Solar Disturbance torques from the space environments along x,y and z axis. P .... control subsystem determines how the solar cells view the sun. .... First, the flight software(FSW) can calculate the amount of attitude change to be ...... The second and fourth terms in equation (2.2) are the Coriolis and centripetal acceleration.
Ministry of Education and Science of the Russian Federation Saint-Petersburg State University of Aerospace Instrumentation
Chair #11
COMPARATIVE DESIGN OF ATTITUDE STABILIZATION AND CONTROL SCHEMES FOR THREE AXES SPACE VEHICLE.
Master Thesis on the direction “Instrumentation” by Adetoro Moshood Adesoye Lanre
Saint-Petersburg 2010
Declaration I confirm that the work presented in this thesis is, except of the instructions and advice of my supervisors are completely on my own achievement and that any information from literatures, journals, books, internet sources used in this thesis are properly acknowledged. Development of this project has been validated with the mass properties and various control system parameters of first Nigeria communication satellite(Nigcomsat-1),however, any opinions, findings, and conclusions or recommendations expressed in this material are those of my extensive finding and simulation results and do not necessarily reflect the views of any organization or agency.
Signature:……………………..
Date:…………………
Supervisors. 1. Prof.Alexander Nebylov 2. Prof.Alexander Panferov
viii
Supervisor’s Review of Work
ix
Abstract of the Master Thesis Various wheels configurations for three axes active attitude control systems as well as their attitude control strategies used nominally for controlling the orientation of various telecommunication satellites in geostationary earth orbit (GEO) are presented in this thesis. Firstly, An active control strategy using frequency domain approach was implemented, The simulation results of each configuration such as one wheel /thrusters, two symmetrically inclined momentum wheels in a V configuration, three reaction wheels and one pitch momentum wheel/two reaction wheels show different performances in term of pointing accuracies, stability, reliability, propellant utilization and cost. The second phase of the study focuses on the optimization of two symmetrically inclined momentum wheels in a V configuration used on Nigeria communication satellite and other similar satellites in geostationary orbit. The result shows that the choice of the inclination angle depends very much on the sensor noise amplification and ease of controllability of both the pitch and the roll angles, while still exploiting the feature of inertial attitude stabilization that keeps the yaw error close to null without measuring it. The analysis starts with a study of the spacecraft dynamics and behavior based on the Euler linearized equations of motion. Then, the implementation of the pitch and roll/yaw control architectures using their respective mathematical models. Subsequently, the control algorithms are tested using the MATLAB® and Simulink®. It is shown that via tuning the control gain values, the pointing errors can be minimized for all the wheels configurations. The responses of the attitude control obtained from the feedback system on different wheels configurations show that good pointing accuracies can be accomplished with dedicated attitude controllers however with varying performances. Mass properties, payload requirements, configurations and control systems parameters similar to Nigeria communication satellite (Nigcomsat-1) was used in the control synthesis.
x
Реферат магистерской диссертации В диссертации рассмотрены различные конфигурации трехосной активной системы управления угловым движением и стратегии построения такой системы, используемой обычно для управления ориентированием различных телекоммуникационных спутников на геостационарной орбите (ГСО). Стратегия активного управления при использовании анализа в частотной области была применена. Во первых, эффективность каждой конфигурации, такой как один маховик + реактивные двигатели, два симметрично наклоненные V-образно маховика + реактивные двигатели, три гиродина и наконец один маховик по углу тангажа + два гиродина была изучена путем моделирования в пределах требуемой точности стабилизации по рысканию, контролируя устойчивость, надежность, расход топлива и стоимость. Вторая фаза исследования была сфокусирована на оптимизации двух симметрично наклоненных V-образно маховиков использованных в нигерийском телекоммуникационном спутнике и в других подобных спутниках на геостационарной орбите. Результат показывает , что выбор угла наклонения сильно зависит от уровня шума измерения и простоты управляемости углов тангажа и крена, в то время как использование свойства инерциальной самостабилизации происходит, которое удерживает ошибку рыскания вблизи нуля без ее измерения. Анализ начинается с изучения динамики спутника и его поведения, основанного на линеаризованных уравнениях движения Эйлера. После этого для осуществления управления по тангажу и крену/рысканию с использованием этих уравнений была найдена соответствующая архитектура системы. Затем алгоритмы управления проверялись путем моделирования с использованием MATLAB и Simulink. Показано, что с помощью настройки коэффициентов усиления указанные ошибки могут быть сведены к минимуму для всех вариантов конфигураций. Характеристики позиционного управления, полученные в системе с обратной связью при различных конфигурациях маховиков, показали, что хорошая указанная точность может быть достигнута с определенными позиционными контроллерами, но с различным качеством управления. Большинство характеристик, требований по массе полезной нагрузки, конфигурации и системе управления, использованных при синтезе законов управления, близки по параметрам к нигерийскому спутнику Nigcomsat-1.
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Acknowledgements For me, working on this thesis was an opportunity to brings together the experience that I have acquired in my almost fifteen months as a KHTT trainee at China academy of space technology (CAST), Beijing, and during my two years master program at Saint Petersburg State University of Aerospace Instrumentation, Saint-Petersburg, Russia. This work however would not have been possible if not for the interest and absolute support provided by Almighty Allah and my sponsor, National Space research and Development Agency (NASRDA), Nigeria. I sincerely dedicated this project to Almighty Allah for seeing me through all the challenging and very difficult situations. My gratitude also goes to the following people, Professor R A Boroffice (Former DG, NASRDA), Dr S.O Muhammed (DG, NASRDA), Dr Olufemi Agboola of the Space System Department of NASRDA who initiated this education collaboration with this University and his help and support throughout the program, The entire members of Expert group of Nigerian space policy, The entire staff of NASRDA-Abuja, CSTP-EPE, CSTD-Abuja and NigComSat nig LTDAbuja. Also, I will like to equally thank the progressive trains of Mr. J.D Ali, Mr. Chizea, Mr. Jonathan Useni Angulu, Mrs Augusta Iheanocho, Engr. Fashade, Mr Conno, Alh Sani Suleiman, Mr Niyi Iwaloye and host of others. All their efforts towards the success of this program are highly appreciated. Special thanks to Mr Otepola who is a former Director of International cooperation from the agency for being part of the team that witness the beginning of this program in Saint Petersburg, Russia. I would like to acknowledge the support of Prof Alexander Nebylov (Director of IIAAT) for being my advisor and supervisor throughout my master program. He mentored me through all of my aerospace engineering projects, specifically during my Master’s Degree research. I would like to also thank Prof Panferov also my supervisor and Dr. Sergey A. Brodsky for their criticism and guide on my work at a number of presentations and group meetings. I would like also to thank my classmates during the master program for making each day interesting. I feel extremely challenged to have received the opportunity to be team leader and part of initiators of the master program and to be surrounded by such entertaining people. I would also like to thank all of my friends that I have made at China Academy of Space Technology, Beijing, Xichang Satellite Launch center, Xichang, Xian satellite tracking telemetry and telecommand, Xian and staff and students of international institute of advanced aerospace instrumentations, Saint Petersburg, Russia and my former co-staff and friends from Nigeria communication satellite Ltd, Nigeria. I would also like to thank the whole extended Adetoro Family and family of Alhaji and Alhaja Ajileye Mudathir for their support and cooperation. Lastly, I would like to thank my cooperative family: My wife, Mrs Latifat Adetoro and my children, Fathia,Hameed and Samad for their steady state cooperation and robustness, Last but not the least, I am grateful for the support of those who read and edited every page of this thesis and cheered and encouraged me throughout and those I might have forgot to mention their names.
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Table of Contents Declaration ............................................................................................................................................... viii Supervisor’s Review of Work....................................................................................................................ix Abstract of the Master Thesis .................................................................................................................... x Реферат магистерской диссертации .....................................................................................................xi Acknowledgements ................................................................................................................................... xii List of Figures ...........................................................................................................................................xvi List of Tables ......................................................................................................................................... xviii List of Acronyms ......................................................................................................................................xix List of Symbols .........................................................................................................................................xxi 1.
2.
INTRODUCTION ............................................................................................................................... 1 1.1.
Background and Motivation .......................................................................................................... 5
1.2.
Goals of Research .......................................................................................................................... 7
1.3.
Relationship of this Research to the Needs of NASRDA, Nigeria. ................................................. 8
1.4.
Control system of Nigeria communication satellite ...................................................................... 8
1.5.
Overview of Previous and Related Work. ...................................................................................13
1.6.
Contribution of the thesis. ..........................................................................................................18
1.7.
Thesis Outline. .............................................................................................................................18
RIGID BODY DYNAMICS. ............................................................................................................ 20 2.1.
Angular momentum of Rigid Body ..............................................................................................21
2.2.
Attitude dynamics and Kinematic equations. .............................................................................23
2.2.1
Euler’s Moment Equation ....................................................................................................23
2.2.2
Gravitational Force. .............................................................................................................25
2.3.
Attitude Description. ...................................................................................................................25
2.3.1 2.4. 3.
Gravity Gradient Moments..................................................................................................29
Space Environments Disturbance torques. .................................................................................30
MATHEMATICAL MODELS FOR THREE AXIS ACTIVE CONTROL SYSTEMS. ........... 33 3.1.
Linearization of the vehicle equation. .........................................................................................33 xiii
4.
3.2.
Linear Stability Analysis. ..............................................................................................................36
3.3.
Momentum bias attitude control system. ..................................................................................39
3.4.
Pointing Error Analysis. ...............................................................................................................40
3.5.
Simulation development and Simulink Models of the control problems. ..................................43
3.6.
Control algorithms. .....................................................................................................................47
3.7.
Linear Open loop Numerical Analysis. ........................................................................................48
3.8.
System configuration of attitude control. ...................................................................................52
3.9.
Requirements of attitude control in this thesis. .........................................................................56
ANALYTICAL PHASE LEAD COMPENSATOR DESIGN. ...................................................... 58 4.1.
Analytical phase lead compensator design for momentum biased spacecraft. .........................58
4.2.
Lead Compensator design for Pitch axis. ....................................................................................59
4.3.
Lead compensator for Roll and Yaw Axes. ..................................................................................63
4.3.1.
Lead compensator for Roll axis. ..........................................................................................64
4.3.2.
Lead compensator for Yaw Axis. .........................................................................................67
4.4.
5.
Comparative design and simulation of all the wheels configurations. .......................................70
4.4.1.
Compensated Simulink control design for type a configuration.........................................72
4.4.2.
Compensated Simulink control design for type b configuration. .......................................76
4.4.3.
Compensated Simulink control design for type c configuration. ........................................82
4.4.4.
Compensated Simulink control design for type d configuration ........................................84
4.5.
Investigation of the two wheels Skewed angles. ........................................................................85
4.6.
Method of Investigation. .............................................................................................................86
RESULTS AND DISCUSSION........................................................................................................ 88 5.1.
Dynamics of solar radiation Torques Model ...............................................................................88
5.2.
Pitch Axis Attitude Response ......................................................................................................89
5.2.1
Pitch Attitude Response for type a, type c and type d........................................................89
5.2.2
Pitch Attitude Response for type b. ....................................................................................90
5.3.
Roll/Yaw Attitude Response ........................................................................................................90
5.3.1
Simulink Simulation results for type a momentum wheel configuration. ..........................91 xiv
5.3.2
Simulink Simulation results for type b momentum wheel configuration. ..........................92
5.3.3
Simulink Simulation results for type c momentum wheel configuration. ..........................93
5.3.4
Simulink Simulation results for type d momentum wheel configuration. ..........................94
5.4.
Comparative Results for type a, type b, type c and type d configurations. ................................95
6.
CONCLUSIONS AND FURTHER STUDIES ............................................................................... 99
7.
APPENDIX A: MATLAB CODES. ............................................................................................... 101
8.
REFERENCES ................................................................................................................................ 110
xv
List of Figures Figure 1.1: Popular wheels configuration of momentum bias vehicle for GEO. Figure 1.2: A typical momentum wheel. Figure 1.3: Space vehicle in geosynchronous orbit with inclination angle to geostationary equatorial plane. Figure 1.4: Single wheel Pitch axis-thrusters configuration. Figure 1.5: Two Skewed momentum wheels/thrusters configuration. Figure 1.6: Three reaction wheels configuration. Figure 1.7: Two reaction wheels/one momentum wheel configuration. Figure 1.8: Computer aided design of the satellite. Figure 1.9: On orbit configuration of Nigeria communication satellite. Figure 1.10: 20 degree skewed wheels configuration of Nigeria Communication satellite. Figure 2.1: One scheme of rotation of Euler angles. Figure 2.2: Description of Euler angles. Figure 2.3: Non spherical shape of the Earth. Figure 2.4: Perturbation torques as a function of altitude. Figure 2.5: Model of solar radiation pressure for geosynchronous orbit Figure 3.1: Model of a three wheels Momentum bias spacecraft. Figure 3.2: Interchange of Yaw and Roll Attitude Components for a Momentum Wheel with Angular Momentum h , Fixed in Inertial Space. Figure 3.3: data linkage diagram of the vehicle control system Figure 3.4: Simulink model of the Momentum bias spacecraft. Figure 3.5: Control block diagram for implementation of the Vehicle simulator. Figure 3.6: Bode plot of the Roll axis transfer function. Figure 3.7: Bode plot of the Pitch axis transfer function. Figure 3.8: Bode plot of the Yaw axis transfer function. Figure 3.9: System configuration of momentum bias spacecrafts. Figure 3.10: Momentum/Reaction Wheel. Figure 3.11: On orbit mode of geostationary communication satellites. Figure 4.1: Pitch axis attitude control block diagram. Figure 4.2: Bode plot of phase-lead controller. Figure 4.3: Open loop Bode plot of Pitch axis to select the frequency at -10log10 αdB . Figure 4.4: Analytical result without solar perturbation. Figure 4.5: Open loop Bode plot of Roll axis to select the frequency at -10log10 αdB . Figure 4.6: Open loop Bode plot of yaw axis to select the frequency at -10log10 αdB . Figure 4.7: Step response for uncompensated Roll and Yaw Dynamics. Figure 4.8: Step response for compensated Roll and Yaw Dynamics. Figure 4.9: Common control solutions to all the wheels configurations. Figure 4.10: Automatic controller switch simulator. xvi
Figure 4.11: Type a wheel configuration. Figure 4.12: Block diagram for a bang-bang plus dead zone controller system. Figure 4.13: Simulink representation of bang-bang control. Figure 4.14: Simulator for compensated type a, type c and type d pitch dynamics. Figure 4.15: Roll/yaw thruster control using time tagged command from ground station. Figure 4.16: Roll/yaw thruster control using on board automatic controller. Figure 4.17: Type b momentum bias configuration. Figure 4.18: Skewed Angle two wheels configuration. Figure 4.19: 2-MW control configuration. Figure 4.20: Dynamics of the two skewed wheels configuration. Figure 4.21: Simulink block for type b roll/yaw and pitch attitude control architecture. Figure 4.22: Three reaction wheels configuration. Figure 4.23: Reaction wheel Roll and yaw axis compensator. Figure 4.24: Redundant based three reaction wheels system. Figure 4.25: Two reaction wheels, one pitch bias momentum wheels configuration. Figure 4.26: Reaction wheel Roll and yaw axis compensator. Figure 5.1a: Simulink Mathematical model of solar radiation torques. Figure 5.1b: Plot of solar radiation pressure on Nigcomsat-1 for one year duration. Figure 5.2: Pitch attitude responses perturbed with solar radiation torques for type a, c and d. Figure 5.3: Pitch axis dynamics perturbed with solar radiation torques for type b. Figure 5.4: Roll/Yaw attitude response using ground command. Figure 5.5: Autonomous Roll/yaw axis dynamics perturbed with solar radiation torques. Figure 5.6: Roll/yaw axis dynamics perturbed with solar radiation torques. Figure 5.7: Roll/Yaw Simulink results for type c wheel configuration. Figure 5.8: Roll/Yaw Simulink results for type d wheel configuration. Figure 5.9: Simulator response using skewed angle of 1.66 degree similar to GEOS N-P Figure 5.10: Simulator response using skewed angle of 2.5 degree similar to Anik E2. Figure 5.11: Simulator response using skewed angle of 20 degree similar to Nigcomsat-1. Figure 5.12: Simulator response using skewed angle of 60 degree. Figure 5.13: Simulator response using skewed angle of 90 degree.
xvii
List of Tables Table.1.1: Performance of Various Attitude control schemes. Table 1.2: Summary of on orbit control loops for Nigcomsat-1. Table 2.1: Comparison for attitude descriptions. Table 3.1: Normal mode pointing error for momentum bias vehicle [12]. Table 3.2: Roots of the polynomial in Pitch and Roll/Yaw open loop transfer functions. Table 4.1: Modifications of Tcx ,Tcy ,Tcz in terms of wheels configurations. Table 5.1: Comparative results of various wheel configurations for momentum bias spacecrafts. Table 5.2: Summary of the plots for various skewed angles.
xviii
List of Acronyms The following definitions are used in the report: AOCS
Attitude and orbit Control System
ADC
Attitude Determination and Control
ADCS
Attitude Determination and Control System
AFM
Apogee firing mode
ANM
Antenna mapping mode
BOL
Beginning-of-Life
CoM
Center of Mass
CAST
China Academy of Space Technology
CGWIC
China Great Wall Industrial Cooperation
CSTD
Center for Satellite Technology Development
CSTP
Center for Space Transport and Propulsion
DC
Direct current
EAM
Earth acquisition mode
ES
Earth Sensor
EPM
Earth pointing mode
EWSK
East/west station keeping
FDI
Fault detection and isolation.
FSS
Fine Sun Sensor
FOV
Field of View
GEO
Geostationary Earth Orbit
GTO
Geostationary Earth Orbit
IIAAT
International Institute of Advanced Aerospace Instrumentation
IRES
Infrared Earth sensor
LM
Launch Mode
LV
Launch Vehicle xix
LVS
Launch Vehicle System
MOS
Mission Operations System
MW
Momentum Wheel
MO
Mission orbit
Nigcomsat-1Nigeria communication satellite-1 NSSK
North/south station keeping
NM
Normal mode
N/S
North/south
NASRDA National Space Research and Development Agency PFM
Perigee firing mode
PID
Proportional integral derivative
PRM
Pseudo rate modulator
RIGA
Rate integrated gyro assembly
RAAN
Right Ascension of the Ascending Node
RW
Reaction Wheel
S/A
Solar Array
SAM
Sun acquisition mode
SUAI
Saint Petersburg State University of Aerospace Instrumentation
SLM
Slant mode
SKM
Station keeping mode
SS
Star sensor
SPF
Single point failure
TRM
Transition mode
TF
Transfer Function
WHECON Wheel control W/E
West/East
WGS-84
World Geodetic System-84
xx
List of Symbols The following symbols are used in the report: I x, y, z −
Moment of Inertias of the space vehicle about its body axis
H ( s ) − Transfer Function of the feedback sensor G ( s ) − Open loop transfer function of the plant
Gcloop − Close loop transfer function
φ ,θ ,ψ − Roll, Pitch, Yaw Euler angles = ω0 7.272 ×10−5 rad/s (Geostationary orbit orbital rate).
ωnut −
Nutation frequency
Tcx ,cy ,cz − Control torques from the control actuator along x,y and z axis
Tdx ,dy ,dz − Solar Disturbance torques from the space environments along x,y and z axis GP −
Transfer function of the plant
Gс −
Transfer function of the controller
GM −
Gain Margin
PM −
Phase Margin
H 1, 2 − Momentum wheel 1 and 2 S' −
Angle between the Sun Line, Earth and polar axis.
φm −
Maximum Phase Lead
ωm −
Frequency at maximum phase lead
ω x , y , z − Angular frequency at x, y and z axes hwx , wy , wz − Angular momentum of the wheel along spin axes hwy 0 −
Nominal Angular momentum of the wheel in pitch axis
H wx , wy , wz − Total momentum along x,y and z axes k−
Phase Lead Gain constant xxi
p−
Pole
z−
Zero
∆−
Determinant
$ −
Skewed angle of the momentum wheel
α−
Gain parameter of phase lead compensator.
xxii
CHAPTER 1 1. INTRODUCTION Spacecrafts are susceptible to many disturbances in space environment that impart an undesirable translation and rotational motions on their dynamics. The primary task of maneuvers and attitude control system was to stabilize the attitude of such vehicle against external torque disturbances such as aerodynamic drag effects, solar radiation and solar wind torques, parasitic torques created by the propulsion thrusters etc. However, Attitude stabilization and control of any space vehicle is one of the most important problems in space vehicle design. This primary task of attitude control is however extensive because mission of various space vehicles, mission orbits and their attitude requirement varies. The basic types of attitude control systems for space vehicles are Gravity-gradient stabilized, Spin stabilization, Dual-Spin stabilization, three axes momentum biased and three axes zero momentum stabilization. Gravity gradient stabilization is passive and at most requires damper. Dampers are designed to damp vehicle rigid body oscillations due to gravity gradient, however if there is sufficient internal damping (due to heat pipes, fuel slosh), the damper may not be required. Gravity gradient stabilization is suitable only for spacecraft with loose slew rate, poor pointing requirements and in the low orbits. Spin stabilization however can be used if one axis of the spacecraft is required to remain inertially fixed and the payload is not affected by spinning, for example a satellite with a long wire antenna along the spin-axis. If the satellite spins about its major axis a passive damping system will suffice. If it has to spin about its minor axis an active nutation control system is needed. The satellite will also need a control system to keep the spinaxis in the same inertial direction if there are significant inertially fixed torques, for example due to a center-of-solar-pressure offset from the center-of-mass along the spin-axis. This method also has a loose pointing accuracy and poor power efficiency. Dual spin spacecrafts on the other hand have two rotating parts. Usually the first part is a platform that rotates at orbit rate or even fixed with respect to its coverage areas (and consequently is nominally pointing at the earth.) The other part rotates at high speed and gives the spacecraft gyroscopic stiffness. This method among other reasons is not suitable for communication satellites in geostationary orbit (GEO) [23]. A bias momentum spacecraft is conceptually the same as a dual-spin spacecraft except that the part that is rotating at high speed is a momentum wheel and rotates at very high rates. Thus, an attitude control torque device such as the momentum wheels and reaction wheels are mounted within vehicles to reduce attitude errors. Wheels control has many advantages. Performance wise, they produce very minimal error as compared to other actuators that is up to 0.01degree of pointing accuracy, but very expensive in terms of cost and mechanically complex though as a function of numbers of wheels implemented per spacecraft [3, 4]. Thus, wheels actuator are normally preferred for high cost missions such as communication satellites in GEO, which require a high pointing accuracy. Up till date, the most feasible and efficient control schemes in GEO are achieved through the use of wheels as primary actuators because of the above stated reasons. A bias momentum 1
spacecraft are the most popular type of spacecraft design today. Some examples are Boeing HS702 Comsat, Lockheed Martin A2100 Comsat, Nigcomsat-1, Lockheed Martin GPS IIR etc. Another version of momentum bias spacecrafts is Zero momentum bias spacecrafts. Zero momentum design has low net inertial momentum. It may be controlled by reaction wheels or by thrusters in all the three axes. In the former case the wheels may each have significant momentum but ideally the vector sum of their momentum is Zero [12].
Table.1.1: Performance of Various Attitude control schemes. The performance comparison of each control schemes is shown in Table1.1.These stabilization techniques are listed in descending order according to their precise pointing ability. For example, in spin-stabilized satellites, the larger mass satellite bus spins (approximately once per second in geosynchronous orbits) and a smaller mass (communications antennas, remote sensors) is despun so that it can point in an assigned direction (a point on the earth or other planetary body being observed or communicated with). Conversely, the 3-axis stabilization system controls the spacecraft body with one or more internal spinning masses under the direction of the control algorithms. Depending on the mission, a space vehicle may have varying requirements of pointing accuracies, at some point, it may be free to tumble and turn, but at others it may have to pinpoint a discrete location on the surface of the Earth or deep space. To do this, the vehicle must be able to determine its own attitude with respect to some reference, and then to modify this attitude to perform the desired mission. Reference devices include earth horizon sensors, sun sensors, star trackers, or even magnetometers which measure the flux lines of the geomagnetic field. Attitude control devices include tiny thrusters, angular momentum storage wheels, gravity-gradient booms, and electromagnetic torque devices. A satellite must determine its attitude with respect to some reference (earth, sun, stars, and/or other satellites) and control its attitude to perform the desired mission. The control logic or computer that accepts input from attitude reference sensors is usually based in space but may include ground-based assets in the control loop, directly or as a backup. The mass and 2
configuration of the control subsystem are determined by mission and payload inputs that include the final orbital attitude, payload pointing accuracy, and satellite estimated on-orbit mass. These factors influence the types of disturbances that will most perturb the spacecraft body, for example, at geosynchronous altitudes; the effects of solar radiation can impart unbalancing torques on the spacecraft body. The attitude control takes its requirements from the pointing accuracies of the single most demanding payload. This determines the available attitude control architectures capable of achieving this accuracy: greater than or equal to 5 degrees for gravity gradient, 1 degree for spin, 0.1 degree for three-axis bias momentum or dual-spin, and less than 0.1 degree for three axis zero-momentum. Having selected attitude control architecture, the designer matches the selected attitude architecture with the minimum components required for implementation. The attitude control design begins with a default/minimum value for the number of components needed as a function of the stabilization method. The designer can make decisions based on special knowledge about the application. For example, the designer may add star mappers to a design that does not require them for pointing to provide, after the fact, pointing knowledge that improves data processing and geophysical application. Another option is to increase or decrease the number of components. Increasing the number of components would build in more redundancy, perhaps extending lifetime [46]. The selection of attitude control architecture impacts the power and thermal subsystems. The most prevalent power subsystems make use of solar cell power generation, and the attitude control subsystem determines how the solar cells view the sun. Spin-stabilized or gravity gradient attitude control designs do not continuously point the entire solar array at the sun as do the 3-axis stabilized attitude control designs. This is taken into account in the power system calculation and determines the number of solar cells, of a particular type, that must be used. Correspondingly, the thermal environment of a 3-axis stabilized attitude control design is more demanding than that created by spin-stabilized attitude control designs, and this will influence the thermal control system mass. As indicated, a spin-stabilized satellite acts like a rotisserie, providing even exposure to hot and cold conditions. A 3-axis stabilized satellite is subject to extreme thermal gradients because particular elements are exposed to hot or cold conditions for extended periods of time. Various systems of three axis attitude control and momentum wheel management for spacecrafts known today use different systems of momentum wheel arrays for various orbit geometries and mission purposes. Momentum bias stabilized spacecraft are very desirable due both to the inherent attitude stability afforded by the gyroscopic stiffness associated with the momentum and to the yaw attitude sensing facilitated by orbital coupling with the momentum bias[50]. The orbital coupling with the momentum bias allows yaw attitude (which is difficult to measure with sensors) to be deduced from a roll attitude sensor. Momentum wheel stabilization systems are used to maintain the attitude by momentum exchange between the spacecraft and the wheel. As a torque acts on the spacecraft along one axis, the momentum wheel reacts, absorbing the torque and maintaining the attitude. As a result, momentum wheels are particularly attractive for 3
attitude control in the presence of cyclic torques or random torques such as in communication satellites. The wheel spins rate increases or decreases to maintain a constant attitude. Over a full period of a cyclic torque, the wheel speed remains constant. Secular torques acting on the spacecraft cause the momentum wheel speed to either increase or decrease monotonically until the wheel speed moves outside operational constraints. A momentum exchange device such as a gas jet must then be used to restore the momentum wheel speed to its nominal operating value. The upper operating limit of a momentum wheel is called the saturation limit. Momentum wheels can be mounted along the pitch, yaw, and roll axes and serve as prime torques for attitude stabilization depending on the number of wheels on the spacecraft.
