1
OPTIMIAL CONTROL OF THE ATTITUDE MANEUVERING FOR RAZAKSAT
®CLASS SATELLITE BASED ON RIGID AND FLEXIBLE
MODEL
by
TEOH VIL CHERD (1140610711)
A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy
School of Mechatronic Engineering UNIVERSITI MALAYSIA PERLIS
2018
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ACKNOWLEDGEMENT
Firstly, I would like to express my sincere gratitude to my advisor Associate Prof. Dr.
Shahriman Abu Bakar for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. My sincere thanks also go Prof. Dr. Sazali Yaacob and Associate Prof. Ir. Dr Ruslizam bin Daud for their insightful comments and encouragement, which incented me to widen my research from various perspectives.
My sincere appreciation also goes to Mr. Norhizam Hamzah and the Aeronautic Technologies Sdn. Bhd (ATSB) team, who have provided me with an insight of the RazakSAT operations which has motivated me in the research works. Without they precious support it would not be possible to complete this thesis. Last but not the least, I would like to thank my parents and to my sisters for supporting me spiritually throughout writing this thesis and my life in general.
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TABLE OF CONTENTS
CONTENTS PAGE
THESIS DECLARATION i
ACKNOWLEDGEMENT ii
TABLE OF CONTENTS iii
LIST OF FIGURES viii
LIST OF TABLES xi
LIST OF ABBREVIATION xii
LIST OF SYMBOLS xiii
ABSTRAK xvi
ABSTRACT xvii
CHAPTER 1 INTRODUCTION 1
1.1 Overview 1
1.2 Research Background 1
1.3 Problem Statement 6
1.4 Research Objectives 8
1.5 Scopes of the Thesis 9
1.6 Thesis Organization 10
1.7 Process Flow 12
CHAPTER 2 LITERATURE REVIEW 13
2.1 Introduction 13
2.2 Satellite System 13
2.3 The Dynamic Modeling 15
2.3.1 Floating Frame Reference (FFR) Formulation 16
2.3.2 Linear Theory of Elasto-dynamics (LTED) 19
2.3.3 Incremental Finite Element (IFE) Formulation 21
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2.3.4 Large Deformation Vector (LDV) Formulation 22
2.4 The Actuator 23
2.4.1 Actuation Limitation 25
2.5 Satellite Kinematics and Maneuver 26
2.6 Time Optimal Control 26
2.7 Satellite Attitude Maneuvering Study 31
2.8 Summary 33
CHAPTER 3 THE SATELLITE DYNAMICS 34
3.1 Introduction 34
3.2 Satellite System Overview 34
3.3 Energy Expressions in Terms of Continuous Variables 37
3.3.1 Kinetic Energy of Flexible Component 38
3.3.1.1 Vector Position in Local Frame 38
3.3.1.2 Rotation of the Panel 40
3.3.1.3 Nonparallel Local and Global Axis 41
3.3.1.4 The Local and Global Reference Frame 42
3.3.2 Kinetic Energy for the Rigid Body 44
3.3.3 Total Kinetic Energy Equation of the System 44
3.3.4 Potential Energy Associated with Bending 46
3.4 Energy Expression in Terms of Discrete Variables 47
3.4.1 Discrete Deformation Variables 48
3.4.2 Kinetic Energy Expressed in Discrete Variables 49 3.4.3 Potential Energy Expressed in Discrete Variables 50
3.5 Application of the Lagrange’s Equation 51
3.5.1 Dynamic Model for X Axis Rotation 52
3.5.2 Dynamic Y and Z Axis Rotation 54
3.6 Euler Rotation Theorem 58
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3.7 The Damping Force 59
3.8 The RazakSAT Class Satellite 63
3.9 Summary 65
CHAPTER 4 THE ATTITUDE CONTROL 68
4.1 Introduction 68
4.2 Satellite Kinematics 68
