and linear mathematical model of EM-CAS is obtained.

2.1 Mechanical System

The models of all mechanical components including stator of BLDC motor are explained. First of all, the kinematic relations between mechanisms are defined and then the equations of motion are written.

18 2.1.1 Kinematic Relations

In order to transfer all moments of inertias and viscous frictions in EM-CAS to the mathematical model, it is necessary to know the relationships between the angular/axial accelerations, speeds and positions of components. For the systems which have no volume restriction, the angle which is shown in Figure 2.1, can be zero and therefore the system can be symmetric for positive and negative aspects w.r.t. hinge axis. Otherwise, if this angle is not zero, the system would be asymmetric, as in this study.

As seen in Figure 2.1 fin-fin shaft-fork and motor shaft-screw are fixed to each other and nut is not allowed to rotate around any axis. Therefore, when motor starts to rotate, screw rotates also and nut of ball screw moves forward/backward along screw’s axis due to rotation of screw. There is a slotted connection between fork and nut components. Thus when nut starts to move, the fork and fin shaft components rotate around hinge axis. Finally, the desired motion of fin is obtained with tilting fork.

Figure 2.1: Assembly State of EM-CAS

19

As shown in Figure 2.2 the fin of EM-CAS is tilted clockwise (- direction). The distance between the fin axis of rotation and the ball screw axis is constant and about

“d”. When the fin moves, the slotted connection between nut and fork is provided the elongation requirements of the fork which avoids locking.

Figure 2.2: Motion State of EM-CAS The kinematic equations are as follows;

( )

(2.1)

( ) (2.2)

(2.3)

( ) (2.4)

Putting equations (2.3) and (2.4) into equation (2.1), the relation between fin and motor angles are found as in equation (2.5).

20

[ ( ) ] (2.5)

By differentiating equation (2.5), the relation between angular speeds of fin and motor are obtained as in equation (2.6) .

̇ [

( )] ̇ (2.6)

By differentiating equation (2.6) which gives ratio of motor and fin angular speeds, the relation between their angular accelerations can be found as equation (2.7).

̈ [ ̈ ( ) ̇ ( )

( ) ] (2.7)

As seen in equation (2.7) the relation between accelerations of motor and fin is not linear and the ratio between them depends on initial angle of fork, instantaneous angular velocities and positions.

2.1.2 Equations of Motion

In order to create all equations of motion, all components of EM-CAS are examined separately, piece by piece and then all equations are combined. First of all, the force couples of motor stator and screw pair are shown and equations are written. As seen in Figure 2.3 motor stator and screw behave as a single piece and act together.

21

Figure 2.3: Free Body Diagram of Motor and Screw Pairs

If the elasticity of the screw is considered to be very low due to its short length, the rotational angles of motor and screw would be almost equal. The total moment of inertia on the motor axis is the summation of inertias of motor and screw as in equation (2.8). The torque applied by screw on the nut is written in equation (2.9).

(2.8)

( ̇ ) ̈ ̇ (2.9)

When screw rotates, nut starts to move along screw’s axis and also nut is not allowed to rotate around any axis (Figure 2.4).

Figure 2.4: Free Body Diagram of Screw and Nut Pairs

22

The force applied by nut on the fork is written in equation (2.10).

̈ (2.10)

̈ ̈ (2.11)

Putting equation (2.11) into equation (2.10), the equation of motion of these components can be found as in equation (2.12).

̈ (2.12)

By moving nut along axis of screw as seen in Figure 2.4, the pins on nut are sliding through slotted connection between fork and nut. Therefore this motion provides fork to rotate around hinge axis as seen in Figure 2.5.

Figure 2.5: Free Body Diagram of Aero Fin and Fork Pairs

The total moment of inertia on the hinge axis is summation of inertias of fork, fin and fin shaft as in equation (2.13).

(2.13)

23

If the equation of motion is written with respect to hinge axis;

( ̇ ) ̇ ̈ (2.14) Taking F term from equation (2.14),

[ ( ̇ ) ̇ ̈ ] (2.15)

So, that all equations of motion of EM-CAS are obtained separately. If these equations are written again by equating similar terms, the equation of motion can be rewritten in terms of a single equation as shown below.

Putting term which is given in equation (2.9) into equation (2.12), equation (2.16) can be obtained.

( ̇ ) ̈ ̇ ̈

(2.16)

By equating equations (2.15) and (2.16), the nonlinear equations of motion of all system can be found as in equation (2.17).

( ̈ ̇ ) [ ̈ ̇ ] ( ̇ ) ( ̇ )

[ ( )]

(2.17)

The directions of rotation are same for both motor and fin, therefore;

( ̇ ) ( ̇ )

24 Linearization of Equation of Motion

By making small angle assumption for fin angle ( ) after assuming that the EM- CAS is symmetric w.r.t. home position (i.e. is zero), the equation (2.5) can be linearized and rewritten as in equation (2.18), its first and second time derivatives can also be obtained as in equations (2.19) and (2.20), respectively.

(2.18)

̇ ̇ (2.19)

̈ ̈ (2.20)

By putting equations (2.18), (2.19) and (2.20) into equation (2.17), equation of motion can be linearized as in equation (2.21).

[ ] ̈ ( ) ̇ ( ) ( ̇ ) [ ( )]

(2.21)

[ ] (2.22)

( ) (2.23)

[( ) ( ̇ )] (2.24)

[ ( )] (2.25)

Term in equation (2.22) and term in equation (2.23) represent “equivalent moment of inertia” and “equivalent coefficient of viscous friction torque”, respectively. Term in equation (2.24) and term in equation (2.25) represent

“Coulomb friction torque” and “equivalent disturbances on motor axis”, respectively.