Figure 1.1.A typical momentum wheel. Figure.1.2 illustrates popular wheels configurations of momentum wheel systems with mass expulsion to control the wheel spin rate (desaturation). OFFSET THRUSTERS
z-axis 3
z-axis α
YA
YAW
OFFSET ANGLE
1
ROLL
2
ROLL
2
x-axis
PITC
x-axis
PITCH
y-axis
y-axis
1. THREE REACTION WHEELS
2. ONE MOMENTUM WHEELS + ROLL/YAW OFFSET THRUSTER VEHICLE
OFFSET THRUSTERS
α
z-axis
z-axis
2
OFFSET ANGLE
3
YA
YAW 3
1
SKEWE D ANGLE
2
ROLL PITCH
x-axis
PITC
y-axis 3. TWO SKEWED MOMENTUM WHEELS + OFFSET THRUSTERS VEHICLE
ROLL x-axis
y-axis 3. TWO REACTION WHEELS + ONE MOMENTUM WHEEL VEHICLE
Figure 1.2 Popular wheels configuration of momentum bias vehicle for GEO [7]. 4
The various wheels configurations however have different performance in term of pointing accuracies of yaw axis, life span, complexity, reliability and cost. It is therefore important to investigate the various wheels configuration in term of stated performances. Four different types of wheels configurations as shown in figure 1.2 above are investigated in this thesis and are stated as follows: •
Pitch momentum wheel with offset roll/yaw thrusters for attitude control (One wheel).
•
Two symmetrically inclined momentum wheels in a V configuration, (Two wheels).
•
Three reaction wheels oriented along roll, pitch and yaw axis(Three reaction wheels)
•
One pitch momentum wheel on y axis and two reaction wheels on roll and yaw axes (One momentum wheel and two roll/yaw reaction wheels).
1.1.
Background and Motivation
The concept of geostationary orbit was first proposed by Herman Potočnik in 1928 and popularized by the science fiction author Arthur C. Clarke in a paper in Wireless World in 1945[1] .Working prior to the advent of solid-state electronics, Clarke envisioned a trio of large, manned space stations arranged in a triangle around the planet. The present day technologies in this aspects has surpassed this concept because modern satellites are now numerous, unmanned, and often no larger than an automobile. Widely known as the "father of the geosynchronous satellite", Harold Rosen, an engineer at Hughes Aircraft Company, invented the first operational geosynchronous satellite, Syncom 2[2]. It was launched on a Delta rocket B booster from Cape Canaveral July 26, 1963. A few months later Syncom 2 was used for the world's first satellite-relayed telephone call. Fortunately, it took place between the then U.S. President John F. Kennedy and Nigerian Prime minister Abubakar Tafawa Balewa. Since then, geostationary earth orbit has being used extensively for global communication purposes over the years. Its primary attribute is that any satellite in this orbit has no dynamic tracking problems because the sub satellite point is fixed at a selected longitude, 0 with 0 latitude. The orbit may therefore provide fixed point communication to any site within the beams of their antennas. For high quality and reliable communication services, various control schemes has being implemented for satellite in this orbit because accurate maintenance of a space vehicle orbit is an essential requirement for maintaining a communication link. This is what is popularly called attitude control. Onboard control of attitude has been a necessary part of all space missions other than a handful of generally low-cost missions utilizing passive control methods such as gravity gradient stabilization. Using attitude sensor measurements to determine the current attitude, the flight software (FSW) compares that attitude to the commanded (i.e., desired) attitude and determines an attitude error. For spacecraft employing gyros (an attitude sensor that measures the change in spacecraft attitude during a set time period, as opposed to measuring the spacecraft’s absolute orientation with respect to inertial space), a Kalman filter usually is utilized both to calculate the current attitude and to calibrate the gyro’s drift bias (which ramps with time) relative to an absolute attitude sensor, such as a star tracker. The attitude error is estimated and fed into a control law that calculates on each control cycle what attitude actuator commands 5
(e.g., reaction-wheel control torques) must be generated in order to null the error. On the next control cycle, feedback from the attitude sensors provides the information needed to determine how good a job of reducing attitude error the previous cycle’s actuator commands did, as well as how much new attitude error has been introduced this cycle by external perturbative torques. Although this description of onboard attitude control has implicitly addressed maintenance of a constant commanded attitude in the presence of perturbative torques, it can equally well be applied to the execution of large desired attitude changes, called slews. Slews can be dealt with two ways. First, the flight software(FSW) can calculate the amount of attitude change to be performed during a given control cycle, and modify the previous commanded attitude to reflect that change, which would then be used directly as part of that control cycle’s attitude error. A second approach is simply to make the commanded attitude the slew target attitude. Although the control law would not be able to null that very large error (say, 90◦) in one control cycle, by limiting the size of the commanded control torques in a given control cycle the FSW could gradually work off the error over a series of control cycles, eventually reaching the slew’s target attitude. Three axis momentum biased is a very effective method for controlling the attitude of a geosynchronous communication satellite orbiting around the earth in a high altitude orbit whose inclination ( α ) to the earth’s equator has at most a value of a few degrees. Figure 1.3 is a diagram showing a typical space vehicle in the form of satellite, travelling in a geosynchronous circular orbit whose radius is about 42,500km and which is inclined at an angle α of a few degrees at most to the earth equator[41]. The satellite is associated with a triad consisting of three orthogonal axes respectively the roll axis Ox, the pitch axis Oy and the yaw axis Oz of the satellite body. Under ideal conditions, when the attitude of the satellite is correct, the roll axis Ox extends parallel to the earth’s equatorial plane in the west-east direction and the yaw axis Oz extends towards the center of the earth and the pitch axis Oy perpendicular to the roll and yaw axes and consequently to the orbit plane. Since the orbit is inclined, the terrestrial latitude of the satellite varies cyclically over a 24-hours period, and the satellite attitude about the roll axis has to be adjusted as it moves along its orbit. Also the effects of major external perturbation torques in geosynchronous orbit (solar radiation pressures moments) and other internal disturbances (thruster’s misalignment, sloshing etc) whose effects when in operation tend to change the attitude of the satellite must be removed.
MODEL OF THE SATELLITE
X - AXIS
GEOSYNCHRONOUS ORBITAL PLANE
NORTH Z - AXIS
Y - AXIS
EARTH EARTH EQUATOR
α
INCLINATION ANGLE
Figure 1.3: Space vehicle in geosynchronous orbit with inclination angle to geostationary equatorial plane.
6
In view of the above, the attitude of such a satellite has to be maintained so that payload carried by the satellite, such as communication antennas, is always oriented towards a particular region on the earth. Various systems of three axis attitude control and momentum wheel management for spacecrafts known today use different systems of momentum wheel arrays for various orbit geometries and mission purposes.Early momentum bias spacecraft (single wheel) were controlled by thrusters and magnetic torquers that apply external torques. Presently different wheel configurations can be found on different spacecrafts in GEO today and these variations motivated this research. The investigation of popular wheels configuration as stated in figure 1.2 to determine variation of levels of accuracy and performances for various wheels consideration will enable the selection of best wheels configurations for future Nigeria communication satellites by the appropriate authority. Additionally, Despite various space vehicles using the same two wheels configurations, the skewed angles for the two wheels placement varies drastically for different satellites for examples, geostationary operational environmental satellite (GOES N-P) has its two momentum wheels skewed at an angle of ±1.66° to the pitch axis [10], Nigcomsat-1 momentum wheels was skewed at angle of ±20°, Anik E2 has its skewed wheels 2.5° [9] etc. The above disparity in the skewed angles for different space vehicles in the same orbit serve as additional motivation to find the effect of this skewed angle on the multiple mission tasks[12].
1.2.
Goals of Research
This research had three goals. One was to design and simulate attitude stabilization and control Systems in frequency domains that is appropriate for use on various wheel configurations used on momentum bias spacecrafts in geostationary orbit. The momentum bias spacecrafts attitude estimation and stabilization systems using wheels as primary actuators are attractive for use on communication satellites in Geostationary Earth Orbit (GEO) because of these devices' light weight, low fuel consumption, and high reliability. The second goal was to prove the techniques by simulation using real spacecraft parameters. MATLAB attitude simulation software was developed in this thesis to mimic and provide a highly flexible means of testing various wheels configurations concepts for attitude determination and control systems. The use of open-source MATLAB .m-files creates this flexibility. The incorporation of solar radiation disturbance torques model causes the simulation to be implemented with perturbed spacecraft environment in space which make it close to reality. Thirdly, various torque models are taken into account so as to give details of the spacecraft's geometry and inertial properties, wheels parameters, skewed angles, sun angle to the Earth and spacecraft, thrusters control and they also include the physics of how they act on the spacecraft's components. 7
1.3.
Relationship of this Research to the Needs of NASRDA, Nigeria.
The results of this research are relevant to the mission statement of National space research and development agency, Nigeria because they advance the state of the art of attitude estimation, stabilization, and control for communication spacecrafts including the one implemented on Nigeria communication satellite. It also shows the benefits of know how technological transfer (KHTT) acquired during the building of Nigcomsat-1 where my specialization is attitude and orbit control design of geostationary spacecrafts. The simulink simulator models input parameter for simulation in the simulink blocks can be changed. Such blocks would therefore be ideal for control synthesis and simulation of small, simple, light-weight LEO spacecrafts using wheels as primary actuators. For Nigeria Communication Satellite Ltd, Nigeria, The results and simulator blocks can be used as training modules to test various control scenarios and trains the young engineers on the behaviors of attitude control subsystem of momentum bias spacecrafts in GEO. Attitude determination and pointing accuracies demonstrated via flight-experiments, post-flight data processing on failed Nigcomsat-1shows similar result with simulation results of this thesis. This similarity therefore shows that we now have a mathematical model of Nigcomsat-1 though without considering solar array flexibility as being implemented in mathematical models implemented in this thesis. The other benefit to the agency has been the development of a MATLAB spacecraft attitude dynamics simulation package. This package has been used in conjunction with the mathematical model of the spacecrafts and actuator dynamics as an evaluation tool for momentum bias attitude control design for different wheels configurations.
1.4. Control system of Nigeria communication satellite Nigeria communication satellite (Nigcomsat-1) is the first operational multipurpose satellite catering to the needs of three different services: Television broadcasting, communications and GPS signal data augmentation that belongs to the Federal Government of Nigeria[12]. The Nigcomsat-1 project envisages the satellite's design, development, placing in orbit, management and control of the satellite during its operational phase and the development and installation of associated ground segment facilities for controlling the satellite, monitoring the performance of its payloads and for full utilization of the satellite's capabilities. It is a class of three axis momentum biased spins stabilized spacecraft which is made-up of skewed two wheels configuration. The spacecraft is also equipped with two large extensive solar arrays. The spacecraft is cuboids shape as shown in figure.1.4. and has small angular velocity i.e. 8
one revolution per orbit in order to maintain one face pointing towards the Earth as shown in on orbit configuration of the satellite in figure.1.5.The solar arrays is designed to have less angular movement because it is required to constantly face the sun and therefore was mounted on a bearing so as to track the sun.
Figure 1.4: Computer aided design of the satellite. The on orbit configuration of the satellite as shown in figure.1.5, Indicates that it is an Earth pointing satellite and first step for its control synthesis is to formulate the mathematical model for its motion in space. In general this model described both translational and rotational motion of the satellite under the influence of external forces and torques. Coordinate system will be used in deriving these equations and adopting the coordinate system implemented for Nigeria communication satellite entails the use of inertial reference frame; local vertical local horizontal frame also called earth pointing frame, spherical coordinate frame and body coordinate frame attached to the satellite body.
9
Figure.1.5. On orbit configuration of Nigeria communication satellite. The use of appropriate reference frame will help in developing the dynamic equations of motion for the communication satellite. Various literatures on control synthesis for similar satellites [11,12,13,and 14] indicated that the resulting dynamic equations are coupled and non linear ordinary differential equations. The objective for mathematical modeling in this thesis is however to simplify the non linear model with reasonable assumptions so as to build a simple but very accurate model. The on orbit configuration of Nigeria communication satellite allows the communication payloads to be Earth pointing, It also allows the solar arrays, the telemetry and telecommand antennas, the sensors, the thermal dissipation parts, the propulsion engines and on board sensors to point in the appropriate directions. To continuously meet the on orbit configuration, the satellite then has to continuously maintain this orientation despite external disturbances using a reliable control system. The satellite uses momentum wheels and thrusters as its actuators. The wheels are momentum exchanger between the spacecraft body and the wheels and combine use of wheels and thrusters prolong the life of the satellite in space. The wheels continuously maintain the required attitude of the satellite based on continuous momentum exchange between the wheels and the satellite, but when it becomes saturated, the thrusters are used to unload the momentum and return the speed of the wheel to its nominal values. The attitude control of the satellite is then complimented with momentum management of the wheels by the control system using thrusters in order to meet the attitude control requirements.
10
On Orbit Configuration of Nigeria communication Satellite. The location of the satellite is a geostationary orbit at an altitude of 35,789 km [10]. The payload is for communication purposes and therefore the spacecraft is pointing to the earth surface with accuracy for yaw axis in normal mission orbit. Attitude control requirements of Nigeria communication satellite are based on those for payload pointing and the satellite bus pointing. The requirements for on-orbit mode will be adopted for this thesis. The on-orbit attitude error of normal mode are about Roll: ±0.06 ̊, pitch: ± 0.06 ̊, Yaw: ± 0.2 ̊; the attitude error of stationkeeping mode about: Roll: ± 0.08 ̊, Pitch: ± 0.08 ̊, yaw: ± 0.2 ̊; The Station keeping errors about: West/East: ± 0.05 ˚, North/South: ± 0.05 ˚. This mission orbit is expected to last for 15 years[11]. The spacecraft coordinate system utilized is such that the z axis is aligned with the nadir direction and the x axis is the forward velocity direction while the y axis is normal to the orbit for on-orbit configuration. The NigcomSat-1 is equipped with 28 transponders on board in four (4) frequency bands (8 Ka-band, 14 Ku-band, 4C-band and 2 L-band). In order to satisfy the complicated coverage area requirements, the satellite is fitted with a total of seven shaped communication antennae. With a lift-off mass of 5,086kg, the NigcomSat-1 satellite will be positioned at 42 degrees east longitude over the equator, with an end-of-life power of over 8KW, and a design orbit life span of 15 years [11]. Mission Orbit Phase (MO). During the mission orbit phase the satellite uses a pitch bias momentum design. Changing the speed of the momentum wheel controls pitch and roll axes while thrusters control yaw error in yaw axis. The earth sensor measures both roll and pitch; though yaw is not directly sensed, but determined from the orbit rate coupling with roll. A yaw gyro is used to sense rate information during station keeping maneuvers. 10N thrusters are used for north-south station keeping maneuvers, east-west station keeping control, for momentum wheel unloading and for backup roll/yaw control [10]. Mode
Description
Controller type
Actuators
Normal mode
Pitch and roll control using momentum wheel, yaw control using 10N thrusters
PID/low bandwidth
V-type MWs, L-types, either of the MW and one RW Yaw control by Thrusters.
SKM
Three axis control
PID/PRM/high bandwidth
Thrusters, Wheel operated at nominal speed.
Table 1.2: Summary of on orbit control loops for Nigcomsat-1. Three controllers are used during normal orbit mode: one normal mode pitch controller, a normal mode roll controller, and the station keeping controller. The momentum wheel control pitch loop and is a proportional integral Derivative (PID) controller which controls the wheels speed 11
demand and the pitch control applied directly to the pitch axis. The normal mode roll controller is a low frequency controller which uses the yaw momentum to control yaw. During station keeping, all three axes are sensed and controlled by a high frequency controller which consists of pseudo rate modulator (PRM) and proportional integral derivative (PID) controller due to magnitude of internal disturbance(propellant slosh) and solar arrays flexibility. The control tasks are explained below; 1. Pitch and roll axis control by momentum wheels in torque mode control where the roll and pitch axis is measured by earth sensor. 2. Yaw control by thrusters by on board observer (yaw estimation) when its error exceeds its dead band using a filter. 3. Momentum management of the wheels, which keeps the momentum of the wheel inside permitted bounds. 4. Station keeping maintains the spacecraft within prescribed limits ( ± 0.05 ˚N/S and E/W) about the nominal longitude position, and also within the same permitted deviation in the inclination of the mission orbit. The station keeping involves east/west station keeping and north/south station keeping.
Description of the On Orbit Modes. Normal Mode (NM)[14] In NM, the infrared earth sensor measures the roll and pitch attitude and the yaw observer estimate the yaw attitude (OBC). The roll and pitch attitude are controlled by a pair of V type installed bias momentum wheels. When one of them is out of order, it can also be controlled by an L type combination of a reaction wheel and the other healthy MW. East- west station keeping can also be performed in NM. The configuration is shown in figure 1.10. Station Keeping Mode (SKM). The SKM mainly provides orbit maneuver for station position acquisition and station keeping. In SKM, infrared earth sensor are used for roll and pitch attitude measurement and sun sensor or gyro integration used for yaw attitude, or gyro integrations used for three-axis attitude measurements. Both north-south and east-west station keeping maneuvers are performed in SKM. During north/south (N/S) station keeping maneuvers, auxiliary active controls are used to maintain pointing accuracy. Since N/S maneuvers produce high disturbances primarily about the roll and yaw axes, rate integrated gyro assembly (RIGA) oriented along each of the three axes are used to provide measurements of attitude error. The RIGA contain three-axis rate integrating gyros and provide in conjunction with attitude control electronics (AOCE) digitized roll and yaw angle increment signals. The filtered and bias-corrected signal is integrated into a yaw or roll angle estimate available for the control logic. In order to determine the bias value needed to correct for gyro drift, the RIGA are always turned on at least 60 minutes prior to a planned 12
station keeping and the drift rate allowed to stabilize. Just prior to the start of station keeping, the integrated roll and yaw estimates will be initialized and the bias value calculated.
Wheels control mode configuration. The normal mode as explained above uses a pair of “V-Wheels” that straddle the pitch axis at a specified angle in a V-shape; each V-Wheel will operates in a single direction. The attitude of the satellite will be controlled by altering the relative speed of the two V-Wheels (“VMode”).The inclination at this angle will give a degree of freedom in desired axis. In addition to the V-Wheels, it also includes a back up yaw wheel centered on the yaw axis of the satellite, referred to as an “L-Wheel.”The L-Wheel will be activated only when one of the two V-Wheels fails. The L-Wheel rotates in one direction if one V-Wheel fails (“L1-Mode”) and the other direction if the other V-Wheel fails (“L2-Mode”). However, the L-Wheel does not operate in both directions during an orbit. That is, the L-Wheel does not slow down to zero speed and reverse direction during an orbital period. Furthermore, while in operation, the L-Wheel never spins more slowly than a certain “bias” speed in order to offset the partial yaw momentum of the functioning V-Wheel[10]. RW MW2 Y MW1 z
Figure.1.6: 20 degree skewed wheels configuration of Nigeria Communication satellite. The OBC will continuously monitor and correct roll and pitch pointing errors detected by the earth sensor. While operational, therefore, the speed of the L-Wheel changes continuously during orbit because it responds to actual errors detected while in orbit. The attitude control system of the satellite is also designed to operate effectively during inclined orbit (slant mode) by compensating for the roll pointing error created by inclined orbit. During inclined orbit, the “roll bias generator” and “yaw momentum bias generator” of the satellite varies the speed of the V-Wheels or the L-Wheel, intentionally causing the satellite to point north or south.
1.5. Overview of Previous and Related Work. The problem of attitude stabilization of a rigid spacecraft, i.e. spacecraft modeled by the Euler’s equations and by a suitable parameterization of the attitude, has been widely studied in recent years. (E.g. Mortensen (1968); Crouch (1984); Salehi and Ryan (1985); Wie et al. (1989); Byrnes and Isidori (1991)). Most of the existing results assume that three torques are available for control purposes, supplied either by gas jet actuators or by momentum exchange devices. If the spacecraft is equipped with three independent actuators, a complete solution to the set point 13
and tracking control problems is available. In Wen and Kreutz-Delgado (1991) and Fjellstad and Fossen (1994) these problems have been solved by means of PD-like control laws, i.e., control laws which make use of the angular velocity and of the attitude, whereas in Akella (2001) and Caccavale and Villani (1999), building on the general results developed in Battilotti (1996) and Lizarralde and Wen (1996) the same problems have been solved using dynamic output feedback control laws. A considerable amount of work has been dedicated in recent years to the problems of analysis and design of momentum bias control laws in the linear case, i.e., control laws for nominal operation of a satellite near its equilibrium attitude. In particular, nominal and robust stability and performance have been studied, using either tools from periodic control theory exploiting the (quasi) periodic behaviour of the system near an equilibrium (see, e.g., Pittelkau (1993), Wisniewski and Markley (1999), Lovera, De Marchi, and Bittanti (2002) and Psiaki (2001)) or other techniques aiming at developing suitable time-varying controllers Steyn (1994) and Curti and Diani (1999). On the other hand, very little attention has been dedicated to global formulations of the momentum bias spacecraft attitude control schemes for various wheels configuration in geostationary orbit especially from practical point of view. Control implementation from a practical point of view have an engineering advantages with reference to different wheels configuration implemented for momentum bias spacecrafts, as demonstrated by the increasing number of inventions and patents in applications of this approach to attitude control schemes for many geostationary communication satellites. In the light of the above discussion, Specific examples of the prior control schemes relating to attitude control of spacecraft are important for reviews as a complete understanding of the investigation in this thesis.
Ref [12,13] each disclose a satellite attitude control system which utilizes reaction wheels aligned, respectively, with the pitch, roll, and yaw axes of the satellite(three wheels). The system is usable in the absence of any inertial yaw attitude reference, such as a gyroscope, and in the absence of a pitch bias momentum. Both the roll-yaw rigid body dynamics and the roll-yaw orbit kinematics are modeled. Pitch and roll attitude control are conventional. The model receives inputs from a roll sensor, and rolls and yaw torques from reaction wheel monitors. The model produces estimated yaw which controls the spacecraft yaw attitude. The model further produces estimates of the constant component of the disturbance torques for compensation thereof. Ref [4, 5] discloses a spacecraft attitude control system using one or more reaction wheels, the speed of which from time to time lie near and pass through zero angular velocity. When operated for extended periods of time at low speeds, the lubrication films are not distributed uniformly on the wheel bearings, leading to reduced lifetime. Reliability is maintained by a threshold comparator coupled to compare wheel speed with a lower limit value, for operating a torque associated with the spacecraft body when the wheel speed drops below the lower limit, in a manner
which
tends
to
raise
the
wheel
speed.
Ref [6, 7] discloses a spacecraft attitude control system which uses one or more momentum or reaction wheels. Wheel bearing viscous (velocity-dependent) friction reduces the actual torque 14
imparted to the spacecraft in response to a torque command signal. Friction compensation is provided by applying the torque command signal to a model of an ideal, friction-free wheel, and calculating the speed which the ideal wheel achieves in response to the torque command. An error signal is generated from the difference between the ideal wheel speed and the actual wheel speed. The error signal is summed with the torque command signal to produce the wheel drive signal. This results in a closed-loop feedback system in which the actual wheel speed tends toward the ideal wheel speed, thereby causing a torque on the spacecraft which is substantially equal to that commanded. Ref [7, 9] discloses an attitude control and stabilization system for momentum biased satellites utilizing a pair of contra-rotating flywheels mounted parallel to its yaw axis. The speed of one of the wheels of the pair is controlled by sensing and correcting the roll axis errors of the satellite. Ref [8] discloses that rolls and yaw attitude errors introduced by orbit inclination deviations from the nominal orbit plane are minimized by sinusoidally varying the momentum produced by a transverse wheel mounted on the spacecraft. The wheel is mounted on the spacecraft such that its axis is parallel to the spacecraft's yaw axis. Sinusoidal variation of wheel momentum is obtained by sinusoidally varying wheel speed in response to a sine wave signal periodically updated from an earth station. In response to the sinusoidal variation of transverse wheel momentum, the spacecraft is rolled to minimize thereby the roll error introduced by the orbit deviation from the nominal orbit plane. Yaw error is minimized by providing sufficient transverse wheel momentum so as to maintain the total spacecraft momentum vector perpendicular to the nominal orbit plane. Refs [10] disclose control of an active three-axis multiple-wheel attitude control system which is provided with a bias momentum to achieve stiffness about the pitch axis (one momentum wheel and two reaction wheels). With this stiffness, the system overcomes the inherent inability of a zero momentum system employing the sun as its yaw reference to be provided with this yaw reference during the periods encompassing satellite high noon and midnight (i.e., approximate co- alignment of the sun line and the local vertical). The satellite attitude is normally maintained by angular momentum exchange developed by three or more reaction wheels positioned on the satellite. The control mechanism is arranged to provide net angular momentum in the pitch axis wheels to achieve the momentum bias or stiffness along the pitch axis, whereas the time average of the angular momentum about the roll and yaw axes is substantially zero. Ref [8] the orientation maneuver of a bias momentum stabilized spacecraft is achieved without the need of attitude determination by sensors and gyros and the like and without the conventional spin-down and spin-up procedures. The spacecraft is provided with a momentum wheel mounted in perpendicular relation to the axis of maximum moment of inertia. The wheel is initially deenergized during the time the spacecraft is initially launched from the ground launching platform until it is desired to orient the spacecraft in the final orbit. The momentum wheel, when energized from zero rotation to increasing rotation speeds, causes the rotation of the spacecraft from spinning about the maximum moment of inertia axis to an axis parallel to the momentum wheel axis with the final convergence of the wheel axis to the momentum vector being effected 15
by
energy
dissipation
in
a
nutation
damper.
