Spacecraft attitude magnetic controller

A Combined Attitude Magnetic Controller for
Remote Sensing Satellite
by
Eng. AHMAD FARRAG EL-SAYED
A Thesis Submitted to the
Faculty of Engineering at Cairo University
in Partial Fulfillment of the
Requirements for the Degree of
Master of Science
in
Electrical Power and Machines
FACULTY OF ENGINEERING
CAIRO UNIVERSITY
GIZA, EGYPT
Jun 2010

II
by
Master of Science
in
Under the supervision of
Dr. Hassan Mohamed Rashad
Electrical Power and Machines Dept.
Faculty of Engineering – Cairo University
CAIRO UNIVERSITY
GIZA, EGYPT
Jun 2010
Prof. Dr. Ahmed Bahgat Gamal Bahgat Dr. Ahmed Yehya El-Raffie
Electrical Power and Machines Dept. National Authority for Remote sensing and
Space Since.

III
by
Master of Science
in
Approved by the
Examining committee:
Prof. Dr. Ahmed Bahgat Gamal Bahgat, Thesis Main Advisor
Prof. Dr. Abed EL-Monem Abed EL-Zaher Wahdan
Faculty of Engineering Ain Shams University
Prof. Dr. Mohamed Mohamed Faheem Saker
Faculty of Engineering Cairo University
Prof. Dr. Ahmed Yehia EL-Raffie
National Authority for Remote Sensing and Space Since
Prof. Dr.Hassan Mohamed Rashad
Faculty of Engineering Cairo University
CAIRO UNIVERSI TY
GIZA, EGYPT
Jun 2010

IV
Abstract
The problem of attitude control of remote sensing satellite using
magnetic actuators is considered in this thesis. Magnetic actuator was used
because of it is a low power consumption, small mass, low cost and reliable
attitude actuator. The attitude control problem of the satellite involves angular
velocity suppression, attitude acquisition and finally attitude stabilization will
be solved by magnetic actuator only. A comparison between the commonly
used controllers for satellite attitude control is presented. The comparison
parameters are the total consumed power, the time required to accomplish the
angular velocity suppression and attitude acquisition, calculation time of the
control algorithm and steady state error in angles and angular velocity. The
simulation is done using the complete non linear model of satellite. Based on
results, a new combined control algorithm was developed to assemble the
advantages of these commonly used controllers. Simulation results showed the
validity of the developed combined algorithm.

V
Acknowledgements
I would like to take the opportunity to express my thanks to the people who
have made my studies an exciting experience of professional and cultural
discovery.
Firstly I would like to thank my senior supervisor Professor Ahmad Bahgat
for his guidance and support in this thesis.
I extend my special thanks to my second supervisor Dr.Ahmad EL-Raffie . for
his kind effort and assistance during this work
Also my deep thank to my supervisor Professor Hassan Rashad for his
support and assistance in this thesis
My appreciation also belongs to Egyptian Space Program (ESP) at which I
am working for assistance provided by my mangers and colleagues. Besides the
technical training during EgyptSat1 project and financial support for this thesis.
I am so grateful to my Automatic control Teacher Dr Mahmoud Kamel for his
support during my undergraduate study
To my thanks also, to Dr Yefemainco ,Mr Badmasteriv and Mr Koshtica
form Khartron Konsat. Zporozhye, Ukraine, for their technical assistance to me
in understanding the attitude control problem during Egptsat1 project.
I want to express my gratitude to My Parents, who always believed that I
would succeed in master. I would like to express my deepest thanks to them for
their patience and support during all those years.
I am deeply indebted to My Wife for her continues support and taking care of
our kids during all those hard years. She always believed that I can go on easily
in this study, I appreciate her support.

VI
Contents
Abstract IV
Acknowledgment V
Table of contents VI
List of figures XI
List of Tables XV
Acronyms XVI
List of symbols XVII
CHAPTER (1) Introduction 1
CHAPTER (2) SPACE SEGMENT OVERVIEW 5
2.1 Space System Composition.................................................................. 5
2.2 Satellite Architecture............................................................................ 6
2.2.1 Satellite Payload ........................................................................... 6
2.2.1.1 Communication......................................................................... 6
2.2.1.2 Positioning and Navigation....................................................... 6
2.2.1.3 Weather ..................................................................................... 7
2.2.1.4 Remote Sensing......................................................................... 8
2.2.2 Satellite Bus .................................................................................. 9
2.2.2.1 The Structural Subsystem ....................................................... 10
2.2.2.2 Attitude Determination and Control Subsystem..................... 10
2.2.2.3 Propulsion Subsystem............................................................. 10
2.2.2.4 Communications Subsystem................................................... 11
2.2.2.5 Command and Data Handling Subsystem .............................. 11
2.2.2.6 Power System.......................................................................... 11
2.2.2.7 The Thermal Subsystem.......................................................... 12
2.3 Satellite Orbits.................................................................................... 12
2.3.1 Special Orbits.............................................................................. 16
2.3.1.1 Low Earth Orbit (LEO)........................................................... 16
2.3.1.2 Medium Earth Orbit (MEO) ................................................... 16
2.3.1.3 Geostationary/Geosynchronous Earth Orbit (GEO) ............... 17

VII
2.3.1.4 Polar Earth Orbit ..................................................................... 17
2.3.1.5 Sun Synchronous Orbits (SSO)............................................... 18
2.3.1.6 Molniya Orbit.......................................................................... 18
CHAPTER (3) Attitude Determination and Control Subsystem ADCS 20
3.1 Internal influence between satellite mission and other subsystems
upon ADCS ................................................................................................... 20
3.1.1 Internal influence between ADCS and Mission requirement..... 20
3.1.2 Internal influence between ADCS and Structure Subsystem ..... 21
3.1.3 Internal influence between ADCS and Power Subsystem.......... 21
3.1.4 Internal influence between ADCS and Communication
Subsystem.................................................................................................. 21
3.1.5 Internal influence between ADCS and Command and Data
Handling Subsystem.................................................................................. 22
3.1.6 Internal influence between ADCS and thermal subsystem ........ 22
3.2 ADCS Tasks....................................................................................... 22
3.3 Satellite operational modes ................................................................ 23
3.3.1 De-tumbling mode (DM)............................................................ 23
3.3.2 Standby Mode (SM).................................................................... 23
3.3.3 High Accuracy Mode (HAM) or Imaging Mode (IM)............... 23
3.3.4 Emergency Mode (EM) .............................................................. 24
3.3.5 Transferring from one operational mode to another................... 24
3.4 ADCS devices .................................................................................... 25
3.4.1 ADCS Sensors ............................................................................ 25
3.4.1.1 Earth’s Horizon sensor............................................................ 26
3.4.1.2 Sun sensor ............................................................................... 26
3.4.1.3 Star mapper ............................................................................. 27
3.4.1.4 Magnetometers........................................................................ 28
3.4.1.5 Inertial Sensor or Gyro............................................................ 29
3.4.2 ADCS Actuators ......................................................................... 32
3.4.2.1 Momentum and Reaction Wheel............................................. 32

VIII
3.4.2.2 Magnetic actuators .................................................................. 33
3.4.2.3 Thruster ................................................................................... 35
3.5 Disturbance Environment................................................................... 36
3.5.1 Gravity Gradient Disturbance..................................................... 36
3.5.2 Magnetic Field Disturbance........................................................ 37
3.5.3 Solar Radiation Pressure Disturbance ........................................ 38
3.5.4 Aerodynamic Disturbance .......................................................... 39
3.6 Attitude Control techniques ............................................................... 39
3.6.1 Passive Control ........................................................................... 40
3.6.1.1 Passive magnetic..................................................................... 40
3.6.1.2 Gravity-gradient stability ........................................................ 41
3.6.1.3 Spin stabilization..................................................................... 42
3.6.1.3.1. Single Spin........................................................................... 42
3.6.1.3.2. Dual Spin............................................................................. 43
3.6.2 Active control techniques ........................................................... 43
3.6.2.1 Momentum exchange Wheels................................................. 44
3.6.2.2 Magnetic actuators .................................................................. 44
3.6.2.3 Thrusters.................................................................................. 44
CHAPTER (4) Control System Modelling 45
4.1 Reference Coordinate Systems .......................................................... 45
4.1.1 Geocentric Inertial Coordinate System....................................... 45
4.1.2 Greenwich Coordinate System ................................................... 45
4.1.3 Orbital Coordinate System.......................................................... 47
4.1.4 Body Coordinate System ............................................................ 47
4.1.5 Device Coordinate System.......................................................... 48
4.2 Modeling of Satellite Rotation Around it's Center of Mass............... 49
4.2.1 Model of Dynamic Equation....................................................... 50
4.2.2 Model of Kinematic Equation..................................................... 57
4.2.3 Linearized Equations of Motion ................................................. 59
4.2.4 Gravity Gradient Stability........................................................... 64

IX
4.3 Satellite center of mass motion model ............................................... 66
4.3.1 Satellite orbits and Keplerian elements ...................................... 66
CHAPTER (5) Analysis of Conventional Controllers and Development of a
new Combined Controller 70
5.1 Attitude Magnetic control concept..................................................... 70
5.2 Controllability .................................................................................... 73
5.3 Angular Velocity Suppression ........................................................... 74
5.3.1 Angular suppression using velocity feed back ........................... 75
5.3.1.1 Energy Considerations ............................................................ 75
5.3.1.2 Lyapunov Stability.................................................................. 77
5.3.2 Angular suppression using B-dot technique............................... 80
5.3.2.1 Angular suppression using B-dot technique No.1 .................. 81
5.3.2.2 Angular suppression using B-dot technique No2 ................... 82
5.3.2.3 Angular suppression using B-dot technique No3 ................... 83
5.3.3 Simulation verification ............................................................... 85
5.3.3.1 Simulations results for angular velocity feedback.................. 87
5.3.3.2 Simulation results for B-dot techniques.................................. 88
5.3.4 Comparison between angular velocity suppression algorithms . 92
5.4 Attitude acquisition and stabilization algorithms............................... 93
5.4.1 PD-Like Controller ..................................................................... 94
5.4.1.1 Checking the stability of PD-Like Controller......................... 95
5.4.2 Sliding Mode Controller............................................................. 96
5.4.2.1 Sliding Manifold Design......................................................... 97
5.4.2.2 Sliding Condition Development.............................................. 98
5.4.3 Linear Quadratic Regulator ...................................................... 100
5.4.4 Simulation verification ............................................................. 102
5.4.4.1 Simulations results for PD-like controller ............................ 104
5.4.4.2 Simulations results for sliding mode controller.................... 106
5.4.4.3 Simulations results for LQR ................................................. 109