4.2.1 The Shortest Attitude Rotation 70
4.3 Quaternion Feedback 72
4.4 The Reaction Wheel 74
4.4.1 The RW’s Orientation 75
4.4.2 The Equation of Motion 78
4.5 Summary 80
CHAPTER 5 SIMULATIONS OF SATELLITE DYNAMICS 81
5.1 Introduction 81
5.2 Finite Element Method 81
5.2.1 Discretize the Domain 82
5.2.2 Derive the simple Finite Element Equations 84
5.2.3 Assemble Element Equations 87
5.3 MATLAB Programming 88
5.3.1 The Satellite System 89
5.3.2 The Flexible Components 90
5.4 The Modal Analysis 91
5.5 The Simulations 94
5.5.1 Case Study 1 95
5.5.2 Case Study 2 101
5.6 Summary 107
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CHAPTER 6 THE OPTIMAL CONTROL 108
6.1 Introduction 108
6.2 Mathematical Background 109
6.2.1 Damped Equation of Motion 109
6.2.2 Hamiltonian Equation 110
6.2.3 Bang-Bang Control 113
6.3 Statement of the Control Problem 116
6.3.1 Cost Function 116
6.3.2 Satellite Kinematics 117
6.3.3 Reaction Wheel 117
6.3.4 Satellite Dynamics 118
6.3.5 The Flexible Panel 118
6.3.6 State and Control Variable 120
6.3.7 Constraints of Actuator 120
6.3.8 Initial and Final Conditions 120
6.4 Optimal Control Problem 121
6.5 Scaling the Problem 122
6.5.1 Designer Units 123
6.5.2 Scaled Optimal Control 123
6.6 Implementation of Optimal Control 124
6.7 Example of Optimal Control Maneuver 126
6.7.1 GPOPS Implementation 127
6.7.2 Optimal Solution 127
6.7.3 Feasibility 130
6.7.4 Scaled States and Co-states 135
6.7.5 Optimality 138
6.8 EQF Maneuver Comparison 143
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6.9 Rigid and Flexible Maneuver Comparison 147
6.10 Summary 149
CHAPTER 7 ANALYSIS AND DISCUSSIONS 148
7.1 Introduction 148
7.2 RW Momentum Envelopes 148
7.3 Pseudo-Inverse 151
7.3.1 Pseudo-Inverse Control 152
7.4 Effective Eigen-axis 155
7.5 Flexibility of the System 160
7.6 Summary 162
CHAPTER 8 CONCLUSIONS AND FUTURE WORKS 163
8.1 Conclusion 163
8.2 Contributions 165
8.3 Future Works 166
REFERENCES 167
LIST OF PUBLICATIONS 173
APPENDIX A 174
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LIST OF FIGURES
PAGE
Figure 1.1: RazakSAT Satellite (ATSB, 2016) 3
Figure 1.2: Configuration of RWs 4
Figure 1.3: Block diagram of rigid satellite dynamic 7
Figure 1.4: Block diagram of satellite dynamic 9
Figure 1.5: Research Process Flow 12
Figure 2.1: Attitude Rotation 14
Figure 2.2: Flexible multi-body system 15
Figure 2.3: Simple flexible satellite model (Ebrahimi et al. 2004) 17 Figure 2.4: Satellite with two symmetrical, hinged, rigid panels (Shahriari et al. 2010) 18
Figure 2.5: Tilted solar panel (Guy et al. 2014) 19
Figure 2.6: Finite element model of flexible satellite (Parman, 2016) 22 Figure 2.7: Configuration for three RW(Ismail et al., 2010) 24 Figure 2.8: Configuration for four RW(Ismail et al., 2010) 24
Figure 2.9: The normal input 29
Figure 2.10: The optimal input 29
Figure 2.11: The normal response 30
Figure 2.12: The optimal response 30
Figure 3.1: Satellite Structure Coordinate System 34
Figure 3.2: Deflected panel 36
Figure 3.3: Coordinate system of the flexible panel 39
Figure 3.4: Diagram of the tilted flexible panel 40
Figure 3.5: Rotation of local axis (XY plane) 42
Figure 3.6: OPi vector 43
Figure 3.7: Strain energy 46
Figure 3.8: Satellite Diagram 64
Figure 3.9:X - Y plane 65
Figure 4.1: Eigen-axis rotations 70