Ref [17,18] discloses a control system for reducing undesirable motion in a spinning, orbiting satellite body. A satellite having an angular momentum stabilizing system aboard is compensated in a manner to substantially reduce undesired motion about an axis, which may be the spin axis. A motion sensor is placed aboard the satellite to derive a signal proportional to the direction and magnitude of the undesired motion. This signal is used to rotate a flywheel, whose axis of rotation is perpendicular to the spin axis of the satellite. The flywheel is rotated either clockwise or counterclockwise depending on the signal from the motion sensor, causing the flywheel to produce an equal and opposite torque to the disturbing torque about the spacecraft's spin axis. This action damps out the undesired motion by counteracting the undesired torque. It was in light of the prior arts as just described above that the present various control schemes was conceived and has now been reduced to specific wheels configurations for practical use. For this thesis report, reference will be made with respect to popular forms of wheels configuration implemented for momentum biased space vehicles in GEO [15]. 1. In one wheel configuration as shown in figure.1.7, a single wheel rotated about an axis oriented along the pitch axis of the spacecraft operating at an average non-zero, or bias, momentum so as to maintain the yaw axis of the main body portion of the spacecraft aligned to the local vertical while the inertial stability of the wheel maintains the pitch axis aligned to the orbit normal such a satellite control system is pitch bias momentum controlled satellite and is embodied in the RCA SATCOM I and RCA SATCOM II satellites as well as several other satellites now in orbit. A significant feature of the bias momentum system is that the angular position of the body needs be measured only about the roll and pitch axes, negating the more complex measurement of yaw. OFFSET THRUSTERS
z-axis α
YAW
OFFSET ANGLE
2
ROLL x-axis
PITCH y-axis
Figure.1.7: Single wheel Pitch axis-thrusters configuration. 2. In another configuration popularly called, two symmetrically inclined momentum wheels in a V configuration.. This configuration provides a system which is relatively simple, light weight and provides satisfactory attitude control about all the three axes. Figure 1.8 is the diagram showing the arrangement of the two wheels configuration constituting the actuators of the 16
vehicle respectively to roll, pitch and yaw axes of the vehicle according to two wheels momentum biased configuration.
OFFSET THRUSTERS
α
z-axis
2
OFFSET ANGLE
YAW 3 SKEWED ANGLE
ROLL PITCH
x-axis
y-axis
Figure.1.8: Two symmetrically inclined momentum wheels in a V configuration. 3. Another prior attitude control system[10] include three reaction wheels respectively associated with pitch, roll and yaw axes and associated sensors popularly called zero momentum bias configuration. Its payload axis pointing towards local vertical is achieved by employing three reaction wheels, one for each axis, yaw, roll, and pitch in a zero momentum configuration. The Earth sensors provide continuous information of roll and pitch errors whereas the yaw information is obtained from a gyro based reference. This gyro is updated twice an orbit near the poles, taking yaw information from the Sun. The controller is a pulse width pulse frequency modulator (PWPFM) employing pseudo rate damping techniques to obtain a highly stable configuration as demanded by the remote sensing payload. z-axis 3
YAW 1
2
ROLL x-axis
PITCH y-axis
Figure 1.9: Three reaction wheels configuration. 4. Another prior attitude control system include one pitch momentum wheel and two roll/yaw reaction wheels respectively associated with pitch, roll and yaw axes and associated sensors.
17
z-axis 3
YAW 1 2
ROLL x-axis
PITCH y-axis
Figure1.10: Two reaction wheels/one momentum wheel configuration. All four types of configuration for attitude control systems provides conventional wheel control for the pitch error by either modulating a single reaction wheel, a body fixed momentum wheel or skewed body fixed momentum wheels ,The control law along the pitch axis are always the same for all the four configurations[21].The main different between the various configurations however lies in the approach to attitude determination and control law for roll and yaw attitude which then leads to different in pointing accuracies and other performance efficiencies[21].
1.6. Contribution of the thesis. Various great control schemes [4,6,8,11,23,12,22] has being implemented practically for various wheels configuration of momentum bias spacecrafts, however the most popular wheels configurations widely use for communication satellite in GEO has not being investigated in a single project. This thesis will investigate their comparative performances. In addition, Despite various space vehicle using the two wheels configurations including Nigeria satellite, the skewed angles for the two wheels placement varies drastically for different satellites for examples, geostationary operational environmental satellite (GOES N-P) has its two momentum wheels skewed at an angle of ±1.66° to the pitch axis [10], Nigcomsat-1 momentum wheels was skewed at angle of ±20°, Anik E2 has its skewed wheels 2.5° [9] etc. It is therefore the contribution of this thesis to investigate this skewed angle in this type of widely accepted fully stabilized momentum biased spacecraft configuration. Lastly, This research is also expected to provide foundation for future momentum bias vehicles control experiment and simulation at National space Research and Development Agency, Nigeria and Nigeria communication satellite Nig Ltd, Nigeria using the simulators developed in this thesis.
1.7. Thesis Outline. Chapter 1 serves as introduction and description of momentum bias spacecrafts with different wheels configurations as will be implemented in this thesis. Various literatures in terms of 18
invention based on these vehicle configurations were reviewed and general description and architecture of first Nigeria communication satellite explained. Chapter 2 was developed based on understanding of dynamics of any space vehicles in two stages. The first stage is to express the dynamics of rotational motion of any space vehicle in term of Dynamics and kinematics. The second stage formulates a non linear model that is suitable for a momentum biased space vehicle using a general equation of spacecraft motions. The mathematical model was formulated on the basis of on orbit configuration of three axes wheels control spacecraft with its auxiliary control system hardware. Basic physical notions such as angular kinetic energy, angular momentum, and moment about the mass center that are used in the derivation of the fundamental laws of angular motion, which are based on Euler's moment equations will be stated here. Chapter 3 consists of the linearized mathematical model of momentum bias spacecraft dynamics, kinematics, controller’s dynamic and their open loop analysis. Appropriate equations of motion for pitch, roll and yaw attitude was presented as well. The analysis was carried out for the Nigeria communication satellite (Nigcomsat-1) pitch, roll and yaw attitude controls where the pitch attitude was treated separately from both roll and yaw attitudes. In chapter 4 and 5, the spacecraft control architectures was implemented in the MATLAB® and Simulink® where a phase lead compensator control loop system was used to damp the three axes and compensate for external disturbances. The performances of the control architectures are presented and discussed based on the type of the wheels configuration. The investigation of disparity in the skewed angles for different space vehicles in the same orbit are also carried out for various satellites such as geostationary operational environmental satellite (±1.66°) [10], Nigcomsat-1 (±20°), Anik E2 (2.5°) [9] etc. In chapter 6,the discussion of comparative results based on various wheels configuration performances and optimization of skewed angles are summarized to give a conclusion about the scope and achievements of this thesis.
19
CHAPTER 2 2. RIGID BODY DYNAMICS. A critical step in any control design is the development of a quantitative mathematical model of the system and in many situations; the key to effective control system design is the ability to develop simple models that captures the essential characteristics of the system. The primary concern here is to develop a simple yet accurate model of the space vehicle. This will be followed by linearization, transfer function model and formulation of control solution to the system. One of the most difficult problems encounter during any control synthesis is formulation of plant equations. While many papers have described and used various approaches to derive the equations, the end result is too theoretical, intimidating, unrealistic and complex to give significant meaning. Extensive research and analysis of various textbooks, papers and journals leads to this simplified step by step approach implemented in this mathematical formulation. The approach adopted is to develop an understanding of dynamics of any space vehicles in two stages. The first stage is to express the dynamics of both translation and rotational motion of any space vehicle in term of linear and angular momentum. The momentum is suitable in these aspects because it is very easy to determine the consequence of forces or moments on the vehicle and their corresponding linear and angular velocity. The linear and angular velocity can then both be manipulated to give linear and rotational velocity, position or acceleration of the space vehicle in desire reference frames using any suitable transformation matrices. The second stage will be to formulate a model that is suitable for a momentum biased space vehicle using general equation as baseline. The mathematical model will be formulated on the basis of on orbit configuration of Nigeria communication satellite and its auxiliary control system hardware. The motion of any space vehicles can be generally described by translation of some reference point plus a rotation about some axis through the reference point. It is often necessary to consider the translational motion and rotational motion as general problem of any spinning space vehicle under external forces, thus, the rigid body in space is a dynamic system with six degree of freedom. The preliminary assumptions are always that, the space vehicle is considered as a rigid body and that its motion is relative to a Newtonian inertial reference frame. This paper will also respect the assumption that the mass variation of the rigid body is considered small enough to be negligible for a specify period of time based on the activities of the mass ejected during this period. The Understanding of the natural motion of an orbiting space vehicle was necessary before one could deal with the control of the vehicle in orbits. Basic physical notions such as angular kinetic energy, angular momentum, and moment about the mass center that are used in the derivation of the fundamental laws of angular motion, which are based on Euler's moment equations will be stated here.
20
2.1. Angular momentum of Rigid Body Determination of Velocity and Acceleration The study of the kinematics and dynamics of motion requires a reference coordinate frame in which position and acceleration of a point mass can be specified. A non rotating and non accelerating reference frame is known as an inertial frame in which laws of mechanics are valid and can most be conveniently expressed. If an orthogonal inertial frame is used to donate the vector position r of a point, then its velocity will be denoted as v = dr / dt an acceleration as a = dv / dt .If the position r is measured in a frame which has an angular velocity ω and translational velocity v0 , then the absolute or inertial velocity of the point is, vinertial = r ' + ω × r + v0
,
(2.1)
Where r ' is the translational velocity of the point measured in the rotating frame. The acceleration of the point with respect to inertial space is , a=
dv , dt
= r' + ω × r ' + ω × r + ω × (r ' + ω × r ) + v0 , = r' + 2ω × r ' + ω × r + ω × (ω × r ) + a0
.
(2.2)
The second and fourth terms in equation (2.2) are the Coriolis and centripetal acceleration respectively. The ω × r is the tangential acceleration. The Coriolis acceleration 2ω × r ' term exist only when there is relative translational velocity r ' in a rotating frame. The Coriolis acceleration causes a particle moving along the Earth’s surface (in a target plane) to drift to the right in the Northern Hemisphere and to the left in the Southern Hemisphere [4]. Angular Momentum and the Inertia Matrix of Rigid body. The references [1] and [2] was used in this general equations and the formulation is applicable to moment of momentum of a rigid body about a fixed point or the moving center of mass .The rotational dynamics of the vehicle will be described in term of angular momentum. In order to outline the general equations, the vehicle described as rigid body is defined as a set of point fixed axes x, y, z rotating with angular velocity ω , and with the origin coinciding with center of mass. If the origin of the body axes x, y, z coincides with the reference frame at any point, the velocity of any point i of the body is then defined by equation (1). The moment of momentum (h0 ) about the origin of the x, y, z ,system is, h0 = ∑ ri × mi (r ' + v0 + ω × ri )
,
(2.3)
i
21
where mi is mass of the ith body v0 is the translational velocity of the body and ri is the position vector of a small (infinitesimal) mass dmi relative to reference frame. Generally r is position vector x, y, z from the reference frame to any point on the line of action of the linear momentum vector and v is the velocity relative to the reference frame. For the orbit of a space vehicle whose mass is M s , it is useful to take the center of gravitational attraction as the reference point. In this reference frame, the moment of momentum will be normal to orbital plane. If the reference point is stationary, v0 = 0 and the term r ' is zero for a rigid body. Thus if reference frame or center of mass is fixed, the angular momentum under these reasonable assumption is given by the following integral, h0 = ∫ r × (ω × r )dm
,
(2.4)
Using the triple vector product identity A × ( B × C ) = ( A.C ) B − ( A..B)C , the equation (2.4) gives h0 = ∫ [(r..r )ω − (r..ω )r ]dm = ∫ [(r 2 − (r.r )]dm.ω .If I (inertia-dyadic) replaces ∫ [(r 2 − (r.r )]dm.ω ,
then h0 = Iω . The integration of equation (4) over the entire body is expected to be close to this result and can be determined as follows, The cross product ω × r = (ω y z − ω z y )i + (ω z x − ω x z ) j + (ω x y − ω y x)k [2].The cross product r × (ω × r )dm which represent the moment about x, y, z axes of the momentum vectors of the body is defines as equation (5),
[
]
r × (ω × r )dm = i ω x ( y 2 + z 2 ) − ω y ( xy ) − ω z ( xz ) dm ,
[
]
[
]
+ j − ω x ( xy ) + ω y ( x 2 + z 2 ) − ω z ( yz ) dm , + k − ω x ( xz ) − ω y ( yz ) + ω z ( x 2 + y 2 ) dm
.
(2.5)
Integrating equation (2.5) over the body, the x, y, z components of the moment of momentum of the body is , h0 = hx i + h y j + hz k .
(2.6)
In general, the total angular momentum for a momentum biased vehicle in circular orbit as will be implemented in this thesis will include additional component hwheel .where hwheel is the wheel’s angular momentum and will be described along the axis in which it was placed relative to the body fixed frame. The moment of inertia of the body about x, y, z can also be described as,
22
I x = ∫ ( y 2 + z 2 )dm, I y = ∫ ( x 2 + z 2 )dm, I z = ∫ ( x 2 + y 2 )dm, and the products of inertia as
I xy = ∫ xydm, I xz = ∫ xzdm, I yz = ∫ yzdm .Substituting this in equation (2.5), The moment of momentum components along the x, y, z axes becomes, hx = Iω x = I x ω x − I xy ω y − I xz ω z ,
(2.7)
h y = Iω y = − I xy ω x + I y ω y − I yz ω z ,
(2.8)
hz = Iω z = − I xz ω x − I yz ω y + I z ω z ,
(2.9)
The equations (2.7, 2.8,2. 9) can be expressed in matrix form as hx I x h = I y xy hz I xz
Or
I xy Iy I yz
I xz ω x I yz ω y I z ω z
,
(2.10)
h0 = Iω ,
[
Where h0 = hx
hy
hz
]
T
[
, ω = ωx ω y
ωz ]
T
Ix and I = I yx I zx
I xy Iy I zy
I xz I yz and I is called the inertia I z
matrix of a rigid body fixed reference frame with its origin as center of mass.
2.2. Attitude dynamics and Kinematic equations. 2.2.1 Euler’s Moment Equation The Euler’s dynamic equation is the equivalent of Newton’s law of motion for rotation about the center of mass. It is of the form of equation (2.11). The moment of momentum of a particle can change if the forces on it have a moment M about the origin. This relationship can be expressed in the form of the Newtonian equation d (h0 ) =M dt
,
(2.11)
And its time derivative is with respect to an inertial reference frame. This is also true for equivalent particle of a space vehicle. The forces on space vehicle are conventionally separated into two categories :( 1) Dominant Central gravitational force and (2) additional Perturbative forces. The gravitational forces acts towards the center of an inertially fixed reference frame(Earth) and contributes nothing towards the moment M .The dominant contributors to moment of inertia in space vehicles is therefore external perturbative forces. The consequence of the perturbative forces when impacted on the vehicle can change either the magnitude of the 23
moment of momentum or the direction of the moment of momentum depending on the direction of application [2 pages 55].The consequence is described in two folds as below. 1.The component of M that is normal to the orbit’s plane i.e. in the same direction as moment of momentum will change the magnitude of moment of momentum but not its direction, so the orbit will remains in the same plane, but will change its shape(eccentricity or semi major axis). 2. The component of M in the orbit plane at right angle to the moment momentum will affects the direction of the normal to the plane, turning it towards M .this represent a rotation of the M orbital plane. The rate of turn is given by . h0 The equation (2.11) shows that the moment about the mass center is equal to the time derivative of the moment about this point. With respect to a body fixed reference frame rotating with angular velocity, The Euler’s equation can be defined as equation (2.12). We can substitute equation (2.6) into equation (2.11) and then differentiate, noting that i, j , k rotate with the body axes.
[ ]
M = h0 + ω × h0
,
(2.12)
= (hx i + h y j + hz k ) + ω × h0
.
(2.13)
The cross product ω × h0 is the rotation of the vector hx i, h y j , hz k due to ω x , ω y , ω z Adding the vectors along x, y, z directions, the equation (2.13) becomes, M = M xi + M y j + M z k
,
(2.14)
The components M x , M y , M z defined by equations (2.15, 2.16, 2.17) is known as Euler’s moment equations that describe attitude motion of any space vehicles. M x = hx + ω y hz − ω z h y
,
(2.15)
M y = h y + ω z hx − ω x hz
,
(2.16)
M z = hz + ω x h y − ω y hx
.
(2.17)
On the basis of equations (2.15, 2.16, 2.17), the joint attitude dynamic equation (space vehicle + momentum wheels actuators) of motion for a non spinning body with a rotating elements (momentum wheels) can be derived as explained below. For notation reasons T will substitute M
and breaking down the external torque T into two
principal parts: the Tc control moment to be used for controlling the attitude motion of the space vehicle i.e. thrusters or momentum wheel and Td ,those moments due to different disturbing 24
space environment perturbations. The total torque vector is thus T = Tc + Td .For a momentum exchange device, The momentum of the entire system is always divided between the momentum hx of the vehicle with reference to its body hb = hy and the momentum of the moment exchange h z devices hwheel
hwx = hwy ,This terms is vector component of the sum of the angular momentum of h wz
all the momentum exchange device. With this addition, the general equation of motion of a space vehicle equipped with rotating device become Tdx = hx + hwx + (ω y hz − ω z hy ) + (ω y hwz − ω z hwy ) ,
(2.18)
Tdy = hy + hwy + (ω z hx − ω x hz ) + (ω z hwx − ω x hwz ) ,
(2.19)
Tdz = hz + hwz + (ω x hy − ω y hx ) + (ω x hwy − ω y hwx ) .
(2.20)
Here hx = I x ω x , hy = I yω y ,Since the plant is already equipped with the wheels though in open loop configuration ,so Tcx , Tcy , Tcz are already part of right hand sides of the equations. These are three coupled non linear ordinary differential equations for state variable ω x , ω y , ω z of a rigid body equipped with rotating elements.
2.2.2 Gravitational Force. The spherical frame is normal used in describing the gravitational forces acting naturally on any satellite in its reference frame. The force induced on by the gravitational field of the Earth is the primary force acting on the orbiting satellite. The gravitational force can therefore not be ignore in orbital dynamics. The Earth is non spherical but assumption of uniformity is normally made µm during mathematical formulation. Gravitational force can be expressed as Fg = 2 . Where r 14 2 µ = 3.986032 × 10 m / s for Earth and m is the mass of the space vehicle and r is the radius of the orbit.
2.3. Attitude Description. There are multiple ways to describe the spacecraft’s attitude once it has been determined by the sensors. For example, Euler angles, sequential rotations about body axes, direction cosines, and the Eulerian parameters (quaternion) are all ways to describe the attitude of a spacecraft. Each description has its own limitations and advantages; therefore control methods used to maintain an attitude rely on the method used to describe the attitude. 25
Euler angles are the most commonly used set of attitude parameters. Euler angles consist of three successive rotation angles that describe the orientation of a spacecraft. The rotations may occur about any of three orthogonal axes, but there cannot be two rotations about the same axis in a row. The order of the rotations is very important to the orientation, and there are twelve possible sets of Euler angles. The most common set is the (3-2-1) Euler angles, which correspond to the yaw-pitch-roll commonly used with aircraft. The (3-2-1) set is considered an asymmetric set since there are no repeated rotations, and it has singularities whenever the second angle has a value of ±90o. The symmetric sets have a repeated rotation, such as the (3-1-3) set, and experience singularities whenever the second angle has a value of 0o or 180o. Clearly, these singularities limit the usefulness of Euler angle descriptions to small rotations. One of the main advantages of Euler angles is the ability to clearly visualize the orientation of the vehicle as it undergoes small rotations. To determine the Euler angles, each single axis rotation is calculated in the matrices.
Table 2.1: Comparison for attitude descriptions. z Z1
θ y1
ф y x
ψ x1
Figure 2.1: One scheme of rotation of Euler angles.
26
If the orientation of a rigid body fixed axes e1 , e2 and e3 associated with unit vectors eˆ1 , eˆ2 and eˆ3 , respectively can be specified in several ways relative a reference frames E1 , E 2 and E3 associated with unit vectors Eˆ 1 , Eˆ 2 and Eˆ 3 ,then each rotation may be expressed in term of the orthogonal rotation matrices as follows: eˆ '1 eˆ1 ' eˆ 2 = R(ψ ) eˆ2 , eˆ ' eˆ 3 3
(2.21)
eˆ '1 eˆ1'' '' eˆ2 = R(θ ) eˆ ' 2 , eˆ ' eˆ '' 3 3
(2.22)
Eˆ1 e''1 Eˆ 2 = R(φ ) e'' 2 . ˆ e'' 3 E3
(2.23)
By combining the preceding sequence we obtained,
Eˆ1 eˆ1 eˆ2 = R(φ ) R(θ ) R(ψ ) Eˆ 2 , ˆ eˆ 3 E3
(2.24)
where
0 1 R(φ ) = 0 cos φ 0 − sin φ
0 cosθ sin φ , R(θ ) = 0 sin θ cos φ
0 − sin θ cosψ 1 0 , R(ψ ) = − sinψ 0 0 cos φ
sinψ cosψ 0
0 0 1
where R(ψ ) R(θ ) R(φ ) is known as rotation matrix of the successive body axis rotation and
φ ,θ ,ψ represent yaw, pitch and roll attitude angles defined about the orbiting reference frame and are the so called Euler angles. After multiplying the matrices together, we can write the transformation to be defined as R(φθψ ) , which takes vectors in the reference frame to vectors in the body frame defines as, cosθ cosψ R(φθψ ) = sin φ sin θ cosψ − cos φ sinψ cos φ sin θ cosψ + sin φ sinψ
cosθ sinψ sin φ sin θ cosψ + cos φ cosψ cos φ sin θ sinψ − sin φ cosψ
27
− sin θ sin φ cosθ cos φ cosθ
For
a
small
infinitesimal
Euler
angles φ ,θ ,ψ ,
i.e.
cos φ ≈ cosθ ≈ cosψ ≈ 1
and
sin φ ≈ φ , sin θ ≈ θ , sinψ ≈ ψ ,the above equations become,
ψ
1 R(φθψ ) ≈ −ψ θ
1 −φ
−θ φ , 1
(2.25)
The angular velocity components ω x , ω y , ω z in vehicle body coordinates for a small Euler angle are also given by, ω x φ − ω0ψ ω y ≈ θ − ω0 , ω ψ + φω 0 z
(2.26)
From equation (2.26), it is easily followed that ,
ω x = φ − ω0ψ , ω y = θ,
,
(2.27)
ω z = ψ + φω0 , Here ω0 is the orbital rate (velocity) or frequency of the body and φ,θ,ψ , φ,θ,ψ are the derivatives of the Euler angles. The details of this formulation on the basis of small Euler angles can be found in [4].
ROLL
PITCH
DIRECTION OF FLIGHT
DOWN
YAW
Figure.2.2: Description of Euler angles.
28
2.3.1 Gravity Gradient Moments. The non spherical shape of the Earth contributes gravitational moment to any orbiting space vehicle. An earth pointing spacecraft, with the y axis along orbit normal with consideration of gravitational torques can be modeled the following dynamical equation
G + T = h + h × ω ,
(2.28)
where G the gravity gradient moment and T is is torques due to other external perturbation most especially the solar radiation moments. This equation shows the influence of gravity gradient in the attitude motion of space vehicle.
Figure 2.3: Non spherical shape of the Earth. As can be seen in figure 2.3, besides being flat at the top and bottom, the Earth has a bulge on Equator. It is important to take this effect into account when determining dynamic equations for Geosynchronous Earth Orbits [5]. A full development of the gravitational moment can be found [8] and an outline of the development can be found in ref[2].The two references give final result of the gravity gradient moment in X , Y and Z coordinate as, Gx =
3µ (I z − I y )sin(2φ ) cos2 (θ ) , 2r 3
(2.29)
Gy =
3µ (I z − I x )sin(2θ ) cos 2 (φ ) , 3 2r
(2.30)
Gz =
3µ (I x − I y )sin(2θ ) cos2 (φ ) . 3 2r
(2.31)
29
The gravitational moment is expressed in term of Euler angles. The equations can be linearized for a body in circular orbit using small angle approximations for φ and θ .The linearization result according to [3] gives, Gx = −3ω0 ( I y − I z )φ ,
(2.32)
G y = 3ω0 ( I z − I x )θ ,
(2.33)
Gz = 0 ,
(2.34)
Where ω0 =
µ r3
.
