Gravity Gradient Stabilization of Nano Satellite Using ...

Gravity Gradient Stabilization of Nano Satellite Using Fuzzy Logic Controller A Thesis Submitted to the College of Science-University of Kufa, in partial fulfillment of the requirements for the degree of Master of Science in physics

By

Mohammed Chessab Mahdi Supervised by Asst.Prof.

Dr.Mohammed Jaafar AL-Bermani

2013 A.D

1434 A.H

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‫ب ِْس ِم ه ِ‬ ‫اَّلل هالر ْ َْح ِن هالر ِح ِي‬

‫هالر ْ َْح ُن عَ ه ََّل الْ ُق ْرَآ َن َخلَ َق ْاْلن ْ َس َان‬ ‫ْ ِ‬ ‫عَل ه َم ُه الْ َب َي َان ال هش ْم ُس َوال َق َم ُر ِ ُِب ْس َب ٍان‬ ‫َوالنه ْج ُم َوال هش َج ُر ي َ ْس ُجدَ ِان‬ ‫صدَ َق هللا العيل ال َع ِظي‬

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DEDICATION

To my parents who always stood behind me and knew I would succeed. Gone now but never forgotten. Thanks for all you did. To my family members who are my partners in happiness and success. To my real brother (Jaafer) who was always supporting me in my life with my sincere gratitude. To all friend and colleagues at work with respect.

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ACKNOWLEDGEMENT

It is a great pleasure for me when presenting this thesis to acknowledge the help of Great Allah Almighty for all His blessing and guidance, which greatly helped me to finish this thesis. I would like to express my special thanks and sincere gratitude to my teacher and supervisor Dr. Mohammed J. Al-Bermani for his patience, motivation, enthusiasm, immense knowledge, continuous direction and support. Without his scientific and technical assistance the thesis never would have been completed. I would like to express my gratitude to all those who gave me the possibility to complete this thesis especially Dr. Adnan Falh Hassan, Dr. Hayder Hamza Hussain and Dr. Adel H.O. Alkhyatt. Finally, I would like to extend my sincere thanks to my family for their patience and endurance.

Mohammed Chessab Mahdi April 2013

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ABSTRACT

Kufasat is a student satellite program and the goal of this program is to design and launch a cube satellite .The purpose of this study is to design and to develop an efficient Attitude Determination and Control System- ADCS for the satellite. The satellite is intended to fly in a low earth orbit at 600km altitude and its mission is to perform scientific measurements. The satellite is to be cubic with 10 cm on all sides and have a total mass of approximately 1kg and be three -axis gravity gradient stabilized. The satellite consists of (1.5) m long gravity gradient boom. The gravity gradient boom has a tip mass of (40) g to improve the gravity gradient stabilization .A gravity gradient stabilized satellite has a limited stability and a pointing capabilities, and a magnetic coils are added to improve both the three axis stabilization and the pointing properties. Magnetic coils around the satellite's XYZ axes can be fed with a constant current-switched in two directions-to generate a magnetic dipole moment which will interact with the geomagnetic field to generate a satellite torque, which is used to control the rotation of the satellite. A problem is that both the direction and the strength of the geomagnetic field change and magnetic control become non-linear and time dependent .The magnetic coils are controlled by using a fuzzy logic controller, based on a combination of membership functions and rules. The controller consists actually of 3 MISO fuzzy control laws, one for each magneto torquer (MX, MY and MZ coils). Each control law embodies a fuzzy rule base to decide on the control desirability and output level when using the corresponding torquer. Magnetic coils allow cheaper satellite, and are an attractive solution to small, inexpensive satellite in low earth orbit. The satellite will be during separation from the launch vehicle is exposed to forces from the release mechanism and tumbling may occur. A detumbling mode is activated in order to calm down the movement .The gravitation boom will be deployed first when the movement of the satellite is sufficient small. This study deals with attitude control after the detumbling mode has successfully been completed and the boom is fully deployed. I UNIVERSITY OF KUFA