X
5.4.5 Comparison between attitude acquisition and stabilization
algorithms ................................................................................................ 112
5.5 Combined attitude control................................................................ 114
5.5.1.1 Simulations results for Combined algorithm ........................ 116
CHAPTER (6) Conclusion and Future Work 119
Appendix A Attitude representation 121
A.1 Introduction 121
A.1 Euler angles 121
A.1 Quaternion 123
A.1 Direction Cosine Matrix 125
Appendix B Transformation Matrices Between Reference Frames 128
B.1 OCS to ICS 128
B.1 GCS to ICS 129
B.1 OCS to BCS 130
References 132

XI
List of figures
Figure (2-1) Space System.................................................................................. 5
Figure (2-2) TV satellite ..................................................................................... 7
Figure (2-3) GPS satellites.................................................................................. 7
Figure (2-4) Weather Satellites in Geostationary orbit....................................... 8
Figure (2-5) Optical remote sensing satellite...................................................... 9
Figure (2-6) Gravitational force and the centrifugal force acting on bodies
orbiting Earth .................................................................................................... 13
Figure (2-7) apogee ,perigee of the orbit and semi-major axis........................ 14
Figure (2-8) Right ascension of the ascending node......................................... 14
Figure (2-9) Keplerian orbital elements............................................................ 15
Figure (2-10) LEO, MEO and GEO ................................................................. 16
Figure (2-11) GEO satellites appear stationary with respect to a point on Earth
........................................................................................................................... 17
Figure (2-12) Sun synchronous orbit ................................................................ 18
Figure (2-13) Molniya orbit.............................................................................. 19
Figure (3-1) Organization of transferring from one operational mode to another.
........................................................................................................................... 25
Figure (3-2) principle of Earth horizon sensor ................................................. 26
Figure (3-3) Sun sensors ................................................................................... 27
Figure (3-4) Start sensor ................................................................................... 28
Figure (3-5) flux-gate magnetometer................................................................ 29
Figure (3-6) Three degree-of-freedom gyroscope construction geometry. .... 30
Figure (3-7) The QRS11Pro gyro used on Rømer............................................ 31
Figure (3-8) The TELDIX Momentum and Reaction....................................... 33
Figure (3-9) Torque Coils ................................................................................. 34
Figure (3-10) Torque rods................................................................................. 35
Figure (3-11) Torque generated thruster mounted to satellite .......................... 36
Figure (3-12) Sunlight and drag effect.............................................................. 39
Figure (3-13) passive magnetic control orientation profile. ............................. 41
Figure (3-14) spin stabilization......................................................................... 43

XII
Figure (4-1) Inertial coordinate system............................................................. 46
Figure (4-2) Greenwich coordinate system....................................................... 46
Figure (4-3) orbital coordinate system.............................................................. 47
Figure (4-4) Body coordinate system................................................................ 48
Figure (4-5) device coordinate system.............................................................. 49
Figure (4-6) Angular motion of rigid body....................................................... 50
Figure (4-7) plane showing regions of stability and instability; adapted
from[12] ............................................................................................................ 66
Figure (5-1) ADCS functional diagram ............................................................ 71
Figure (5-2) Magnetic Torque Direction .......................................................... 72
Figure (5-3) Magnetic Control Torques[23] ..................................................... 73
Figure (5-4)Control torque is always perpendicular to the geomagnetic field
vector................................................................................................................. 74
Figure (5-5) Function diagram of velocity suppression using velocity feed
back algorithm.................................................................................................. 75
Figure (5-6 ) Block diagram of velocity suppression using B-dot technique
No1.................................................................................................................... 82
Figure (5-7 )Block diagram of velocity suppression using B-dot technique No2
........................................................................................................................... 83
Figure (5-8) Bock diagram of velocity suppression using B-dot technique No3
........................................................................................................................... 84
Figure (5-9) Cyclogram of angular velocity suppression algorithm................. 86
Figure (5-10) Satellite angular velocity suppression using angular velocity
feedback ............................................................................................................ 87
Figure (5-11) satellite energy during angular suppression using angular velocity
feedback ............................................................................................................ 87
Figure (5-12) required dipole moment to suppress the satellite angular velocity
using angular velocity feedback........................................................................ 88
Figure (5-13) Satellite angular velocity suppression ....................................... 89
Figure (5-14) satellite energy during angular suppression ............................... 90
31

XIII
Figure (5-15) required dipole moment to suppress the satellite angular velocity
........................................................................................................................... 91
Figure (5-16) Function diagram of attitude acquisition and stabilization using
PD-like controller.............................................................................................. 95
Figure (5-17) Block diagram of attitude acquisition and stabilization using
sliding mode.................................................................................................... 100
Figure (5-18 )Block diagram of attitude acquisition and stabilization using
LQR................................................................................................................. 102
Figure (5-19) Cyclogram of attitude acquisition and stabilization algorithm 103
Figure (5-20) Satellite response during angular velocity suppression using
angular velocity feedback, followed by, attitude acquisition and stabilization
using PD-like controller.................................................................................. 104
Figure (5-21) Zoom in for satellite response during angular velocity
suppression using angular velocity feedback, followed by, attitude acquisition
and stabilization using PD-like controller....................................................... 105
Figure (5-22) Required dipole moment during angular velocity suppression
using angular velocity feedback, followed by, attitude acquisition and
stabilization using PD-like controller ............................................................. 106
using sliding mode controller.......................................................................... 107
and stabilization using sliding mode controller. ............................................. 108
stabilization using sliding mode controller..................................................... 109
Figure (5-26 Satellite response during angular velocity suppression using
using LQR....................................................................................................... 110

XIV
and stabilization using LQR............................................................................ 111
stabilization using LQR .................................................................................. 112
Figure (5-29) S Cyclogram of the developed combined algorithm................ 115
using combined algorithm............................................................................... 116
and stabilization combined algorithm............................................................. 117
stabilization using combined algorithm.......................................................... 118

XV
List of Tables
Table 3-1 Ranges of ADCS sensors accuracy .................................................. 31
Table 5-1initial data used for satellite simulation............................................. 85
Table 5-2the comparison result between angular velocity suppression
algorithms.......................................................................................................... 92
Table 5-3the comparison result between attitude acquisition and stabilization
algorithms........................................................................................................ 113
Table 5-4 the combined algorithms results..................................................... 118

XVI
Acronyms
ADCS - Attitude Determination and Control Subsystem
BCS - Body Coordinates System
DCS - Device Coordinates System
FCC Flight Control Center
GCS - Greenwich Coordinates System
OCS - Orbit Coordinates System
MM - Magnetometer
AVM - Angular Velocity Meter
SC - Satellite
SS - Star Sensor
RW - Reaction Wheel
MT - Magnetorquer
LV Lunch Vehicle
RCS Reference Coordinate System
BCS Body Coordinate System
GPS Global Position Satellites
DM - Detumbling mode
SM Standby Mode
IM Imaging Mode
EM Emergency Mode
TM Telemetry Information

XVII
List of symbols
a Orbit Semi-Major Axis.
e Orbit Eccentricity
Orbit Right Ascension of The Ascending Node
i Orbit Inclination
W
Argument of The Perigee
fo True Anomaly of The Satellite
coili The Current Passing Magnetic Coil In The
N The Number of Windings In The Coil
A Cross Sectional Area
Relative Permeability
Tgg Gravitational Torque
μ The Gravitational Constant of The Earth
R The Distance Between Satellite Center of Mass And Earth
Center of Mass
J The Moment of Inertia Tensor For The Satellite
Deviation Angel From The Nadir Pointing
mT The Magnetic Torque
D Vector of Total Satellite Magnetic Dipole
B Earth Geomagnetic Field Vector
psc The Center of Pressure
SpT Solar Radiation Torque
So Solar Constant
c Speed of Light
si The Angle of Sun Light Incidence
cg The Center of Gravity
q The Surface Reflectance Factor
adT Aerodynamic Torque
The Density

XVIII
cD The Coefficient of Drag
vc The Orbital Velocity
IZ Z Axis of ICS
IY Y Axis of ICS
IX X Axis of ICS
gZ Z Axis of GCS
gY Y Axis of GCS
gX X Axis of GCS
OZ Z Axis of OCS
OY Y Axis of OCS
OX X Axis of OCS
DZ Z Axis of DCS
DY Y Axis of DCS
DX X Axis of DCS
w The Angular Velocity of Rotating Co-Ordinate System With
Respect To ICS
H The Satellite Angular Momentum
a Acceleration
extT Sum of External Torques Acting On The Satellite
disT Total External Disturbance Torque
cT Control Torque
Satellite Absolute Angular Velocity
y Satellite Relative Angular Velocity
B
IA Transformation Matrix From ICS To BCS
wH The Angular Momentum of The Wheels
O
The Orbit Rate
er The 3rd
Column In The Rotation Matrix From OCS To BCS

XIX
Quaternion Describes The Orientation of BCS With Respect
To OCS
0 Scalar Part of Quaternion
Vector Part of Quaternion
Y The Instantaneous Angular Velocity of The Satellite In
Quaternion Form
F Input Matrix For Control
U Vector of Input Control Torque
X State Vector
L Dipole Moment
m Mass
f Force
Me Earth Mass
G The Universal Gravitational Constant
e Earth Rotation Velocity
KinE Kinetic Energy
ggE Potential Energy Associated With The Gravity Gradient
gyroE Potential Energy Due To The Revolution of The Satellite
About The Earth
totE Total Energy
V Lyapunov Candidate Function
Way Angel That Describe The Rotation About Z Axis
Roll Angel That Describe The Rotation About X Axis
Pitch Angel That Describe The Rotation About Y Axis
m
kR Rotation Matrix From k Coordinate System To m Coordinate
System
[R] Direction Cosine Matrix
Quaternion Multiplication
The Angle Between The Line of Aries And Greenwich Line