2.4. Space Environments Disturbance torques. Attitude prediction requires a model of the environmental disturbance torques acting on the spacecraft. To numerically integrate Euler's equations, the torque must be modeled as a function of time and the spacecraft's position and attitude. Assessment of expected disturbance torques is an essential part of rigorous spacecraft attitude control design. The principal source of perturbation to space vehicles are solar radiation pressure, gravity gradient, aerodynamics, magnetic and meteoroid or manmade debris [12].Other perturbing force which may be important include the effects associated with internal moving parts, thrust misalignments, propellants sloshing, thermal emissivity, propellant leakage, outgassing etc. Solar radiation is generally a significant source of attitude and trajectory errors for high altitude above 1000km.Gravity gradients which result from the extended dimension of space vehicle may either cause disturbing or perturbing torques or provide restoring torques when the effect is used for attitude control. Gravity gradients as well as magnetic torques caused by the interaction of vehicle magnetic materials with the planetary magnetic field are most important at low altitude [12]. Similarly, aerodynamic effects are significant only below 500kms altitude and are generally negligible above 1000kms.Aerodynamic torques acting on vehicle are functions of vehicle geometry and altitude. The torques arising from internal moving parts such as the rotating wheels, circulating fluids, scanning devices etc must be included in the general equation of motion of the vehicle. Thrust misalignment torques is caused by the thrust line of action not passing through center of mass. Torques from propellant leakage and or outgassing of vehicle materials are of similar nature. Finally, metric and manmade debris environment must also be considered in certain cases as a possible source of external torques acting on vehicle. The variation as a function of altitude of the typical disturbances torques acting on space vehicle are shown in figure 2.4 below,
30
5
10
Altitude (Km)
Radiation Pressure
Gravity Gradient
4
10
Magnetic Effects
3
10
Aerodynamic Effects
2
10 -6 10
-4
-5
-3
10
10
-2
10
10
Figure 2.4 Perturbation torques as a function of altitude. The figure above shows that solar radiation pressure and gravity gradient torques has dominant effect in geostationary orbit on the attitude a satellite [8].On this basis the two will be considered as the major perturbation torques in this thesis. The bias momentum provides gyroscopic stiffness to environmental disturbances, primarily to the solar radiation torques [42] while the gravity torques can be coupled with the mathematical model of the attitude motion. The model of the solar radiation torques as a function of time will be defined as follows; Considering geostationary orbit, typical significant disturbance is the solar radiation pressure about the body axes as shown in figure 2.4 and is expressed as, Tdx ≈ a cos S ' + b sin S ' cos µ ,
(2.35)
Tdy ≈ c sin S ' cos µ ,
(2.36)
Tdz ≈ d cos S ' − e sin S ' cos µ ,
Where
S ' is
the
angle
(2.37) between
µ = ωot , 66.55 ≤ S ≤ 360 , 0 ≤ µ ≤ 360 0
'
0
0
the 0
sun
line
and
the
Earth’s
polar
axis.
and typical values for the coefficient a, b, c, d , e are
on the order of 10−6 Newton − meter [5].
31
North pole
S
(to SUN) S1
e1
e3 TO THE DIRECTION OF VERNAL EQUINOX
s/c
µ
e2 66.55⁰ ≤ S1 ≤ 113 0⁰ ≤ µ ≤ 360
Figure 2.5: Sun, Earth and spacecraft configuration in geosynchronous orbit.
32
CHAPTER 3 3. MATHEMATICAL MODELS FOR THREE AXIS ACTIVE CONTROL SYSTEMS. The dynamics of an earth pointing spacecraft will be described by the linearized Euler equation of motion with respect to the Local-Vertical-Local-Horizontal coordinate frame. Various assumptions as listed below are used for modeling of momentum bias space vehicle. 1. The spacecraft can be treated as a rigid body with a constant inertia matrix and center-ofmass that doesn’t move. This means that the rotation of the solar panels has no effect on centerof-mass or inertia. 2. the space vehicle is equipped with sensors that measure roll, pitch and yaw. Roll is the angle about the x-axis (the axis going from west to east). Pitch is the axis going from north to south (y) and yaw is the axis pointing at the earth (z). 3. There are no moving parts on the spacecraft. The rotation of the solar panels with respect to the core of the spacecraft will be ignored as discussed above. 4. Momentum wheels, Reaction wheels or thrusters can be used as actuators depending on the wheels configurations. 5. The solar radiation pressure torques is considered as the major disturbance torques. The solution of most space vehicle dynamics and control problems requires a consideration of gravitational forces and moments. When a body is in a uniform gravitational field, its center of mass becomes the center of gravity and the gravitational torques about its center of mass is zero. The gravitational field is not uniform over a body in space, however, a gravitational torques exists about the body’s center of mass. We model the equation of attitude motion of rigid body in circular orbit in local vertical and local horizontal (LVLH) reference frame [19].
3.1. Linearization of the vehicle equation. The schematic of various wheels configuration is presented in figure 1.2.Numerous assumptions that are manageable for design process was adopted in various literatures and will be subsequently adopted to make reasonable mathematical models suitable for control design synthesis of this thesis. Several papers [2, 4, 5, 8, and 14] exclude the equation of motion for translational motion from rotational motion for the control and momentum management control design. The models used in this paper will be formulated based on the basis of the general equations of motion of a space vehicle equipped with rotating device i.e. equations (2.18,2.19, 2.20), the linearized Euler angle equations (2.26) and gravitational moments equation (2.32,2.33,2.34).The time rate of change of the angular momentum of a body about its center of mass is equal to the sum of the external torques acting on the body. The main coupled external torques acting on the vehicle is due to gravity. This is due to the variation in gravity over the vehicle which generate moments on the vehicle (gravity gradient torque).The gravity gradient 33
torques acting on the vehicle is defined by equation (2.32,2.33, 2.34).Torques caused by gravitational of other planetary bodies, magnetic fields, solar radiation pressure and winds and other less significant phenomenon will be ignored in the dynamics model and will be view as disturbances. We consider a generic model of a spacecraft equipped with three wheels aligned along roll, pitch and yaw axes. The general models that will be used on the basis of the various wheels configuration for control design are; The general equation of motion of a wheels + space vehicle i.e. Space vehicle equipped with rotating device in term of ω and h in the entire three axes; Tdx + Gx = hx + hwx + (ω y hz − ω z hy ) + (ω y hwz − ω z hwy ) = I xω x + hwx + ( I z − I y )ω yω z + ω y hwz − ω z hwy ,
(3.1)
Tdy + G y = hy + hwy + (ω z hx − ω x hz ) + (ω z hwx − ω x hwz ) = I yω y + hwy + ( I x − I z )ω zω x + ω z hwx − ω x hwz ,
(3.2)
Tdz + Gz = hz + hwz + (ω x hy − ω y hx ) + (ω x hwy − ω y hwx ) = I zω z + hwz + ( I y − I x )ω xω y + ω x hwy − ω y hwx ,
(3.3)
1. The coupled gravitational moment;(The external torques acting on the vehicle)
Gx = −3ω02 ( I y − I z )φ ,
(3.4)
G y = 3ω02 ( I z − I x )θ ,
(3.5)
Gz = 0 ,
(3.6)
2. The angular velocity components ω x , ω y , ω z in vehicle body coordinates for a small Euler angle given by [4]
ω x = φ − ω0ψ ω y = θ − ω0 ,
(3.7)
ω z = ψ + ω0φ Equations above hold for small disturbances and are appropriate for a rigid spacecraft with flexible solar panels and antennas and, possibly, three momentum wheels aligned along each of the roll, pitch and yaw axis within the rigid structure, as shown in Figure 3.1 .For the other wheels configuration, the number of wheels can be reduced in the equations accordingly and the thrusters substituted as control torques as will be implemented during attitude control design in this thesis, 34
Figure 3.1: Model of a three wheels Momentum bias spacecraft [14].
For the mathematical model in this problem for geostationary orbit, the gravitational moment is considered and the torques Tсx , Tсy and Tсz from the wheels are all modeled together with the space vehicle attitude dynamics equations(2.15,2.16,2.17) while the solar radiation torques will be considered as the major external disturbance components Tdx , Tdy and Tdz that acts on pitch, roll and yaw respectively. On this basis, linearized equations are formulated from equation (3.1, 3.2, 3.3), equations (3.4) and equations (3.5, 3.6, 3.7) as follows: Tdx + Gx = I xω x + hwx + ( I z − I y )ω yω z + ω y hwz − ω z hwy ,
(3.8)
Tdy + G y = I yω y + hwy + ( I x − I z )ω zω x + ω z hwx − ω x hwz ,
(3.9)
Tdz + Gz = I zω z + hwz + ( I y − I x )ω xω y + ω x hwy − ω y hwx ,
(3.10)
Substituting equations (3.4-3.7) in the above equations, Tdx − 3ω02 ( I y − I z )φ = I xω x + hwx + ( I z − I y )(θ − ω0 )(ψ + ω0φ ) + (θ − ω0 )hwz − (ψ − ω0φ )hwy , Tdy + 3ω 02 ( I z − I x )θ = I yω y + hwy + ( I x − I z )(ψ + ω0φ )(φ − ω0φ ) + (ψ + ω0φ )hwx − (φ − ω0ψ )hwz T + 0 = I ω + h + ( I − I )(φ − ω ψ )(θ − ω ) + (φ − ω ψ )h − (θ − ω )h , dz
z
z
wz
y
x
0
0
0
wy
0
wx
where ω0 is the orbital rate, hwy 0 is the nominal bias momentum along the negative pitch axis, and
φ , θ and ψ are called the roll, pitch and yaw attitude angles of the vehicle relative to the Local vertical local horizontal reference frame. The equations of motion of the vehicle can be simplified further as: I xφ + [a + ω0 hwy ]φ + [b + hwy 0 ]ψ = Tdx + Tcx ,
(3.11)
I yθ + dθ = Tdy + Tcy ,
(3.12) 35
I zψ − [b + hwy 0 ]φ + [c + ω0 hwy 0 ]ψ = Tdz + Tcz
(3.13)
where, = a 4ω02 ( I y − I z ) b= −ω0 ( I x − I y + I z ) = c ω02 ( I y − I x ) = d 3ω02 ( I x − I z )
, φψ and ψθ has being dropped. Dropping this In the process of linearizing, a seculiar term φθ term simplifies the equation without loosing physical significance of the equations. Equations (3.11, 3.12, and 3.13) are the general equation for three axes stabilized geostationary vehicle in any wheels configuration in term of Euler angles. The dynamics of the system in any of the three axes is actively controlled, either by reaction wheel, momentum wheel or thrusters as tabulated below. The entire wheels configuration is assumed to use thrusters for momentum unloading.
3.2. Linear Stability Analysis. It is clearly shown in equation (3.11, 3.13) that the roll and yaw motions ( φ ,ψ ) are coupled and both decoupled from the pitch motion θ of equation (3.12) .To carry out the open loop linear stability analysis, the external control inputs are set as T= T= T= 0 so that only the dx dy dz momentum wheels torques are taken as inputs. In view of these the pitch axis equation was considered separately and subsequently coupled roll and yaw motions ( φ ,ψ ) Stability about the Y Body Axes Pitch control system architecture is simple because the pitch motion is decoupled from the coupled roll/yaw motion. Hence, the linearized Euler equation of motion for the pitch channel from Equation (3.12) can be explained as follows: Considering equation of pitch axis given by:
I yθ + 3ω02 ( I x − I z )θ = Tcy Taking Laplace transformation assuming zero initial conditions
I y s 2 Θ( s ) + 3ω02 ( I x − I z )Θ( s ) = Tcy ( s ) Where the capital letter denote Laplace transform variable of the pitch attitude angle as 36
Θ( s ) = L[θ (t )] ,
(3.14)
Θ( s ) 1 = 2 Tcy ( s ) I y s + 3ω02 ( I x − I z )
(3.15) The characteristics equation of the pitch axis is then defined as,
I y s 2 + 3ω02 ( I x − I z ) = 0 The three cases for stability analysis of pitch axis are defined as follows: Condition 1: If I x < I z then one of the characteristics root is positive real number i.e. has one unstable root. The pitch angle will diverge exponentially with time. Condition 2: If I x = I z which is the neutral case of stability. Condition 3:If I x > I z ,then the root of the characteristics equation are double pure imaginary numbers and is therefore oscillatory[13]. There is no damping factor in this second-order equation, so the harmonic motion will be undamped. It follows that for any initial condition or nonzero disturbance Tdy , the spacecraft will oscillate in a stable motion about the pitch axis with amplitude proportional to the initial condition and the level of the disturbance Tdy and inversely proportional to the difference between moment of inertia I x , I z about this axis. This means that the only way to limit the amplitude of oscillation is by choosing appropriate values for the satellite's moments of inertia. In the design of any momentum biased vehicle, it will be necessary to implement an active control system that will introduce damping into the pitch motion. Stability about the X and Z Body Axes For the roll/yaw coupled open loop stability analysis. The coupled mathematical models are defined by equations (3.11) and (3.13). Furthermore, for the spacecraft orbit under consideration in this thesis, the constant orbits a, b, c, d can be neglected compared to the relatively large value of pitch bias momentum, Model of roll-yaw attitude are then simplified as
Tcx = I xφ + ω0 hwy 0φ + hwy 0ψ ,
(3.16)
Tcz =I zψ − hwy 0φ + ω0 hwy 0ψ ,
(3.17)
Taking Laplace of both sides assuming zero initial conditions;
Tcx ( s )= ( I x s 2 + ω0 hwy 0 )Φ ( s ) + shwy 0 Ψ ( s ) Tcz ( s ) = − shwy 0 Φ ( s ) + ( s 2 I z + ω0 hwy 0 )Ψ ( s ) Where the capital letters denote Laplace transform variable of the roll/pitch attitude angle as Ψ (s) = L[ψ (t )] ,
and Φ ( s ) = L[φ (t )] ,
(3.18) 37
The roll/yaw axes have two input variables and two output variables. Using transfer function equation of the output variables, The roll/yaw axes can be can be represented in matrix form as below,
Tcx ( s ) / I x G11 ( s ) G12 ( s ) Φ ( s ) T ( s ) / I = G ( s ) G ( s ) Ψ ( s ) , z 21 22 cz
(3.19)
where = G11 ( I x s 2 + ω0 hwy 0 ) / I x , G12 = shwy 0 / I x , G21 = − shwy 0 / I z ,= G22 ( s 2 I z + ω0 hwy 0 ) / I z . The simultaneous solution of the above equation (3.21) using Cramer’s rule can be used to find the output variables Φ ( s ) and Ψ ( s ) as follows; −1
Φ ( s ) G11 G12 Tcx ( s ) / I x , Ψ ( s ) = G 21 G22 Tcz ( s ) / I z
(3.20)
The solution to the open loop roll and yaw axis can be solved as below;
Φ(s) 1 G22 Ψ ( s ) = ∆ ( s ) −G 21 ΦΨ
−G12 Tcx ( s ) / I x , G11 Tcz ( s ) / I z
(3.21)
where ∆ ΦΨ ( s ) = G11G22 − G12G21 is the determinant of the matrix. 2 i.e. G11G22 − G12G21 = ( s 2 + ω0 hwy 0 / I x )( s 2 + ω0 hwy 0 / I z ) + s 2 hwy 0 / IxIz = 0 2 2 2 2 ∆ ΦΨ ( s ) =+ s 4 s 2ω0 hwy 0 / I x + s 2ω0 hwy 0 / I z + ω02 hwy 0 , (3.22) 0 / I x I z + s ω0 hwy 0 / I x I z =
=s 4 + s 2 (ω0 hwy 0 (
1 1 2 2 2 + ) + hwy 0 / I x I z ) + ω0 hwy 0 / I x I z =0 Ix Iz
2 2 The practical relation of hwy 0 ω0 I x I z and hwy 0 ω0 ( I x + I z ) are defined in ref[12], The
characteristics equation above then becomes, 2 2 2 2 2 0 , ∆ ΦΨ ( s ) = s 4 + s 2 hwy 0 / I x I z + ω0 hwy 0 / I x I z + s ω0 =
(3.23)
Factorizing the above equation, ∆ ΦΨ ( s ) =( s 2 + ωn2 )( s 2 + ω02 ) =0 ,
Where ω = 2 n
(3.24)
2 hwy 0
IxIz
The simultaneous solution of the equations (3.16,3.17) using Cramer’s rule can be used to find the output variables φ ( s ) and ψ ( s ) as follows; 38
= Φ(s)
G22 ( s )Tcx ( s ) / I x G12 ( s )Tcz ( s ) / I z − ∆ ΦΨ ( s ) ∆ ΦΨ ( s )
(3.25)
= Ψ (s)
G11 ( s )Tcz ( s ) / I z G12 ( s )Tcx ( s ) / I x − ∆ ΦΨ ( s ) ∆ ΦΨ ( s )
(3.26)
This characteristics equations for roll/yaw axis is fourth order and undamped which consists of two (double) second-order poles: the first located at the orbital frequency i.e. s1,2 = ± jω0 and the second undergoing nutational motion at angular frequency ωnut i.e. s1,2 = ± j
hwy 0 IxIz
.The nutation
frequency as shown is proportional to the momentum bias hwy 0 and the moments of inertia I x , I z about the two transverse axes X and Z axis of the vehicle. The pair of the poles are natural frequencies of the roll/yaw axes of the plant, i.e. the two ways in which one can excite the h system. The first mode has a larger natural frequency of wy 0 rad / s and is called the shortIxIz period mode (because the time-period of the oscillation, i.e. T1 =
2π
ωn
is smaller for this mode).
The second characteristic mode with a smaller natural frequency (orbital frequency) ω0 rad/s-has a longer time-period and is called the long-period mode. While an arbitrary input will excite a response containing both of these modes, The two modes will be studied separately. The zeros of the roll and pitch axis are also at imaginary axis as a characteristics of minimum-phase systems. As stated above, the roll axis and yaw axes are interrelated to each other i.e. the roll attitude always influences the yaw axis and vice versa [6]. The earlier presented roll and yaw dynamics show the coupling effects. These equations are used in the control system to investigate the behavior of roll and yaw dynamics in the presence of the solar radiation disturbance torques.
3.3. Momentum bias attitude control system. In a momentum bias control system, a momentum wheel is spun up to maintain a large angular momentum relative to disturbance torques. This design is common in Earth-oriented spacecraft where the momentum wheel is along the pitch axis, nominally parallel to orbit normal. The advantages of the momentum bias design are: (I) short-term stability against disturbance torques, similar to spin stabilization; (2) roll-yaw coupling that permits yaw angle stabilization without a yaw sensor for pitch axis pointing; (3) a momentum wheel that may be used as an actuator for pitch angle control; and (4) a momentum wheel that may be used to provide scanning motion across the celestial sphere for a horizon sensor. Thus, momentum bias systems can provide threeaxis control with less instrumentation than a three-axis reaction wheel system. By incorporating two axis horizon earth sensors into the momentum wheels configuration, roll and pitch error signals may be provided to the control system. Yaw control can be achieved without a yaw 39
sensor through the kinematics of quarter-orbit gyroscopic coupling as shown in Figure 3.2. Here, a yaw error, ξ y at one point in the orbit becomes a roll error, ξ r a quarter of an orbit later.
Figure 3.2: Interchange of Yaw and Roll Attitude Components for a Momentum Wheel with Angular Momentum h , Fixed in Inertial Space. The yaw error when the spacecraft is at A becomes a roll error when the spacecraft moves to B [22]. The linearized model describing the roll, pitch and yaw axes dynamics of a momentum bias spacecraft is now represented as follows. Three different output variables Φ ( s ), Θ( s ), Ψ ( s ) (in the Laplace domain) are of interest when the spacecraft is displaced from the equilibrium point. The input variable in the Laplace domain is the Tcx , Tcy , Tcz and perturbed external disturbances Tdx , Tdy , Tdz , the three transfer functions separately defining the relationship between the inputs,
and the three respective outputs, are as follows: 2 Φ ( s ) ( s I z + ω0 hwy 0 ) / I z I x = ( s 2 + ωn2 )( s 2 + ω02 ) Tcx ( s )
,
Tcz (s) = 0 Iz
(3.27)
1/ I y Θ( s ) = 2 2 Tcy ( s ) s + 3ω0 ( I x − I z ) / I y 2 Ψ ( s ) ( s + ω0 hwy 0 ) / I z I x = 2 Tcz ( s ) ( s + ωn2 )( s 2 + ω02 )
(3.28)
,
Tcx (s) = 0 Ix
(3.29)
3.4. Pointing Error Analysis. This analysis will be used to determine pointing budget of the pointing error of attitude control. Pointing budget quantify the effect of each error source on the overall vehicle pointing. There are a number of competing methodologies for doing pointing budgets for satellites. One of the most popular is to divide the error contributions into four temporal categories: The errors are grouped into one of the four following areas, with each applicable error component individually identified and supported by measurement and/or analysis: •
Constant errors including components due to alignment errors control bias resolution, etc. 40
• •
•
Diurnal errors including components due to thermal distortion effects, orbit inclination errors, orbit longitudinal position errors, etc. Short-term errors including components due to sensor noise, dynamic coupling of wobble into pitch, motor friction torque disturbances, nutation coupling into pitch via damper control loops, servo dead bands, control thresholds, antenna stepping, truncation of control laws, etc. Long-Term (e.g. seasonal variation in the yaw body fixed torque on a momentum bias spacecraft without yaw sensing).
The total error for each group shall be equal to the root of the sum of the squares of each 3 σ component and the total steady-state error about each axis shall be equal to the arithmetic sum of the individual group errors. For the momentum bias vehicle in GEO, This type of satellite points at the earth for its entire mission life. The yaw axis is aligned with nadir, the pitch axis points south and the roll axis points east. If the spacecraft has single circular beams about nadir , only roll and pitch errors matter. If it has off-nadir spot beams or if the main beam has a non-zero elevation, yaw couples into the beam errors. Once roll, pitch, and yaw error values have been computed, they must be combined to form the beam pointing accuracy. The beam is the radio frequency wavefront that touches the satellite footprints. The beam pointing error of communication payload is directly connected to the attitude error through the equations below: Α Azimuth = φ+
δ az
180 / π
Ε Elevation = θ+
ψ ,
(3.30)
δ el
(3.31) ψ , 180 / π where Α Azimuth is beam pointing error for azimuth, Ε Elevation is beam pointing error for elevation,
δ el
and
δ az are the beam center offsets in elevation and azimuth angle respectively.
Using the reference[12],The normal mode pointing error budget are determined by taking the square root of the sum of the square of the sources of error in normal mode and may be assign as follows,
41
Table 3.1: Normal mode pointing error for momentum bias vehicle [12]. where MRC is the master reference cube which is the master alignment reference for the spacecraft. This is a cube that reflects laser light from a laser measuring device. CEP is the Circular Error Probability. The CEP in degrees says that for normally distributed random errors the 3 σ probability is that the error will be within the CEP. This budget was generated for a momentum bias spacecraft that uses an earth sensor (ESA) to measure roll and pitch. The quantity of interest is the beam alignment (antenna boresight) with respect to a target on the earth. The temporal quantities are: bias, assumed fixed for the life of the spacecraft; diurnal, which vary with the orbit period; and short term, which change at rates faster than orbit rate. Yaw is not sensed and is controlled solely by the spacecraft momentum. The largest component, in this case, is the orbit rate disturbance so the only entries for yaw are orbit inclination and this disturbance related error. In roll and pitch there are many errors due to misalignments. Alignments are measured using an optical system. ESA RC is the reference cube for the earth sensor.ESA errors appear in all three temporal categories. These are taken from the manufacturers specifications. All alignment errors are determined by measuring post vibration errors. The spacecraft is aligned and then placed on a shaker to simulate launch. The change in alignments is considered the alignment uncertainties.
42
3.5. Simulation development and Simulink Models of the control problems. The control synthesis of this thesis report is implemented analytically and using simulink block. Simulink will are used to develop all the simulations of the control system including all the neglected parts a, b, c, d .This will enable the quantification of the performances using various wheels configuration. If necessary one can also vary important system parameters (mass properties) and check the resulting performance over time. Matlab scripts were also used to enter some valuable parameters into simulink block diagrams. The model has been developed in Simulink with the goal of creating a modular model, where sub models can easily be reused. The top level model consists of four major blocks containing logically grouped sub blocks. The vehicle control software for the simulator was designed so that it could be had the following functions: manipulation and renewal of system parameters, revision of controller information, simulated real-time system monitoring, download of the running mode file, and saving and output of experimental results. It was designed in consideration of convenience and flexibility for the use so that if any vehicle information is changed, the controller information might be revised only by manipulating control parameters. Figure 3.3 is a control block diagram for implementation of the software. The top level model consists of four major blocks containing logically grouped sub blocks, as depicted in figure 3.3:
Figure 3.3: data linkage diagram of the vehicle control system.
43
Functions of Each sub model. Spacecraft dynamics and kinematics: Simulates the dynamics of roll, yaw and pitch axis of spacecraft attitude, the gravitational torques and Euler angles. Attitude control algorithms: Simulates the Attitude Control System. Wheels configuration: Simulates the two skewed wheels configuration used on Nigcomsat-1 and spacecraft with similar wheel configuration. Bang -bang control: simulates the on and off of the thrusters of the spacecraft during thruster controlled roll/yaw attitude dynamics. Ephemerides: Calculates positions and solar angle to spacecraft and earth from the sun as a function of time. The model presented in this for the sun position is based on [16]. The position of the Sun in the ECI-frame depends only on time and a number of constant parameters and can be described by a Keplerian orbit as the Earth’s orbit around the Sun. The dynamics of the spacecraft, the Euler angles, gravitational torques defined the complete spacecraft dynamics model in geostationary orbit. The attitude and angular rate of the spacecraft is calculated in this block based on the initial conditions, the disturbances and actuator torques and the angular momentum. The inputs and parameters to the block are listed below. • Initial attitude • Initial angular rate • Spacecraft inertia • Disturbance torque • Actuator torque • Wheels angular momentum 1. The gravitational moment ;( The external torques acting on the vehicle) Gx = −3ω02 ( I y − I z )φ ,
G y = 3ω02 ( I z − I x )θ ,
Gz = 0 ,
44
2.The angular velocity components ω x , ω y , ω z in vehicle body coordinates for a small Euler angle given by [4]
ω x = φ − ω0ψ ω y = θ − ω0 ω z = ψ + ω0φ
, 3.Mathematical model wheels momentum biased vehicle for the control synthesis.