Contents SERIES

page

Abstract

I

List of Contents

II

List of Symbols

VI

List of Abbreviations

IX

List of Tables

X Chapter One: General Introduction and Review

1.1 Introduction

1

1.2 Stabilization of spacecraft

1

1.2.1 Passive stabilization

1

1.2.2 Active stabilization

2

1.2.2.1 Spin Stabilization

2

1.2.2.2 Three Axis Stabilization

3

1.3 Mode of stabilization of spacecraft

4

1.3.1 Detumbling mode

4

1.3.2 Boom deployment

5

1.3.3 Three axis controller

5

1.4 Attitude Determination and Control System (ADCS) 1.4.1 Attitude Determination System (ADS)

5 6

1.4.1.1 Earth Sensors

6

1.4.1.2 Sun Sensor

7

1.4.1.3 Star Trackers

7

1.4.1.4 Magnetometer

8

1.4.1.5 Gyroscopes

8

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1.4.1.6 GPS

8

1.4.2 Attitude Control System (ACS)

9

1.4.2 .1 Active actuators

10

Thrusters

10

Reaction wheels

10

Magnetic torques

11

1.4.2 .2 Passive actuators

11

Gravity gradient

11

Magnetic dipole

12

Aerodynamic

12

Solar radiation

12

1.5 Fuzzy Logic Controller

12

1.6 Orbit Dynamics

13

1.7 Earth Magnetic Field

16

1.8 Literature Survey

18

1.9 Objective of Work

23

1.10 Layout of The Thesis

23

Chapter Two: Modeling of Satellite Dynamic 2.1 Introduction

24

2.2 Reference Frames

25

2.2.1 Earth-Centered Inertial Frame- ECI

25

2.2.2 Body Fixed Frame

25

2.2.3 Spacecraft Orbit Frame- SCO

25

2.2.4 Earth-Centered Earth Fixed Frame- ECEF

25

2.2.5 North-East-Down Frame-NED

26

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2.3 Equations of Motion

27

2.3.1 Dynamic Equations

27

2.3.2 Kinematics Equations

29

Euler Angles

30

Quaternion

32

2.4 Linearized Dynamic Model

34

2.5 Disturbance Torques

35

2.5.1 Gravity Gradient Torque

35

2.5.1.1 The Stability Analysis

39

2.5.1.2 The Effect of Yaw Spinning

40

2.5.2 Magnetic Field Torque

42

2.6 Complete Linearized Mathematical Model

43

2.7 Test for Satellite Motions

44

2.7 State Space Modeling

49

Chapter Three: Fuzzy Logic and Fuzzy Controller Design 3.1 Introduction

52

3.2 Fuzzy logic Control

53

3.2.1 Fuzzification

54

3.2.2 Rule base

54

3.2.3 Inference

55

3.2.4 Defuzzification

55

3.3 Fuzzy Controller Design

56

3.4 Fuzzy logic attitude control

57

3.5 Three-axis Fuzzy Controller Design

59

IV UNIVERSITY OF KUFA

Chapter Four :Simulation and Results 4.1 Introduction

67

4.2 PID controller

68

4.2.1 PID control for attitude problem 4.3 LQR controller

70 79

4.3.1 Introduction

79

4.3.2 Basic concept of LQR

80

4.3.2.1 Controllability

81

4.3.3 LQR problem

81

4.3.4 Weighting matrices Q and R determination

83

4.3.5 LQR design and simulation

83

4.4 Fuzzy controller

92

4.4.1 Attitude Control Maneuver (ACM) test

102

Chapter Five : Conclusions and Suggestions for Future Work

5.1 Conclusions

106

5.2 Recommendations for Future Work

109

Bibliography

111

Appendix A :Direction Cosine Matrix

A.1

Appendix B : Coil Design

B.1

Appendix C : Matlab code and Simulink diagrams

C.1

V UNIVERSITY OF KUFA

List of Symbols Symbol

Meaning

a

Practical acceleration vector (N/m2).

a

Mean distance between two masses.

Ax ,Ay ,Az

Area of the magnetic coil (m2).

A

Plant matrix of the attitude dynamic system of the satellite.

B

Input matrix representing by the earth’s magnetic field.

Bb

Local geomagnetic field vector.

Bφ ,Bθ ,Bψ

Earth’s magnetic field affects the Roll, Pitch, Yaw axis.

C

Output matrix.

F

Gravitational force corresponding to a differential element (N).

F

Force effecting on particle (N).

Fb

Body frame

Fo

Orbit frame

G

Newton’s gravitational constant.

G

Gravity gradient vector.

Gx, Gy, Gz

Components of gravity gradient vector

,

Gaussian coefficients.

H

Total angular momentum vector of the rigid body(kg.m2.rad/sec).

im

The inclination of the spacecraft’s orbit with respect to magnetic equator (degree).

ix, iy, iz

The currents passing through magnetic coils (Amp).

I

The inertia tensor matrix of the spacecraft (kg.m2).

Ix,Iy,Iz K kx,ky,kz

Moments of inertia for Roll, Pitch, Yaw axes (kg.m2). Control gain. Direction cosines of the Euler axis relative to reference frame. VI UNIVERSITY OF KUFA

m

Mass of particle (g).

M

Mass of the Earth.

M

The summation of the external moments exerted about the center of mass of the rigid body (Nm).

mb

Generated magnetic moment inside the body

mx,my,mz

Magnetic moment for each principal axis (Nm).

Mx,My,Mz

The external torques for each principal axis (Nm).

Nk

Number of windings in the magnetic coil.

N

The period of one orbit that the satellite needs to complete one revolution around the earth.

P

Mutual period of revolution. Schmidt quasi-normalized

q

Quaternion parameters vector.

r

The position vector of the mass measured from the center of the spacecraft mass(km)

Ro

The radius vector from the center of the Earth to the center of the spacecraft (km).

t TG TGxTGyTGz Tm Tmx,Tmy,Tmz

Time index (sec). The gravity gradient torque vector(Nm). The gravity gradient torque about each principal axis (Nm). The magnetic torque vector (Nm). The magnetic torques about the Roll ,Pitch, Yaw axes(Nm).

u

Input magnetic moment vector .

v

Velocity of the particle(m/sec).

V

Scalar potential function .

Xb,Yb,Zb

Satellite coordinate frame. VII UNIVERSITY OF KUFA

Xi,Yi,Zi

Inertial coordinate frame.

Xo,Yo,Zo

Orbital coordinate frame.

y

Output vector.

ωo

Mean orbital motion

ω b/i

The angular velocity of body frame relative to an inertial frame (rad/sec).

ω b/o

The angular velocity of body frame relative to orbital frame (rad/sec).

ω o/i

The angular velocity of the orbital frame with respect to the earth (rad/sec) .

ωx,ωy,ωz x φ, θ, ψ

The angular velocity of Roll, Pitch, and Yaw axes (rad/sec). Attitude and its rate of change state vector. Roll, Pitch and Yaw angles (degree).

φb

Roll angle bias due to any z-spin rate (degree).

μ

Membership function.

μo

Permeability of free space.

α

The angle between the magnetic moment and the Earth’s magnetic field (rad).

Ω

Skew-symmetric matrix.

γ

Vernal equinox.

Ω

Longitude of ascending node.

ω

Argument of perigee.

VIII UNIVERSITY OF KUFA

List of Abbreviations Abbreviation ADCS

Abbreviation Attitude determination and control system.

ADS

Attitude Determination System.

ACM

Attitude Control Maneuver

ACS

Attitude Control System.

AOCS ARE

Attitude and Orbit Control System Algebraic Riccati Equation

CG

Center of gravity.

ECI

Earth Centered Inertial

ECEF

Earth Centered Earth Fixed

FLC

Fuzzy logic controller.

GG

Gravity gradient

GPS

Global Positioning System.

IAGA

International Association of Geomagnetism and Aeronomy

IGRF

International Geomagnetic Reference Field.

LEO

Low Earth Orbit.

LQR

Linear Quadratic Regulator.

MIMO

Multi Input Multi Output.

MISO

Multi Input Single Output.

NED

North East Down.

PID

Proportional Integral Derivative.

SCO

Spacecraft Orbit.

SISO

Single Input Single Output. IX UNIVERSITY OF KUFA

List of Tables Page

Table (2.1) Satellite parameter

45

(3.1) Rule base for the controller of roll, pitch and yaw angles

61

(4.1) kufasat parameters

70

(4.2) Initial condition values for attitude response

71

(4.3) PID controller parameters

71

(4.4) kufasat parameters

84

(4.5) Scaling factors values

93

(4.6) System analysis of (PID , LQR & FLC) for case 1

100


101


101

(4.9) ACMs with different reference command

102

(B.1) Coil Design Constraints

B.3

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Chapter One

1

General Introduction and Review

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Chapter One

General Introductions and Reviews


1.1 Introduction The attitude of spacecraft is its orientation in the space [1] .The motion of spacecraft is specified by its position, velocity, attitude and attitude motion. The first two quantities describe the translation motion of the center of mass of the spacecraft and are the subject of what is called “Orbital Determination”, depending on the aspect of the problem that is emphasized. The later two quantities describe the rotational motion of the body of the spacecraft about the center of mass which is called “Attitude Dynamics”, and attitude analysis may be divided into: Attitude determination and Attitude control. 1.2 Stabilization of spacecraft Stabilization of spacecraft is the process of maintaining an existing orientation with respect to some external frame (coordinate system) .Attitude control systems can be divided into two categories: passive and active systems. 1.2.1 Passive stabilization The only completely passive attitude control which has been used with any success is the gravity gradient method, which uses the change in gravity with altitude to create a torque when the principal axes are not aligned with the orbit reference frame. Long booms are usually extended to create the torque. This type has been used only early low earth orbit satellites maintaining the earth pointing of antennas or other instruments. 1 UNIVERSITY OF KUFA