Chapter1 Introduction
1
CHAPTER (1)
INTRODUCTION
Remote sensing satellites are used to observe features on the ground, the
behavior of the oceans, or the characteristics of the atmosphere from space.
Observation instruments are installed on satellites for remote sensing purposes.
That satellite has the advantage of being able to give up-to-date information
(satellite remote sensing can be programmed to enable regular revisit to object
or area under study) , observe wide areas, with good spectral resolution and it
give continuous acquisition of data. Data collected by the satellites are
transmitted to ground stations where images of earth's surface are reconstituted
to obtain the required information.
In order to help the satellite to keep continues earth observation and
nadir pointing all over its life time starting from in orbit injection, attitude
determination and control subsystem (ADCS) is used to provide the required
pointing accuracy.
ADCS uses different types of sensor to determine or estimate the current
attitude of satellite such as; star sensor, sun sensor, earth sensor, magnetometer
and gyros. In addition it used different type of actuators to keep continues
observation of earth against the external disturbances such as; magnetorqure,
thruster and momentum exchange devices.
Since the satellite needs different pointing requirements all over its
mission life time, starting from separation from launcher , high accurate nadir
pointing during imaging periods, passing with low accuracy pointing intervals

2
during non imaging periods. Therefore ADCS operation is divided into
different operation modes. These modes are, angular suppression or detumbling
mode (DM), which, is used to suppress the high angular velocity obtained due
to separation from launcher. Non imaging mode or Standby Mode (SM) where
the main target is to save the system resources (i.e. power and devices life time)
Imaging mode (IM), where high accurate pointing is needed for the purpose of
earth imaging. Finally in any case of failure for sensor or actuator, ADCS will
enter emergency mode (EM) to diagnose the failure reason.
There are different techniques to apply control torque for disturbance
compensation and to maintain the required orientation. For these purposes, two
types of control techniques are often employed, passive and active control.
Because Attitude control system is highly mission dependent, so the decision to
use a passive or an active control technique or a combination of them depends
on mission pointing and stabilization requirements. Since for imaging remote
sensing satellite three axes attitude stabilization is needed, so active control will
be used.
The main trend now for satellite design, is to achieve low cost, weight
and power satellite. Since magnetic actuators and sensors are considered as
lowest cost ,weight and power device used in satellite attitude control, that is
why they are used now widely used in three axis attitude control in designing
of new satellites such as CITCH, Oresat ,SunSat and EgyptSat1
Magnetic actuators are suitable in practice for low Earth orbit (LEO)
satellites. Such actuators operate on the basis of the interaction between a set of
three orthogonal, current-driven magnetic coils and the magnetic field of the
Earth (Wertz, 1978; Sidi, 1997) and therefore provide a very simple solution to
the problem of generating torques on board a satellite. The major drawback of
this control technique is that the torques which can be applied to the satellite

3
for attitude control purposes are constrained to lie in the plane orthogonal to the
magnetic field vector. In particular, three axes magnetic stabilization is only
possible if the considered orbit a variation of the magnetic field which is
sufficient to guarantee the stability of the satellite (Bhat & Dham, 2003).
For angular suppression phase commonly two types of control are used,
angular velocity feedback and B-dot technique, but for attitude acquisition and
attitude stabilization state (i.e. angular velocity and quaternion) feedback, linear
quadratic regulator and sliding mode controllers are used.
In this thesis, a comparison study between the above mentioned
controllers is made in the corresponding operation modes, taking into account
the following parameter as comparison criteria
Settling time
Steady stat error
Required dipole moment
Required calculation time for one cycle of the used control algorithm
Cost
The comparison study showed that each one of the above mentioned
algorithms has advantage and disadvantages referring to the comparison
parameters, the motivation here is to develop a combined control algorithm that
assembles the advantages of these different controllers, according to the
corresponding operation mode.
A Simulation was done for the developed controller and compared with
the previous mentioned control algorithm during all operation modes.
Outline of Thesis
Chapter 2 gives introduction about the satellite. Architecture of satellite
was presented where the different application of satellite payload and the main
satellite subsystems were described. In addition to the different types of
satellite orbits was discussed

4
Chapter 3 focuses on attitude determination and control subsystem. The
internal influence between ADCS and other satellite subsystem was discussed.
Then ADCS tasks and operational modes for remote sensing satellite were
described. In addition, the extern environments disturbances that affect ADCS
operation were introduced. More over the principle of operation for commonly
used sensors and actuators for ADCS were introduced. Finally the different
control techniques used in ADCS were briefly discussed
Chapter 4 provides definitions of coordinate systems used throughout
the thesis. Detailed description of the satellite motion is given, and linearized
model for the satellite dynamics/kinematics was introduced finally the model of
satellite motion in orbit was presented
Chapter 5 discusses the magnetic control problem. First, the
controllability and stability of magnetic actuated satellite was described, and
then the commonly used control algorithms were introduced. The development
of the combined algorithm was discussed in details. Finally A complete
simulation for full scenario of ADCS operation mode was applied for specified
satellite parameter. The results are compared for the commonly used controller
and the combined controller
Chapter 6 concludes this thesis. It summarizes the results and suggests
areas for further research.

Chapter2 Space Segment Overview
5
CHAPTER (2)
SPACE SEGMENT OVERVIEW
2.1 Space System Composition
The block diagram of a space system is shown in Figure (2-1). Space
system can be broken down into three main physical parts; the space segment,
the launch segment, and ground segment. The space segment may be a single
satellite or a constellation of satellites. The satellite contains the payloads that
will accomplish the main mission, as well as the satellite bus that provides the
supporting services for operation of the payload. The launch segment is the
launch vehicle which injects the satellites into its orbit. The ground segment
consists of gateways where the commands are up linked to satellite and data
(i.e. health of satellite and payload data) is down linked from satellites as well
as processing and distribution facilities to put the raw data in the appropriate
form and location for users.[1]
Figure (2-1) Space System
Space System
Space
Segment
Launch
Segment
Ground
Segment

6
2.2 Satellite Architecture
In general, space segment consists of the satellite; with its main two
parts, the payload and the satellite bus. The satellite bus comprises the other
supporting subsystems whose functions needed to allow the satellite to perform
its mission. Examples of those subsystems are attitude and orbit control, power
generation and data handling. The Payload Module houses the payload sensors,
the facilities needed for data handling and interfaces with satellite subsystems
[2].
2.2.1 Satellite Payload
The payload is dependent upon the mission of the satellite, and is
typically regarded as the part of the satellite "that pays the bills". Typical
payloads are listed below.
2.2.1.1 Communication
Communication satellites provide broadcast (i.e. DirecTV) or point-to-
point (i.e. Iridium) communication services to users around the globe, as well
as data and voice relay between satellite in orbit and controllers on the ground
(i.e. TDRSS). Broadcast missions typically have a set region on the Earth, to
which they are broadcasting, and typically utilization geostationary orbits and a
single satellite to cover a single region, or four satellites in GEO to provide
worldwide broadcast coverage, see Figure (2-2). Point-to-point missions are
typically accomplished with either one or several GEO satellites (like the
broadcast mission).Communication missions that relay data between space and
the Earth typically use GEO satellites.
2.2.1.2 Positioning and Navigation
Positioning and navigation (POS/NAV) missions typically provide near
global coverage and use triangulation as a strategy to provide the POS/NAV
service. Thus, multiple satellites need to be in view of a ground receiver at any
point in time, leading architects to use MEO orbits. Currently, the U.S. fields

7
GPS, and the Russians field Glonass. The European community is in the
planning stages of their Galileo POS/NAV satellite system, and will likely field
it later this decade.
Figure (2-2) TV satellite
Figure (2-3) GPS satellites
2.2.1.3 Weather
Weather satellites are referred to as the third eye of meteorologists, as
the images provided by these satellites are some of the most useful sources of
data for them. Satellites measure the conditions of the atmosphere using
onboard instruments. The data are then transmitted to the collecting centers
where they are processed and analyzed for varied applications.

8
All the weather satellites are placed into either of the two types of orbits
around the Earth, namely the polar sun-synchronous low Earth orbit and the
geostationary orbit (GEO) Figure (2-4). Polar orbit weather satellites, due to
their low altitudes, have better spatial resolution as compared to the GEO
satellites Hence they help in a detailed observation of the weather features like
the cloud formation, wind direction, etc. However, these satellites have a
poorer temporal resolution, visiting a particular location only one to four times
a day. Hence, only a few weather satellite systems have satellites in these
orbits. Most weather satellites employ a geostationary orbit as it offers better
temporal resolution as compared to that provided by the polar satellites.
Geostationary weather satellites are the basis of the weather forecasts that are
seen on television [3]
Figure (2-4) Weather Satellites in Geostationary orbit
2.2.1.4 Remote Sensing
Remote sensing satellite is used for acquiring information about the
Earth's surface by sensing reflected or emitted energy by the Earth's surface
with the help of sensors on board the satellite. Based on the source of radiation,
remote sensing can be classified to passive and active.
Passive remote sensing refers to the detection of reflected solar
radiation by the objects on the Earth or the detection of thermal or microwave
radiation emitted by them. The most common passive sensors are imaging
sensors include multi-spectral and panoramic cameras .Camera systems are

9
optical sensors that use a system of lenses to form an image of Earth’s surface
due to the detected radiation at the focal plane of the camera.
Active remote sensing involves the use of active artificial sources of
radiation which mounted on board the satellite. These sources comprise both a
transmitter as well as a receiver. The transmitter emits electromagnetic
radiation of a particular wavelength band, depending upon the intended
application. The receiver senses the same electromagnetic radiation reflected or
scattered by the ground. One of the most common active sensors used is the
synthetic aperture radar (SAR). In SAR imaging, microwave pulses are
transmitted by an antenna towards the Earth's surface and the energy scattered
back to the satellite is measured SAR makes use of the radar principle to form
an image by utilizing the time delay of the backscattered signals.
Figure (2-5) Optical remote sensing satellite
2.2.2 Satellite Bus
Here we will briefly summarize satellite bus subsystems. These
subsystems support payload mounting, correct pointing of the payload, and
maintain payload in the right orbit. Besides, receiving commands from ground,
forming and transmitting telemetry to FCC, and provide data storage and
communications. Also, provide the required electric power and control the
payload temperature.

