STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS USED

IN SPACECRAFTS ATTITUDE CONTROL

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

THE MIDDLE EAST TECHNICAL UNIVERSITY

BY

EMRE YAVUZOĞLU

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN

THE DEPARTMENT OF AEROSPACE ENGINEERING

NOVEMBER 2003

Approval of the Graduate School of Natural and Applied Sciences

Prof. Dr. Canan Özgen

Director

I certify that this thesis satisfies all the requirements as a thesis for the degree of

Master of Science

Prof. Dr. Nafiz Alemdaroğlu

Head of the Department

This is to certify that we have read this thesis and that in our opinion it is fully

adequate, in scope and quality, as a thesis for the degree of Master of Science.

Assoc. Prof. Dr. Ozan Tekinalp

Supervisor

Examining Committee Members

Prof. Dr. Mehmet Akgün

Prof. Dr. Cevdet Çelenligil

Prof. Dr. Kemal Özgören

Assoc. Prof. Dr. Ozan Tekinalp

Dr. Volkan Nalbantoğlu

iii

ABSTRACT

STEERING LAWS FOR CONTROL MOMENT

GYROSCOPE SYSTEMS USED IN SPACECRAFT ATTITUDE

CONTROL

Yavuzoglu, Emre

M. S., Department of Aerospace Engineering

Supervisor: Assoc. Prof. Dr. Ozan Tekinalp

November 2003, 191 pages

In this thesis, the kinematic properties of Single Gimballed Control Moment

Gyroscopes (SGCMGs) are investigated. Singularity phenomenon inherent to them

is explained. Furthermore, existing steering laws with the ir derivations are given.

A novel steering law is developed that may provide singularity avoidance or

may be used for quick transition through a singularity with small torque errors. To

avoid singularity angular momentum trajectory of the maneuver is to be simulated

iv

in advance for the calculation of singularity free gimbal histories. The steering law,

besides accurately generating required torques, also pushes the system to follow

trajectories closely if there is a small difference between the planned and the

realized momentum histories. Thus, it may be used in a feedback system. Also

presented are number of approaches for singularity avoidance or quick transition

through a singularity. The application of these ideas to the feedback controlled

spacecraft is also presented. Existing steering laws and the proposed method are

compared through computer simulations. It is shown that the proposed steering law

is very effective in singularity avoidance and quick transition through singularities.

Furthermore, the approach is demonstrated to be repeatable even singularity is

encountered.

Keywords: Attitude Control System, SGCMG, Singularity, Steering Law,

Momentum Trajectory, Torque Trajectory, Gimbal History

v

ÖZ

UZAY ARAÇLARININ YÖNELİM KONTROLÜNDE

KULLANILAN KONTROL MOMENTİ JİROSKOBU

SİSTEMLERİ İÇİN SÜRME YÖNTEMLERİ

Yavuzoğlu, Emre

Yüksek Lisans, Havacılık ve Uzay Mühendisliği Bölümü

Tez Yöneticisi: Doç. Dr. Ozan Tekinalp

Kasım 2003, 191 sayfa

Bu tezde, tek çerçeveli kontrol momenti jiroskoplarının kinematik özellikleri

ve tekillik problemi incelenmiştir. Varolan sürme yöntemlerinin formüllerinin nasıl

çıkarıldığı gösterilmiştir.

Tekilliğe girme riskini tamamen ortadan kaldıran veya tekilliğe girilse bile

çok küçük tork hataları oluşturarak çok hızlı bir geçiş için kullanılabilen yeni bir

vi

sürme tekniği geliştirilmiştir. Tekillikten tamamen kaçınmak amacıyla, manevranın

ileriye yönelik ideal momentum profili simüle edilerek tekillikten bağımsız çerçeve

açısı profilleri hesaplanmaktadır. Geliştirilen bu teknik; manevra sırasında gerekli

tork seviyelerinin tam olarak üretilmesinin yanısıra, çerçeve açılarının da

hesaplanan çerçeve açısı profilini yakından takip etmesini sağlamaktadır. Bu

özellikler, metodun geribesleme sistemi içerisinde kullanılmasına olanak

vermektedir. Ayrıca tekillikten kaçınmak veya çabuk geçiş sağlamak için birtakım

yaklaşımlar önerilmiştir. Bu fikirlerin geribeslemeli olarak kontrol edilen uzay

aracına yönelik uygulamaları incelenmiştir. Varolan diğer sürüş teknikleri ve

önerilen yöntemler yapılan simülasyon çalışmalarıyla kıyaslanmıştır. Geliştirilen

sürme yöntemlerinin tekillikten kaçınma ve çabuk geçiş sağlama konularında çok

etkili olduğu gözlenmiştir. Yöntem, tekillikle karşılaşılsa bile tekrar edilebilirliği

sağlamaktadır.

Anahtar Kelimeler: Yönelim Kontrol Sistemi, Tek Çerçeveli Kontrol

Momenti Jiroskobu, Tekillik, Sürme Tekniği, Momentum Profili, Tork Profili,

Çerçeve Açısı Profili

vii

To my wonderful parents …

viii

ACKNOWLEDGEMENTS

I cannot thank my advisor, Assoc. Prof. Dr. Ozan Tekinalp, enough for the

effort and time that he has spent for me. His guidance, encouragement, precious

friendship and his generousity in sharing his talents and experience with me are

greatly appreciated.

The greatest thanks goes to my parents, Enise and Nihat, who have always

supported me in every step of my life. Their love, vision, understanding and caring

are inestimable. Also, I would like to thank to my cute dog Kekik for her patience,

unique love and friendship.