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The advantages for this method are simplicity, reliability, cheap and long lifetime. 1.2.2 Active stabilization The active stabilization methods are divided into two categories: spin stabilization and three-axis stabilization. 1.2.2.1 Spin Stabilization A body spinning about its major or minor axis will keep the direction of its spinning axis fixed with respect to the inertial space. This direction, according to Euler's moment equations of angular motion, will change only if the external moments are applied about its center of mass and perpendicularly to the spin axis [2]. This category includes two types: a- Single Spin Single-spin attitude stabilization is a very simple concept from the perspective of attitude control, the whole body rotates about the axis of maximum moment of inertia to provide a gyroscopic stiffness and stability. The cost of this method is generally low and it has along system life .The disadvantage of this method is that, the systems need to be earth-pointing, such as communications antennas which are spinning. In addition to, power obtained from solar cells which are not efficient due to spin. b- Dual Spin Dual -spin attitude stabilization is an extension of the singlespin stabilization principle .A dual spin spacecraft consists of rotor, which rotates at high speed to give the gyroscopic stability to the spacecraft .The payload (antenna and communication equipment) is mounted at the platform, which can be fixed or rotated at different speed from the rotor 2 UNIVERSITY OF KUFA

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.This method of stabilization has few disadvantages. However, this system is much more complex which leads to an increase in cost. 1.2.2.2 Three axis stabilization Some spacecraft require autonomous control of all three axes during their missions. Three-axis stabilization is complete control of the spacecraft's orientation along all three axes. Three axis control systems include wheels, magnetic control devices, and thrusters. Advantages to these systems include good pointing accuracy, and a non-inertial pointing accuracy. However, the hardware is often expensive, and complicated, leading to a higher weight and power [3, 4]. In general there are two types of three-axis stabilization system:-

a- Moment Biased System With a moment wheel along the pitch axis while the angular momentum along the pitch axis provides gyroscopic stiffness. The control torque along the pitch axis is provided by the change in the speed of momentum wheel. In these systems, the pitch and roll axis are controlled directly and the yaw axis is controlled indirectly due to gyroscopic coupling of yaw and roll errors, which lead to eliminating the need for the yaw sensor.

b- Zero Momentum Satellite employing three reaction wheels, one directed along each axis, to control the attitude via variations in individual wheel speeds which are centered around zero momentum.

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Chapter One


1.3 Mode of stabilization of spacecraft It can be divided into three modes: - Detumbling mode, Boom deployment mode and pointing mode.

1.3.1 Detumbling mode The most important operation mode of the Attitude Determination and Control System (ADCS) is the detumbling mode .The purpose of the detumbling mode is to detumble the satellite after it has been released from the launch vehicle. During detumbling, the initial spin of the satellite is slowed down until the gravity boom can be safely deployed. The controller used for detumbling is a B.dot law ̇

.

The principle of a ̇ -controller is to minimize the derivative of the magnetic field vector measured by a magnetometer. As the spacecraft orbits the earth, the magnetic field vector in the spacecraft reference frame changes depending on the position of the spacecraft. However, the dominant rate of change in direction of the field vector is caused by the tumbling of the satellite as it may tumble with angular rates much larger than the orbital rate.

Minimizing the change in the measured field vector by means of actuation causes the spacecraft to approach an angular rate close to the orbital rate which is achieved by forcing the derivative of the measured B-field, ̇ , to zero [5].


Chapter One


1.3.2 Boom deployment Before boom deployment the Zb-axis has to be aligned with the Z oaxis to ensure that the boom not is deployed in a wrong direction, which can make it difficult to restore to the proper attitude. The requirements for boom deployment is that the Zb must be less than |30◦| from the Zo -axis. The boom will be deployed directly by a command from the ground station [6].

1.3.3 Three-axis stabilization mode (pointing mode) The stabilization mode can only be initiated after the boom is deployed. The mode consists of attitude estimation and actuation. The output of the attitude estimator is a quaternion, which describes the orientation between the orbit and body reference frame, and the angular velocities of the satellite. In this mode the actuators are used to control the orientation of the satellite. Estimation of the attitude is necessary for the controllers to calculate

the

magnetic

dipole

moment

to

generate

through

the

electromagnetic coils [5].

1.4 Attitude Determination and Control System (ADCS) The ADCS can be perceived as two different systems, Attitude Determination System (ADS) and Attitude Control System (ACS). The ADS is used to inform the satellite’s present attitude. This information is then used as input for ACS that rotates the satellite into a desired attitude. The combination of ADS and ACS cane be summarized in a block diagram of attitude determination and control system ADCS as shown in Figure (1.1)


Chapter One


Fig (1.1) Block diagram of ADCS

1.4.1 Attitude Determination System (ADS) Attitude determination is the calculation of the relative orientation between two reference frames, two objects, or a reference frame and an object. Attitude determination uses a combination of sensors and mathematical models to collect vector components in the body and inertial reference frames, typically in the form of a quaternion, Euler angles or rotation matrix. In order to be able to determine the attitude of the satellite, sensors are needed to sense the orientation of the satellite. The main sensors available on satellite area are: 1.4.1.1 Earth Sensors Earth sensors tell the satellite which way the Earth is? Conventional Earth Sensors use infra-red cameras and telescopes to locate the position of the Earth’s horizon and hence to calculate the vector to the center of the Earth. There are two new generations of Earth sensors: one based on