25

By using these abbreviations, equation of motion can be rewritten in the familiar form as in equation (2.26).

̈ ̇ (2.26) The parameters which are used in equation (2.24) are Coulomb friction torques. This torques will be found by making friction measurement tests and used to feed output of inner-loop velocity controller as friction compensation.

The disturbances can be separated in three different classes. One of them is thrust caused , other gravity caused and last one is aero dynamic load caused . These disturbances can change with depending on accelerations, velocities and positions of the systems, also environmental conditions.

In order to see how big these loads are, some calculations can be done. For example, in order to calculate aerodynamic loads , some CFD analysis of the flight must be done.

The loads on the components of the EM-CAS which are caused by gravity and accelerations of the system can be found by using static equations [10]. The representations of both Earth’s Fixed Axis and the Missile’s Local Axis are shown in Figure 2.6.

Figure 2.6: Representation of Earth’s Fixed and Missile’s Local Axes

26

In order to pass from Earth’s Fixed Axis to the Missile’s Local Axis, the basic rotation matrices for the rotations about the yaw, pitch and roll axes are defined as in equations (2.27), (2.28) and (2.29), respectively [10].

[

] (2.27)

[

] (2.28)

[

] (2.29)

Gravitational acceleration is in the ⃗⃗ direction of Earth’s Fixed Axis.

⃗ [ ] (2.30)

To separate gravitational acceleration into the components of missile’s local axis, three rotational matrices need to be used. So, that weight vectors of fin, fork and fin shaft are given in equations (2.31), (2.32) and (2.33), respectively.

⃗ ( ) ( ) ( ) ⃗⃗⃗ (2.31)

⃗ ( ) ( ) ( )⃗⃗⃗ (2.32)

⃗ ( ) ( ) ( ) ⃗⃗⃗ (2.33)

By cross multiplying the weight vectors of components and distance vectors between center of masses and hinge axis, the torque vectors which occur on hinge axis can be obtained as in equation (2.34) [10].

27

⃗⃗ ( ⃗ ⃗ ) ⃗ ( ⃗ ⃗ ) ⃗ ( ⃗ ⃗ ) ⃗ (2.34)

However, it needs to take into account second element of the torque vector as in equation (2.35), because only the component in the ⃗⃗ direction can provide fin to rotate.

⃗⃗ (2.35)

On the other hand, some disturbances also occur on the components of EM-CAS because of missile’s acceleration by thrust force and vibrations.

There are only one DOF between EM-CAS and missile itself which is pitch rotational angle. Therefore, in order to see the effects of missile’s accelerations, only pitch rotational matrix, needs to be used in the equations.

⃗ [ ] (2.36)

Due to missile’s accelerations in equation (2.36), the loads on fin, fork and fin shaft can be calculated as in equations (2.37), (2.38) and (2.39), respectively.

⃗ ⃗⃗⃗ ( ) (2.37)

⃗ ^⃗⃗⃗ ( ) (2.38)

⃗ ⃗⃗⃗ ( ) (2.39) By combining these three equations, torque vector which occurs due to missile’s thrust, can be written as in equation (2.40). However, it needs to take second member of the torque vector as in equation (2.41), because only the component in the ⃗⃗

direction can cause fin to rotate.

28

⃗⃗ ( ⃗ ⃗ ) ⃗ ( ⃗ ⃗ ) ⃗ ( ⃗ ⃗ ) ⃗ (2.40)

⃗⃗ (2.41)

The force vectors that acting on the nut due to gravitational acceleration and missile’s accelerations can be calculated as in equations (2.42) and (2.43), respectively.

⃗ ⃗⃗⃗ ( ) ( ) (2.42)

⃗ ⃗⃗⃗ (2.43)

It needs to take first member of the force vector of the equations (2.42) and (2.43) as given in the equations (2.44) and (2.45), because only the component in the ⃗⃗

direction can provide the nut to move.

⃗⃗ (2.44)

⃗⃗ (2.45)

2.2 Electrical System

The electrical section of EM-CAS consists of brushless DC motor and servo drive amplifier. The brushless DC motor and servo drive amplifier couple can be modeled as a brushed DC motor. The equations that can be applied to such electric motors are given below.

̇ (2.46)

(2.47)

The current drawn by the motor can be found by using the relation in the equation (2.46). The motor torque is proportional to the current drawn by the motor as seen in the equation (2.47) and the ratio between them is the torque constant ( .

29

The servo drive amplifier has two modes named current and velocity. The mode which is used on the motor drive varies according to the mechanism and the application used. In this study motor drive is used in the current mode, because the units of controller parameters that are used in drive’s software are not known. In order not to make so many tests to figure out the controller units, drive is used in the current mode which has only one loop. Otherwise there would be more than one loop. Therefore velocity and position loops are constructed out of the servo drive amplifier by Speedgoat^® real time target machine (RTTM) controller. The controller parameters and their units of the current loop will be identified in the section 3.2.2.

2.3 Block Diagram of EM-CAS

The mechanical and electrical portions are combined and therefore the block diagram of EM-CAS is obtained as in Figure 2.7. The current loop of motor drive and velocity and position inner loops of controller are not yet in this block diagram. In the block diagram the flow of force and torque can be followed easily.

Figure 2.7: Nonlinear Block Diagram of EM-CAS

As seen in the Figure 2.7 on the feedback paths there are some inertia that we are not familiar with. The reason is that the mechanical components (i.e. motor shafts, screw, fin etc.) are modeled as rigid bodies instead of elastic modeling. The maximum operating frequency demand is expected to be lower than natural frequency of the