10
2.2.2.1 The Structural Subsystem
The structural subsystem carries, supports, and mechanically aligns the
satellite equipment. It also cages and protects folded components during boost
and deploys them in orbit. The main load-carrying structure or primary
structure is sized by either the strength needed to carry the satellite mass
through launch accelerations and transient events during Launch or stiffness
needed to avoid dynamic interaction between the satellite and the launch
vehicle structures. Secondary structure, which consists of deployable and
supports for components is designed for compact packaging and convenience
of assembly. [4]
2.2.2.2 Attitude Determination and Control Subsystem
The attitude determination and control subsystem measures and controls
the satellite's angular orientation (pointing direction).The simplest satellite are
either uncontrolled or achieve control by passive methods such as spinning or
interacting with the Earth's magnetic or gravity fields. These may or may not
use sensors to measure the attitude or position. More complex systems employ
controllers to process the satellite attitude information obtained from sensors
and actuators torquers to control attitude, velocity, or angular momentum. SC
may have several bodies or appendages, such as solar array or communication
antennas, that required certain direction pointing. The complexity of the
attitude control subsystem depends on the number of body axes and appendage
to be controlled, control accuracy, and speed of response, maneuvering
requirements and the disturbance environment. [4]
2.2.2.3 Propulsion Subsystem
Propulsion subsystem is used to change orbital parameters in order to
transfer from one orbit to another, maintain the satellite in the required orbit all
over its life time. In addition it is also used in attitude maneuver and
stabilization against environmental disturbance forces (e.g. drag), correct
satellite angular momentum and satellite attitude control. The equipment in the

11
propulsion subsystem includes a propellant supply (propellant, tankage,
distribution system, pressurization and propellant controls) [4].
2.2.2.4 Communications Subsystem
The communications subsystem links the satellite with the ground or
other satellite. Information sent to the satellite (i.e. uplink or forward link),
consists of commands and needed data to satellite (i.e. satellite control
commands and new SW version). Information received from the satellite (i.e.
downlink or return link) consists of satellite status telemetry and payload data.
The basic communication subsystem consists of a receiver, a transmitter, and a
wide-angle (hemispheric or omni-directional) antenna. Systems with high data
rates may also use a directional antenna [4].
2.2.2.5 Command and Data Handling Subsystem
The command and data handling subsystem distributes commands to
assigned subsystems. It also stores data from the satellite and payload. For
simpler systems, we combine these functions with the communications
subsystem as a tracking, telemetry, and command subsystem. This arrangement
assumes that distributing commands and formatting telemetry are base upon
extensions of communications modulation and demodulation. In its more
general structure, it comprises a central processor (computer), data buses,
remote interface units, and data storage units to implement its functions. It may
also handle sequenced or programmed events [4].
2.2.2.6 Power System
Satellites must have a continuous source of electrical power-24 hours a
day, 365 days a year. The two most common power sources are high
performance batteries and solar cells. Solar cells are an excellent power source
for satellites. They are lightweight, resilient, and over the years have been
steadily improving their efficiency in converting solar energy into electricity.
There is however, one large problem with using solar energy. If solar energy

12
were the only source of power for the satellite, the satellite would not operate
during eclipse period. To solve this problem, batteries are used as a
supplemental on-board energy source. Initially, Nickel-Cadmium batteries were
utilized, but more recently Nickel-Hydrogen batteries have proven to provide
higher power, greater durability, and the important capability of being charged
and discharged many times over the lifetime of a satellite mission. [5]
2.2.2.7 The Thermal Subsystem
The thermal subsystem controls the satellite equipment's temperatures.
Normally thermal control is done through either passive or active techniques.
In passive control, It does so by the physical arrangement of equipment, using
thermal insulation and coating. This is done to balance heat from power
dissipation, absorption from the Earth and Sun, and other radiation sources in
the space. Sometimes passive, thermal-balance techniques are not enough. In
this case, active control in the form of electrical heaters, high-capacity heat
conductors, and/or heat pipes, are employed [4].
2.3 Satellite Orbits
After a satellite is separated from launching vehicle, it moves in a path
around the Earth called an orbit. Satellite orbiting Earth due to the balance
between two forces, gravitational force which attracts the satellite towards the
Earth and centrifugal force (due to linear velocity of the satellite in orbit )
which causes repulsion of the satellite out from Earth [3],see Figure (2-6.)
During satellite mission design, the orbit is chosen which is appropriate to its
mission. So, a satellite that is in a very high orbit will not be able to see objects
on Earth as many details as orbits that are lower, and closer to the Earth's
surface. Similarly, the satellite velocity in orbit, the areas observed by the
satellite, and the frequency with which the satellite passes over the same
portions of the Earth are all important factors in satellite orbit selection.
Essentially, there are six orbital parameter called classical Keplerian orbital
elements define the orbit as shown in Figure (2-8) [5].

13
Figure (2-6) Gravitational force and the centrifugal force acting on bodies
orbiting Earth
1. Semi-major axis. a This is a geometrical parameter of the elliptical
orbit. It can, however, be computed from known values of apogee and
perigee distances as [3], for definition of apogee and perigee see Figure
(2-7).
2
perigeeapogee
a (2.1)
2. Eccentricity.e The orbit eccentricity is the ratio of the distance between
the centre of the ellipse and its focus to the semi-major axis of the
ellipse [3] see Figure (2-7).
3. Right ascension of the ascending node . it tells about the orientation of
the line of nodes, which is the line joining the ascending and descending
-nodes, with respect to the direction of the vernal equinox [3] See
Figure (2-8).
Vernal equinox is the line that intersects the Earth's equatorial plane and
the Earth's orbital plane, which passes through the centre of the Earth with
respect to the direction of the sun on 21 March [3].
4. Inclination i . is the angle that the normal to the orbital plane of the
satellite makes with the normal to the equatorial plane [3] , Figure (2-9).
5. Argument of the perigee W. This parameter defines the location of the
major axis of the satellite orbit. It is measured as the angle between
the line joining the perigee and the focus of the ellipse and the line of

14
nodes in the same direction as that of the satellite orbit [3], see Figure
(2-9).
6. True anomaly of the satellite fo. This parameter is used to indicate the
position of the satellite in its orbit. It is defined as the angle, between the
line joining the perigee and the centre of the Earth with the line joining
the satellite and the centre of the Earth [3], see Figure (2-9)
Orbits can be classified according to different criteria [3], such as
1. According to orbit Altitude
o Low Earth Orbit (LEO): orbit altitude ranging in altitude from
200–1000 km
o Medium Earth Orbit (MEO): orbit altitude ranging from 1000 km
to just below geosynchronous orbit at 35786 km.
o High Earth Orbit (HEO): orbit altitude above 35786 km.
(a) (b)
Figure (2-7) apogee ,perigee of the orbit and semi-major axis
Figure (2-8) Right ascension of the ascending node

15
Figure (2-9) Keplerian orbital elements
2. according to inclination
o Equatorial orbit : an orbit that co-planed with the equator i.e.
orbit with zero inclination
o Polar orbit: An orbit that passes above or nearly above both poles
of the Earth on each revolution. Therefore it has an inclination of
about 90 degrees
o Inclined orbit: An orbit whose inclination between 0 and 90
degrees.
3. according to Eccentricity
o Circular orbit: An orbit that has an eccentricity of 0 and whose
path traces a circle
o Elliptic orbit: An orbit with an eccentricity greater than 0 and less
than 1 whose orbit traces the path of an ellipse

16
2.3.1 Special Orbits
An important consideration in space mission design is determining the
type of Earth Orbit that best suits the design goals and purpose of the mission.
A brief description for the special orbits which frequently used such as; low
Earth orbit, medium Earth orbit, geostationary orbit, polar orbit, Sun-
synchronous orbit and Molniya orbit, is presented.
2.3.1.1 Low Earth Orbit (LEO)
Orbiting the Earth at roughly 200-1000 Km altitude [6]: Almost 90
percent of all satellites in orbit are in LEO [6]. LEO is often utilized because of
the low launch requirements that are needed to place a satellite into orbit. LEO
satellites orbit the Earth in roughly 90 minute periods. This means that they are
fast moving, and sophisticated ground equipment must be used to track the
satellite, LEO is used for such missions as flight tests, Earth observations,
astronomical observations, space stations and scientific experiments [8], [6].
Figure (2-10) LEO, MEO and GEO
2.3.1.2 Medium Earth Orbit (MEO)
MEO sometimes called Intermediate Circular Orbit (ICO), is the region
of space around the Earth above low Earth orbit (1,000 kilometers) and below
geostationary orbit (35,786 Km).The most common use for satellites in this

17
region is for navigation, such as the GPS (20,200 Km) and Galileo
(23,222 Km) constellations. Communications satellites that cover the North and
South Pole are also put in MEO [6]. The orbital periods of MEO satellites
range from about 2 o 12 hours. Telstar, one of the first and most famous
experimental satellites, orbited in MEO [9]
2.3.1.3 Geostationary/Geosynchronous Earth Orbit (GEO)
Satellite in geostationary orbit appears to remain in the same spot in the
sky all the time. Really, it is simply traveling at exactly the same speed as the
Earth is rotating below it, but it looks like it is staying still regardless of the
direction in which it travels, east or west. A satellite in geostationary orbit is
very high up, at 35,850 km above the Earth. Geostationary orbits, therefore, are
also known as high orbits; GEO is used for communications satellite
Figure (2-11) GEO satellites appear stationary with respect to a point on Earth
2.3.1.4 Polar Earth Orbit
For full global coverage of the Earth, a ground track would have to
cover latitudes up to 90o
. The only orbit that satisfies this condition has an
inclination of 90°. These types of orbits are referred to as polar orbits. Polar
orbits are used extensively for the purpose of global observations.