Chapter One


imaging oxygen airglow, the other based on a direct measurement of the gradient of the gravitational attraction of the Earth [1]. 1.4.1.2 Sun Sensor A sun sensor is an instrument which measures the direction from the satellite to the sun. The direction to the sun can be measured in two different ways, both of them relying on photocells. The first one, the analog sun sensor, also called cosine sensor, measures the intensity of the sun, and the second one, the digital sun sensor, uses a pattern where different photocells is exposed depending on the direction of the sun. A digital sun sensor is built up of a pattern of photocells. The photocells are placed inside an installation that restricts which photocells that are illuminated and make this depended on the direction of the sun. The angle of the sun can be found in the output of the sensors, since the photocells that is illuminated generate a higher energy level, either higher voltage or higher current, than the sensors that is in the shadow [7]. 1.4.1.3 Star Trackers They are the most accurate sensors used for attitude determination at the present but they are also the most expensive attitude determination instrument. It is important that they deliver a full attitude determination, meaning that they don’t need another vector measurement since the measurement of the stars in the field of view already provides an attitude solution. The process consists in taking a picture of the sky, comparing this picture with a star map stored on board of the spacecraft and based on some specific algorithms identifying the stars found and generating an attitude solution [3, 7]. 7 UNIVERSITY OF KUFA

Chapter One


1.4.1.4 Magnetometer Magnetometers are inexpensive, lightweight, and highly reliable sensors that are carried on most low Earth orbit spacecraft. As they provide us with information about the attitude of the spacecraft and therefore, they become interesting for small satellite systems [8]. A magnetometer measurers the flux density of the magnetic field it is placed in. A three axis magnetometer placed inside a satellite, will measures the geomagnetic intensity and direction surrounding the satellite. In a low orbiting satellite this can be used as a low cost, low weight, and reliable attitude sensors, with an accuracy of 0.5 to 3 degrees. The most common magnetometer used in space is the flux-gate magnetometer, it has two parts: the magnetic sensors (one for each axis of spacecraft) and an electronics unit that transforms the sensors measurement into a usable format. 1.4.1.5 Gyroscopes Gyroscopes determine the attitude by measuring the rate of rotation of the spacecraft. Gyroscopes have a high accuracy for limited intervals. Some disadvantages exist with gyroscopes .Since they measure a change instead of absolute attitude, gyroscopes must be used with other attitude hardware to obtain full measurements, and there is more complexity since they have moving parts [4]. 1.4.1.6 Global Positioning System (GPS) The utilization of phase difference measurements from Global Positioning System (GPS) receivers provides a novel approach for three-axis attitude determination. A GPS receiver can be used to determine the satellite attitude. By placing two antennas a distance apart from each other and 8 UNIVERSITY OF KUFA

Chapter One


measuring the difference in carrier wave phase between the two antennas, the attitude, except for the rotation around the axis on which the two antennas is placed, and determined with an integer ambiguity. This requires an additional antenna and continuous measuring which obviously requires more power [9, 10]. 1.4.2 Attitude Control System (ACS) The main purpose of an ACS is to orientate the main structure of the satellite to the desired attitude with sufficient accuracy in the space environment. Some form of attitude control is required to change the attitude of a spacecraft or keep it in a stable position [11]. The attitude control of a spacecraft can be considered being either actively controlled (that meaning a controller calculates necessary control torques and acting on the satellite to adjust its attitude to a desired position) or passively controlled (that meaning the satellite uses external torques that occurs due to its interaction with the environment and thus they cannot be avoided. In this case the disturbances are being used for forcing the attitude of the satellite). The attitude control system has the following elements: 1- Actuators: The actuator applies the desired torque to adjust the attitude .There are two types of actuators: active and passive. 2- Control Law: the controller takes the estimated attitude and its rate , the desired attitude and its rate and then computes the desired torque for the actuators and converts this torque to the desired electrical commands to be send to the actuators . 3- The Spacecraft Dynamics: this is really not an element of attitude control, it is concerned with the dynamics and kinematics of the 9 UNIVERSITY OF KUFA

Chapter One


satellite orientation about the center of mass under effects of internal and external torques. Chapter two includes the mathematical model of spacecraft attitude dynamics. 4- Disturbance Torque: There are two classes of disturbance torque; internal and external. The first results from moving parts, reaction wheels, fuel slosh. The second is the environmental torques and these torques arise from gravity gradient, earth’s magnetic field, solar pressure, and aerodynamic effect [2]. Therefore the attitude must be corrected from time to time with respect to the inertial frame. 1.4.2 .1 Active actuators include: 1) Thrusters: The torques generated by the thrusters is considered as external torques since the angular momentum of the entire satellite changes. The accuracy of the attitude control depends on the minimum impulse of the type of thruster used. Depending on the size of the satellite and taking in consideration the complexity of this solution and different types of thrusters are normally being used: gas jets, ion jets or even nuclear propulsion. 2) Reaction wheels: A reaction wheel is a rotor with high inertia which is accelerated. This acceleration will produce a torque on the reaction wheel .The torque on the wheel will generate a torque with opposite sign on the satellite that can be used to control the angular velocity of the satellite. The main advantage of these devices is that they have very high accuracies, and their main disadvantage is that they have big power consumptions and masses. There


Chapter One


are three types of such devices: momentum wheels, reaction wheels and fly wheels [12]. 3) Magnetic torquers : Magnetic torquer (also known as magnetotorquer, torque rod or torque bar) is an actuator used for satellite attitude control. The principle is to produce a controllable magnetic moment which interacts with the Earth's magnetic field to produce a mechanical torque onto the satellite. However, the use of such a device is limited to the low Earth orbits where the Earth’s magnetic field strength has usable values and should take in consideration that the generation of torques can be done just for the one perpendicular to the magnetic field vector. Considering this cosine dependency must be very easily intuited the main disadvantage of this method- named that the absolute torque which is produced is very small [13]. 1.4.2 .2 Passive actuators include: 1) Gravity gradient : This method is based on the fact that the gravitational force decreases with the square of the distance. By building the satellite in a certain configuration, parts of the satellites which are closer from the earth’s center are subject to a bigger force than the ones which are further away and in consequence this effect can be used to produce torques that adjust the satellite to a certain position [12].