Figure 4.2: RW schematic 74
Figure 4.3: Initial orientation 75
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Figure 4.4: First rotation around Ĉ3 75
Figure 4.5: Second rotation around ŵ2 76
Figure 4.6: Configurations of RWs 77
Figure 5.1: Types of element 82
Figure 5.2: Mesh model for a single panel 83
Figure 5.3: Mesh model for three panels 84
Figure 5.4: Tetrahedron element 85
Figure 5.5: Block diagram for satellite model (DM=Dynamic Model) 90
Figure 5.6: Block diagram for DM Panel 1, 2 and 3 91
Figure 5.7: Mode shapes of flexible panel 94
Figure 5.8: Control Input for Case Study 1 96
Figure 5.9: Attitude of satellite for Case Study 1 97
Figure 5.10: Deflection of Panel 1 for Case Study 1 98
Figure 5.11: Deflection of Panel 2 for Case Study 1 99
Figure 5.12: Deflection of Panel 3 for Case Study 1 100
Figure 5.13: Control Input for Case Study 2 102
Figure 5.14: Attitude of satellite for Case Study 2 103
Figure 5.15: Deflection of Panel 1 for Case Study 2 104
Figure 5.16: Deflection of Panel 2 for Case Study 2 105
Figure 5.17: Deflection of Panel 3 for Case Study 2 106
Figure 6.1: Vector field of the damped model 110
Figure 6.2: bang-bang control 114
Figure 6.3: Switching curve - example 116
Figure 6.4: Optimized block diagram of satellite dynamic 124
Figure 6.5: GPOPS control solution 127
Figure 6.6: Optimal quaternion trajectories– GPOPS 128
Figure 6.7: State quaternion propagation 130
Figure 6.8: State satellite body rate propagation 131
Figure 6.9: State R Wrate propagation 132
Figure 6.10: State flexible panel propagation 133
Figure 6.11: Scaled state propagation 134
Figure 6.12: Scaled state propagation (cont.) 135
Figure 6.13: Scaled co-state propagation 136
Figure 6.14: Scaled co-state propagation (cont.) 137
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Figure 6.15: Hamiltonian propagation 138
Figure 6.16: Switching Function 140
Figure 6.17: Switching Function (cont.) 141
Figure 6.18: EQF comparison 142
Figure 6.19: EQF comparisons - Euler angles 143
Figure 6.20: Flexibility comparison (EQF vs GPOPS) 144
Figure 6.21: Flexibility comparisons (Rigid vs Flexible) 146
Figure 7.1: Momentum envelope 149
Figure 7.2: Momentum Space of a 4 RW System 150
Figure 7.3: Momentum Space of pseudo-inverse RW 153
Figure 7.4: Momentum Space of a 4 RW System (pseudo-inverse) 154
Figure 7.5: Pseudo Inverse in-scripted sphere 155
Figure 7.6: Effective Eigen axes 156
Figure 7.7: Effective Eigen Axis for Time-Optimal Maneuver 157 Figure 7.8: Random maneuvers for Time-Optimal Maneuver 158 Figure 7.9: Comparison of Effective Eigen axis maneuver 159 Figure 7.10: Effect of panel length on time optimal control 160 Figure 7.11: Effect of panel lengths (Rigid vs flexible model) 161
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LIST OF TABLES
PAGE
Table 2.1: Summary of selected literature review 31
Table 5.1: Parameters for flexible panel 93
Table 5.2: Modes of bending natural frequency 93
Table 5.3: Parameters of RazakSAT® 95
Table 5.4: Parameters of Case Study 1 95
Table 5.5: RMSE for Case Study 1 100
Table 5.6: Parameters for Case Study 2 101
Table 5.7: RMSE for Case Study 2 107
Table 6.1: Example Parameters 126
Table 6.2: Performance comparison 144
Table 7.1: Momentum Vertices 151