U x = I xφ + [a + ω0 hwy 0 ]φ + [b + hwy 0 ]ψ U= I yθ + dθ y
,
U z = I zψ − [b + hwy 0 ]φ + [c + ω0 hwy 0 ]ψ , where a = 4ω02 ( I y − I z ) , b = −ω0 ( I x − I y + I z ), c = ω02 ( I y − I x ), d = 3ω02 ( I x − I z )
45
Figure 3.4: Simulink model of the Momentum bias spacecraft. 4. The configuration of the solar radiation torques as a function of time about the body axes are shown in figure 2.4 and is expressed as, Tdx ≈ a cos S ' + b sin S ' cos µ ,
Tdy ≈ c sin S ' cos µ , Tdz ≈ d cos S ' − e sin S ' cos µ ,
where
S ' is
the
angle between the sun line and the Earth’s polar axis. µ = ωot , 66.550 ≤ S ' ≤ 3600 , 00 ≤ µ ≤ 3600 and typical values for the coefficient a, b, c, d , e are
on the order of 10−6 Newton − meter [5].
46
Figure 3.5: Simulink model of solar radiation torques.
3.6. Control algorithms. The Attitude determination and control system block simulates the controllers developed for the spacecraft. This includes sensor simulation, attitude determination system and filters/algorithms, controller and actuator simulation. To simulate the behavior of the spacecraft the spacecraft dynamics block uses inputs from the control algorithms and solar radiation torques simulator. The overall simulink simulator is implemented as in figure 3.6,
47
Figure 3.6: Control block diagram for implementation of the Vehicle simulator. In the normal mode of spacecraft operation in geostationary orbit, a geostationary communication satellite requires no rapid maneuvers and associated settling time requirements, so the control problem is one of stabilizing the attitude against very low frequency disturbances such as solar radiation torques and gravity gradient torques. The stability criteria in this thesis is dependent of the class of design concept chosen for Nigeria communication satellite where actuators (Momentum wheels or thrusters), sensors (earth and sun sensors) are collocated at the center core cylinder of the vehicle with no movement of the communication antennas. A momentum wheel is added to the satellite equipment to provide inertial stability about the pitch axis (YB axis), which is perpendicular to the orbit plane. Acceleration and deceleration of the momentum wheel could be used to produce corrective torques to control the satellite in the roll-yaw plane [9,10].The linearized analysis will be carried out to determine the stability criteria of momentum bias spacecraft using the parameters of the spacecraft.
3.7. Linear Open loop Numerical Analysis. Effort is made to synthesis the various wheels configurations using the material properties of Nigeria communication satellite described by the following parameters: 0 0 14,542 I = 0 4,396 0 (kg.m 2 ) 0 0 12,356
Thrusters=10Newton. = ω0 7.272 ×10−5 rad/s (Geostationary orbit orbital rate). The following numerical analysis is derived for characteristics equations for pitch and roll/yaw motions. 48
•
Pitch axis.
The characteristics equation of pitch axis defined by equation (3.15) is stated below, s2 +
3ω02 ( I x − I z ) 0 = Iy
(I x − I z ) = (14,542 − 12,356) / 4,396 = 4.972 ×10−1 , Iy 3ω02 ( I x − I z ) = 3* 4.972 ×10−1ωo2 =14.918 ×10−1ω02 Iy s 2 + 3* 4.972 ×10−1ωo2 = 0
− j 0.122 ×10−1ωo2 , s2 p = + j 0.122 ×10−1ωo2 ; This means that the pitch axis is marginally s1 p = stable because of its complex roots. •
Roll/Yaw axis
The characteristics equation of the roll and yaw axis is defined as, ( s 2 + ωn2 )( s 2 + ω02 ) ,
Substituting for the parameters on Nigcomsat-1, the numerical values of roots of characteristics equations of roll/pitch axis for Nigcomsat-1 is defined as below, s1 , s2 = ± j 7.273 ×10−5 radian .
hwy 0 50 radian . s1n , s2 n = ±j = ±j 14,542*12,356 IxIz
EULER ANGLES
ROOTS OF CHARACTERISTICS EQUATIONS
Pitch
±8.840 ×10−5 j
Roll/Yaw
±7.273 ×10−5 j , ±0.373 ×10−4 j
Table 3.2: Roots of the polynomial in Pitch and Roll/Yaw open loop transfer functions. As shown in Table 3.2,the roots of the characteristics equation for open loop pitch axis of Nigcomsat-1 is ±8.840 ×10−5 j ,while for roll/yaw axis consists of orbital ±7.273 ×10−5 j , and nutational ±0.373 ×10−4 j , frequencies. 49
The open loop Bode plots for each of the three transfer functions of the roll ( Φ ( s ) Tcx ( s ) ), pitch ( Θ( s ) Tcy ( s ) ) and yaw ( Ψ ( s ) Tcz ( s ) ) axis are plotted to see their frequency responses. Figures 3.7, 3.8 and 3.9 show the magnitude and phase Bode plots for the three transfer functions in the limit s = jω (they are the frequency responses of the concerned output variable). Using Matlab Control Systems Toolbox (CST), these plots are directly obtained by the command bode, after constructing each transfer function using the linear time invariant(LTI) object transfer function(tf). Bode plot for open loop response of Roll axis
100
Magnitude (dB)
System: Roll_axis Peak gain (dB): 93.5 At frequency (rad/sec): 7.26e-005
0
System: Roll_axis Frequency (rad/sec): 0.00374 Magnitude (dB): 61
-100
-200
Magnitude (deg)
-300 0
Roll_axis
System: Roll_axis Phase Margin (deg): -180 Delay Margin (sec): 5.25e+003 At frequency (rad/sec): 0.000598 Closed Loop Stable? No
-45
-90 System: Roll_axis Phase Margin (deg): 0 Delay Margin (sec): -0 At frequency (rad/sec): 0.000492 Closed Loop Stable? No
-135
-180 -5 10
-4
-3
10
System: Roll_axis Phase Margin (deg): 0 Delay Margin (sec): -0 At frequency (rad/sec): 0.00908 Closed Loop Stable? No
-2
10
-1
10
10
0
1
10 Frequency (rad/sec)
2
10
3
10
4
10
5
10
10
Figure 3.7: Bode plot of the Roll axis transfer function. Bode plot for open loop response of Pitch axis
Magnitude (dB)
100
System: Pitch_axis Peak gain (dB): 123 At frequency (rad/sec): 8.93e-005
0 -100 -200 -300 0
Magnitude (deg)
Pitch_axis -45 -90 System: Pitch_axis Phase Margin (deg): 0 Delay Margin (sec): -0 At frequency (rad/sec): 0.0151 Closed Loop Stable? No
-135 -180 -5 10
-4
10
-3
10
-2
10
-1
10
0
10 Frequency (rad/sec)
Figure 3.8: Bode plot of the Pitch axis transfer function.
50
1
10
2
10
3
10
4
10
5
10
Bode plot for open loop response of Yaw axis
100
Magnitude (dB)
System: yaw _axis Peak gain (dB): 93.2 At frequency (rad/sec): 7.26e-005
0
System: yaw _axis Frequency (rad/sec): 0.00373 Magnitude (dB): 61.1
-100
-200
Magnitude (deg)
-300 0 System: yaw _axis Phase Margin (deg): -180 Delay Margin (sec): 5.89e+003 At frequency (rad/sec): 0.000533 Closed Loop Stable? No
-45
yaw_axis
-90 System: yaw _axis Phase Margin (deg): 0 Delay Margin (sec): -0 At frequency (rad/sec): 0.000451 Closed Loop Stable? No
-135 -180 -5 10
-4
10
-3
10
System: yaw _axis Phase Margin (deg): 0 Delay Margin (sec): -0 At frequency (rad/sec): 0.00973 Closed Loop Stable? No
-2
10
-1
10
0
10 Frequency (rad/sec)
1
10
2
10
3
10
4
10
5
10
Figure 3.9: Bode plot of the Yaw axis transfer function. The bode diagrams reflects the result of the table of characteristics equations. From the Bode plots (Figures 3.7,3.8,3.9), we can note the natural frequencies of the long and the short period modes for roll/yaw axes respectively, as either the peaks or changes of slope (called breaks) in the respective gain plots. The peaks due to one of the complex poles disappear due to the presence of zeros in the vicinity of the poles. As expected, the natural frequencies agree with the values already calculated from the characteristic polynomial. As expected, near each natural frequency the phase changes by 180° (pitch axis), except for the roll/yaw axis that alternate (Figure 3.7,3.9). The latter strange behavior is due to the fact that, the transfer function Φ ( s ) Tcx ( s ) and Ψ ( s ) Tcz ( s ) has its numerator quadratics (i.e. a pair of complex zeros) almost cancels out the quadratic corresponding to the long period mode in the denominator polynomial (i.e. a pair of complex poles), indicating that there is essentially no noticeable change in the output in the long period mode for roll/yaw dynamics. It should be noted also that the gains of all three transfer functions decay rapidly with frequency at high frequencies. Such decay in the gain at high frequencies is a desirable feature, called roll-off, and provides attenuation of high frequency noise arising due to unmodeled dynamics in the system. Hence, a controller transfer function must be selected to provide not only good gain and phase margins, which indicate the closed-loop system's robustness to process noise, but also a large decay (or roll-off) of the gain at high frequencies, indicating robustness due to the measurement noise. However, plot of roll and yaw axis shows that considerable magnitude (gain) exists at both short period and the long period natural frequencies. Hence, both modes essentially consist of oscillations in the roll and yaw angles. The bode plot for open loop analysis shows how we have obtained an insight into the system's behavior just by analyzing the frequency response of its transfer function(s).On this basis we designed a compensators that will shape the loop so as to obtain desired responses.
51
3.8. System configuration of attitude control. The configuration of the sensors, actuators and other hardware of the compensator/controller for any wheels configuration are shown in figure 3.10 and are discussed. This is done by defining some basic design parameters such as the actuation strategy and which sensors to use. Following this, the physical configuration of sensors and actuators is set and considerations on control algorithms design which is the basis of this thesis are made. Target Attitude
Controller
Attitude determination
Actuators
S/C Dynamics
Practical Attitude
Sensors
Figure 3.10: System configuration of momentum bias spacecraft. Actuator strategy To be able to control the attitude of any heavy weight momentum bias spacecraft, either a wheels or reaction jets is needed as actuator system. Actuators must be able to cause an angular acceleration and exerts torques on the spacecraft. As the torque is the derivative of the angular momentum, the actuator must change the angular Momentum of the spacecraft, which according to Newton’s law must be constant when the Spacecraft is not affected by external forces. As a result there are only two ways to alter the attitude of the spacecraft. 1. By transferring angular momentum to an external object. 2. By transferring angular momentum to another part of the spacecraft. They can be achieved by a number of different methods. The ones used for the various wheels configuration in this thesis are discussed below. Chemical thrusters. Chemical thrusters utilize a chemical reaction which accelerates a propellant and expels it from the spacecraft. In other words, momentum is transferred from the spacecraft to the propellant, thus chemical thrusters belong to the second group of actuators. They can be used to exert a torque on the spacecraft .All the wheels configuration nominally use chemical thrusters for momentum unloading while wheels configuration of type a and type b use thrusters to control roll and yaw axis. Type a and b therefore have operational time dependent on the amount of on 52
board fuel. Small monopropellant and bipropellant thrusters are used for attitude control on many satellites. They are used for attitude control during orbit change maneuvers or for momentum unloading, as an alternative to magnetic torquers. Most small thrusters are on-off thrusters. That is, their valves have only two positions, on and off. Momentum wheel. Momentum wheels consist of a motor and a flywheel. When the flywheel is accelerated by the motor it picks up angular momentum, which is transferred from the satellite frame (on which the motor is mounted). With three momentum wheels it is possible to transfer angular momentum from the satellite to the momentum wheels. Instead of using three momentum wheels it is also possible to use a single momentum wheel or two skewed wheels in a gyroscopic suspension. However there is a limit of how much angular momentum can be transferred to the momentum wheels as the motor has a saturation limit, and momentum wheels are thus often used in conjunction with another actuation system. The momentum wheels belong to the second group of actuators as the angular momentum is transferred to the momentum wheels. Momentum wheels are a viable actuation system for the communication satellites as it is possible to find motors of very small sizes and with low power consumption.
Reaction Wheel. Reaction wheels are momentum exchange devices. A motor is fixed to the spacecraft and the shaft of the motor is attached to a flywheel. When a control voltage is applied to the motor, the motor generates a torque spinning the wheel in one direction and the spacecraft in the other. Hence the term reaction wheel. Since the torque is internal to the spacecraft system, the reaction wheel cannot change the total inertial angular momentum of the spacecraft system. Rather, it transfers momentum between the flywheel and the rest of the spacecraft. If the external torque on the spacecraft is periodic (with a sufficiently high frequency) with respect to the inertial frame, then the reaction wheel can completely control the spacecraft. However, if there is a constant inertial torque, the wheel will spin up and eventually saturate. The wheel can also be used to maneuver the spacecraft. When the wheel spins up the spacecraft will develop an angular rate. When the wheel spins down it will absorb the angular momentum in the body and stop the motion.
53
Figure 3.11: Momentum/Reaction Wheel. Sensors analysis The performance of a spacecraft control system is limited by the performance of its sensors and actuators. Many of the sensors are used for determining the attitude or attitude rates of the spacecraft. Others are used for determining the relative orientation or position of components on a spacecraft. Typical sensors for momentum bias spacecrafts in GEO are: • • • •
Earth sensors (scanning or static) Star sensors Sun sensors (one or two axis) Gyros (many types)
Many satellites are earth pointing and need to point a payload at a target on the earth. Star sensors and sun sensors give an inertial reference directly and can, through means of a table lookup, give an earth reference. Earth sensors give an earth-fixed reference, but are much less accurate than star and sun sensors. Gyros measure inertial rotation rates. Rate integrating gyros give the integral of the body rates. Since each gyro integrates about a body axis gyros cannot give an inertial reference directly and are always used in conjunction with star, sun or earth sensors. Selection of sensors is a process of trading cost against sensor life, accuracy, mass, power consumption and processing software complexity. An important factor in selecting sensors is their radiation hardness. Radiation will degrade many analog circuits and most CCD, CMOS or CID arrays thus limiting the life of optical sensors. Other considerations include sensor placement and interference with the payload. We describe briefly the possible sensors that can be used on the momentum bias spacecrafts in GEO. The is based on information from [32]. 54
Sun sensor. A sun sensor is a reference sensor which measures a direction in a known reference coordinate system. Sun sensors are visible light detectors which measure one or two angles between their mounting base and the incident sunlight. Sun sensors are popular, accurate and reliable but require a clear field of view and are very expensive. Star sensor. Star sensors can be either scanners or trackers. Scanners work on spinning satellites where the attitude is determined by the light of stars passing through multiple slits in the field of view of the scanner. Star trackers recognize star patterns in the field of view of the sensor. Camera technologies as Charged-Coupled Devices (CCD), Active Pixel Sensors (APS) and CMOS could be considered for this task. The location of two or more stars is enough to determine the attitude of the satellite. This means that a star tracker alone can determine a three axis attitude when pointing towards the sky. For recognizing star patterns, an on-board star database is necessary. However, the Problem with star trackers is they require the satellite to be stabilized to some extent before the tracker works. This stabilization may require extra sensors which increases the overall cost of the mission. Rate integrated gyroscope A gyro is a sensor which measures the angular rate of the satellite or the angle of rotation from an initial reference, without any knowledge of external references. A gyro measures the angular velocity by means of the Coriolis effect which is defined by Newton’s first law of motion: A body in motion continues to move at a constant speed along a straight line unless acted upon by an unbalanced force. The advantage using gyros is that they can provide high frequency angular rate information (up to hundreds of Hertz), which the other types of sensor may not be able to do. Also, gyros measure angular velocity directly, which e.g. sun sensors and star sensors cannot. When grouped together gyros can provide full 3-axis information. Horizon Earth sensor Horizon sensors detect the threshold between infrared light emitted from the atmosphere of the Earth and space by utilizing an infrared diode and a lens. There are two kinds of horizon sensors; scanners and horizon crossing indicators [12]. A horizon crossing sensor is fixed in the satellite structure and can provide valuable attitude information only when the sensors line of sight crosses the mentioned threshold.
55
3.9. Requirements of attitude control in this thesis. Mission orbit are usually divided into normal mission mode (NM) and station keeping mode (SKM) with a transition mode (TRM) connecting the two modes as shown in figure 4.7. During the mission orbit, a momentum bias spacecraft is modeled as a rigid body with a rotor and flexible solar arrays.
Command
SKM
Position Acquisition
Command
TRM
Eliminating residual rate
Command Automatic
NM
Figure 3.12: On orbit mode of geostationary communication satellites. During normal operation part of the mission orbit, the solar array flexibility does not impact the control system performance as such the control synthesis here did not consider solar arrays dynamics. Consequently it is convenient to synthesis the dynamics of the normal mode operation without considering solar arrays flexibility. The roll/yaw and pitch dynamics can be decoupled for analysis of normal mode operations. The pitch dynamics are a double integrator. The roll/yaw dynamics are a coupled fourth order plant with two pure imaginary pole pairs. The first at orbit rate is due to the kinematics, the second is the nutation mode and is due to the bias momentum. The main diagonal channels each have a pure imaginary zero pair. This zero pair is located between the orbit rate and nutation mode providing 180 degrees of phase shift (Figure 3.7,3.9) and making the control problem easier. Disturbances at orbit rate, or at the nutation frequency, will cause uncontrolled growth in the attitude errors. Since many of the disturbance sources are at orbit rate, it is necessary to add an automatic control system. The nutation pole causes large oscillatory responses whenever a sudden torque is applied to the spacecraft. A common method for spacecraft attitude control is a Proportional-Derivative (PD) controller and a lead compensator [15, 20, 22, 30]. If both attitude and attitude rate are sensed e.g. using earth sensor, sun sensor or three axis rate gyro, then stabilization about each principal axis may be obtained by feeding back a linear combination of attitude deviation and attitude rate to the torque using PD controller. The use of gyro however is very expensive, consume power, reduce control reliability and heavy. However if only attitude is sensed, then stabilization about each principal axis may be obtained by feeding back attitude deviation with lead compensation to torque about each of the respective axis. It is 56
on the basis of the better advantage of lead compensator over PD controller that we use lead compensator throughout this control synthesis. The purpose of phase lead compensator design in the frequency domain generally is to satisfy specifications on steady-state accuracy, and ensure good transient response with sufficient damping. A specification on bandwidth or crossover frequency can represent a requirement on speed of response in the time domain or a frequency-domain requirement on which sinusoidal frequencies will be passed by the system without significant attenuation. The overall philosophy in the design procedure presented here is for the lead compensator to adjust the system’s Bode phase curve so as to establish the required phase margin at a specified frequency, without reducing the zero-frequency magnitude value. The design procedure presented here is basically graphical in nature. All of the measurements needed can be obtained from accurate Bode plots of the uncompensated system. The results and plots presented in this thesis are all done in MATLAB, and the various parameters used for the control synthesis are adopted from the mass properties of first Nigeria communication satellite launched in 2007. The primary references for the procedures described in this paper are [12, 23, 24, 29]. All techniques are applied to single-input single-output control systems in this thesis and the parameter tested on both pitch axis and coupled roll/yaw simulink blocks. The normal mode will be designed with the underlisted performances for roll, pitch and yaw axes. Therefore, the specification taken for this study is given by root mean square (rms) antenna pointing error of 0.10 or less, which is accomplished by allowing the attitude errors (proportional to bandwidth) in the presence of the disturbances to be as: 0 0 0 φss = ±0.05 , θ ss = ±0.05 ,ψ ss = ±0.2 .Other performance design include;
Fine accurate pointing during the satellite life time, low propellant consumption, Simplicity and high reliability Minimal Steady state error due to step input. Minimal Settling time.
57
CHAPTER 4 4. ANALYTICAL PHASE LEAD COMPENSATOR DESIGN. 4.1. Analytical phase lead compensator design for momentum biased spacecraft. The mathematical model of the Plant dynamics previously derived by equations (3.11, 3.12, 3.13) is stated below,
I xφ + ω0 hwyφ + hwy 0ψ = Tdx + Tcx ,
(4.1)
I yθ + dθ = Tdy + Tcy ,
(4.2)
I zψ − hwy 0φ + ω0 hwy 0ψ = Tdz + Tcz
(4.3)
The basic lead compensator consists of a gain K , a pole p , and a zero z , with transfer function expressed as,
α ( s + 1/ α T )
α >1 ( p > z ) , ( s + 1/ T ) where= p 1/= T z / α , z = 1/ α T , K = α 1 T= , Gc ( s ) =
(4.4)
(4.5)
ωm α 1 + sin φm , (4.6) α= 1 − sin φm where φm maximum phase lead to be introduced by the phase lead compensator and ωm is the frequency at which it occurs. The bode plot of phase lead compensator is shown in figure 4.1 ,
58
Figure 4.1: Bode plot of phase-lead controller[13].
4.2. Lead Compensator design for Pitch axis. The model of pitch motion though undamped, is a minimum phase system. Our goal is to provide for a well damped response to step input command. Therefore, it is required to add damping to the double imaginary pole. We have three degrees of freedom in the lead compensator. One can adjust the gain, Can choose the maximum amount of phase lead it can add, and can select the frequency where the phase lead is at its maximum. In the normal mode, we were not concerned with damping the solar array flexible mode; we would put our maximum phase shift at the gain crossover. The analytical block diagram for closed loop momentum wheel control about pitch axis is shown in Figure 4.2.The system is simplified to the extent that limiter, dead zone, wheel dynamics and other non linear effects that occur in reality are neglected. Details of the non linear effects can be found in [14]. In Figure 4.2, the lead compensation introduces a phase lead into the control loop to ensure good transient response and sufficient damping. The two axis earth sensor will provide the pitch error (same sensor for roll) signal however a unity feedback attitude sensor was assumed. The wheel as actuator will provides fine control about the pitch axis. Momentum unloading may be allowed at 10 percent speed variation [13]. The saturation block at the output of the controller limits the torques from the wheel between the lower and upper limit. The objective of the system is to maintain the error between desire response θ r (t ) and actual response
θ y (t ) close to null. The parameters of Nigeria communication satellite (Nigcomsat-1) stated earlier are used for various wheels configurations. The orbit attitude is 35,768Km with near zero 59
degree inclination. Simulation time of one orbital period (24hours) was considered and the Disturbance torques model as solar torques are defined in equations (2.35-2.37) and its simulink model simulator described in figure 3.5 was used.
Figure 4.2: Pitch axis attitude control block diagram. The forward path transfer function of the uncompensated system is
Θ( s ) 1 , = G= p (s) 2 Tcy ( s ) I y s + 3ω02 ( I x − I z )
(4.7)
Using the numerical values of Nigeria communication satellite, The poles at −5 −5 s1 = −8.840 ×10 j and s2 = +8.840 ×10 j , describe the marginal stability response of the pitch dynamics momentum.
between the actual pitch angle and the wheel rate of change of angular
The forward path transfer function of the compensated plant is written as, α ( s + 1/ α T ) 1 (4.8) Gc ( s )G p ( s ) = 2 , 2 ( s + 1/ T ) I y s + 3ω0 ( I x − I z ) The first step in the design of the lead compensator is to plot the bode diagram of uncompensated process with acceptable gain constant. This acceptable gain constant was skipped here because our focus is to damp the pitch axis dynamics for attitude stabilization. The bode plot of the process shows the system has infinity phase margin and the amount of positive phase shift that must be added at a specified frequency in order to satisfy the phase margin specification will be specified. If the compensator is to have a single-stage lead compensator, then the amount that the phase curve needs to be moved up at the gain crossover frequency in order to satisfy the phase margin specification must be less than 90ο ,and is generally restricted to a maximum value in the range 55ο 65ο [19].The pitch axis open loop dynamic is oscillatory. Phase margin (PM) is defined as PM = 180 + ∠G ( jω x ) , 60
(4.9)
Therefore, the uncompensated phase margin for pitch axis is PM = 180 − 180 = 0ο • Determination of φm and α Given the value of the uncompensated phase margin from the previous step, we can now determine the amount of positive phase shift that the lead compensator must provide. The compensator must move the phase curve of G p ( jω ) at ω = ω x upward from its current value of −180ο to the value needed to satisfy the phase margin specification. On this basis, phase margin of 65ο was chosen. The value of φm = 65ο chosen in this thesis is at (or near) the upper limit for a single stage of lead compensation.
Knowing φm , we can compute the value of α • The corresponding value for the parameter α is = α
1 + sin φm 1 + sin 65ο 1.906 = = = 20.495 , 1 − sin φm 1 − sin 65ο 0.093
(4.10)
The value α = 20.495 corresponds to a maximum phase shift of 65ο . z The compensator’s pole–zero combination is now related by the ratio = α= 20.495 . p • Determination of p and z from T . Once α is determined, we now determine the value of T . This is accomplished by placing the 1 corner frequencies of the phase lead controller 1 α T and T such that φm is located at new gain-crossover frequency ωт so the phase margin of compensated system is benefitted by φm . Ref[] shows that the high frequency of the phase lead controller is 20 log10 α dB ,thus to have the new gain cross over at ωm which is the geometry mean of 1
1 α T and T as shown in
figure(4.1),we need to place the ωm at the frequency where the magnitude of uncompensated G p ( jω ) is −10 log10 α dB so that adding the controller gain of 10 log10 α dB to this makes the magnitude curve go through 0dB at ωm . The following steps are taken to ensure that φm occurs at ωm . The high frequency gain of the phase lead controller of equation (4.4) is = = 20 log10 α dB 20 log10 20.495 26.232dB ,
(4.11)
The geometry mean ωm of the two corner frequencies 1
1 α T and T should be located at the frequency at which the magnitude of the uncompensated process transfer function G p ( jω ) in dB is equal to the negative value in dB of one-half of this gain i.e. −10 log10 α dB .This way, the 61
magnitude curve of the compensated transfer function will pass through the 0dB axis at ω = ωm .Thus, ωm should be located at frequency where,
G p ( jω )
dB
= −10 log10 α = −10 log10 20.495 = −13.100dB .