Chapter One


2) Magnetic dipole : This method depends highly on the strength of the magnetic field in a specific point of spacecraft’s orbit and this is the reason why it can be used only for the low Earth orbits where the values of the magnetic field strength are big enough. The method is based on installation on board of the satellite of a strong constant magnetic dipole –or a permanent magnet, which will interact with the Earth’s magnetic field vector causing an adjustment of the two axes. 3) Aerodynamic : The aerodynamics of the satellite (influenced by the geometry of the shape) can be used in flight to adjust the position with respect to the flight direction (based on the forces caused by the atmospheric drag). However this method can only apply where the atmospheric drag is big enough (i.e. a low Earth orbit) and only for a short period of time, as this will lower the altitude and will shorten the decay period. 4) Solar radiation : They can be used to generate torque on the spacecraft based on the solar radiation pressure which is created by the sun. The orientation is then an axis that points to the sun. 1.5 Fuzzy Logic Controller Fuzzy control is a practical alternative for a variety of challenging control applications since it provides a convenient method for constructing nonlinear controllers via the use of heuristic information. Such heuristic information may come from an operator who has acted as a “human-in-the12 UNIVERSITY OF KUFA

Chapter One


loop” controller for a process. In the fuzzy control design methodology, we ask this operator to write down a set of rules on how to control the process. Then we incorporate these into a fuzzy controller that emulates the decisionmaking process of the human [14]. There are two main characteristics of fuzzy systems that give them better performance for specific applications. 1- Fuzzy systems are suitable for uncertain or approximate reasoning, especially for the system with a mathematical model that is difficult to derive. 2- Fuzzy logic allows decision making with estimated values under incomplete or uncertain information [15]. Fuzzy controllers consist of an input stage, a processing stage, and an output stage. The input stage maps sensor or other inputs, such as switches, thumbwheels, and so on, to the appropriate membership functions and truth values. The processing stage invokes each appropriate rule and generates a result for each, then combines the results of the rules. Finally, the output stage converts the combined results back into a specific control output value. 1.6 Orbit Dynamics The way that a spacecraft moves is described by using orbit dynamics, which determines the position of a body orbiting another body. In 1609, Johann Kepler discovered that heavenly bodies orbit in elliptical paths. He determined three laws which describe all planetary motion [1, 16]. Kepler’s First Law: If two objects in space interact gravitationally, each will describe an orbit that is a conic section with the center of mass at one focus.


Chapter One


If the bodies are permanently associated, their orbits will be ellipses; if they are not permanently associated, their orbits will be hyperbolas. Kepler’s Second Law: If two objects in space interact gravitationally (whether or not they move in closed elliptical orbits), a line joining them sweeps out equal areas in equal intervals of time. Kepler’s Third Law: If two objects in space revolve around each other due to their mutual gravitational attraction, the sum of their masses multiplied by the square of their period of mutual revolution is proportional to the cube of the mean distance between them. Hence (

)

(1.1)

where P is their mutual period of revolution. a is the mean distance between them. m and M are the two masses. G is Newton’s gravitational constant. Out of the two revolving objects the one with the greatest mass is called the primary, and the less massive object is called the secondary. If the mass of the satellite is denoted m, and the mass of the Earth is denoted M the mass of the satellite is considered negligible, m+M ≈ M Thus in the case of a satellite orbiting the Earth it follows from Kepler’s laws, that the trajectory of the satellite is an ellipse with the center of the Earth at the one focus. In order to mathematically describe orbits the following terms are used:


Chapter One


Vernal equinox (γ) is the line connecting the center of the Earth and the center of the sun where the ecliptic, which is the plane of the Earth’s orbit around the sun, crosses the equator, i.e., spring equinox. Due to the precession and nutation of the Earth the direction of vernal equinox must be associated with a specific year to give an unambiguous definition of the direction in space. Line of nodes is the line through the two points of intersection between the satellite orbit and the reference plane, typically the equatorial plane of the earth. The point in the orbit where the satellite crosses from south to north is called the ascending node. Longitude of ascending node (Ω) is the angle from vernal equinox to the ascending node. Note that this angle lies in the equatorial plane. Perigee direction is the point in the orbit closest to the center of mass of the primary (the center of the earth) this is often referred to as the barycenter. Argument of perigee (ω) is the angle from the ascending node to the perigee direction. Inclination (i) is the angle from the equatorial plane to the orbit plane[2,16].

Fig (1.2) Standard Orbital Elements


Chapter One


1.7 Earth Magnetic Field The earth’s magnetic field generally resembles the field around a magnetized sphere, or a tilted dipole seen in Figure (1.2) . As of 1999, the dipole axis was tilted approximately 11.5˚ from the spin axis, and drifting approximately 0.2˚/yr. Its strength at the Earth’s surface varies from approximately 30000nT near the equator to 60000nT near the poles. Further, there exists a low magnetic intensity field at approximately 25˚S and 45˚W known as the Brazilian Anomaly. A high exists at 10˚N and 100˚E, and the two of these together suggests that not only is the dipole axis tilted, but it does not quite pass through the center of the Earth. The accepted model for Earth’s magnetic field is the International Geomagnetic Reference Field[17].

Fig (1.3) Magnetic Field Model


Chapter One


The International Geomagnetic Reference Field (IGRF) is a global model of geomagnetic field. It allows spot values of the geomagnetic field vector to be calculated anywhere from the Earth’s core out into space. The IGRF is generally revised once every five years by a group of modelers associated with the International Association of Geomagnetism and Aeronomy (IAGA)[18]. According to physics, the magnetic field, B, is defined as the negative gradient of the scalar potential function V, such that (1.2) Although a simple dipole model gives a good approximation of the geomagnetic field, it can be modeled more closely using a spherical harmonic model of the scalar potential as given in the following equation. This is the equation about which the IGRF is based. (

)

∑

( )

∑

(

)

( )(1.3)

Where: a is the reference radius of the earth (a = 6371.2km). r, θ, and φ are the geocentric coordinates ( r is the radius in kilometers, θ is the co-latitude (θ = 90˚− latitude), and φ is the longitude [18, 19]. The coefficients

and

IAGA for the IGRF, and

are Gaussian coefficients put forth by the ( ) represents the Schmidt quasi-normalized

associated Legendre functions of degree (n) and order (m). The input to this function is actually cosθ, rather than θ, but this has been dropped for brevity.


Chapter One


1.8 Literature Survey A number of possible approaches to the control of attitude dynamics has been developed through the last years, a particularly effective and reliable one is constituted by the use of electromagnetic actuators, which turn out to be especially 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 Sidi [2] and therefore provided 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 spacecraft for attitude control purposes are constrained to lie in the plane orthogonal to the magnetic field vector. White, Shigemoto, and Bourquin [20] were among the first to mention using magnetic torques for spacecraft control in 1961. Their analysis examined the feasibility of using the interaction of the earth’s magnetic field and current-carrying coils in a fine-control attitude system. This research developed control laws to track the spin axis of the orbiting astronomical observatory. They determined that it is possible to obtain torque about all three axes of a spacecraft on an intermittent basis, which changes the angular velocity of the spacecraft and can be used to change the orientation. The first implementation of magnetic control is found in spin-stabilized spacecraft.