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LIST OF ABBREVIATION
ADCS Attitude Determination and Control System ANSYS Analysis System
AOCS Attitude and Orbit Control System APDL ANSYS Parametric Design Language DAE Differential Algebraic Equation
DM Dynamic Model
DOF Degree of Freedom
EMR Effective Mass Representation EOM Equation of Motion
EQF Eigen-axis Quaternion Feedback ERT Euler Rotation Theorem
FE Finite Element
FEM Finite Element Method FFR Floating Frame Reference
GPOPS General Purpose Optimal Control Software IFE Incremental Finite Element
LDV Large Deformation Vector
LTED Linear Theory of Elasto-dynamics MED Momentum Exchange Device PMP Pontryagin Maximum Principles PZT Lead Zirconate Titanate
RMSE Root Mean Square Error
RW Reaction Wheel
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LIST OF SYMBOLS
𝛾𝛾𝑖𝑖 Angle of Rotation of qi-ri Plane αi Angle of Rotation of X-Y Plane αRW Angle of Ŵ1, Ŵ2 Plane Rotation βRW Angle of Ŵ1, Ŵ3 Plane Rotation
h Angular momentum
hr,j Angular Momentum of RW
hout Angular Momentum With Reference To The Satellite Body
Ωj Angular Velocity of RW
C Arbitrary Constant
ŵ1, ŵ2,ŵ3 Body Frame of Coordinate System of RW
β Cant Angle
xi, yi, zi Component of Vector Position of The Panel β2,i Constants With the Unit S
β1,i Constants With the Unit S-1
u(t) Control Function
J Cost Function
λ Co-State
Ii Cross Sectional Moment of Inertia
ζ𝑖𝑖 Damping Ratio
µ Deflection of the Flexible Panels
Cc Derivative Gain
ηi Energy Dissipation Per Stress Cycle
tf Final Time
EiIi Flexural Rigidity
εi Generalized
Aj Generalized Coordinate of the System
Q Generalized Forces
X, Y and Z Global Coordinate System
H Hamiltonian
hi Height of Panel
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Ĉ1, Ĉ2, Ĉ3 Inertia Frame of Coordinate System of RW
t0 Initial Time
Tdi Kinetic Energy Due To Flexible Motion Ti Kinetic Energy Due To Rotation
TR Kinetic Energy of The Rigid Body
L Lagrangian
Li Length of Panel
pi, qi, ri Local Coordinate System of Flexible Panel ai, bi, ci Magnitude of OPi Vector
µi Magnitude of The Deflection M, Mk,i Mass Matrix
ρi Mass Per Unit Volume of Flexible Panel
Wi Matrix of The Velocity of Unit Mass of The Flexible Panel
N Moment
Ix, Iy, Iz Moment of Inertia of The Satellite Body ωn,i Natural Frequency of i-th Panel
n Number of Panel
m Number Or Modes
u* Optimal Control
Ri Position Vector (Local Coordinate)
Vi Potential Energy Is Associated Strain Energy
kc Proportional Gain
q,q123, q4 Quaternion
𝜽𝜽̇𝒔𝒔 Rate of Angular Rotation of The Satellite τr,i RE Generated Torque
𝜃𝜃̇𝑋𝑋,𝜃𝜃̇𝑌𝑌,𝜃𝜃̇𝑍𝑍 Satellite Angular Velocity 𝜃𝜃𝑋𝑋,𝜃𝜃𝑌𝑌,𝜃𝜃𝑍𝑍 Satellite Attitude Rotation
Øi,j Shape Function
K, Ki, Kk,i Stiffness Matrix σi Stress In The System
S(t) Switching Function
Is The Total Moment of Inertia τreq Torque Required
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T Total Kinetic Energy
BGA Transformation Matrix of Qi-Ri Plane
OFP Transformation Matrix of X-Y Plane
CRw Transformation Matrix That Relates The Wheel Frame To The Satellite Body Frame
I, j, k Unit Vector Notation 𝑂𝑂𝑃𝑃𝚤𝚤
���� Vector of Pi from O
wi Width of Panel
Ei Young Modulus
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Kawalan optimum pemutaran sikap untuk satelit kelas RazakSAT® berdasarkan model tegar dan fleksibel.