18
2.3.1.5 Sun Synchronous Orbits (SSO)
A Sun-synchronous orbit (SSO) is a nearly polar orbit where the
ascending node precesses at 360 degrees per year or 0.9856 degrees per day.
SSO orbital plane has a fixed orientation with respect to the_Earth-sun
direction and the angle between the orbital plane and the Earth-sun line remains
constant throughout the year as shown in Figure (2-12), so this type of orbit
assures that the local solar time (LST) at the ascending node is nearly constant
throughout the life of the mission. Satellites in sun-synchronous orbits are
particularly suited to applications like passive remote sensing, meteorological
and atmospheric studies,[6].
Figure (2-12) Sun synchronous orbit
2.3.1.6 Molniya Orbit
Highly eccentric, inclined and elliptical orbits are used to cover higher
latitudes, which are otherwise not covered by geostationary orbits. A practical
example of this type of orbit is the Molniya orbit. It is a widely used satellite
orbit, used by Russia and other countries of the former Soviet Union to provide
communication services. Typical eccentricity and orbit inclination figures for
the Molniya orbit are 0.75 and 65° respectively. The apogee and perigee points
are about 40000 km and 400 km respectively from the surface of the Earth. It
has a 12-hour orbit and a satellite in this orbit remains near apogee for

19
approximately 11 hours per orbit [4] before diving down to a low-level perigee.
Usually, three satellites at different phases of the same Molniya orbit are
capable of providing an uninterrupted service.
Figure (2-13) Molniya orbit

Chapter3 Attitude Determination
and Control Subsystem
20
CHAPTER (3)
ATTITUDE DETERMINATION AND CONTROL
SUBSYSTEM ADCS
In this chapter more detailed explanation about ADCS is introduced.
The impact of other subsystems requirements on ADCS and impact of ADCS
requirements on the other subsystems are presented. In addition, the tasks that
ADCS must perform all over the satellite lifetime and the ADCS operational
modes are describe. Then, an illustration for the physical concepts and
functions of ADCS devices such as sensors and actuators are exhibited.
Besides, different disturbances affecting rotational motion of the satellite are
demonstrated. Finally, the general control methods applied with ADCS are
presented. The control methods and
3.1 Internal influence between satellite mission and other subsystems
upon ADCS
ADCS is very closely coupled with other subsystems; it is interactively
influences and being influenced by other satellite’s subsystems. In the
following section, a briefer description for interaction between ADCS and
other subsystem is presented.
3.1.1 Internal influence between ADCS and Mission requirement
Main mission of the satellite imposes the main requirements on ADCS.
Normally, the requirements associated with the mission are
Earth pointing or inertial pointing ( this will affect in ADCS control
techniques)

21
Accuracy /stabilization requirements (this will affect in accuracy of
selected ADCS sensors).
Slewing requirements (this will affect in selection of actuators types)
Mission life time (this will affect in life time of selected ADCS devices)
Orbit parameters (this will affect in the magnitude of environment
disturbance which will perturb ADCS)
3.1.2 Internal influence between ADCS and Structure Subsystem
The ADCS Subsystem directly interacts with the structure subsystem.
The structure of the satellite affects the space craft moment of inertia and
location of its center of mass, which is affecting the dynamics and stability of
the satellite. Also, the rigidity of the structure determines whether the model of
the satellite will be a rigid body or a flexible one. In addition, mounting
accuracies of ADCS devices are one of the main constrains upon the structural
design of the satellite.
3.1.3 Internal influence between ADCS and Power Subsystem
The ADCS and the power subsystem are influencing each other. The
power budget of the satellite must take into account the requirements of the
ADCS sensors and actuators during different operational modes. For satellite
using solar panels, there are additional pointing requirements placed on the
ADCS, if solar panels must be kept aligned with the Sun for optimal
performance
3.1.4 Internal influence between ADCS and Communication Subsystem
If the satellite antenna is required to be pointed within a given accuracy
in order to communication with ground station, the Communication subsystem
will add pointing requirements on the ADCS Subsystem during communication
session.

22
3.1.5 Internal influence between ADCS and Command and Data
Handling Subsystem
Since the Command and data handling subsystem is the main brain that
organizes the data flow between satellite subsystems; so it imposes
requirements on the volume and rate of data transfer to ADCS or from ADCS
to other subsystems.
3.1.6 Internal influence between ADCS and thermal subsystem
In order to keep temperature of the satellite’s components within
specific range the thermal subsystem may impose maneuver requirements on
ADCS, by pointing the hot side to deep space and pointing the cold side
towards the sun
3.2 ADCS Tasks
According to the previous mutual impacts of ADCS with other
subsystems, ADCS has the following tasks must to be executed all over the
satellite life time. That is, ADCS executing the following tasks from the
moment of separation up to de-orbiting or discarding of the mission.
1. Damping the satellite angular velocity, obtained from LV after satellite
separation.
2. Attitude acquisition of the satellite where the BCS is oriented to be
coincide with the assigned RCS (in Earth observation missions OCS will
be this RCS). In this attitude acquisition the satellite is initially oriented
towards the RCS supports the mission requirements.
3. The satellite three-axis stabilize in the RCS with the required accuracy
during the imaging sessions.
4. Three-axis stabilization in nadir pointing with low accuracy during non-
imaging periods
5. Attitude determination with the required accuracy during all ADCS
operational modes

23
3.3 Satellite operational modes
According to the above required tasks from ADCS, the ADCS
operational mode will be.
3.3.1 De-tumbling mode (DM)
This mode occurs after the satellite is released from the LV or after
loosing of orientation due to any failure. During this mode the ADCS suppers
the satellite angular velocity that received from the LV, Because of power
limitation this process should be completed within specified period.
3.3.2 Standby Mode (SM)
After DM satellite can have arbitrary attitude Automatically so after
finishing DM, ADCS transfers to SM in order to make attitude acquisition of
satellite (i.e. Orient the satellite BCS to be co-onside with OCS to get
stabilization at nadir pointing with low accuracy) and stay in this case
whenever there is no imaging tasks assigned to the satellite. In this mode the
satellite attitude should be kept even with a low accuracy to avoid loosing the
satellite’s attitude, it is a low accuracy mode. In this mode, the most important
thing is to save the system resources (i.e. lifetime of ADCS devices) and reduce
the consumed power. ADCS stay in SM about 95% of the whole satellite
lifetime
3.3.3 High Accuracy Mode (HAM) or Imaging Mode (IM)
In this mode, ADCS should provide the required control to achieve the
pointing of the payload requirements. As an example, for imaging remote
sensing satellite using magnetic actuator the satellite must be stabilized at nadir
with high accuracy during imaging periods, so this mode called imaging mode
(IM)..

24
3.3.4 Emergency Mode (EM)
In case of any failure in ADCS (e.g. loosing satellite attitude or any
failure of ADCS devices ) ADCS automatically transfer to EM .In this mode
ADCS switch off all ADCS devices and make diagnostic for ADCS devices
according to command from ground and send TM to ground in order to take
the suitable decision.
3.3.5 Transferring from one operational mode to another
The organization of transfer from one mode to another is shown in
Figure (3-1).ADCS operational cyclogram and conditions for transferring
between modes are as follows:
1. After separation from LV and starting of satellite operation ADCS
enters DM.
2. When DM is finished, ADCS directly transfers the satellite to SM and
stay in SM.
3. Before imaging time, within specified period (i.e. Period sufficient to
stabilize the satellite at the required attitude with the required
accuracy),ADCS transfers the satellite to IM.
4. After finishing of imaging task, ADCS transfers the satellite again to
SM
5. In normal cases, the sequence of items 3-4 are repeated.
6. In case of any failure (i.e. failure in ADCS devices or attitude
orientation ), ADCS directly transfers the satellite to EM.

25
Figure (3-1) Organization of transferring from one operational mode to
another.
3.4 ADCS devices
A satellite in space must point to a given direction as assigned by the
mission requirements. Many satellites are Earth orientated while others are
inertial space object oriented such as sun or a star of interest. The orientation
of the satellite in space is known as its attitude. In order to achieve control and
stabilization of the satellite, attitude sensors are used to determine the current
attitude and actuators are used to generate required torque to maintain the
required attitude. This section gives brief description of the most common
used ADCS sensors and actuators.
3.4.1 ADCS Sensors
Sensors generally determine the attitude and pointing direction of
satellite with respect to reference objects, this object could be inertial space or a
body of known position. The most commonly used reference objects, Earth,
Sun, stars, geomagnetic field and inertial space.
DM
finishing
ADCS
failure
DM SM IM
EM
Imaging
command
Finishing imaging
session
ADCS
failure
ADCS
failure
Fixing of ADCS
failure

26
3.4.1.1 Earth’s Horizon sensor
For near-Earth satellites the Earth covers a large proportion of the sphere
of view and presents a large area for detection. The presence of the Earth alone
does not provide a satisfactory attitude reference hence the detection of the
Earth’s horizon is widely used.
Horizon sensor is infrared device that detect the contrast between the
cold of deep space and the heat of the Earth’s see Figure (3-2). Horizon sensors
can provide pitch and roll attitude knowledge for Earth-pointing satellite. For
the better accuracy in low Earth orbit (LEO), it is necessary to correct the data
for the Earth oblateness and seasonal changes in the apparent horizon
[10].Earth’s Horizon sensor is used in AEROS-I,-2, MAGSAT, SEASAT [15].
Figure (3-2) principle of Earth horizon sensor
3.4.1.2 Sun sensor
Sun sensor is widely used with satellite mission due to the special
features of sun as a space object. One of these features is the brightness of the
sun, which makes it easy to be distinguished among other solar and stellar
objects. also the Sun-Earth distance makes it appear as nearly a point source
(0.25 º). Those factors urge ADCS designer to rely upon sun sensors in high
pointing accuracy missions.