(4.12)
This can be determine either from data array in MATLAB using the bode function of uncompensated open loop of the pitch axis or obtained graphically from the plot as done in this thesis. From pitch axis bode plot shown in figure 4.3, this frequency is found to be 0.0321 rad/s, now using equation (4.5),we have, 1 • The value of T can be determined from 0.0321* 20.495 = ω= = 0.145 rad/s m α T 1 ⇒ = 0.145 / 20.495= 7.091e − 3 Rad/s. αT
Figure 4.3: Graphical determination of frequency at −10 log10 αdB magnitude. The transfer function of the phase lead controller for this pitch axis dynamics with numerical values of the controller parameter is then represented as α ( s + 1/ α T ) 20.495( s + 0.007) , ( s + 1/ T ) = ( s + 0.145)
• The close loop transfer function is Gc ( s )G p ( s ) , Gcloop ( s ) = 1 + H ( s )Gc ( s )G p ( s )
(4.13)
62
α ( s + 1/ α T ) 1 ( s + 1/ T ) I s 2 + 3ω 2 ( I − I ) y x z 0 Gcloop ( s ) = α ( s + 1/ α T ) 1 1+ 2 2 ( s + 1/ T ) I y s + 3ω0 ( I x − I z )
,
Gcloop ( s ) =
α ( s + 1/ α T ) , ( s + 1/ T ) I y s + 3ω02 ( I x − I z ) + α ( s + 1/ α T )
Gcloop ( s ) =
(20.495s + 0.145) . I y s + 0.145 I y s 2 + 20.495s + 0.155
(4.14)
2
3
Time response of compesated pitch axis 0.06
Steady state response(degree)
0.05
0.04
0.03
0.02
0.01
0
0
100
200
300
500
400
600
700
800
900
1000
time(sec)
Figure 4.4: Analytical result without solar perturbation. Result: Checking the time domain performance of the compensated pitch axis, The pitch axis is now stable and we have the settling time of about 500 secs for zero solar radiation torques. This result is understandable because we use the parameters of spacecraft which is over 5tonnes weight. Ref [20,21, 22] has produced similar results. In a real mission, the pitch attitude equation of the satellite is much more complicated .Consequently, there will be side effects such as the structural dynamics sloshing effects in the fuel tanks, sensor noise, solar radiation torques etc. However, the basic form of pitch dynamics will be perturbed by solar radiation torques alone in this thesis.
4.3. Lead compensator for Roll and Yaw Axes. The roll and yaw dynamics was separated in the analytical control synthesis for simplification but the parameters of the controller was simulated with coupled simulink block for the coupled roll/yaw dynamics. Separation of the roll/yaw dynamics for easy analysis lead to a simplify equations (4.15, 4.16) below, 63
I xφ + ω0 hwyφ = Tcx ,
(4.15)
I zψ + ω0 hwy 0ψ = Tcz ,
(4.16)
This control scheme also requires the measurement of the roll and estimation of yaw Euler angles, problem which is not covered in this thesis but will be assumed to be available for feedback control. The forward path transfer function of the uncompensated roll and yaw axis respectively is system is,
Φ(s) 1 1 = = , 2 2 Tcx ( s ) I x s + ω0 hwy 14,542 s + 0.00345
(4.17)
Ψ (s) 1 1 = = , 2 2 Tcx ( s ) I z s + ω0 hwy 12,356 s + 0.00345
(4.18)
4.3.1. Lead compensator for Roll axis. The forward path transfer function of the compensated plant is written as α ( s + 1/ α T ) 1 Gc ( s )G p ( s ) = 2 , ( s + 1/ T ) I x s + ω0 hwy
(4.19)
Reference to procedure for pitch axis, the phase margin of 65 was chosen for roll and pitch axis. • Determination of φm and α
φm = 65ο , Knowing φm , we can compute the value of α , • The corresponding value for the parameter α is = α
1 + sin φm 1 + sin 65ο 1.906 = = = 20.495 , 1 − sin φm 1 − sin 65ο 0.093
(4.20)
The value α = 20.495 corresponds to a maximum phase shift of 65ο .. z The compensator’s pole–zero combination is now related by the ratio = α= 20.495 . p • Determination of p and z from T .
64
Once α is determined, we now determine the value of T . This is accomplished by placing the 1 corner frequencies of the phase lead controller 1 α T and T such that φm is located at new gain-crossover frequency ωт so the phase margin of compensated system is benefitted by φm . Ref[] shows that the high frequency of the phase lead controller is 20 log10 α dB ,thus to have the new gain cross over at ωm which is the geometry mean of 1
1 α T and T as shown in
figure(4.1),we need to place the ωm at the frequency where the magnitude of uncompensated G p ( jω ) is −10 log10 α dB so that adding the controller gain of 10 log10 α dB to this makes the magnitude curve go through 0dB at ωm . • The following steps are taken to ensure that φm occurs at ωm .The high frequency gain of the phase lead controller of equation(4.4) is , = 20 log10 α dB 20 = log10 20.495 26.232dB ,
(4.21)
The geometry mean ωm of the two corner frequencies 1
1 α T and T should be located at the frequency at which the magnitude of the uncompensated process transfer function G p ( jω ) in dB is equal to the negative value in dB of one-half of this gain i.e. −10 log10 α dB .This way, the magnitude curve of the compensated transfer function will pass through the 0dB axis at ω = ωm .Thus, ωm should be located at frequency where,
G p ( jω )
dB
= −10 log10 α = −10 log10 20.495 = −13.100dB .
(4.22)
This can be determine either from data array in MATLAB using the bode function of uncompensated open loop of the roll axis or obtained graphically from the plot as done in this thesis. From figure (4.4), this frequency is found to be 0.0176 rad/s, now using equation (4.5), we have, 1 • The value of T can be determined from 0.0176* 20.495 = ω= = 0.0796 rad/s m α T 1 ⇒ = 0.0796 / 20.495= 3.888e − 3 rad/s. αT
65
Bode plot for uncompensated Roll axis
Magnitude (dB)
100
System: Roll_axis Frequency (rad/sec): 0.0176 Magnitude (dB): -13.1
0
-100
Magnitude (deg)
-200 0 Roll_axis -45 -90 -135 -180 -5 10
-4
10
-3
10
-2
-1
0
10 10 Frequency (rad/sec)
10
1
10
Figure 4.4: Graphical determination of frequency at −10 log10 αdB magnitude. The transfer function of the phase lead controller for this pitch axis dynamics with numerical values of the controller parameter is then represented as, α ( s + 1/ α T ) 20.495( s + 0.004) ( s + 1/ T ) = ( s + 0.0796) ,
(4.23)
• The original uncompensated system has the following transfer function: Φ(s) 1 = 2 Tcx ( s ) 14,542 s + 0.00345
• The close loop poles of the compensated system are as follows; Gcloop ( s ) =
Gc ( s )G p ( s ) 1 + H ( s )Gc ( s )G p ( s )
,
(4.24)
α ( s + 1/ α T ) 1 ( s + 1/ T ) 14,542 s 2 + 0.00345 Gcloop ( s ) = α ( s + 1/ α T ) 1 1+ 2 ( s + 1/ T ) 14,542 s + 0.00345
Gcloop ( s ) =
Gcloop ( s ) =
,
(4.25)
α ( s + 1/ α T ) ( s + 1/ T )14,542 s 2 + 0.00345 + α ( s + 1/ α T ) (20.495s + 0.0796) , 14,542 s + 1157.54 s 2 + 20.50 s + 0.0798 3
66
(4.26)
2
10
4.3.2. Lead compensator for Yaw Axis. The transfer function of the uncompensated yaw axis is system,
Ψ (s) 1 1 = = , 2 2 Tcx ( s ) I z s + ω0 hwy 12,356 s + 0.00345
(4.27)
The forward path transfer function of the compensated plant is written as α ( s + 1/ α T ) 1 Gc ( s )G p ( s ) = 2 , ( s + 1/ T ) I z s + ω0 hwy
(4.28)
The phase margin of 50ο was chosen for Yaw axis. • Determination of φm and α
φm = 50ο Knowing φm , we can compute the value of α • The corresponding value for the parameter α is 1 + sin φm 1 + sin 50ο 1.766 = = = = 7.612 , α 1 − sin φm 1 − sin 50ο 0.232
(4.29)
The value α = 7.612 corresponds to a maximum phase shift of 50ο . z The compensator’s pole–zero combination is now related by the ratio = α= 7.612 p • Determination of p and z from T . Once α is determined, we now determine the value of T . This is accomplished by placing the 1 corner frequencies of the phase lead controller 1 α T and T such that φm is located at new gain-crossover frequency ωт so the phase margin of compensated system is benefitted by φm . Ref[] shows that the high frequency of the phase lead controller is 20 log10 α dB ,thus to have the new gain cross over at ωm which is the geometry mean of 1
1 α T and T as shown in
figure(4.1),we need to place the ωm at the frequency where the magnitude of uncompensated G p ( jω ) is −10 log10 α dB so that adding the controller gain of 10 log10 α dB to this makes the
magnitude curve go through 0dB at ωm . 67
The following steps are taken to ensure that φm occurs at ωm . The high frequency gain of the phase lead controller of equation(4.4) is = 20 log10 α dB 20 = log10 7.612 17.630dB ,
(4.30)
The geometry mean ωm of the two corner frequencies 1
1 α T and T should be located at the frequency at which the magnitude of the uncompensated process transfer function G p ( jω ) in dB is equal to the negative value in dB of one-half of this gain i.e. −10 log10 α dB .This way, the magnitude curve of the compensated transfer function will pass through the 0dB axis at ω = ωm .Thus, ωm should be located at frequency where
G p ( jω )
dB
= −10 log10 α = −10 log10 7.612 = −8.815dB .
(4.31)
This can be determine either from data array in MATLAB using the bode function of uncompensated open loop of the yaw axis or obtained graphically from the plot as done in this thesis. From figure (4.5), this frequency is found to be 0.015 rad/s, now using equation (4.5),we have, 1 • The value of T can be determined from 0.015* 17.630 = ω= = 0.0630 rad/s m α T 1 ⇒ = 0.0630 /17.630= 3.572e − 3 rad/s. αT Bode plot for uncompensated Yaw axis
100
Magnitude (dB)
50 0
System: Yaw_axis Frequency (rad/sec): 0.015 Magnitude (dB): -8.84
-50 -100 -150 -200 0
Magnitude (deg)
Yaw_axis -45
-90
-135
-180 -5 10
-4
10
-3
10
-2
10
-1
0
10
10
1
10
2
10
Frequency (rad/sec)
Figure 4.5: Graphical determination of frequency at −10 log10 αdB magnitude. The transfer function of the phase lead controller for this pitch axis dynamics with numerical values of the controller parameter is then represented as, α ( s + 1/ α T ) 17.630( s + 0.003) ( s + 1/ T ) = ( s + 0.0630) ,
(4.32)
68
The close loop transfer function is Gc ( s )G p ( s ) , Gcloop ( s ) = 1 + H ( s )Gc ( s )G p ( s )
(4.33)
Ψ (s) 1 1 = = 2 2 Tcz ( s ) I z s + ω0 hwy 12,356 s + 0.00345 α ( s + 1/ α T ) 1 ( s + 1/ T ) I s 2 + ω h z 0 wy Gcloop ( s ) = α ( s + 1/ α T ) 1 1+ 2 ( s + 1/ T ) I z s + ω0 hwy
,
(4.34)
Gcloop ( s ) =
α ( s + 1/ α T ) ( s + 1/ T )( I z s 2 + ω0 hwy ) + α ( s + 1/ α T )
Gcloop ( s ) =
(17.630 s + 0.0630) , 12356 s + 778.428s 2 + 17.633s + 0.0632
(4.35)
3
Analytical Plot for uncoupled for Roll and Yaw Axes. Step Response of uncompensated Roll/Yaw Dynamics 12
Time response for yaw dynamics for 0.2 degree step input
10
Roll/Yaw axis(Degree)
8
6
4
2
Time response for Roll dynamics for 0.05 degree step input 0
uncompensated Yaw axis uncompensated Roll axis -2
0
5000
10000
Time(secs)
Figure 4.6 : Step response for uncompensated Roll and Yaw Dynamics.
69
15000
Step Response of compensated Roll/Yaw Dynamics 0.25
Time response for yaw dynamics for 0.2 degree step input
Roll/Yaw axis(degree)
0.2
0.15
Compensated Yaw axis Compensated Roll axis
0.1
Time response for Roll dynamics for 0.05 degree step input 0.05
0
0
500
1000
1500
2000
2500
3000
Time(secs)
Figure 4.7: Step response for compensated Roll and Yaw Dynamics. Result: Checking the time domain performance of the compensated Roll and Yaw axis dynamics, The Roll and Yaw axis is now stable and we have the settling time of about 600secs for zero solar radiation torques. This result is understandable because of the parameters of Nigcomsat-1 which is over 5 tonnes weight. Ref [20, 21, 22] has produced similar results.
4.4.
Comparative design and simulation of all the wheels configurations.
Simulation studies are to be conducted here because available analytical results from phase lead compensator for roll, pitch and yaw are inadequate for our comparism analysis. On this basis close to reality simulation studies are to be used as a design tool as well as to demonstrate system performance, stability, and compliance with design resign requirements and specifications. To achieve realistic system response, close to each wheel configuration, respective wheel dynamics, thruster and space environment are included in the simulator. Four different types of wheels configurations investigated in this thesis will be simulated using their corresponding mathematical model implemented as simulink block simulator and are stated as follows: •
Type a: Pitch momentum wheel with offset roll/yaw thrusters for attitude control (one wheel).
•
Type b: Two symmetrically inclined momentum wheels in a V configuration (Two wheels).
•
Type c:Three reaction wheels oriented along roll, pitch and yaw axis(three reaction wheels)
70
•
Type d: One pitch momentum wheel on y axis and two reaction wheels on roll and yaw axes (One momentum wheel and two roll/yaw reaction wheels).
All the four types of configuration for attitude control systems provide conventional wheel control for the pitch error by either modulating a single reaction wheel, a body fixed momentum wheel or skewed body fixed momentum wheels ,The control law along the pitch axis are always the same for all the four configurations[ 22 ].The main different between the various configurations however lies in the approach to attitude determination and control torques for roll and yaw attitude which when implemented leads to different pointing accuracies and other performance efficiencies[20]. The analytical tuning parameters for the roll, pitch and yaw compensators as shown in figure 4.8 will now be implemented in simulink block simulator so as to compares all the wheels configurations.
Figure 4.8:Common control solutions to all the wheels configurations. we investigate the designed controllers on the coupled and more complex simulink block simulator interfaced with solar radiation torques disturbances and other actuator simulators. For pitch wheel bias momentum spacecraft, the roll/yaw axis control will be performed independent of pitch axis. Gyroscopic coupling effect by momentum wheel is exploited for roll/yaw control. Since we consider a generic model of a spacecraft equipped with three wheels aligned along roll, pitch and yaw axes similar to type c and type d. The general model that will be used on the basis of the various wheels configuration for control design will now be modified as shown in table 4.1 below. Possible control schemes for various wheels momentum bias configuration.
71
Wheels configuration type
a
type
Total wheels
Pitch control
Roll control
Yaw control
Sensors
1MW
Tcy = -h wy(MW)
Tcx(thrusters)
Tcz(thrusters)
Earth sensor
hwx =0
hwz = 0
b
2MW+1RW(backup)
type
c
Skewed angle of 20 degrees 3RW+1RW(backup)
type
d
1MW+2RW
Tcy = -h wy(MW)
Tcx = 0 -h wx = 0
Tcz = -h wz(MW)
Earth sensor
Tcy = -h wy(RW)
Tcx = -h wx(RW)
Tcz = -h wz(RW)
Tcy = -h wy(MW)
Tcx = -h wx(RW)
Tcz = -h wz(RW)
Earth/Star sensor Earth/Star sensor
Table 4.1: Modifications of Tcx , Tcy , Tcz in terms of wheels configurations. In order to compare the performance of each control system in the presence of solar radiation torques disturbances, The control signal is turned on after 200 seconds for conveniences from uncompensated open loop plant to compensated momentum biased controlled plant. The time response of each wheels configuration will then be compared in term of settling time, pointing accuracies and other qualitative factors. This time can be adjusted in the automatic switch implemented in this control thesis as shown below by changing the value in the constant block.
Figure 4.9: Automatic controller switch simulator.
4.4.1. Compensated Simulink control design for type a configuration. A single momentum wheel as shown in figure 4.11 was the primary control element responsible for ensuring gyroscopic stability of the satellite. Additional control equipment, such as reaction jets was used to fine-tune the attitude accuracy by controlling nutational motion about the rollyaw satellite axes. An earth sensor was the only sensor used to implement the control laws and the roll and yaw attitude angles can be stabilized close to null in this configuration. Type a concept wheel configuration has being implemented successfully on some space vehicles in GEO including Intelsat V, Satcom and TV sat for three axis stabilization. 72
OFFSET THRUSTERS
z-axis α
YAW
OFFSET ANGLE
ROLL
2
x-axis
PITCH y-axis
Figure 4.10:Type a wheel configuration. In this configuration, A single momentum wheel is usually aligned along the orbit normal direction while the wheel itself retains a certain angular momentum level. The wheel speed in this configuration is usually adjusted about the nominal value to control the pitch error. The angular momentum of the wheel result in gyroscopic stiffness effect about the orthogonal body axes of roll and yaw. The gyroscopic stiffness contributes to maintaining pointing accuracies with respect to external disturbance inputs. The roll attitude is corrected by corresponding offset thrusters to remove the momentum accumulation on the wheel. The roll/yaw control system must accomplish two requirements. First, it must attenuate the external disturbances on the spacecraft. These are at harmonics of the orbit rate and generally have no significant components above twice orbit rate. The second requirement is nutation damping. Since there is no passive source of nutation damping, the control system must damp the nutation. Since the nutation frequency is much higher than orbit rate the controller is broken into two parts, one to attenuate low frequency disturbances and the second to damp nutation. The type a wheel configuration uses reaction jets to stabilize the attitude motion of roll and yaw axis. Unlike momentum and reaction wheels, a reaction jet consists of two values; on or off. Furthermore, most on-off reaction jet control systems are in practice pulse modulated. However for simplification of our analysis we consider a common control law called bang bang proportional thrusters for the stabilization of roll and yaw axes [22]. A bang-bang control is defined by,
T=
−θ
θ
Tmax = −Tmax signθ ,
(4.36)
where Tmax the maximum is control torque and θ is the angular error. A block diagram for a bang-bang plus dead zone controller system is shown in figure 4.11.
73
Figure 4.11: block diagram for a bang-bang plus dead zone controller system.
Here the control torque is characterized by a dead zone followed by a maximum torque. The law in functional form is, T = f (θ )Tmax ,
(4.37)
Where
f (θ ) = −1 for θ > θ max , ≡ 0 for −θ max ≤ θ ≤ θ max , ≡ 1 for θ < −θ max ,
and θ max is the half-width of the dead band. The simulink representation of bang-bang control is represented as figure 4.12,
Figure 4.12: simulink representation of bang-bang control. The control torque depends only on the sigh of the difference between the desired and the actual output. So far, the pitch control via a momentum wheel and roll/yaw control using on-off bangbang actuators are applied to Type a bias momentum spacecraft using linearized and simplified models with control torques defined as in table 4.1. For further analysis in this configuration, we simulated autonomous roll/yaw attitude control with thrusters or ground based control. The normal mode will start after 200 seconds as implemented in this thesis for ground based roll/yaw attitude control or will start immediately with simulator for autonomous control system using on board computer.
74
Type a, Type c and Type d Pitch control configuration.
Figure 4.13: Simulator for compensated type a, type c and type d pitch dynamics.
Type a Roll/yaw control from ground station.
Figure 4.14: Roll/yaw thruster control using time tagged command from ground station. 75
Type a Roll/Yaw On board automatic control.
Figure 4.15: Roll/yaw thruster control using on board automatic controller.
4.4.2. Compensated Simulink control design for type b configuration. The evolution of the inclination vector for geostationary satellites caused changes in the orbit's inclination angle of about 0.7°-0.9° per year. The task of the station keeping process is to restrict this inclination to within ±0.05°. However, some satellite missions require pointing the payload antennas at different ground targets, a task that can be accomplished by appropriate changes in roll and pitch attitudes. Using two symmetrically inclined momentum wheels in a V configuration as shown in figure 4.17 allows control of both the pitch and the roll angles, while exploiting the feature of inertial attitude stabilization that keeps the yaw error close to null without being forced to measure it (Wie et al. 1985). Moreover, sensor noise amplification of type a configuration is no longer a problem because the torque control capabilities of momentum exchange devices are much higher than any amplified noise levels.
76
OFFSET THRUSTERS
α
z-axis
2
OFFSET ANGLE
YAW 3 SKEWED ANGLE
ROLL PITCH
x-axis
y-axis
Figure 4.16: Type b momentum bias configuration. Two skewed momentum wheels configurations was used on Nigeria communication satellite launched in 2007(Nigcomsat-1), SEASAT and Indians communication satellite series (INSAT) and some other popular communication satellites in GEO. The control system for this configurations are made up of Sensors, controllers and actuators., The components of the control systems are as follows, the sensor for normal orbit mode are two axis infrared earth sensors that measure roll and pitch attitude references and the actuators are made up of two momentum wheels skewed at an angle ($) with respect to the pitch axis and the yaw reaction wheel (H3) for back up mode in case of failure of any of the nominally operating wheels (H1 or H2). The angular momentum of the two wheels is represented as H1 and H2 respectively and yaw wheel as H3. There is also a yaw wheel desaturation thrusters to unload the wheel when it exceeded its nominal speed. The x axis in nominally in the flight direction, the y axis is normal to the orbit plane, and z is directed towards the nadir (Earth).Such roll, pitch and yaw control axes are nearly coincident with the axes of the vehicle. The configuration of the wheels in its body coordinate is shown in Figure.4.17.
77
Figure 4.17: Two symmetrically inclined momentum wheels in a V configuration. The two skewed angle of 20 degree used on Nigcomsat-1 was adopted for simulation and the generalized dynamic equation of motion, equations (2.18, 2.19, 2.20), will be modified for the configuration of the two wheels. The two wheels are the primary sources of angular momentum bias. Their momentum axes lie in the y-z body plane, and they deviate from the y axis by an angle $, as shown in the figure. The net effect is that a momentum bias of Hty = ( H 1 + H 2) cos $ is aligned along the y axis and Htz = ( H 1 − H 2) sin $ aligned along z axis.
Hty H1
I
$
$
0
Htz
Z - AXIS
Y- AXIS
Figure 4.18: 2-MW control configuration. Using the torque capabilities of the two momentum wheels, the torque command equations can be written in the following form: = Tcx Htx = 0 (No wheels component in x direction),
(4.35)
, Tcy = H 1cos $ + H 2 cos $ = Hty
(4.36) 78
, Tcz = H 1sin $ − H 2sin $ = Htz
(4.37)
This can be represented in matrix notation as:
Tcy cos $ cos $ H 1 T = sin $ − sin $ , H 2 cz
(4.38)
For the required torque commands determined by any adequate attitude controller such as lead compensator in this thesis, the wheels should provide the following torques: The total Momentum and corresponding derivative produced by the momentum wheels to the body axes is defined respectively as,
hx hx h = hy + hwy = hy + ( H 1 + H 2) cos $ , hz + hwz hz + ( H 1 − H 2) sin $
hy + hwy ≈ hwy hz + hwz ≈ hwz
(4.39)
,
(4.40)
where , h = hx
hy
T
hz ,
hwy H 1cos $ + H 2 cos $ , =
= hwz H 1sin $ − H 2sin $ , = hwy H 1cos $ + H 2 cos $ , = hwz H 1sin $ − H 2sin $ , can be and H2 Consider the linear system which in matrix format is defined as above, then, H1 found with Cramer's rule as, −1
cos $ cos $ Tcy H1 = , H2 sin $ − sin $ Tcz
(4.41)
where H 1, H 2 can further be simplify as
= H1
Tcy T cz
− cos $ cos $ , ∆
(4.42) 79
− sin $ Tcy = − sin $ Tcz , H2 ∆
(4.43)
cos $ cos $ Where ∆ is the determinant of = −2sin $ cos $ , sin $ − sin $ = H1 = H2
Tcy 2 cos $ Tcy 2 cos $
+
Tcz , 2sin $
(4.44)
−
Tcz , 2s in$
(4.45)
The control of pitch axis in primary mode is implemented by modulating the pitch bias momentum hwy of the two momentum wheels about their nominal bias momentum say hwy 0 in response to the error signal θ from pitch channel of the Earth sensor. The roll/yaw axes are controlled by differentially modulating the yaw angular momentum hwz of the two momentum wheels in response to the error signal φ from roll channel of the Earth sensor. The momentum command distribution matrix is used to convert the pitch and yaw momentum commands to the wheel momentum commands.The simulink block for the wheels dynamics as represented by the above mathematical model is defined by figure 4.19 below,
Figure 4.19: Dynamics of two symmetrically inclined momentum wheels.