ABSTRAK
Peningkatan keperluan terhadap prestasi satelit telah menyebabkan penggunaan tenaga yang semakin meningkat. RazakSAT-2 adalah program satelit baru yang akan dilengkapi dengan panel suria yang lebih besar untuk menghasilkan tenaga yang mencukupi bagi tujuan tersebut. Ini akan menyebabkan fleksibiliti menjadi menonjol dalam sistem satelit. Operasi satelit adalah sangat sensitif terhadap gangguan gerakan kenyal dan had masa. Oleh itu, pemahaman terhadap kelakuan sistem adalah penting untuk mendapat penyelesaian kepada situasi masa operasi yang terhad dan masalah fleksibiliti. Kaedah Bingkai Rujukan Terapung telah digunakan untuk menperoleh model matematik sistem satelit yang terdiri daripada tiga panel solar. Di samping itu, model dinamik untuk sistem empat roda reaksi dan kawalan maklum balas Quaternion paksi Eigen juga diperolehi. Seterusnya, simulasi model dijalankan dengan perisian MATLAB dan ANSYS untuk tujuan pengesahan model. Melalui simulasi tersebut, peratusan ralat min persegi yang diperoleh adalah rendah iaitu di antara kadar peratusan 2.015% hingga 4.841%. Ini bermaksud model yang diperoleh itu adalah mencukupi untuk menggambarkan dinamik sistem satelit tersebut. Dengan menggunakan model dinamik tersebut, kawalan optimum digunakan untuk meminimumkan masa bagi mencapai orientasi yang dikehendaki. Di samping itu, kaedah tersebut juga dapat meminimumkan amplitud kenyal panel suria. Perisian GPOPS telah digunakan untuk tujuan tersebut. untuk mengarahkan satelit ke arah yang diingini sambil meminimumkan amplitud kenyal panel suria. Kawalan optimum telah menunjukkan pengurangan masa manuver sebanyak 3.49% hingga 25.11% untuk perbandingan antara maneuver Axis Eigen dengan pengawal konvensional maklum balas Quaternion paksi Eigen. Fenomena ini disumbangkan oleh dua factor utama, iaitu, kawalan optimum dapat menggunakan sepenuhnya keupayaan roda reaksi manakala pengawal maklum balas Quaternion paksi Eigen dipengaruhi oleh batasan pseudo-inverse. Ini menyebabkan peningkatan prestasi roda reaksi maksimum sebanyak 35%. Kedua, penggunaan kawalan optimum membolehkan trajektori untuk menyimpang dari paksi Eigen kepada paksi yang mempunyai kelisan yang lebih tinggi bagi mencapai manuver yang lebih pantas. Dari segi prestasi penggunaan model tegar dan fleksibel dalam kawalan optimum, ia telah menunjukkan bahawa pergerakkan fleksibel dapat dihapuskan pada kadar 10.53% lebih cepat untuk model fleksibel. Faktor utama yang mempengaruhi masa manuver adalah frekuensi semula jadi sistem satelit. Kesan frekuensi semula jadi terhadap kawalan optimum diperhatikan menunjukan bahawa masa manuver meningkat apabila frekuensi semula jadi berkurangan. Untuk kerja-kerja masa depan, parameter tambahan seperti kesan pengikatan, gangguan luaran dan pengagihan jisim ketidakseimbangan pada badan satelit yang tegar dan fleksibel akibat pemesongan perlu dikaji. Ini boleh menyumbang kepada model fleksibel yang lebih baik dan meningkatkan ketetapan model.
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Optimal control of attitude maneuvering for RazakSAT® class satellite based on rigid and flexible model.
ABSTRACT
The increase in demand for performance for satellite capabilities has pushed the design of the system to be more and more power consuming. This is the case for RazakSAT-2, which is a new satellite program that will be equipped with bigger solar panel to generate sufficient power. Thus, this translates to a higher flexibility in the satellite.