In 1965, Ergin and Wheeler [21] developed controlling laws for spin orientation control by using a magnetic torque coil and discussed advantages of magnetic control. This control law is able to align the spin axis normal to 18 UNIVERSITY OF KUFA

Chapter One


the orbit plane. A similar analysis is also conducted by Wheeler [22] for active nutation damping, as well as spin-axis precision of rigid, axially symmetric, spinning satellites in circular earth orbits. The feasibility of a single magnetic dipole aligned with the spin axis is investigated. Wheeler determines that theoretical stability is provided by control laws that direct the spin axis of any axially symmetric spinning satellite in a circular earth orbit to any direction in space. Renard [23] developed control laws for magnetic attitude using an averaging method. This work is examine by having a control coil axis parallel to the spin axis and inverting the polarity of the control torque every quarter of orbit period, then any desired orientation is obtainable, although for certain motion it may be necessary to wait and to take advantage of the orbital eccentricity at certain times of the day. Sorenson [24] examined the magnetic attitude control system to point the spin axis of a spacecraft in a highly eccentric orbit and maintain a constant spin speed. The control is based on minimum energy considerations, and uses a Kalman filter to decrease energy requirements and to provide active damping. This method is effective for full control of spinning spacecraft in Earth orbits between 20± and 70± and eccentricities of up to 0.7. Shigehara [25] further examined the problem of magnetic control of spinning spacecraft by developing a control law which uses a switching function instead of averaging techniques. The switching function, which is derived from the condition of asymptotic stability selects the pattern of the magnetic dipole achieving the maximum effective torque and minimum transverse torque at every instant. The switching point is different from the averaged quarter orbit point, and attains the desired attitude direction faster. 19 UNIVERSITY OF KUFA

Chapter One


This method is proven through simulation runs which have an increase in performance over averaging methods. In 1975, Schmidt [26] described using magnetic attitude control on three-axis stabilized, momentum-biased satellites. Here, a momentum wheel is mounted along the pitch axis to provide bias, or nominal angular momentum that is not zero. Schmidt shows up that this system requires momentum switching of the closed-loop controller, and thus is reliable for long duration missions. This work is used towards the RCA SATCOM geosynchronous satellite, which are three-axis stabilized using air core coils. Sticker and Alfriend [27] further examined by using magnetic control with momentum bias. They develop a three-axis closed-loop attitude control system which was fully autonomous. Analytical expressions of system response are compared with numerical solutions of the governing equations. The two solutions of equations are in agreement, suggesting a feasible threeaxis control system. Goel and Rajaram [28] developed a closed-loop control law which performs both attitude corrections and nutation damping for three-axis stabilized spacecraft with momentum bias. In this system, a magnetic torquer is placed along the roll axis of the spacecraft, and yaw control was obtained by the roll/yaw coupling from the momentum wheel. Simulation results were matched with analytical results, and indicated that there was adequate damping of the system. Junkins, Carrington, and Williams [29] discussed the use of timeoptimal magnetic attitude maneuvers with spin-stabilized spacecraft. They suggested a nonlinear bang-bang switching function which is used with a single electromagnet aligned with the spin axis, and determines that their method is practical for rapidly determining maneuvers for a spacecraft. This 20 UNIVERSITY OF KUFA

Chapter One


method was used to determine optimal maneuvers for the NOVA navigational satellite. Martel, Pal, and Psiaki [30] examined by using magnetic control for gravity-gradient stabilized spacecraft in 1988. Whereas previous spacecraft used momentum wheels to augment the magnetic control, Martel, Pal, and Psiaki claimed that the proper ratio of moments of inertia, causing gravitygradient stabilization, along with magnetic control could provide three-axis stabilization. Simulations show that the algorithm performed well over a large range of orbital inclinations and attitude angles. In 1989, Musser and Ebert [31] are among the first to attempt to use a fully magnetic attitude control system for three-axis stability. They claim that this becomes possible due to the increase in on-board computer computational power in spacecraft. Musser and Ebert develop linear feedback control laws which use a linear quadratic regulator to obtain the value of the magnetic control torque. The control laws as a function of time are replaced with laws that are a function of orbital position. Musser and Ebert perform the simulations showing that their technique is a good candidate for on-board attitude control systems. Wisniewski [32, 33, and 37] further develops the ideas of Musser and Ebert. He used a combination of linear and nonlinear system theories to develop control laws for three-axis stabilization of the spacecraft. Linear theory is used to obtain both time-varying and constant-gain controllers for a satellite with a gravity-gradient boom. His analysis uses the fact that the geomagnetic field varies nearly periodically at high inclination orbits. In addition, he develops a nonlinear controller for a satellite without appendages based on sliding mode control theory. He showed that three-axis


Chapter One


control can be achieved with magnetic torques only, and implemented this idea on the Danish Orsted satellite. Grassi [34] developd a three-axis, fully autonomous, magnetic control system for use on a small remote sensing satellite. This control could be carried out solely with magnetometer measurements and orbital location information. Arduini and Baiocco [35] examined controlling laws for magnetic control of a gravity-gradient stabilized spacecraft. They discuss the challenges that exist due to magnetic torques only applied perpendicular to the magnetic field. Their control algorithm is based on first determining the ideal torque, and then generating the actual torque through a series of suggested approaches. They also discuss the relationship between stability and the change in energy of the system. In [36], a fuzzy controller has been implemented but actuators have not been used optimally and high accuracy has not been achieved. In 2000 Lin and Wen-Su introduce an intelligent control theory, which involves the subjects of fuzzy logic control and neural control. In June 2003 the first six CubeSats were launched by means of the Rocket launch vehicle. Since then the interest in CubeSats has grown and a number of launches have taken place. More recently, fuzzy control has been proposed as an alternative approach to conventional control techniques for the complex nonlinear systems (Tsai et al., 2008; Chen, 2009; Wang et al., 2009; Chen, 2010; Lee et al., 2010; Lendek et al., 2010). The primary advantage of the fuzzy controller is the ability to easily incorporate heuristic rule-based knowledge from experts in the control strategy.