80
Type b control Simulink simulator The result of analytical design of phase lead compensator for momentum bias spacecraft will be implemented in simulink block similar to type a configuration but with modifications of skewed momentum wheels dynamics and the skewed angle and the torques in x,y and z directions are produced by wheels torques as indicted in equation().The technical specifications of momentum wheels used on Nigcomsat-1 will be adopted for this control synthesis. According to the model of skewed wheels dynamics, the choice of the skewed angle depends very much on the control torque capabilities about the Z axis, which will be sufficient to deal with the anticipated level of external disturbances. The control loop architecture as in figure shows that, during the on orbit normal mode operations, the yaw attitude error is neither measured nor is estimated. The yaw error in a limit cycle however controlled indirectly by yaw momentum loop with the measurement of wheel yaw momentum. The yaw momentum control loop provides active roll but passive yaw control with secularly increasing yaw error due to external disturbances [4]. Two controllers are implemented in this investigation because the roll/yaw and pitch axis are decoupled: one pitch controller, a normal mode roll/yaw controller according to ref[21] has a relation i.e. Tcx = −aTcz since the torques due to skewed wheels is zero as shown in equation(4.35).here a is the proportional tuning parameter for reducing the roll error. The roll control gains are used to control the roll/yaw of the spacecraft. The design approach is to only use the roll controller and roll feedback to control both roll and yaw axes. The roll and yaw attitude performances were then monitored. The simulink block diagram based on the models of the compensated control problems in this thesis and methodology of feedback control is shown in figure below.
Figure 4.20: Simulink block for type b roll/yaw and pitch attitude control architecture.
81
4.4.3. Compensated Simulink control design for type c configuration. The payload axis of spacecraft with this configuration pointing towards local vertical is achieved by employing three reaction wheels, one for each axis, yaw, roll, and pitch in a zero momentum configuration. The Earth sensors provide continuous information of roll and pitch errors whereas the yaw information is obtained from a gyro based reference. This gyro is updated twice an orbit near the poles, taking yaw information from the Sun. The controller is a pulse width pulse frequency modulator (PWPFM) employing pseudo rate damping techniques to obtain a highly stable configuration as demanded by the remote sensing payload. A skew wheel is provided, making an angle tan pie/2 with all the three axes (Figure. 4.24). This wheel provides redundancy to the failure of any of the three wheels.
z-axis 3
YAW 1 2
PITCH
ROLL x-axis
y-axis
Figure 4.21: Three reaction wheels configuration. This is also called zero momentum bias configurations because the three reaction wheels are oriented equally along the roll, pitch and yaw axis. Because the disturbance torques in high GEO are very small (solar radiation torques), it is possible to use small reaction wheels to absorb them with an active control system to maintain three-axis stability. In such a system, gyroscopes are generally used to sense and feedback anybody motion to the wheel torque motors on each axis. The torque motors then apply a compensating torque to each reaction wheel, which effectively absorbs the disturbance torques. Thus, the angular momentum vector changes slowly with time, and the attitude remains fixed in inertial space. When the wheels near saturation, the angular momentum is adjusted using gas jets or magnetic coils. Ideally, the attitude is controlled to the same steady-state value during desaturation, although in practice transient attitude errors are induced. A slew, or attitude reorientation maneuver, can be executed using the set of reaction wheels to rotate the body about a commanded axis, usually one of the wheel axes, the angular momentum vector remains inertially fixed, although the attitude angles change as do the angular momentum components in a body-fixed coordinate system. For example, the x axis wheel might approach saturation at the final attitude just to absorb the larger momentum component. In addition to estimating the attitude, it is also necessary to keep track of the wheel momenta for calculating 82
momentum dumping commands and slew execution times, the effect which is not considered in this thesis. Type c configuration provides active and continuous control of roll by modulating the roll wheel, pitch wheel by modulating the pitch wheel and yaw by the yaw wheel. It also has low net inertial momentum because it can be controlled by reaction wheels in all the three axes, though the wheels may each have significant momentum but ideally the vector sum of their momentum is zero. Using the torque capabilities of the roll, pitch and yaw reaction wheels, the torque command equations can be written in the following form: Wheels Dynamics for type c configuration [22].
Tcx = hwx − ω0 hwz ,
(Reaction wheel)
Tcy = hwy ,
(Reaction wheel)
Tcz = ω0 hwx + hwz ,
(Reaction wheel)
,
(4.46)
,
(4.47) ,
(4.48)
The simulink block diagram based on the models of the compensated type c configuration for roll and yaw axis in this thesis and methodology of feedback control is shown in figure below.
Figure 4.22: Reaction wheel Roll and yaw axis compensator. For Type c configuration, three orthogonal reaction wheels will be the minimum requirement.A redundant fourth is normally added at an equal angle to the other three as shown in figure .so as to avoid a single point failure.
83
Fig 4.23: Redundant based three reaction wheels system.
4.4.4. Compensated Simulink control design for type d configuration
z-axis 3
YAW 1 2
PITCH
ROLL x-axis
y-axis
Figure 4.24: Two reaction wheels, one pitch bias momentum wheels configuration. This configuration is a generic model similar to type c configuration but equipped with two reaction wheels in aligned along roll and yaw axes and pitch momentum wheel(type c uses pitch reaction wheel) as illustrated figure. This body fixed momentum wheel with two reaction wheels oriented along roll and yaw axis has similar control structure and simulation results with type c configuration except momentum wheel oriented in pitch axis instead of reaction wheel. There is therefore limit to maneuverability in the roll and yaw axis due to gyroscopic stiffness in pitch axis because of the momentum wheel 84
bias. This configuration provides active and continuous control of roll by modulating the roll wheel, and yaw by the yaw wheel as in type c configuration. Wheels Dynamics
Tcx = hwx − ω0 hwz , Tcy = hwyo ,
(Reaction wheel)
(4.49)
(Modulating momentum wheel)
Tcz = ω0 hwx + hwz , (Reaction wheel)
(4.50) (4.51)
The simulink block diagram based on the models of the compensated control problems in this thesis and methodology of feedback control is shown in figure below.
Figure 4.25: Reaction wheel Roll and yaw axis compensator.
4.5. Investigation of the two wheels Skewed angles. Mmomentum bias mechanization is only useful for small attitude motion control about a nominal attitude with the bias nominally orbit normal [14]. However, for selected spacecraft applications, it can be often useful to operate the spacecraft in an upright orientation (pitch axis anti-orbitnormal) at some times during the mission and in an inverted orientation (pitch axis orbit-normal) during other portions of the mission. For example, this capability is useful in establishing new antenna coverage areas or in providing for other new missions. 85
Further, there are a large number of momentum bias stabilized spacecraft presently on orbit. As a result of proliferation in the number and uses of spacecraft, coupled with the aging and replacement of such satellites, the desire to move them and invert the attitude to establish new antenna coverage areas is becoming more frequent. The two momentum wheels skewed configuration (type b) is therefore invented [10] to satisfy that need. In symmetrical skewed two wheels architecture, the vehicle can be operated in upright or inverted attitude with no change to the control architecture or implementation. This allows the many momentum bias spacecrafts of this architecture on orbit to perform new and unforeseen missions. Despite various space vehicles using the same two wheels configurations, the skewed angles for the two wheels placement varies drastically for different satellites for examples, geostationary operational environmental satellite (GOES N-P) has its two momentum wheels skewed at an angle of ±1.66° to the pitch axis [10], Nigcomsat-1 momentum wheels was skewed at angle of ±20°, Anik E2 has its skewed wheels 2.5° [9] etc. The above disparity in the skewed angles for different space vehicles in the same orbit serve as motivation to find the effect of this skewed angle on the multiple mission tasks. The two wheels configuration must fulfilled the following properties, the two wheels should be momentum wheels which have the same inertia of the same order of magnitude, the two wheels are to be located so that the actuator momentum has a component directed opposite to the positive direction of the pitch axis [2]. In this investigation, it was assumed that all the spacecrafts with this configuration has fulfilled this requirement.
4.6. Method of Investigation. (Wie et al. 1985) has previously states that using two symmetrically inclined momentum wheels in a V configuration (type b) allows control of both the pitch and the roll angles, while still exploiting the feature of inertial attitude stabilization that keeps the yaw error close to null without being forced to measure. It is on this basis that the skewed angle will be investigated. The torques on two symmetrical wheel configurations has being previously defined as follows form equations (4.35, 4.36, 4.37), = Tcx Htx = 0 (No wheels component in x direction),
, Tcy = H 1cos $ + H 2 cos $ = Hty , Tcz = H 1sin $ − H 2sin $ = Htz
The total Momentum and corresponding derivative produced by the momentum wheels to the body axes is defined respectively by equation (4.39) as, 86
hx hx h = hy + hwy = hy + ( H 1 + H 2) cos $ , hz + hwz hz + ( H 1 − H 2) sin $ The dynamics of the skewed wheels was used to implement the simulink block as in figure and for this thesis the external torques are simulated with solar radiation torques of figure() . It can be shown from the equations above that all the equations depends on the skewed angle ($).This dependency is exploited here by building a simulator where the skewed angle can be varied for Nigcomsat-1( 200 ),GEOS N-P( 1.660 ), Anik E2( 2.50 ), 600 and 900 skewed angles. The 900 was chosen specially because at this angle the effect of the wheels on the vehicle is zero as shown in the analytical solution of the wheel dynamics below. 0 (No wheels component in x direction), = Tcx Htx =
= Tcy = H 1cos 90 + H 2 cos 90 = Hty 0 , ( H 1 + H 2) cos 90 = 0 , ( H 1 − H 2) sin 90 ≈ 0
hx hx h , h = hy + hwy = y hz + hwz hz
Figure 4.26: Simulator for investigation of skewed angle. 87
CHAPTER 5 5. RESULTS AND DISCUSSION. 5.1. Dynamics of solar radiation Torques Model This section presents the attitude performances of the type a, b, c and d wheel configurations based on the time response plots of the respective pitch and roll/yaw architectures as a function of the wheel configurations. The numerical evaluation is done using defined mission parameters and constraints of Nigcomsat-1. For the attitude performance evaluation, the nature of the solar radiation torques disturbances model used for simulation is shown in Figure 5.1a .The open loop response of solar torques using mass properties of Nigcomsat-1 was also shown in figure 5.1b for one year duration.
Figure 5.1a: Simulink Mathematical model of solar radiation torques [42].
88
Magnitude of solar radiation torques on x,y and z axis(Nm)
One year Solar radiation torques on Nigcomsat-1
-6
6
x 10
4
2
0
Tdx Tdy Tdz
-2
-4
-6
0
1
2
3
4
5
6
7
8
9 4
Time (s) from January to December
x 10
Figure 5.1b: Plot of solar radiation pressure on Nigcomsat-1 for one year duration.
5.2. Pitch Axis Attitude Response
5.2.1 Pitch Attitude Response for type a, type c and type d. The pitch control architecture for type a, type b and type c is the same as shown in their respective simulink blocks. The result from their plot shows similar response as in figure 5.2. The plot indicates that the pitch attitude is stabilized with minimal steady state error for 24 hours attitude stabilization and control i.e. before the wheel becomes saturated.
compesated pitch axis with solar torques 0.06
Steady state response(degree)
0.05
0.04
0.03
0.02
0.01
0
0
100
200
300
400
500
600
700
800
900
time(sec)
Figure 5.2: Pitch attitude responses perturbed with solar radiation torques for type a, c and d.
89
1000
5.2.2 Pitch Attitude Response for type b. By tuning the skewed angle of the momentum wheels in type b configuration, the maximum pitch attitude pointing accuracy is improved close to the response of type a, c and d configuration. The skewed angle of 20° was found to have being optimized to give good pitch axis attitude response as shown in the plot in figure 5.3 for Nigcomsat-1. Time Response for Two skewed momentum wheels Pitch axis dynamics 0.06
Pitch axis steady state Output(Degree)
0.05
Simulation Perturbed with solar radiation torques.
0.04
Pitch axis Dynamics
0.03
0.02
0.01
0
0
500
1000
1500
Simulation time(seconds)
Figure 5.3: Pitch axis dynamics perturbed with solar radiation torques for type b.
5.3. Roll/Yaw Attitude Response The results from all the four wheels configuration obtained from their respective simulink block were analyzed. Cases involve are: i.
ii. iii. iv.
Roll/yaw attitude performance based on the roll/yaw control using reaction jet(thrusters) either by actuating the thruster from satellite control station or by automatic means using on board computer.(Type a) Roll/yaw attitude performance based on the Roll control through roll error and yaw actuator and controller but without sensing yaw (Type b). Roll/yaw attitude performance with a separate controller for roll and yaw and both axis sensed. (Type c). Roll/yaw attitude performance with a separate controller for roll and yaw and both axis sensed (Type d).
90
5.3.1 Simulink Simulation results for type a momentum wheel configuration.
Simulated Roll/Yaw Compensated Dynamic response using ground command 250
Roll/Yaw Dynamic response(Degree)
200
Solar radiation torques was simulated as sources of space disturbance torques.
150
Yaw Response Roll Response
100
50
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
8000
9000
10000
Simulation time(seconds)
Figure 5.4:Roll/Yaw attitude response using ground command. Autonomous Roll/Yaw thrusters Attitude Control using bang bang system 0.35
Roll Response Yaw Response
Roll/Yaw Dynamic response(Degree)
0.3
Simulation perturbed with solar radiation torques. We can notice the effects of continous firing of the thrusters in this automatic mode. 0.25
0.2
0.15
0.1
0.05
0
0
1000
2000
3000
4000
5000
6000
7000
Simulation time(seconds)
Figure 5.5: Autonomous Roll/yaw axis dynamics perturbed with solar radiation torques. Discussion. We simulated roll/yaw attitude control from ground station and automatic control using on board controller. For ground based attitude control, the control signal is turned on 200 seconds after the attitude control loop simulator starts. It can be shown from the plot that initial attitude is controlled from unstable to stable immediately after 200 seconds, the attitude error is also eliminated in the entire three axes. The phase lead compensator therefore achieved the design objective both using ground command and using on board computer. Another noteworthy point is nutational motion induced by the wheel angular momentum. The initial nutational motion is 91
also controlled quickly after activation of the controller. The phase lead controller cancelled gyroscopic terms due to pitch wheel so that nutational motion is controlled quickly. The effect of the thruster continuous firing and mis-firing in autonomous bang-bang controller for roll/yaw is visible due to non smooth of the roll/yaw dynamics response even though stable. The response also indicates that attitude stabilization scheme using reaction thrusters with achievable accuracies depend largely on the minimal impulse bit that a thruster can deliver. Also, since the torques delivered is with constant amplitude, the reaction pulses must be width- or frequency-modulated to achieve a better attitude response. The thruster actively damped the roll/yaw dynamics quickly but at the expense of fuel consumption. The result of this simulation shows that the use of rocket jet to control roll and yaw attitude for autonomous attitude control is fuel consuming, non smooth output, and less reliable due to thruster bang-bang continuous activations. Autonomy is no panacea for this type of configuration in rolls and yaw axis also because of the added hardware cost, complexity, and weight. The ground base control for roll/yaw reaction jet actuator on the other hand, though very smooth output, is susceptible to large attitude deviation if there is delay in attitude control command from ground station as clearly shown in the plot of ground based band-bang roll/yaw attitude control. The single-momentum wheel scheme has two principal drawbacks. First, with an earth sensor as the only sensor used to implement the control laws, amplification of sensor noise precludes adequate damping of the nutation pole. The second inherent drawback is that the roll and yaw attitude angles can be stabilized close to null [].
5.3.2 Simulink Simulation results for type b momentum wheel configuration. Simulation Result for Roll and Yaw axis . Time Response for Roll Dynamics of two Skewed momentum wheels configuration Roll Response(Degree)
0.05 0.048
Simulation perturbed with solar radiation torques and roll error detected with earth sensor.Roll and Yaw axis has coupling effect and at every one quarter of the orbit the roll error translate to yaw error and vice versa.Ref [22]
0.046 0.044 0.042 0.04
Roll axis response 0.038 0.036
0
0.5
1
1.5
2
Simulation Time(Seconds)
4
x 10
Time Response for Yaw Dynamics of two Skewed momentum wheels configuration Yaw Response(Degree)
0.15 0.1 0.05 0
Simulation perturbed with solar radiation torques and yaw error not measured but estimated using on board observer.The yaw error is neither estimated nor measured in this thesis.This is visible in this plot as the yaw response is loosely control with high pointing error.
-0.05 -0.1 -0.15
Yaw axis Response -0.2
0
1
2
3
4
5
6
Simulation Time(Seconds)
Figure 5.6: Roll/yaw axis dynamics perturbed with solar radiation torques. 92
7
8
9 4
x 10
Discussion. The phase lead compensator achieved the design objective along roll and pitch axis for type b configuration. Another noteworthy point is nutational motion induced by the wheel angular momentum as seen in the initial stage of roll and yaw responses. The phase lead controller cancelled gyroscopic terms due to pitch wheel so that nutational motion is controlled quickly. The effect of the skewed wheels controller for roll/yaw is visible due to poor error of the yaw dynamics output. This configuration has few attitude controls on board equipments but poor pointing accuracies in the yaw axis as shown in the plot. An on board observer can be implemented to estimate the yaw error so as to improve the yaw pointing accuracies, this is however not investigated in this thesis. The configuration also has very poor maneuverability in roll and yaw axis. On this basis, this configuration requires that all the three axes should be sufficiently damped before switching on the normal mode using this controller. In practical case, a mode known as transition mode (TRM) are normally implemented as shown in figure 4.7 as a transition between high altitude error and normal mode steady state response error. We also observed that, Implementation of 200 seconds automatic switching does not work in this configuration because when the attitude error is more that the control authority of the controller, the plant becomes unstable especially in roll and yaw axis. The type b configuration spacecrafts are therefore sensitive to disturbance in roll and yaw axis. They however consume less fuels compare with type a configuration.
Roll axis steady state response(degree)
5.3.3 Simulink Simulation results for type c momentum wheel configuration.
Time Response for Reaction wheel Roll control 0.1
Simulation perturbed with solar radiation torques.We can see the smooth output produced by Reaction wheel torques compared with thruster torques as in type a wheel configuration.This configuration is therefore higly efficient for attitude control in the roll,pitch and yaw axis with very high pointing accuracies.
0.08
0.06
0.04
0.02
0
0
500
1500
1000
2500
2000
Simulation Time(seconds)
Yaw Steady state response(Degree)
Time Response for Reaction wheel yaw control 0.4 0.3 0.2 Simulation perturbed with solar radiation torques.We can see the smooth and high pointing accuracy of the yaw axis compare with type a and type b wheel configuration.This configuration indeed has higher efficiency though at the expence of complex attitude control scheme and expensive star sensor.
0.1 0 0
Yaw Response
500
1500
1000 Simulation Time(seconds)
Figure 5.7: Roll/Yaw Simulink results for type c wheel configuration.
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2000
2500
Discussion. It can be shown from the plot that reaction torques from the wheels eliminate the attitude error in the entire three axes. This configuration therefore has low net inertial momentum because despite the fact that each reaction wheel have significant momentum in their respective axis but ideally the vector sum of their momentum is zero. The entire three axis in the type c configuration are actively equally controlled. The phase lead compensator achieved the design objective in the three axis reaction wheels control. The initial nutational motion is also controlled quickly after activation of the controller. The result of simulation shows that the advantages of a three-axis stabilized reaction wheel system are: (I) capability of continuous high-accuracy pointing control, (2) large-angle slewing maneuvers without fuel consumption and (3) compensation for cyclic torques. The result shows that closed-loop control can provide a significant improvement in accuracy compare with Type a and type b system, which requires frequent, complex, and expensive ground-based operational support to maintain yaw pointing accuracy. Overall this configuration is highly efficient for various performance design specifications including the above mention performances.
5.3.4 Simulink Simulation results for type d momentum wheel configuration.
Figure 5.8: Roll/Yaw Simulink results for type d wheel configuration. Discussion. This configuration has similar performance with type c configuration except the momentum bias induced by the pitch momentum bias in the pitch axis. The result of type d configuration demonstrate that body fixed momentum wheel along pitch axis and two reaction wheels along roll and yaw axes is the most robust and closely followed by type c. 94
5.4. Comparative Results for type a, type b, type c and type d configurations. The following results are from qualitative analysis of the various wheels configuration using their respective simulink block. The time response of the wheel configuration are shown in the plots and the control law parameters for roll, pitch and yaw axis are adequately designed using phase lead compensators. It is shown in the respective plots that despite the enormous size and mass properties of the spacecraft considered in this thesis, the dynamics loop of all the wheels configuration in their respective axis was adequately shaped to the desired output with the phase lead compensators. This stability is guarantee as long as the actuators and the sensors are collocated at the center core cylinder as in Nigcomsat-1 primary structure module. In order to compare the performances, we try to introduce various dummy actuator schemes i.e. bang-bang to simulate thrusters, skewed wheels to simulate two wheels modulation, time tagged so simulate automatic switch on and dead band to constrain the output of the various actuators. All these dummies differentiate between various wheel configurations. Results are shown in table for the entire wheel configurations as a function of weight, cost, complexity, pointing accuracy, reliability, fuel consumption, and stability margin. Wheel Type
Pointing accuracy
Stability margin
Robustness
Response Time
Fuel
Type a
pointing accuracy in roll, pitch and yaw axis but at the expense of fuel
Lower than type c and type d.
Fast
Type b
Pointing accuracy in roll, and pitch but poor in yaw axis. Performance schemes are always implemented to reduce yaw error.
Same performance as in type a
Sensitive to vibration due to thruster activities, care must be taken to avoid structural resonance. Sensitive to vibration due to thruster activities, care must be taken to avoid structural resonance.
Type c
Pointing accuracies in roll pitch and yaw.
Close to Type d
Moderate depend on wheel torques
Fuel Efficient. Fuel only used for wheels desaturation
Type d
Pointing accuracies in roll, pitch and yaw.
Best
Robust because wheels torques is the only source of vibration. A wheel damper can eliminate this during spacecraft assembly. Most robust and reliable.
Moderate depends on wheel torques.
Fuel Efficient. Fuel only used for wheels desaturation
High fuel consumption, so Spacecraft lifespan very dependent on it. Average
Slow
Reliability
cost
Weight
Moderate
Control instrument less expensive than In type c and type d.
Heavy due to on board propellant.
Moderate. Wheel Configuration usually equipped with back up reaction wheel along z axis failure of any of nominal momentum wheel however reduces control performance. Reliable
Less instrument compared with type a
Not so heavy when compared with type a.
Expensive because number control instrument
Heavier than in type b but less than type a.
Reliable
Expensive because number control instrument
of of
of of
Heavier than in type c because of pitch momentum wheel.
Table 5.1: Comparative results of various wheel configurations for momentum bias spacecrafts. It is shown that type c can withstand much lower structural frequencies than type a and type b as seen from the transient responses. Time response of type c and type d are found to be nearly the same. The stability margin of type d was however found to be different from type c because of the added stiffness due to bias momentum wheel.
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The bang-bang thruster control as in type a and thruster mode for momentum desaturation implemented in this thesis could be improved with a better thruster pulse duration period that is long enough so that no structural resonance or control interaction is induced. The results shows that the body fixed momentum wheel along pitch axis and two reaction wheels along roll and yaw axes is the most robust in term of sensitivity to disturbances and gyroscopic rigidity. The stability of type a is found to be very sensitive to structural resonance due to thruster activities and if the type b also complement yaw control with thruster as actuator, it also has sensitivity to structural frequency variations. The qualitative performance of type c is found to be very close to type d.
5.5. Response for Various Skewed angles. Solar Disturbance on Open Loop Spacecraft Dynamics using Different skewed angles. Solar Disturbance on Open Loop Spacecraft Dynamics using skewed angle of 1.66 degree similar to GEOS N-P 20
ROLL AXIS(2.874) PITCH AXIS(15.63) YAW AXIS(-2.962)
Spacecraft Euler Angles(Degree)
15
X: 8.64e+004 Y: 15.63
10
5
X: 8.628e+004 Y: 2.874
0 X: 8.634e+004 Y: -2.962
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-20
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7
8
9 4
One day solar time(s)
x 10
Figure5.9: Simulator response using skewed angle of 1.66 degree similar to GEOS N-P
Solar Disturbance on Open Loop Spacecraft Dynamics using skewed angle of 2.5 degree similar to Anik E2 20
ROLL(2.9 DEGREE) PITCH(15.63 DEGREE) YAW(-2.927 DEGREE)
Spacecraft Euler Angles(Degree)
15
X: 8.64e+004 Y: 15.63
10
5
X: 8.624e+004 Y: 2.9
0 X: 8.624e+004 Y: -2.927
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One day solar time(s)
6
7
8
9 4
x 10
Figure 5.10:Simulator response using skewed angle of 2.5 degree similar to Anik E2 96
Solar Disturbance on Open Loop Spacecraft Dynamics using skewed angle of 20 degree similar to Nigcomsat-1 20
Spacecraft Euler Angles(Degree)
15
X: 8.64e+004 Y: 15.63
ROLL(3.092 DEGREE) PITCH(15.63 DEGREE) YAW(-3.164 DEGREE)
10
X: 8.64e+004 Y: 3.092
5
0 X: 8.64e+004 Y: -3.164
-5
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One day solar time(s)
x 10
Figure5.11:Simulator response using skewed angle of 20 degree similar to Nigcomsat-1.
Solar Disturbance on Open Loop Spacecraft Dynamics using skewed angle of 60 degree 20
Spacecraft Euler Angles(Degree)
15
X: 8.64e+004 Y: 15.63
10 X: 8.64e+004 Y: 5.661
5
0
X: 8.64e+004 Y: -5.658
-5
-10
ROLL AXIS(5.661) PITCH AXIS(15.63) YAW AXIS(-5.658)
-15
-20
0
1
2
3
4
5
6
One day solar time(s)
Figure5.12:Simulator response using skewed angle of 60 degree.