Satellite mission is known to be highly sensitive to the flexible motions and it is time constrained. Hence, understanding the behavior of the system is required to solve the time constrain flexibility problem. The Floating Reference Frame is applied to obtain the mathematical model of the system which consists of three solar panels. In addition, the model for the actuator is also developed for a four-reaction-wheel system and the Eigen-axis Quaternion Feedback control is also derived. The obtained model is simulated using the MATLAB and ANSYS software for verification of the model. The obtained Percentage Root Mean Square Error falls between 2.015% to 4.841% which is low. Hence, this signifies that the model is sufficient to describe the dynamic of the system. From the model, the control of the minimum time optimal control is developed to minimize the time to achieve desired orientation while minimizing the amplitude of the flexible solar panel. GPOPS toolbox is applied to obtain the optimal control solution. The optimal control is shown to decrease the maneuver time by 3.49% to 25.11% depending on the Eigen Axis of the rotation compared to the conventional Eigen-axis Quaternion Feedback controller. This phenomenon is contributed by two factors. Firstly, the optimal control is able to fully utilize the all the capacity of the reaction wheel while the Eigen-axis Quaternion Feedback controller is plagued by the pseudo-inverse limitation which allows a maximum 35% increases in performance.
Secondly, the application of optimal control allows the trajectory to deviate from the effective Eigen axis to achieve faster maneuver by utilizing the torque that is unavailable to the effective Eigen axis maneuver. In terms of the performance of the rigid and flexible model in the optimal control, it shown that the flexible motion converges at 10.53% faster for the flexible model. The primary factor that affects the maneuver time is the natural frequency of the system. The effect of the natural frequency is observed in this section and is shown that maneuver times increase when the natural frequency decreases. For future works, additional parameters such as the stiffening effect, external disturbances and the imbalance mass distribution on the rigid and flexible due to the deflection are studied. This can contribute to a more refined flexible model that would further increase the accuracy of the model.
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1 CHAPTER 1
INTRODUCTION
1.1 Overview
This chapter presents an introduction of the thesis that includes the research background, problem statement, objectives, scopes and the thesis organization. An overview flow chart of the research is illustrated in Section 1.7 of this chapter.
1.2 Research Background
RazakSAT, as shown in Figure 1.1 is launched into low Earth orbit on 14 July 2009. It is placed into a near-equatorial orbit that presents many imaging opportunities for the equatorial region. It weighs over three times as much as TiungSAT-1 and carries a high resolution Earth observation camera. The satellite is intended to provide greatly increased coverage of Malaysia, compared to most of the other earth observation satellites. RazakSAT 2 Satellite Program is a continuation of the strategic satellite technology development of RazakSAT. The program aims to strengthen the satellite technology of Malaysia in the aspect of infrastructure, human capital and industry’s capabilities enhancement. The ever increasing demand for the complexity and advancement of space missions leads the satellite to be designed with deployable appendages such as solar panels, booms or antennas that are flexible in the nature. In 1958, the first spacecraft was launched by the United States which incorporated solar panels appendages was the Vanguard 1 satellite. The spacecraft design was largely
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influenced by Dr. Hans Ziegler who is regarded as a pioneer to the spacecraft solar power (Perlin, 2004).
Solar panel designed on the satellite supplies power for two primary purposes, namely the power to operate the sensors, active heating, cooling and telemetry while the power for the satellite propulsion is the electric propulsion, or sometimes it is also called the solar-electric propulsion (Doody, Stephan, & Fisher, 2015). Hence, the solar panel requires a large surface area that allows it to be pointed towards and exposed to the sun as the satellite moves to collect the sunlight and turn it into power for the system. Generally, satellite is built so that the solar panel is pivoted on the satellite body as it moves. Thus, the panel can be shifted to remain in the direct path of the sun light regardless of what the attitude of the satellite is pointed even if the rest of the body of the satellite moves around. The larger surface area of the solar panel, the more electricity can be converted from light energy from the Sun. Since satellite is designed to be compact and small, this has posed a limitation on the amount of power that can be generated (Doody, 2015).