Chapter One


1.9 Objective of Work The main objectives of this work which are presented in this thesis are: 1- To design a suitable attitude determination and control system (ADCS) of Nano gravity gradient stabilized satellite. 2- To develop a nonlinear mathematical model of a spacecraft with assumption the satellite is a rigid body, then to derive linear model. 3- Study magnetic coils which added in order to improve the accuracy of attitude control. 4- To

investigate,

simulate

and

evaluate

possible

controller

configurations (PID, LQR and fuzzy logic controller FLC) on a linear model in order to control the currents of magnetic coils when magnetic coils behave as the actuator of the system. 5- To compare the results of the simulations obtained from the PID controller, LQR controller and fuzzy logic controller. 1.10 Layout of the Thesis Chapter one gives a general introduction to define terms used in the spacecraft field, the attitude control system requirements and the available literature survey. Chapter two introduces the equation of motion of threeaxis stabilized satellite under the effect of gravity gradient torque, and the magnetic actuators used with propose model .Chapter three describe fuzzy logic controller design for the attitude control of a small spacecraft three-axis stabilization. Chapter four includes simulation and results .Chapter five is conclusions and suggestions for future work. Finally all necessary mathematical details and all Simulink block diagram in addition to initial values Matlab file are presented in the appendices. 23 UNIVERSITY OF KUFA

Chapter Two

2

Modeling of Satellite Dynamic

UNIVERSITY OF KUFA

Chapter two

Modeling of Satellite Attitude Dynamics


2.1 Introduction Spacecraft system design relies on the modeling and simulation tools. Modeling and simulation are a critical component of the system design which verifies the vehicle and operational design parameters that are difficult to verify with groundbased testing. The development of a simulation tool which achieves more accurate results will aid the design of more reliable and capable satellites. [38, 39]

A satellite can be regarded as an ideal rigid body [6], so the chapter concisely presents important mathematical background for kinematics of the attitude of a general rigid body in space. The dynamic model of the satellite is derived using a Newton-Euler formulation, as well as the kinematic equations of motion.

Attitude dynamic which describe the orientation of a body in space, required a clear description of the reference frames being used to give a basis for the rotations .Hence this chapter includes definitions of the different useful coordinate systems.

The chapter will also present four different torques which result from gravity gradient, magnetic field, aerodynamic drag, and solar radiation pressure effects which are representing the environmental disturbance torques and influence satellite orientation.


Chapter two


2.2 Reference Frames The main idea about kinematics is to describe the motions as relative to some Reference frames [40] .Several coordinate frames are used to define the position of a satellite in space and each has particular property, which makes it appropriate to a limit number of applications. The first three are used to describe the orientation, or attitude of the spacecraft in Law Earth Orbit (LEO).

2.2.1 Earth-Centered Inertial Frame- ECI This frame is a non-rotating reference frame and is assumed fixed in space. The origin of the ECI coordinate frame is located at the center of earth with axes xi, yi and zi. 2.2.2 Body Fixed Frame The body fixed reference frame is fixed to the vessel, and the origin is usually placed in the center of gravity (CG) with the body-axis chosen to coincide with the principal axes of inertia. 2.2.3 Spacecraft Orbit Frame- SCO The orbital frame has the origin located at the center of gravity of the satellite. The z-axis points towards the center of Earth. The x-axis points along the orbit trajectory and is perpendicular to the vector towards the center of the orbit. 2.2.4 Earth-Centered Earth Fixed Frame- ECEF This frame has its origin fixed to the center of earth, coinciding with the ECIframe origin. However the axes x and y rotate relative to the ECI frame about the axis z (It rotate together with the earth).


Chapter two


2.2.5 North-East-Down Frame- NED This frame is a geographical reference frame we refer to in our daily life. The origin is placed relative to the Earth's reference ellipsoid. The x-axis points towards true north, the y-axis points east, while the z-axis points downwards to the center of Earth.

Fig (2.1) (ECEF-ECI-NED and BODY Fixed) frames.


Chapter two


2.3 Equations of Motion The equations of motion of satellite are categorized into the dynamic and kinematic equations of motion [41]. 2.3.1 Dynamic Equations of Rotational Motion A set of three differential equations expressing relations between the force moments, angular velocities, and angular accelerations of a rotating rigid body. These equations are based on Newton’s Second Law, which states that “The total force acting on an object is equal to the time rate of change of its linear momentum in an inertial reference frame “, or (

∑

)⁄

⁄

(2.1)

Where (ΣF) is the vector sum of forces acting on mass (m) moving with velocity (v), the product (m.v) is the linear momentum, and (a) is the vector acceleration of (m) relative to an inertial frame ⁄

∑

where (r) is the distance vector from the point of rotation to the object. The righthand side of the this equation equals the time derivative of the angular momentum, (

)⁄

( ⁄

(

)

(

(

)

)

(

(

⁄ ) ⁄ )

⁄ ⁄

∑ ∑

)

⁄

(2.2) 27 UNIVERSITY OF KUFA

Chapter two


where (H) is the total angular momentum of the rigid body about its center of mass , ( M ) is summation of external moments exerted about the center of mass of the rigid body[42] . The total angular momentum vector H of the satellite with respect to an inertial fixed reference frame is determined by: ⁄

(2.3)

Where (ωb/i) is the angular velocity of body frame relative to an inertial frame ,and (I) is the moment of inertia of the spacecraft. The moment of inertia of the satellite in a current study is in the form of

[

]

(2.4)

The rigid body has a reference frame, which is fixed in body but rotates with respect to the inertial frame, hence equation (2.2) becomes [

]

[

]

(2.5)

The rotational equation of motion of rigid body about its center of mass can be expressed as: ⁄

(

)

(

)

(2.6)

Where ω = ωb/i


Chapter two


̇ [

]

][ ̇ ] ̇

[

[

]

[

] [

]

(2.7)

Computing the cross product and matrix multiplication yields ̇ [

[ ̇

]

]

[

]

̇

Or, in components we have three dynamic equations for the roll, pitch, yaw axes respectively as follows: ̇ ̇ ̇

(

)

(2.8a)

(

)

(2.8b)

(

)

(2.8c)

These three equations are known as Euler’s equations of motion for a rigid body. 2.3.2The Kinematics Equations of Rotational Motion Kinematics can be defined as the study of the motion of objects without regard to the mechanisms that cause the motion. Kinematics equation is used to describe the orientation of rigid body that is in rotational motion .It is mathematical in nature because it does not involve any forces associated with the motion [43]. Depending on the specification of the system such as the pointing accuracy and the value of the angle, there are two different modes of operation; the first is suitable for small angles and the second for large angles.


Chapter two


2.3.2. a Euler Angles This method is used when high pointing accuracy is required and for small angle requirements. The Euler angles are three angles; roll (φ), pitch (θ)and yaw (ψ);which are rotated around intermediate coordinate axes to obtain an attitude matrix.