27
Sun sensor measures one or two angles between their mounting base
and incident sunlight. Categories of sensors are ranging from just sun presence
detector, which detects the existence of sun, rather accurate analogue sensor
measuring sun incidence angle, up to high accuracy digital instrument, which
measure the sun direction to accuracy down to one arc-minute. Typical digital
sun sensor is shown Figure (3-3).
Sun sensor is accurate and reliable, but require direct line of sight to the
sun. Since most low-Earth orbits include eclipse periods, the attitude
determination system should provide some way of handling the regular loss of
Sun vision. Sun sensor is used in AEROS-1,2 , GEOS-3, MAGSAT, SAGE,
SEASAT [15].
Figure (3-3) Sun sensors
3.4.1.3 Star mapper
Star mapper provides the most accurate absolute pointing information
possible for a satellite attitude. It contains Charged-Coupled Device (CCD)
sensors or Active Pixel Sensors (APS) which provides a relatively inexpensive
way to image the sky. It extracts information about satellite attitude by
mapping the obtained stars image with the stored stars pattern catalog. Any

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29
used to provide information about satellite orientation. Employing estimation
techniques such as Kalman filter, allows magnetometer to work as standalone
device for attitude determination [11]. The Earth’s magnetic field also varies
with time and can't be calculated precisely, so a magnetometer is often used
with another sensor such as a sun, horizon or star sensor or a gyroscope in
order to improve the accuracy. Magnetometer is used in AEROS-1, Egyptsat1,
GEOS-3, SEASA [15].
Figure (3-5) flux-gate magnetometer
3.4.1.5 Inertial Sensor or Gyro
By definition, a gyroscope, is any instrument, which uses a rapidly
spinning mass to sense and respond to changes in the inertial orientation of its
spin axis. There are types of attitude sensing gyros: mechanical and optical
gyro. These sensors measure satellite orientation change.
Mechanical Gyroscopes
The angular momentum of a gyro, in the absence of an external torque,
remains constant in magnitude and direction in space. Therefore, any rotation
of the satellite about the gyro's input axis results in a precession of the gimbal

30
about the output axis. Figure (3-6) shows the basic principles of how
mechanical gyros operate
Figure (3-6) Three degree-of-freedom gyroscope construction geometry.
Optical Gyroscopes
Optical gyros are gyroscopes that utilize a light ring instead of a
mechanical rotor as the main component to determine rotational changes. All
optical gyros work on the same principle, the Sagnac effect, This effect works
on relativistic principles but can be described in "normal" terms. Two light
beams are traveling through circular paths of the same length but in opposite
directions around in an optical coil. If the optical coil is rotating, one of the
light beams will take a longer period of time to travel the circumference of the
coil. This time lag is measured and converted into a rotational rate for the coil.
Thus, the rotation the gyro is feeling can be measured. The length changes
associated with the light beam are of nuclear dimensions and are difficult to
measure. However, great accuracy can be achieved through the use of this type

31
of gyroscope. The most common devices of this type is the Ring Laser Gyro
(RLG) and Fiber Optic Gyros (FOG) .Gyros are used in ATS-6,
Egyptsat1,LANDSAT-D·, MAGSAT [15].
Figure (3-7) The QRS11Pro gyro used on Rømer
Typical values for accuracy of ADCS sensors are shown in the
following table
Table 3-1 Ranges of ADCS sensors accuracy
Sensor Accuracy
Earth’s Horizon sensor
0.05 deg. (GEO)
0.1 deg. (LEO)
Sun sensor 0.01 deg.
Star mapper 2 arc. sec.
Magnetometers
1.0 deg. (5,000 Km altitude)
5.0 deg. (200 Km altitude)
Gyro 0.001 deg./hr

32
3.4.2 ADCS Actuators
ADCS actuators are used to generate the required torque for correction
of satellite attitude. The generated torque is operated against the environmental
disturbance or to force the satellite to point to a cretin direction according to the
control system requirement. A brief description of the commonly used
actuators is presented in this section.
3.4.2.1 Momentum and Reaction Wheel
Momentum wheels and reaction wheels are similar in construction; they
are simply motor with a flywheel mounted on the motor shaft, the difference in
terminology resulting primarily from the speed at which they operate. A
momentum wheel typically operates at constant speed, providing a means of
momentum storage, which in turn provides gyroscopic stabilization to the
satellite. Reaction wheels generally operate at varying speed, providing means
of reacting torque. According to Newton's third law, as a torque is electrically
applied on the motor shaft to cause the wheel to accelerate, an equal and
opposite torque is generated on the satellite, causing the attitude to change.
Momentum wheels are commonly used singly or in pairs to provide spin
stabilization. Normally, reaction wheel system consists of four wheels. Three
reaction wheels are aligned to the satellite pitch, yaw and roll control axes. The
fourth wheel is skewed symmetrically with respect to the orthogonal control
axes. This commonly used configuration provides full redundancy for roll or
pitch or yaw in case of wheel failure. An image of typical reaction wheel is
shown in Figure (3-8)

33
Figure (3-8) The TELDIX Momentum and Reaction
Momentum and reaction wheels have the advantage of providing quick
and accurate attitude control. Also, they can be used at any altitude. Their
disadvantage is that they can be costly, massive, and require large amounts of
power. However, wheels may saturate since the RW is a motor that has
maximum speed, since the angular momentum that can be stored in the wheels
is limited, so a secondary control system is used to prevent the stored
momentum from reaching the maximum limit. The secondary control system
can be thrusters system or magnetorquers. Momentum and reaction wheels are
used in Egyptsat1, FLTSATCOM, MAGSAT and SEASAT [15].
3.4.2.2 Magnetic actuators
Magnetic actuators enforce a torque on the satellite by generating a
dipole moment, which interacts with the Earth's magnetic field. Generally,
there are two types of magnetic actuators, torque coils and magnetic rods or
magnetorqure.
1. Torque Coils

34
The torque coil is simply a long copper wire, winded up into a coil.
Generally, three coils are used, one coil in each axis as shown in Figure (3-9
The generated dipole moment L by each coil is calculated by
ANiL coil (3.1)
Where, coili is the current in the coil, N is the number of windings in
the coil, and A is the area spanned by the coil.
Figure (3-9) Torque Coils
2. Torque Rods
Torque rods operate on the same principle as torque coils, but instead of
a large area coil the windings is spun around a piece of ferromagnetic material
with very high permeability as shown in Figure (3-10). Ferromagnetic
materials, have a relative permeability, , of up to 106. the generated dipole
moment L is calculated by the following formula
ANiL coil (3.2)

35
Hence, generating specified dipole moment from magnetic rod needs
current much lower than that needed to magnetic coil. However, the weight of
magnetic rod increases drastically because of the metal core in the rods.
Another inconvenience of the torque rods is the hysteresis effect associated
with ferromagnetic core which add nonlinearity to the control loop. Advantages
and disadvantages of using magnetic actuator will be discussed in details in
CHAPTER (5). Magnetic actuators are used with Egyptsat1, MAGSAT,
TIROS-IX, LANDSAT-D and AEROS-1, 2[15].
Figure (3-10) Torque rods
3.4.2.3 Thruster
Thruster works on the principle of Newton's third law, according to
which "for every action, there is an equal and opposite reaction". Referring to
this principle, if gas is propelled out of a nozzle, the satellite will accelerate in
opposite direction. However, if the nozzles are not pointed directly away from
the center of mass this will lead to cause rotational of satellite as well. In
addition, if two thrusters in opposite direction but not co-lined rotation only
will be generated. The source of the used gas defines the type of thruster .
Cold gass thrusters use high pressure storage tank. Hot gas thrusters use the
combustion of either monopropellant or bipropellant.
Six thrusters are needed to be mounted in pairs to generate the torque
needed for three-axis control. Thruster as actuator is highly accurate and

36
generate higher torque than RW and magnetic rods. On the other hand, the
structure used with the thrusters is large and heavy. Besides, run out of either
gas or propellant will lead to stop functioning of thrusters. Thrusters are used in
ATS-3,6 , FLTSATCOM, GOES-I and SKYNET[15].
Figure (3-11) Torque generated thruster mounted to satellite
3.5 Disturbance Environment
In an Earth orbit, the space environment imposes several external
torques that the ADCS system must tolerate. According to orbit altitude, three
or four sources of disturbing torques are affecting the space craft[4]. These
torques are; gravity gradient, magnetic field effect, solar radiation pressure, and
aerodynamic forces. Those disturbances are affected by the satellite’s
geometry, orientation, and mass properties in addition to satellite orbital
altitude.
3.5.1 Gravity Gradient Disturbance
Any object with nonzero dimensions orbiting Earth will be subjected to
a “gravity-gradient” torque. In short, the portions of the satellite that are closer
to the Earth are subjected to a slightly larger force than those parts farther away
[7]. This creates a force imbalance that has a tendency to orient the satellite
towards the center of Earth in order to compensate this imbalance. According
to [15] the gravity gradient torque can be determined by equation (3.3) . The
worst case torque arises at
o
90

37
)2sin(
2
3
3 iiZZgg JJ
R
T (3.3)
Where,
Tgg: is the resulting gravitational torque [Nm]
μ: is the gravitational constant of the earth [m³/s²] (μ = 3.896*1014
m³/s²)
Jii :is the moment of inertia tensor for the satellite in i axis.(in body
coordinate system) [kgm²] (i=x,y,z)
Is the maximum deviation angel from the local vertical [rad]
R: is the distance between satellite center of mass and earth center of
mass [km]
The previous formula for calculation of gravity gradient is used to give
course estimation of gravity gradient disturbance torque but an accurate
formula given in (4.28) is used in calculation of satellite mathematical model
3.5.2 Magnetic Field Disturbance
Magnetic field torques are generated by interactions between the
satellite magnetic dipole and the Earth’s magnetic field. This satellite magnetic
dipole is the summation of two components; first component is the induced
magnetic dipole, which is caused by current running through the satellite
wiring harness and second component is the residual dipole moment, which is
caused due to magnetic properties of the satellite components. The satellite
magnetic dipole exhibits transient and periodic fluctuations due to power
switching between different subsystems. These effects can be minimized by
proper placement of the wiring harness. The magnetic torque is calculated by
following formula
BDTm (3.4)
Where
D = the vector of total satellite magnetic dipole.
B = local geomagnetic field vector.