Attitude maneuvering is the most fundamental function of satellite that is essential to complete mission requirements. The attitude maneuvering of the satellite is usually done by rotating around the three primary axes, namely the roll, pitch and yaw axis. In the rotational maneuvers of satellite system with large flexible panel, elastic deformations in the flexible appendages are often present (Elmadany et al., 2013). The solar panel is an essential satellite element as well as the potential sources of vibratory motion. In space operation, problems with minuet vibrations are particularly acute where precision is a critical aspect of completing any space mission due to the fact that
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satellite missions usually focus their attention on small objects at very great distances away so that any minor local disturbances can greatly become accentuated. The flexibility will become prominent when the size of the panel increases, the lower Young’s Modulus material is used for the structure of the appendages and the larger angle of rotation. Nevertheless, the weight of the satellite is a critical issue that must be kept to a minimum while an increase in the strength of high-performance materials does not match in terms of their stiffness where it has evolved to be lighter, flexible and vibration-prone.
Figure 1.1: RazakSAT Satellite (ATSB, 2016)
The structure of the satellite attitude maneuvering system consists of two important sections, namely the attitude control and the satellite dynamics. The attitude control confines the controller and the actuator which provide the torque into the satellite system to obtain the desired angle of rotation. The satellite dynamics describes the response of the satellite system to the given input provided by the actuator.
Generally in a satellite system, control engineers typically employ different types of actuators to perform spacecraft attitude control. Satellite applies the gravity gradient stabilization or magnetic control technique to control the satellite’s attitude because they are simple and cost less. However, these methods are only able to achieve low accuracies and limited control torques because they are dependent on the gravitational field and the geomagnetic field condition. Hence, Momentum Exchange Device (MED)
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becomes preferable to cope with these limitations. One of the most common MED is the Reaction Wheel (RW), which generates a reactionary body torque and momentum by counter-rotating a small rotor. RWs are simple to control and have a high momentum capacity, which translates to high angular velocity rates for the satellite. It is common for a satellite to be designed with three or more RWs (S. Nudehi et al., 2008). The RWs are prevalent, cost less, are mechanically simpler, weigh less, and are easier to control compared to other types of actuator (Crews, 2013). An example of a four RWs system is shown in Figure 1.2.
Figure 1.2: Configuration of RWs
Satellites are sometimes required to reorient or reposition as soon as possible with the condition that the structural vibrations are minimal. Achieving such control task for the systems becomes difficult when many structural flexible modes exist in the satellite. Vibration control can be classified as passive control where devices lack feedback capability. The active control involves the real-time recording instrumentation integrated with input processing equipment and actuators within the structure. Hybrid control devices have combined features of active and passive control
Y RW3 RW2
X
RW1 RW4
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systems (Chu et al., 2005). However, due to the constrain of power supply, active and hybrid control is not ideal in satellite mission. Hence, a time optimal control is applied to achieve the desired output where optimal control function for the atttiude change torque is applied in the satellite system. The time optimal solutions are usually obtained using numerical methods that are generally computationally extensive where time- optimality Pontryagin Minimum Principle (PMP) is applied (Wie, 2008). The Minimum Principle is a set of necessary conditions for optimality formulated in 1956 by Lev Pontryagin (Naidu et al., 2002). PMP provides necessary conditions for obtaining the time-optimal control solutions. In the case of the satellite attitude maneuvering for a rest-to-rest rotation, bang-bang control is applied for the PMP because of its simplicity which only consists of switching the excitation between the two boundary values to achieve the desired attitude change. The time-optimality for a bang-bang control only depends on the times when the switches take place while maintaining the load constant between the positive and negative value (Clarke, 2013).
The solution to the optimal control is observed via virtual and physical experimental simulation. Physical experiment involves the scaled or exact reproduction of the processes in the laboratory that the material is subjected to the actual in space.