Fig (2.2) Roll Pitch and Yaw Angles

There are twelve possible sets of body-axis rotations. Thus, there are twelve possible sets of Euler angles. Start by assuming both the orbital frame (xo ,yo,zo) and the body frame (xb,yb,zb) coincide when the relationship between these two frames can be found ,first spacecraft is rolled (φ) degrees about the x-axis, then it is pitched(θ) degrees about the new y-axis, and final it is yawed(ψ) about the new z-axis .By combining the results of these rotations. This process leads to direction cosine matrix which allows the transformation of any vector from orbit frame to body frame. Appendix (A) includes the complete derivation of the direction cosine .Hence from Appendix (A) Fb = C1(φ) .C2(θ) .C3(ψ) Fo

(2.9 )


Chapter two


where ,Fb and Fo represents the body and orbit frame respectively [ ]

[

] [ ] (2.10)

The angular velocity of the orbit frame Fo with respect to the earth is ⁄

(2.11)

where ,ωo is the mean orbital motion and it is given by √

(2.12)

Where Ro is the radius vector from the center of the earth to the center of the spacecraft. This means that orbital motion equals to angular rate of the spacecraft moving in its orbit. ωo is equal to 1.107×10-3 rad/s for 500km orbit altitude[44]. Hence, from equation (2.9) ωo/i yields to ωo/i = – ωo[

]

(2.13)

The angular velocity of body-fixed reference frame Fb with basis vectors (

)is given by : ⁄

⁄

⁄

(2.14)

Also, for the sequence of C1(φ) ← C2(θ) ← C3(ψ), the angular velocity of the body frame relative to the orbital frame, ωb/o can be represented as


Chapter two


̇ [

].[ ̇ ] ̇

]= [

(2.15)

then by substituting equations (2.13) and (2.15) into (2.14) , this yields ̇ [

].[ ̇ ]–ωo[ ̇

]=[

] (2.16)

Finally, the kinematic differential equations of an orbiting rigid body can be found as ̇ [ ̇ ]= ̇

[

].[

]+

[

]

(2.17)

2.3.2. b Quaternion Parameters called the quaternion are used to describe the orientation of rigid body. A quaternion is defined with three imaginary parts and one real part. [ ] , form a vector part of the quaternion The first three components, [ ] may be and the quantity, q4 is a scalar part. Thus the quaternion [ ] . written as Where ( ) is a vector describes the axis of rotation and ( amount of rotation.

) is a scalar describes the

Quaternion defines the rigid body attitude as a single Euler axis rotation angle .The vector part of the quaternion indicates the direction of the Euler axis and the scalar part of the quaternion is related to the rotation angle about the Euler axis .The four quaternion parameters can be defined as follows[45]:


Chapter two


() ()

(2.18)

() () ]

Where, (θ) is the rotation angle about the Euler axis, and (kx ,ky ,kz) are the direction cosines of the Euler axis relative to reference frame .The quaternion kinematics differential equation described by

̇

(2.19)

[

Where

]

and Ω is a skew –symmetric matrix defined as:

(2.20) [

]

Thus equation (2.11) will be: ̇ ̇

[

] ̇

̇

[ [

]

(2.21)

]

The quaternion parameters are dependent on each other, but constrain by the relationship: (2.22)


Chapter two


The advantages of quaternion over the direction cosine matrix are:1- Quaternions have no inherent geometric singularity as do Euler angles. 2- Quaternions are well suited for real-time computation because only products and no trigonometric relations exist in the quaternion kinematic differential equations. 3- Its ability to define the rotational relationship between two coordinate systems using only four numbers as opposed to the nine elements of a direction cosine matrix ; so the computation time is reduce by more than 40%

over the

direction cosine matrix [43]. 2.4 Linearized Dynamic Model Many applications require linear equations. In order to apply a linear optimal controller on the system, the satellite equation needs to be linearized. Functions of Euler angles depend on trigonometric primitives such as the sine and cosine. As a consequence, it is useful to consider the linearized versions of these functions [2]. In this context, for small angle linearization involves substituting: Sin ζ ≈ tan ζ ≈ ζ , cos ζ ≈1 . Then equation (2.16) becomes; ̇ [

]

[ ̇ ̇

]

(2.23)

where ωo is the orbital angular velocity , by substituting these equation into equation (2.8) the following linearized dynamic equations are obtained:


Chapter two

̈


(

)

) ̇

(

(2.24a)

̈

(2.24b) ̈

(

)

) ̇

(

(2.24c)

Equations (2.24.a,b,c) show that the motion in roll and yaw is coupled through the orbit rate, whereas motion in pitch is independent, so that the motion in roll and yaw axes is treated as Multi Input Multi Output (MIMO) system, while the motion in pitch is treated as Single Input Single Output (SISO) system. Equations (2.24.a, b, and c) can be rewrite in the forms ̈

(

(

)

) ̇

(

)

̈ ̈

(

(

)

(

) ̇

)

(Roll)

(2.25a)

(Pitch)

(2.25b)

(Yaw)

(2.25c)

2.5 Disturbance Torques The total torque acting on the satellite body is made up from several types. The four major types of external disturbance torques are gravity gradients, magnetic fields, aerodynamic and solar radiation [45]. Factors such as the orbit altitude, mass properties, spacecraft geometry, and spacecraft orientation affect the magnitude of each disturbance torque and therefore which type of torque is dominant [46]. 2.5.1 Gravity Gradient Torque Gravity gradient torque depends on the variations in the gravity field, as explained by Newton’s Law of Universal Gravitation. The spacecraft experiences a 35 UNIVERSITY OF KUFA

Chapter two


larger gravitational attraction on the side closest to Earth, while the portion of the spacecraft further away from Earth experiences a smaller gravitational attraction. Torques are generated from the differences in gravitational pull along the spacecraft, and results in the spacecraft rotating until it is aligned along its minimum inertia axis. Irregularly shaped spacecraft are especially affected by gravity gradient torques, as gravity gradients will cause the spacecraft to rotate into a specific orientation [47]. We suppose that the moving satellite is at a distance Ro from the center of mass of the earth. In Figure (2.3) (iR, jR, kR) are the unit vectors of the reference axis frame. The origin of the reference frame is located in the center of ρ mass of the body (cm).

Fig (2.3) Gravitational moments on an asymmetric spacecraft. The attracting gravity force is aligned along the kR axis; ρ is the distance between the cm of the body and any mass element dm in the body; and iB,jB, kB are the unit vectors of the body coordinates axis frame .


Chapter two


We can find the components of the vector R = -RokR in the body axes by using anyone of the Euler angle transformations. The components of the vector R in the body axes will be labeled Rx , Ry and Rz. we have

[

]= [A

]

φ][

(2.26)

It follows that (

) ( (

φ φ

)] )

(2.27)

Define the gravity gradient vector as [

]

The force exerted on amass element due to gravity is [(

)⁄| | ] ,

where r = R + ρ is the distance from the earth's cm to the mass dm. Since ρ