38
In the worst case, the vectors are perpendicular to each other and the
cross product turns into a product of scalar values.
3.5.3 Solar Radiation Pressure Disturbance
Solar radiation pressure is a result of the transfer of momentum from
photons of light to the surface of the satellite. The result of this pressure across
the satellite surface is a force that acts through the center of pressure, psc , of the
satellite. In most cases, the center of pressure is not co-onside with the center of
mass of the satellite, thus a torque will be generated around the center of mass
cm see Figure (3-12). For Earth-orbiting satellite, where the distance from the
satellite to the Earth is small compared to the Earth-Sun distance, the mean
solar flux acting on the satellite is considered a constant (regardless of orbital
radius or position).
The solar radiation torque is calculated using the following equation [4] .
)()cos()1( gpssSp cciqA
c
So
T
(3.5)
Where
So is solar constant [W/m²] = 1428 W/m² (max)
c is speed of light [m/s] = 3*108
m/s
A is the cross sectional area subjected to solar radiation pressure [m²]
q is reflectance factor (0: perfectly absorbing, 1: perfectly reflecting)
si is the angle of sun light incidence [rad]
cps is the center of pressure [m]
cg is the center of gravity [m]
Referring to the previous assumptions, the solar pressure disturbance
torque is the only one that is not dependent of the orbit altitude. However, it is
dependent of the sun incidence angle i. The worst case torque arises at i = 0°.

C
3.5.
as
esp
torq
cha
to t
sign
effe
adT
Wh
is
cD i
A is
vc i
cps
cg i
3.6
com
Chapter3
.4 Aerod
Aerody
shown in
ecially at l
que is alm
anging of s
the aerodyn
nificantly
ects is calcu
Dc
2
1
here
s the densit
is the coeff
s the cross
s the orbita
is the cent
is the cente
Attitu
There
mpensation
dynamic D
ynamic torq
Figure (3
low altitud
ost negligi
ome param
namic drag
with solar
ulated by (
C cvA
2
ty [kg/m³]
ficient of d
sectional a
al velocity
ter of press
er of gravit
Fig
ude Contr
are differe
n and to ma
Disturbanc
ques are du
3-12. Aero
des (less th
ible. These
meters, such
g during ti
r activity.
(3.6) .
gpa cc
drag
area subjec
[m/s]
sure [m]
ty [m]
gure (3-12)
rol techniq
ent techniq
aintain the
39
ce
ue to atmo
odynamic
an 500). A
e torques i
h as cross
lting. In ad
The gene
cted to atm
Sunlight a
ques
ques to app
required or
A
a
ospheric dr
torques ca
At higher a
is difficult
sectional a
ddition, atm
erated torq
mospheric d
and drag ef
ply contro
rientation .
Attitude De
and Contro
rag acting o
an be qui
ltitudes the
to be calc
area of sate
mospheric
que due to
drag [m²]
ffect
ol torque fo
. For these
eterminatio
l Subsystem
on the sate
ite signific
e aerodyna
culate beca
ellite subje
density va
o aerodyna
(3.6)
for disturba
purposes,
on
m
ellite
cant,
amic
ause
cted
aries
amic
ance
two

40
types of control techniques are often employed , passive and active control
[4][12]. Since Attitude control system, is highly mission dependent, so the
decision to use a passive or an active control technique or a combination of
them depends on mission pointing and stabilization requirements.
3.6.1 Passive Control
For missions with rather coarse orientation requirements, passive control
techniques are used for attitude control. The main advantageous of these
techniques are saving resources concerning both mass and power and the
associated cost. In addition, they provide longer lifetime for the space mission.
However, a poor pointing accuracy is obtained. The most common passive
control techniques are passive magnetic system (i.e. Permanent magnate),
gravity gradient and spin stabilization [4].
3.6.1.1 Passive magnetic
In this method, the concept of magnetic compass is applied, that is, the
satellite is equipped with permanent magnet that will keep the alignment
between certain axis of the satellite with geomagnetic field vector .As a result,
the south pole of the magnet will be drawn towards the magnetic north pole of
the Earth, and vice versa. This will lead to a slight tumbling motion with two
revolutions per orbit and no possibilities of controlling spin around the magnets
axis as shown in Figure (3-13) so continues nadir pointing will not be possible.
Permanent magnet technique is used in AZUR-1 [15].

41
Figure (3-13) passive magnetic control orientation profile.
3.6.1.2 Gravity-gradient stability
Gravity-gradient stability uses the mass characteristics of the satellite to
maintain the nadir pointing towards Earth (as described in 3.5.1). The
magnitude of gravity-gradient torque decreases with the cube of the orbit
radius, and symmetric around the nadir vector, thus not influencing the yaw of
satellite. Therefore, the gravity gradient stability is used in simple satellite in
LEO without yaw orientation requirements [4].
Yet, stability in the gravity gradient case depends upon the the
configuration of the mass characteristics of the space craft. The following
condition is necessary for gravity-gradient stability [12]:
JzzJxxJyy&JzzJxxJyy (3.7)
Where Jii :is the moment of inertia tensor for the satellite in i axis.(in
body coordinate system) (i=x,y,z)

42
As a result, the gravity gradient stability can be achieved by
manipulation of lay out of the satellite's components to grantee the above
mentioned condition (3.7). Other solution is to add a sufficient mass on a
deployed boom to reach the stability condition. This will increase the moment
of inertia in the directions transverse to the boom, and the satellite will be
stable with the mass pointed toward or away from the earth. Gravity gradient
stability is suffering from continuous oscillation about nadir due to lack of
damping. Hence, gravity-gradient stabilization should be supported with
damping system to reduce the small oscillation around the nadir vector.
Gravity-gradient stabilization technique is used in DODGE, GEOS-3, and
RAE-2 [15].
3.6.1.3 Spin stabilization
Spin stabilization technique applies the gyroscopic stability to passively
resist the effect of disturbance torques about the spinning axis. Spin-stabilized
satellites spins about their major or minor axes, so angular momentum vector
remains approximately fixed with respect to inertial space. [15]. Spinning
satellite is classified according to spinning object to single or dual spin. The
stability criteria and the corresponding spinning axis is predicted according to
the following analysis.
3.6.1.3.1. Single Spin
In single spin satellites, the whole satellite spins about the angular
momentum vector as shown in Figure (3-14) This method of stabilization is
simple and has a high reliability. The cost is generally low, and it has a long
system life. However, Spin-stabilized satellite are subject to nutation and
precession, but have a gyroscopic resistance which provides stability about the
transverse axis.
On the other side, spinning satellite will have poor maneuverability.
Beside, it will not be suitable for systems that need to be Earth pointing, such

43
as payload scanners and communication antennas. Single spin stabilization
technique is used in AEROS-I,2, ALOUETIE-I,2and ARIEL-I [15].
Figure (3-14) spin stabilization
3.6.1.3.2. Dual Spin
In satellite with dual spin, a major portion of the satellite is spun, while
the payload section is despun. This technique is favorable because fixed inertial
orientation is possible on the despun portion. This method of stabilization has a
few disadvantages, however. This system is much more complex, which leads
to an increase in cost and a decrease in reliability. In addition, the stability is
sensitive to mass imbalances. Duel spin stabilization technique is used in ANS,
ATS-6, SEASAT and SMM [15].
3.6.2 Active control techniques
For complex mission requirements, satellite requires continues
autonomous control about the three axes during the mission. In general, active
control systems employ momentum exchange wheels, magnetic control
devices, and thrusters. Advantages of these systems are high pointing accuracy,
and a not constrained to inertial pointing like spin stabilization technique.

44
However, the hardware is often expensive, and complicated, leading to a higher
weight and power consumption.
3.6.2.1 Momentum exchange Wheels
Three-axis stabilization through momentum exchange wheels applies
reaction wheels, momentum wheels, and control moment gyros. This is to
provide three axis stabilization. Advantages and disadvantages of this wheel
system are discussed in 3.4.2.1. Three-axis stabilization technique using wheels
is used in Egyptsat1, FLTSATCOM, MAGSAT and SEASAT [15].
3.6.2.2 Magnetic actuators
Magnetic actuators devices use the interaction of the satellite magnetic
dipole moment and the Earth’s magnetic field to provide a control torque.
Magnetic control torques work better in low Earth orbits than higher orbits,
such as geostationary, because as the distance from the Earth increases, the
geomagnetic strength decreases. Advantage and disadvantage of magnetic
actuators is discussed in 3.4.2.2 Three-axis stabilization technique using
magnetic actuators is used in Egyptsat1, MAGSAT, TIROS-IX, LANDSAT-D
and AEROS-1, 2[15].
3.6.2.3 Thrusters
Mass propulsive devices, such as thrusters, can be used for three-axis
stabilization. These often consist of six or more thrusters located on the satellite
body. The strength of the obtainable torque is dependent on the thrust level as
well as the torque-arm length about the axis of rotation. Advantage and
disadvantage of thrusters is discussed in 3.4.2.3 3.4.2.2. Three axis stabilization
technique using thrusters is used in ATS-3,6 , FLTSATCOM, GOES-I,
SKYNET[15].