However, the cost of physical simulator is high due to the requirement to emulate the space environment which is near vacuum and no gravitational field. Alternatively, a virtual simulation is able to model a real-life situation in a computer so that it can be studied to observe how the system works by changing variables in the simulation and the predictions may be made about the behavior of the system. Virtual simulation is selected for observing the response of the satellite system. It is defined as the imitation of the response or operation (Banks, 2005). In order to simulate the
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satellite system, the dynamic model must be obtained which is the representation of the key characteristics or behaviors of the physical or abstract system. The developed dynamic model represents the satellite system itself while the simulation is the representation of the operation of the satellite system. Simulation can be used to illustrate the effects of desired alternative conditions and courses of action of the dynamic motion. In addition to that, simulation is ideal because the actual operational environment of the satellite is difficult to engage because it is not easily accessible or it may simply not exist (Sokolowski et al., 2008).
1.3 Problem Statement
The current RazakSAT Satellite System applies a rigid model which is shown in Figure 1.3. A rigid model is a simple assumption that idealizes the behavior of the structure as a non-deformable block of body. This has provided simplified solution to the attitude rotation and much of the satellite relies on this rigid body assumptions.
However, a rigid body never exists in the physical world, and it is essentially just idealized assumption (Hughes, P.C., 2012). The model has a very little physical representation of the real behavior of the satellite structure. In the form of multi-body satellite where flexible deformable bodies make up a significant part of the system, then flexibility must be considered into the dynamic model (Azadi et al., 2015). An ill- defined dynamic model inherits inaccuracy and unpredictability to the attitude maneuvering of the satellite system. This threatens a critical operation such as satellite imaging which is highly sensitive to the vibration and leads to a decline in imaging quality of space camera in satellite remote sensing imaging (Haghshenas, 2015). In
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order to cater to these problems, understanding the dynamics of the structure is the primary step to describe the system.
Figure 1.3: Block diagram of rigid satellite dynamic
An Eigen-axis Quaternion Feedback (EQF) controller is developed for satellite Eigen-axis rotational maneuver. The controller basically consists of a linear feedback of error and body rates with the natural gyroscopic coupling torque. It works by continuously taking measurements using a sensor and making calculated adjustments to rotate the satellite attitude to the desired state. Through the EQF controller, the attitude maneuvering is able to perform a multiple-axis rotation with a single axis of rotation via the Eigen-axis. This is an advantage to the attitude maneuvering provided that the controller is able to access to the full capacity of the actuator. However, for the satellite system in order to provide for redundancy, the RW assembly includes at least one additional RW. This configuration of the RW is applied in RazakSAT satellite (Lee, H.
et al., 2002). The four RWs are configured in the pyramidal form. The use of four or more wheels presents a control allocation problem. This is caused when torque or momentum required in the body frame must be produced by the redundant set as well. A common method of torque and momentum allocation is to use the Moore-Penrose pseudo-inverse, which provides a least squares solution. According to this approach, the full magnitude of the torque capacity generation of RW may not be available in every direction. The pseudo-inverse does not take physical wheel limitations into account and
Satellite Attitudes Satellite
Kinematics Satellite
Dynamics Actuator
Controller
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may request more torque than is available. Once any wheel is saturated, the remaining allocated torque is simply lost. This is because to remain in Eigen-axis, the integrity of the least squares solution must be maintained. Thereby, proportional saturation across all wheels is necessary. Because of the nature of the Moore Penrose pseudo-inverse, maximum momentum and torque will be unavailable for certain axes.
1.4 Research Objectives
This research aims to study on the dynamic model and the control of a flexible satellite structure in the quaternion kinematic space. The overview of the satellite system is shown in Figure 1.4. The configuration of RazakSAT class satellite is applied throughout the research. The list of the research objectives is as follows:
1. To develop the flexible dynamic equation of an attitude maneuvering of a satellite system based on the RazakSAT model;
2. To obtain the mathematical models of the attitude control system consisting of the controller and actuation module of a satellite system;
3. To validate and analyze the developed mathematical dynamic models for the satellite systems by using the Finite Element Method (FEM).
4. To establish a constrained optimal control for satellite system for the shortest time to achieve desired attitude while vibratory motions are minimized.
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