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7
8
9 4
x 10
Solar Disturbance on Open Loop Spacecraft Dynamics using skewed angle of 90 degree 100
ROLL AXIS(94.36 DEGREE) PITCH AXIS(15.63 DEGREE) YAW AXIS(-106 DEGREE)
X: 8.64e+004 Y: 94.36
Spacecraft Euler Angles(Degree)
50
X: 8.64e+004 Y: 15.63
0
-50
X: 8.64e+004 Y: -106.6
-100
-150
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7
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9 4
One day solar time(s)
x 10
Figure 5.13: Simulator response using skewed angle of 90 degree. Discussion. SKEWED ANGLE IN DEGREE GEOS N-P (1.66) Anik E2(2.5) Nigcomsat-1(20) For further analysis(60) For further analysis(90)
MAXIMUM ROLL ANGLE FOR 24HRS 2.874 2.9 3.092 5.661 94.36
MAXIMUM PITCH ANGLE FOR 24HRS 15.63 15.63 15.63 15.63 15.63
MAXIMUM YAW ANGLE FOR 24HRS -2.96 -2.927 -3.164 -5.658 -106
WHEEL ANGULAR MOMENTUM Not available Not available Not available Not available
Table 5.2: Summary of the plots for various skewed angles. The result of the investigation is summarized in figure 5.2 .The result shows that the choice of the inclination angle depends so much on ease of controllability of both the pitch and the roll angles, while still exploiting the feature of inertial attitude stabilization that keeps the yaw error close to null without measuring it. The ease of controllability enable the adaptability of spacecraft for change in mission, For example, this capability is useful in establishing new antenna coverage areas or in providing for other new missions. On this basis, the skewed angles vary because of the different and ease of adaptability of spacecraft to different on orbit reconfiguration modes. Nigeria communication satellite has this capability i.e. Antenna Mapping mode for reorienting the antenna for antenna mapping and slant mode for saving fuel at the close of satellite lifetime or if there is serious anomaly where fuel is excessively expended. The results however shows that the wheel size will increase as the skewed angle increases in order to perform similar attitude control task. This is not confirmed in this thesis by comparing the wheel size used for Anik E2 and GEOS N-P because there is no available data to proof this. The result of skewed angle of 90 degree validate this simulator because the effect of the wheels on the spacecraft is zero and the open loop dynamics due to solar radiation torques becomes excessively large because all the external disturbance from the solar torques are transfer to spacecraft body since there is no momentum exchange device (momentum wheels) to absorb it.
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CHAPTER 6 6. CONCLUSIONS AND FURTHER STUDIES The objective of this thesis is to design control architecture for a momentum bias spacecraft using different type of wheels configuration for pitch, roll and yaw actuations and compare their performances. From the pitch attitude performance evaluation based on the design configuration of the platform, the desired pitch angle pointing accuracy is attained for the entire wheels configuration using the same control law. It is shown that a good pointing accuracy can be achieved with configuration of Nigcomsat-1 where the mass properties principal axis has the relations I x > I z > I y . The roll attitude with the yaw control (type b) has the least pointing accuracy especially in yaw axis. This indicates that yaw attitude has poor pointing accuracy with existence of the solar radiation disturbance. It is recommended to design a more complex filtering technique and yaw axis on board observer if the external disturbances need to be rejected completely in this axis. It is known that, there are two most important elements influencing the pointing accuracy during attitude control design, which are the torque controller and attitude sensors of the control system. This effect was noticeable in type a configuration when we use thrusters to control roll/yaw attitude dynamics. Thus, it is desirable to improve design on them so as to increase the performance of the control system thoroughly. The use of three axis star sensors and wheels actuator in type c and type d also give a very smooth and high pointing accuracy in all the three axes with type d showing the highest performance design after series of performance comparism between them. The active control method that produced control torques along the satellite roll and yaw axes in type c and type d was proven to be more efficient and able to stabilize the satellite around its equilibrium position. For the study on the optimization of two symmetrically inclined momentum wheels in a V configuration used on Nigeria communication satellite and other similar satellites in geostationary orbit. The result shows that the choice of the inclination angle depends very much on the ease of controllability of both the pitch and the roll angles, while still exploiting the feature of inertial attitude stabilization that keeps the yaw error close to null without measuring it. The skewed angle therefore varies according to ease of adaptability of the spacecraft to different mission tasks and orientation. The underlisted are not covered in this thesis and will therefore be considered for further studies; 1. The effect of the thruster continuous firing and mis-firing in autonomous bang-bang controller for roll/yaw is visible due to non smooth of the roll/yaw dynamics. For better efficiency for spacecraft applications in automatic control mode, a popular thrusters control schemes such as rate modulator and pulse width phase modulator etc are normally used and this will be investigated in the future research.
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2.
The effect of the skewed wheels controller for roll/yaw is visible due to poor error of the yaw dynamics output in this thesis. This configuration has few attitude controls on board equipments but poor pointing accuracies in the yaw axis as shown in the plot. An on board observer can then be implemented to estimate the yaw error so as to improve the yaw pointing accuracies, This can be investigated in future work.
3. For future research a more robust control algorithms will be implemented so as to have a plant with less sensitivity to parameter changes, sensor noise. The model of a flexible satellite model will also be investigated in the future so as to account for solar array dynamics and fuel sloshing.
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7. APPENDIX A: MATLAB CODES. openloopPlant_bodeplot.m clc clear all format long Ix=14542;Iy=4396;Iz=12356;%mass properties of Nigcomsat-1 w0=7.237e-5;% Geostationary orbital frequency hwy1=50;%Angular momentum of the wheel % calculation of the coefficients of the transfer functions hww0=w0*hwy1; Ixz= (Ix-Iz)/Iy; Iyz=Iy-Iz; Ixyz=Ix-Iy-Iz; Iyx=Iy-Ix; IxIz=Ix*Iz; hwsq=hwy1*hwy1; hwsixiz=hwsq/IxIz; numc=hww0/IxIz; denum1=[1 0 hwsixiz]; denum2=[1 0 w0*w0]; numz=[1/Iy 0 numc];%numerator of yaw axis transfer function denum=conv(denum1,denum2);% denumerator of roll/yaw axis transfer function numx=[1/Ix 0 numc]; %numerator of roll axis transfer function % denum=[1 0 s2 0 s0 ];%characteristics equation Roll_axis=tf(numx,denum);% Transfer function of Roll angle due to Tcx alone Yaw_axis=tf(numz,denum);% Transfer function of Roll angle due to Tcx alone w=logspace(-5,2,1000);%frequency range of the plot. %Step response and bode plot of uncompensated Roll axis figure(1) step(numx,denum) grid on xlabel('Time') ylabel('Amplitude(Radian) ') legend('Roll Axis') title('Step response for uncompensated Roll axis') hold on %.......................................................................... %.......................................................................... figure(2) bode(Roll_axis,w);%bode plot for x axis grid on xlabel('Frequency') ylabel('Magnitude ') legend('show') title('Bode plot for uncompensated Roll axis') hold on %Step response and bode plot of uncompesated Yaw axis figure(3) step(numz,denum) grid on xlabel('Time') ylabel('Amplitude(Radian) ') legend('Yaw Axis') title('Step response for uncompensated Yaw axis') hold on %.......................................................................... %.......................................................................... 101
figure(4) bode(Yaw_axis,w);%bode plot for z axis grid on xlabel('Frequency') ylabel('Magnitude ') legend('show') title('Bode plot for uncompensated Yaw axis') hold on
pitch_Controller.m clc clear all format long Ix=14542;Iy=4396;Iz=12356;%mass properties of Nigcomsat-1 w0=7.237e-5;% Geostationary orbital frequency hwy1=50;%Angular momentum of the wheel % calculation of the coefficients of the transfer functions hww0=w0*hwy1; Ixz= (Ix-Iz)/Iy; Iyz=Iy-Iz; Ixyz=Ix-Iy-Iz; Iyx=Iy-Ix; a=4*w0*w0*Iyz; b=-w0*Ixyz; c=w0*w0*Iyx; d=3*w0*w0*(Ix-Iz); IxIz=Ix*Iz; cw0=(c+hww0)/IxIz; aw0=(a+hww0)/IxIz; s2=((w0*w0)+(hwy1*hwy1))/IxIz; s0=((w0*w0)*(hwy1*hwy1))/IxIz; PITCHLEAD_ANALYTICALPLOT; %Link to simulink block to plot pitch axis.
Roll_yaw_Controller.m clc clear all %uncompensated roll plant Numo=[1]; Denuo=[14,542 0 0.0345]; %compensated Roll plant Numc=[20.495 Denuc=[14542
0.0796]; 1157.54
20.50
%uncompensated yaw plant Numo1=[1]; Denuo1=[12356 0 0.0345]; %compensated yaw plant Numc1=[17.630 0.0630]; Denuc1=[12356 778.428 17.633
0.0798];
0.0632];
t=0:0.1:3000; c1=step(0.05*Numo,Denuo,t); c2=step(0.05*Numc,Denuc,t); c11=step(0.2*Numo1,Denuo1,t); c21=step(0.2*Numc1,Denuc1,t); 102
figure(1) plot(t',c11, '--','linewidth',2 ) hold on plot(t',c1,'r','linewidth',2 ) title('unit step of uncompensated Roll/Yaw systems') xlabel('t secs'); ylabel('Roll axis'); grid on legend(' uncompensated Roll axis','uncompensated Yaw axis','Location','NorthEast');
figure(2) plot(t',c21, '--','linewidth',2) hold on plot(t',c2,'r','linewidth',2) grid on title('Step Response of compensated Roll/Yaw Dynamics') xlabel('t secs'); ylabel('Roll/Yaw axis(degree)'); legend('Compensated Roll axis','Compensated Yaw axis','Location','NorthEast');
Solarradiation.m %Run this file after simulation of solar radiation simulink block clc format long plot(t,Tdx,'b','linewidth',3); grid on hold on plot(t,Tdy,'r','linewidth',3); plot(t,Tdz,'g','linewidth',3); title('One year Solar radiation torques on Nigcomsat-1') xlabel('Time (s) from January to December'); ylabel('Magnitude of solar radiation torques on x,y and z axis(Nm))'); legend('Tdx','Tdy','Tdz','Location','NorthEast');
Other codes %importing parameters from simulink block and plot on workspace e.t.c plot(TE,PH,'r','linewidth',2); title('Two wheels actuators open loop response for Nigcomsat-1 with solar radiation torques') xlabel('Time (s)'); ylabel('Magnitude of the pitch axes(degree)'); legend('pitch angle','Location','NorthEast');grid on plot(TE,RL,'r','linewidth',2);hold on;plot(TE,YW,'b','linewidth',3); title('Two wheels actuators roll/yaw axis open loop response for Nigcomsat-1 perturbed by solar radiation torques') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(degree)'); legend('Roll angle','Yaw angle','Location','NorthEast'); plot(TE,RL,'r','linewidth',2);hold on;plot(TE,YW,'b','linewidth',3); 103
title('Two wheels actuators roll/yaw axis open loop response for Nigcomsat-1 perturbed by solar radiation torques') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(degree)');grid on legend('Roll angle','Yaw angle','Location','NorthEast'); comet(TE,RL,'r','linewidth',2);hold on;plot(TE,YW,'b','linewidth',3); title('Two wheels actuators roll/yaw axis open loop response for Nigcomsat-1 perturbed by solar radiation torques') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(degree)'); legend('Roll angle','Yaw angle','Location','NorthEast'); comet3(TE,RL,'r','linewidth',2); comet3(TE,RL); plot(TE,PH,'r','linewidth',3); title('24hours roll/yaw axis open loop response for Nigcomsat-1') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(radian)'); legend('Roll angle','Yaw angle','Location','NorthEast'); plot(TE,RL,'r','linewidth',3); title('24hours roll/yaw axis open loop response for Nigcomsat-1') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(radian)'); legend('Roll angle','Yaw angle','Location','NorthEast'); plot(TE,YW,'r','linewidth',3); title('24hours roll/yaw axis open loop response for Nigcomsat-1') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(radian)'); legend('Roll angle','Yaw angle','Location','NorthEast'); plot(TE,YW,'r','linewidth',3); title('24hours roll/yaw axis open loop response for Nigcomsat-1') xlabel('Time (s)'); ylabel('Magnitude of the Roll/Yaw axes(radian)'); legend('Roll angle','Yaw angle','Location','NorthEast'); plot(TE,PH,'r','linewidth',3); plot(TE,YW,'r','linewidth',3); plot(TE,PH,'r','linewidth',3); plot(TE,RL,'r','linewidth',3); plot(TE,YW,'r','linewidth',3); plot(TE,RL,'r','linewidth',3); plot(TE,PH,'r','linewidth',3); bode(y) bode(y2);grid on nyquist(y) rlocus(y) rlocus(y);grid on;hold on;rlocus(y2) rlocus(y);grid on;hold on; rlocus(y) rlocus(y2) bode(y) bode(y);grid on plot(TE,RL) plot(TE,YW) plot(TE,PH);grid on plot(TE,YW) plot(TE,PH);grid on plot(TE,PH);grid on;hold on;plot(TE,RL);plot(TE,YW) grid on;hold on;plot(TE,RL);plot(TE,YW) plot(TE,PH);grid on plot(TE,YW) plot(TE,RL) plot(TE,PH);grid on 104
plot(TE,RL) plot(TE,YW) plot(TE,RL) plot(TE,PH);grid on plot(TE,RL) plot(TE,PH);grid on plot(TE,PH);grid on plot(TE,RL) plot(TE,YW) Roll_axis=tf(num,denum) minreal(Roll_axis=tf(num,denum,.1) minreal(Roll_axis,.1) minreal(Roll_axis,.01) minreal(Roll_axis,.0001) minreal(Roll_axi) zpk(Roll_axis) bode(zpk(Roll_axis)) de = 0.2; num = [1.151 0.1774]; den = [1 0.739 0.921 0]; pitch = tf(num,den); step(de*pitch) y=0.02(1+s/0.05)(1+s/0.01) y=0.02(1+s/0.05)*(1+s/0.01) y=0.02(1+s/0.05) y=0.02*(1+s/0.05) sym s y=0.02(1+s/0.05)(1+s/0.01) y=0.02*(1+s/0.05)*(1+s/0.01) sym s y=0.02*(1+s/0.05)*(1+s/0.01) syms s y=0.02*(1+s/0.05)*(1+s/0.01) y=0.02*(1+s/0.05)*(1+s/0.01)/(1+s/0.3)^4 plot(TE,RL) plot(TE,YW) plot(TE,PH) plot(TE,PH) plot(TE,YW) plot(TE,RL) plot(TE,PH) plot(TE,PH);GRID ON plot(TE,PH) y=plot(TE,PH) nichols(y) bode(y) y=plot(TE,PH) plot(TE,PH) plot(TE,RL) plot(TE,RL);grid on plot(TE,YW) plot(TE,RL) plot(TE,RL);grid on;plot(TE,RL) plot(TE,RL);hold on;grid on;plot(TE,YW) plot(TE,RL);grid on;plot(TE,RL) plot(TE,RL) plot(TE,PH) format long plot(TE,PH) plot(TE,PH) plot(TE,PH);grid on 105
plot(TE,WH);grid on plot(TE,PH);grid on w=logspace(-1,4); G=i*w*0.5. / ( - w . *w+30*i*w+1e6); syms w; G=i*w*0.5. / ( - w . *w+30*i*w+1e6); G=i*w*0.5. G=i*w G=i*w*0.5 G=i*w*0.5. / ( - w . *w+30*i*w+1e6); G=( - w . *w+30*i*w+1e6); G=( - w *w+30*i*w+1e6); G=i*w*0.5. / ( - w *w+30*i*w+1e6); G=i*w*0.5 / ( - w *w+30*i*w+1e6); w=logspace(-1,4); G=i*w*0.5 / ( - w *w+30*i*w+1e6); G=i*w*0.5. / ( - w . *w+30*i*w+1e6); G=i*w*0.5. G=i*w*0.5 G=i*w*0.5 / ( - w . *w+30*i*w+1e6); G=i*w*0.5 / ( - w.*w+30*i*w+1e6); G=i*w*0.5./ (- w.*w+30*i*w+1e6); 20log(abs(G)) abs(G) phase(G) 20log10(abs(G)) y=abs(G) 20log10(y) 20log10(y') 20*loglO(abs(G)) 20*log10(abs(G)) 180*phase(G)/180 180*phase(G)/pi gain=20*log10(abs(G)) semilog(w,gain) semilogx(w,gain) semilogx(w,gain);gain semilogx(w,gain);grid on phase=180*phase(G)/pi semilogx(w,phase) semilogx(w,phase);grid on w=[logspace(-1,2.5) 350:2:1500 logspace(3.18,5)]; w=logspace(-1,2.5) w=logspace(-1,2) w=logspace(-1,2.5) w=[logspace(-1,2.5) 350:2:1500 logspace(3.18,5)]; w=logspace(3.18,5) plot(t,P) plot(t,P);grid on G=tf(1,[convs([0.1 1],[0.02 1],[0.01,1],[0.005 1])]); G=tf(1,[conv([0.1 1],[0.02 1],[0.01,1],[0.005 1])]); damp(denu) plot(t,Tdx) plot(t,Tdy) plot(t,Tdz) comet(t,Tdz) plot(t,Tdz);hold on;plot(t,Tdx),plot(t,Tdy) plot(t,Tdz);hold on;plot(t,Tdx),plot(t,Tdy);legend('show') step(Pitch_axis) step(0.5*Pitch_axis) rlocus(Pitch_axis) 106
nyquist(Pitch_axis) nchols(Pitch_axis) lti(Pitch_axis) damp(yaw_axis) damp(Roll_axis) margin(Roll_axis) plot(tp,pitch) plot(TE,PH) plot(TE,PH) plot(TE,PH);grid on [mag2,phase2,w]=bode(Pitch_axis); [mag2,phase2,w]=bode(Pitch_axis) plot(TE,PH);grid on clc plot(TE,PH);grid on plot(tp,pitch) %-- 07/08/10 17:26 --% grid on xlabel('Frequency(rad/sec)') ylabel('Magnitude ') legend('show') title('Bode plot for open loop response of Roll axis') hold on plot(tp,pitch);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') plot(TH,PH);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') plot(TE,PH);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') plot(TE,PH);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title(' compesated pitch axis with solar torques') plot(tp,pitch);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') d=3*w0*w0*(Ix-Iz); d d+0.155 num4=[20.495 0.145];denu4=[4396 637.45 20.495 0.155] Gp=tf(num4,denu4) bode(Gp) w=logspace(-5,5,200); bode(Gp,w) bode(Gp,w);grid on plot(TE,PH);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') bode(Gp,w);grid on w=logspace(-8,5,1000); bode(Gp,w);grid on plot(tp,pitch);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') step(Gp);grid on roots(Gp) root(Gp) Gp roots(denu) format long roots(denu) plot(tp,pitch);grid on;xlabel('time(sec)');ylabel('Steady state response(degree)');title('Time response of compesated pitch axis') %-- 10/08/10 13:39 --% %-- 10/08/10 13:39 --%1e-3 1e-3 107
1e-3*1000 plot(TE,RL);grid on plot(TE,YW);grid on plot(TE,PH);grid on %-- 11/08/10 11:07 --% help conv a=[1 -70]; b=[1 0 . 5 ] ; num=-0.0005*conv(a,b) a=[1 -70]; b=[1 0 . 5 ] ; b=[1 0.5 ] ; num=-0.0005*conv(a,b) num=conv(a,b) Roll_axis bode(Roll_axis) bode(Roll_axis);grid on sqrt(numc) sqrt(Ix*numc) initial(Roll_axis);grid on step(Roll_axis);grid on bode(Roll_axis);grid on step(Roll_axis);grid on Roll_axis=tf(num,denum) step(num,denum) Roll_axis=tf(num,denum) bode(Roll_axis);grid on Roll_axis Yaw_axis=tf(numz,denum) Roll_axis=tf(num,denum) plot(TE,PH);grid on Roll_axis=tf(numx,denum) clc plot(tp,roll) plot(tp,roll);grid on clc plot(TE,PH) clc num=50000 den=[1 600 500 0] den=[1 60 500 0] gp=tf(num,den) bode(gp) bode(gp);grid on step(gp);grid on clc bode(gp);grid on num=[20.495 0.145]; denum=[1 0.145]; yu=tf(num,denum) bode(yu) plot(TE,PH) figure(2);plot(TE,PH) clf figure(2);plot(TE,PH) figure(2);plot(TE,PH);grid on clc (a+hww0) num=[1] denum=[14542 0 0.0034517] bode(num,denum) w=logspace(-5,5, 1000); bode(num,denum);grid on 108
bode(num,denum,w);grid on %-- 13/08/10 11:30 --% clc denum=[12356 0 0.0034517] bode(num,denum,w);grid on num=[1] denum=[12356 0 0.0034517] format long denum=[12356 0 0.0034517] bode(num,denum,w);grid on w=logspace(-5,5, 1000); bode(num,denum,w);grid on plot(TE,RL) plot(TE,RL);grid on plot(TE,RL) clc plot(TE,RL) plot(TE,RL);grid on t' size(t') size(c1) plot(t',c1) plot(t',c2) plot(TE,RL) plot(TE,RL);grid on plot(TE,YL);grid on plot(TE,YW);grid on plot(TE,RL);grid on %-- 22/08/10 20:44 --% plot(TE,RL);grid on plot(TE,RL);grid on;hold on; plot(TE,YW) plot(TE,PH);grid on plot(TE,RL);grid on;hold on; plot(TE,YW) clc plot(TE,RL);grid on;hold on; plot(TE,YW) clc subplot(1,1);plot(TE,RL);grid on;subplot(1,2); plot(TE,YW);grid on subplot(2,1);plot(TE,RL);grid on;subplot(2,2); plot(TE,YW);grid on subplot(2,1,1);plot(TE,RL);grid on;subplot(2,1,2); plot(TE,YW);grid on plot(TE,PH);grid on subplot(2,1,1);plot(TE,RL);grid on;subplot(2,1,2); plot(TE,YW);grid on clc subplot(2,1,1);plot(TE,RL);grid on;subplot(2,1,2); plot(TE,YW);grid on %-- 28/08/10 01:21 --% plot(TE,PH);grid on %-- 29/08/10 13:23 --% plot(TE,YW) plot(TE,SKW) clc plot(TE,SKW);grid on figure(2);plot(TE,SKW);grid on figure(3);plot(TE,SKW);grid on figure(4);plot(TE,SKW);grid on figure(5);plot(TE,SKW);grid on plot(TE,RL,'r','linewidth',3);hold on plot(TE,YW,'r','linewidth',3);hold on
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8. REFERENCES [1] Adetoro L., and Kunle Fashade, “Transfer orbit trajectory controller design for a typical spacecraft Launching from Nigeria”, IFAC conference 2010, Saint Petersburg, Russia. [2] Adetoro L., “Post Launch Campaign of Nigeria Communication Satellite”, 2007. [3] Agrawal, B. N., Design of Geosynchronous Spacecraft. Englewood Cliffs, NJ:PrenticeHall,1986. [4] Arduini, C. and Baiocco, P., “Active Magnetic Damping Attitude Control for Gravity Gradient Stabilized Spacecraft”, Journal of Guidance Control and Dynamics, Vol.20, No.1, January-February 1997.Jurnal Mekanikal, Jun 2004 47. [5] Azor, R. , "Solar Attitude Control Including Active Denutation Damping in a Fixed Momentum Wheel Satellite," AIAA Guidance Navigation and Control Conference (10-12 August,1992 Hilton Head Island, SC). Washington, DC: AIAA, pp. 226-35. [6] Benoit, A., and Bailly, M., "In-Orbit Experience Gained with the European OTS/ECS/TELECOM 1 Series of Spacecraft" (AAS 87-054), Proceedings of the Annual Rocky Mountain Guidance and Control Conference (31 January - 4 February 1987, Keystone, CO). San Diego, CA: American Astronautical Society,pp.525-42 . [7] Bingham, N., Craig, A., and Flook, L., "Evolution of European Telecommunication Satellite Pointing Performance," Paper no. 84-0725, AIAA, New York,1984. [8] Bittner, H., Bruderle, E., Roche, Ch., and Schmidts, W., "The Attitude Determination and Control Subsystem of the Intelsat V Spacecraft" (ESA SP-12), Proceedings 258 8/ Momentum-Biased Attitude Stabilization of AOCS Conference (3-6 October, Noordwijk, Netherlands). Paris: European Space Agency,1977. [9] Brodsky,S,Nebylov A.V,and Panferov,A,Mathematicl Models and Software for Flexible Vehicle Simulations , 16th IFAC Symposium on Automatic Control In Aerospace 2004, Saint Petersburg, Russia. [10] Deutsch, R., Orbital Dynamics of Space Vehicles. Englewood Cliffs, NJ: Prentice-Hall, 1963. [11] Devey, W. J., Field, C. F., and Flook, L., "An Active Nutation Control System for Spin Stabilized Satellites," Automatica 13: 161-72,1977. [12] Dougherty, H. J., Lebsock, K. L., and Rodden, J. J., "Attitude Stabilization of Synchronous Communications Satellite Employing Narrow-Beam Antennas,"Journal of Spacecraft and Rockets 8: 834-41,1971. [13] Dougherty, H. J., Scott, E. D., and Rodden, J. J., "Analysis and Design of WHECON – An Attitude Control Concept," Paper no. 68-461, AIAA 2nd Communications Satellite Systems Conference (8-10 April, San Francisco),1968. [15] Duhamel, T., and Benoit, A., "New AOCS Concepts for ARTEMIS and DRS,"Space Guidance, Navigation and Control Systems (Proceedings of the First EAS International Conference, ESTEC, 4-7 June, Noordwijk, Netherlands). Paris:European Space Agency,pp. 33-9,1991. [16] Fleeter, R., and Warner, R., “Guidance and Control of Miniature Satellites”, Automatic Control in Aerospace, IFAC (Tsukuba, Japan), Oxford, UK: Pergamon, 1989, pp. 243248. 110
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[40] Thomson, W. T,Introduction to Space Dynamics. New York: Dover, 1986. [41] [wens, R. P., Fleming, A. W., and Spector, V. A. "Precision Attitude Control with a Single Body-fixeded Momentum Wheel," Paper no. 74-894, AIAA Mechanics and Control of Flight Conference (5-9 August, Anaheim, CA), (1974). [42] Wertz, J. R. ,Spacecraft Attitude Determination and Control. Dordrecht: Reidel, 1978. [43] Wertz, J.R., Spacecraft Attitude Determination and Control, Kluwer Academic Publisher, 1995. [44] Wie, B., Byun, K.W., Warren, Y.W., Geller, D., Long, D. and Sunkel, J., New approach to attitude/momentum control for the space station, Journal of Guidance, Navigation and Control, 12, (5), 714-722,1988 [45]Wie, B., Space Vehicle Dynamics and Control, AIAA Education Series, 1998.
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