Chapter 4 Control System Modeling
45
CHAPTER (4)
CONTROL SYSTEM MODELLING
4.1 Reference Coordinate Systems
Several different reference coordinate systems or reference frames are
used to describe the attitude of a satellite in orbit. The most utilized coordinate
systems employed in attitude control problem are the inertial, Greenwich,
orbital, body, and device frames.
4.1.1 Geocentric Inertial Coordinate System
The Geocentric Inertial Coordinate System or Earth-Centered Inertial
(ECI)coordinate system has its origin in the Earth center The IZ -axis points is
the axis of rotation of Earth. The IX -axis is in the direction of the vernal
equinox, and the IY -axis completes the right-hand rule for the coordinate
system. A demonstration for the geocentric inertial coordinate system is shown
in Figure (4-1).
4.1.2 Greenwich Coordinate System
The Greenwich Coordinate System or Earth-centered Earth-fixed
reference frame also has its origin at the center of the Earth, but it rotates
relative to inertial space, shown in Figure (4-2) The GZ -axis direction is the axis
of rotation of Earth. The GX -axis points to the Greenwich Meridian, and the GY
-axis completes the right-hand rule for the coordinate system

46
Figure (4-1) Inertial coordinate system
Figure (4-2) Greenwich coordinate system

47
4.1.3 Orbital Coordinate System
The orbital coordinate system (OCS) is located at the mass center of the
satellite. This frame is non inertial because of orbital acceleration and the
rotation of the frame.
The motion of the frame depends on the orbit altitude. The -axis in the
direction from the satellite to the Earth , OZ -axis in the direction opposite to
the orbit normal, and the OX -axis is perpendicular to the OZ -axis and OY -axes
according to the right-hand rule . In circular orbits, OX is the direction of the
satellite velocity. The three directions OX , , and are also known as the roll,
pitch, and yaw axes, respectively. Figure (4-3) shows a comparison of the
inertial and orbital frames in an equatorial orbit.
Figure (4-3) orbital coordinate system
4.1.4 Body Coordinate System
Like the OCS frame, the body coordinate system has its origin at the
satellite’s mass center. This coordinate system is fixed in the body. The -axis in
the direction from the satellite to the Earth , -axis in the direction opposite to
the orbit normal, and the -axis is perpendicular to the -axis and -axes
according to the right-hand rule . In circular orbits, is the direction of the

48
satellite velocity. The relative orientation between the orbital and body frames
is the satellite attitude, when the satellite is nadir pointing OCS is co-onside
with BCS
Figure (4-4) Body coordinate system
4.1.5 Device Coordinate System
The device coordinate system is fixed at the device body (i.e. sensor or
actuator …). It define the orientation of the device with respect to satellite BCS
.As shown in Figure (4-5) the ZD- axis is Z-axis of the device 's body and XD-
axis is X-axis of the device 's body and YD-axis is perpendicular to ZD-axis and
XD-axis

49
Figure (4-5) device coordinate system
4.2 Modeling of Satellite Rotation Around it's Center of Mass
In this section, derivation of the equations used for modeling the
kinematics and dynamics of satellite rotational motion. These equations are
borrowed of Bhanderi, 2001.
In general, kinematics equations involved in the satellite model are
represented through three types of parameters. The direction cosine matrix has
the disadvantage of having nine parameters to represent three degrees of
freedom motion. Due to this redundancy, numerous ways of representing the
satellite attitude with a minimum set of parameters have been developed.
Euler angles describe the rotation around the principal axes and use
therefore only three parameters. However some singularities arise for some
rotations, which is why Euler angles are commonly used when the attitude of
the object involved, is known to be within a certain margin to avoid this
singularities [Wertz, 1978].
Quaternions use four parameters with a single constraint, to represent
attitude, and are subjected to no singularities. This is useful when considering
that the attitude of a satellite is usually unknown after the release from the
launcher. For this reason quaternions are commonly used in space. Appendix A

50
gives a brief description of the Euler angles, direct cosine matrices and
quaternions.
The modeling of a satellite’s rotational motion is divided into the
kinematic equation and the dynamic equation. The kinematic equation
describes the change in the attitude parameters of the satellite, regardless of the
forces acting on it. The dynamic equation describes the time dependent
parameters as functions of external forces.
4.2.1 Model of Dynamic Equation
Here we assume that satellite is a rigid body moving in ICS. Then
satellite motion can be described by the translation motion of its center of mass,
in addition to a rotational motion about some axis through its center of mass. In
the following analysis, depend on the well-known operator equation acting on a
given vector [13][14].
w
dt
d
dt
d
B (4.1)
Where
B vector defined in BCS
I vector defined in ICS
Figure (4-6) Angular motion of rigid body

51
which simply states that "the rate of change of the vector A as observed
in the fixed coordinate system I (e.g. ICS) equals the rate of change of the
vector A as observed in a rotating coordinate system B (e.g. BCS) with angular
velocity w , plus the vector product Aw ". In Figure (4-6), it is assumed that,
an orthogonal three axis frame has its origin O located at the center of mass of
the satellite body B ; i, j, k are the respective unit vectors along BCS. So for
any particle im , in the satellite body B, rRR , hence
iioiioi rwvvrwrRR (4.2)
Where
w is the angular velocity vector of the body B with respect to ICS.
The angular momentum H of a body particle im , using (4.2) can be
expressed as
)( iioiiiiii rwrRmrRmrH (4.3)
However, by definition we have 0ii vr in a rigid body, so in this case
it follows that
)()( iiiiioioiii rwmrrmvrwvmrH (4.4)
To find the angular momentum of the entire body, we shall sum the
momentum components of all the mass particles
i i
iiiiio rwmrrmvH )(
(4.5)
Since the angular motion is about the center of mass
i
iirm 0 so (4.5) can be
rewritten as

52
i
i
ii mrwrH )(
(4.6)
After performing the vector triple product, we get the following equations
i i
iiiyiiix
i
iiiz
i i
iiiziiix
i
iiiy
i i
iiiziiiy
i
iiix
mzywmzxwmxywk
mzywmyxwmzxwj
mzxwmyxwmzywiH
)(
)(
)(
22
22
22
(4.7)
Where iii zyx ,, are the coordinates of a particle i in the BCS and
zyx www ,, are the angular velocity comportments around kji ,, body axes. The
summations of the squared coordinate components are easily identified as the
three moments of inertia of the body about its three orthogonal axes. The
summations of the products of the coordinate components are identified as the
products of inertia. With these definitions equation (4.7) can be rewritten as
zyyzxxzzzyzzyxxyyyxzzxyyxxx JwJwJwkJwJwJwjJwJwJwiH
(4.8)
zyx kHjHiHH
(4.9)
If we define the angular velocity vector which describe the angular
velocity of BCS with respect to ICS as T
zyx ][ then (4.9) can be rewritten
as
I
JJJ
JJJ
JJJ
H
z
y
x
zzyzxz
yzyyxy
xzxyxx
(4.10)

53
Where J the inertia tensor or inertia matrix
The dynamic equation of motion can be derived through applying
Newton’s second law for rotational motion, where the rate of change in angular
momentum of the satellite is equal to the sum of all external torques. and
recalling (4.6)
ii
k
i
i
k
i
i
vmr
HH
1
1
(4.11)
Where ir is the position of the i th
particle with mass im and velocity iv .
Taking the time derivative of Equation (4.11), yields
k
i
iiiiii amrvmvH ).( (4.12)
ia is the acceleration of the i th
particle. The first term under the
summation of Equation (4.12) is a cross product of two parallel vectors, which
is zero. Realizing that iami is the force acting on the i th
particle yields
extTH (4.13)
Where extT is the sum of external torques acting on the satellite. Such as
controlling torques cT , gravity gradient ggT and other external disturbance disT
(i.e. aero drag, magnetic disturbance….)
cdisggext TTTT (4.14)
Equation (4.13) holds only, if the internal torques sum up to zero (i.e.
the body is rigid)[10]. An expression of the derivative of the angular
momentum in terms of the satellite’s angular velocity is sought, in order to
obtain the dynamic equation.

54
In ICS frame, denoted I , the angular momentum of the satellite can be
expressed as a function of the angular velocity and the moment of inertia
matrix J of the satellite, by
II
JH (4.15)
Since the moment of inertia is more conveniently expressed in the BCS
frame, denoted B , the angular momentum is found in the body frame. The
transformation matrix B
IA represents the rotation from the inertial frame to the
satellite body frame, which is used to represent Equation (4.15) in the body
frame, yielding
IB
I
B
HAH (4.16)
The derivative of
B
H is given by
IB
I
IB
I
IB
I
B
HAHA
HA
t
H
(4.17)
In order to obtain an expression for B
IA consider the kinematic equation
for rotating systems, which for the angular momentum vector H is
BBIB
BBBI
HHH
HHH
(4.18)
Since IB
I
BI
HAH , combining Equations (4.17) and (4.18), gives
IB
I
BIB
I
HA
HHA
(4.19)
Defining the cross product matrix function as
0
0
0
)(
12
13
23
vv
vv
vv
vS (4.20)
Where vis an arbitrary vector, Equation (4.19) is written

55
IB
I
IB
I HASHA (4.21)
Since Equation (4.21) holds for all the sought expression for the
derivative of the attitude matrix is
B
I
B
I ASA (4.22)
Inserting Equation (4.22) into Equation (4.17), gives
IB
I
IB
I
B
HAHASH (4.23)
Recalling from Equation (4.13), that the derivative of the angular
momentum is the external torques and applying the attitude matrix rotations in
Equation (4.23), yields
B
ext
BB
THSH (4.24)
Finally the angular momentum is expressed in terms of the moment of
inertia and the angular velocity, as given in Equation (4.15). Solving with
respect to gives the sought nonlinear differential equation, written in the
form
JTJ
B
ext
1
(4.25)
Where the superscript of frame is left out, since all vectors and matrices
are given in the BCS frame.
If the satellite is equipped with reaction wheels, the total angular
momentum is given by
wHJH (4.26)
Where wH is the angular momentum of the wheels.
Using Equation (4.26) and (4.14)in the derivation of the dynamics
above, yields
wwcdisgg HHJTTTJ 1
(4.27)
Where wH is the sum of the internal torques, generated by the
momentum wheels.

Spacecraft attitude magnetic controller

Spacecraft attitude magnetic controller

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