I would like to thank specially to my dearest love Cansu Gürbüz whose love

and encouragement gave me the strength to complete this work on time.

Very special thanks are addressed to Dr. Vaios J. Lappas for his support,

encouragement and guidance.

Thanks to Ömer Onur, Ebru Sarıgöl, Mustafa Kaya, Özgür Demir, Burak

Seymen and other friends in the department, and especially to my dear roommate

Gizem Karslı.

ix

TABLE OF CONTENTS

ABSTRACT............................................................................................................. iii

ÖZ ............................................................................................................................. v

DEDICATION ........................................................................................................ vii

ACKNOWLEDGEMENTS ...................................................................................viii

TABLE OF CONTENTS......................................................................................... ix

LIST OF TABLES .................................................................................................. xii

LIST OF FIGURES................................................................................................xiii

NOMENCLATURE............................................................................................... xvi

LIST OF ACRONYMS......................................................................................... xvii

CHAPTER

1. INTRODUCTION................................................................................................. 1

1.1 Motivation ....................................................................................................... 1

1.2 Background ..................................................................................................... 3

1.3 Original Contributions..................................................................................... 5

1.4 Scope of the Thesis ........................................................................................ 6

x

2. ATTITUDE DETERMINATION AND CONTROL SYSTEMS OF

SPACECRAFTS ....................................................................................................... 8

2.1 Introduction ..................................................................................................... 8

2.2 ADCS Hardware ............................................................................................. 9

2.3 Advantages of CMG ..................................................................................... 13

2.4 CMG Based Attitude Control Systems ......................................................... 16

3. CHARACTERISTICS OF SINGLE GIMBAL CMGS...................................... 21

3.1 Introduction ................................................................................................... 21

3.2 Formalism ..................................................................................................... 23

3.3 Pyramid Configuration.................................................................................. 25

3.4 Singularity and Singularity Types................................................................. 27

3.5 Conclusion..................................................................................................... 37

4. OVERVIEW OF THE STEERING LAWS........................................................ 38

4. 1 Introduction .................................................................................................. 38

4.2 Classification of Existing Steering Laws ...................................................... 40

4.3 Unified Steering Law .................................................................................... 49

5. SIMULATION STUDY I ................................................................................... 53

5.1 Introduction ................................................................................................... 53

5.2 Description of the simulation work............................................................. 54

5.3 Results of the Existing Steering Laws........................................................... 56

xi

5.4 Results of USL .............................................................................................. 79

6. CMG BASED ATTITUDE CONTROL MODEL SIMULATION.................. 115

6.1 Introduction ................................................................................................. 115

6.2 Simulation parameters................................................................................. 116

6.3 Results of existing Steering Laws ............................................................... 118

6.4 USL Simulations ......................................................................................... 128

6.5 Attitude hold simulations ............................................................................ 150

7. CONCLUSION ................................................................................................. 163

APPENDIX ........................................................................................................... 165

A.1 Basic Coordinate Systems .......................................................................... 165

A.2 Attitude Representation and Equations of Motion ..................................... 167

A.3 Null Motion ................................................................................................ 171

A.4 Simulated Annealing Hide and Seek Algorithm ........................................ 179

REFERENCES...................................................................................................... 185

xii

LIST OF TABLES

TABLE

2.1 SSTL actuator comparison table ............................................................. 15

5.1 Nodal locations, desired gimbal angles, and gimbal angles attained with

USL steering law in constant torque simulation. ............................................ 81

5.2 Nodal locations and desired gimbal angles used in corner maneuver,

together with the corresponding singularity measures.................................... 86

6.1 Simulation parameters............................................................................. 117

xiii

LIST OF FIGURES

FIGURES

1.1 BILSAT satellite to be launched on Fall of 2003 ....................................... 2

2.1 General Block Diagram of ADCS.............................................................. 8

2.2 Attitude Determination and Control System Hardware ............................. 9

2.3 SSTL actuator volume comparison.......................................................... 14

2.4 CMG based attitude control system ......................................................... 16

3.1 Single Gimbal CMG................................................................................. 22

3.2 Double Gimbal CMGs ............................................................................. 22

3.3 Pyramid mounting arrangement ............................................................... 25

3.4 Momentum Envelope. .............................................................................. 29

3.5 Classification of all singularity types ....................................................... 30

3.6 Singular Direction. ................................................................................... 31

3.7 2H Singular Surfaces................................................................................ 36

4.1 Classification of Steering Laws................................................................ 40

5.1 Ideal angular momentum and torque profiles .......................................... 55

5.2 Constant torque simulation with MP inverse ........................................... 57

5.3 Constant torque simulation with MP with preferred initial gimbal set. .. 60

5.4 Constant torque simulation with MP combined with IG .......................... 62

5.5 Constant torque simulation with SR inverse ............................................ 67

xiv

5.6 Constant torque simulation with SR inverse combined with IG.............. 69

5.7 Constant torque simulation with GSR inverse ......................................... 73

5.8 Constant torque simulation with GSR combined with IG method........... 76

5.9 Constant torque simulation results using optimized gimbal set............... 82

5.10 Corner maneuver simulation results using optimized gimbal set .......... 86

5.11 Maneuver with desired gimbal angles to demonstrate repeatability. ..... 89

5.12 Simulation results with dynamic torque input........................................ 93

5.13 Constant torque maneuver using USL with null vector. ........................ 97

5.14 Corner maneuver simulation using USL with null vector.................... 101

5.15 Constant torque simulation using USL with an arbitrary and constant

rate vector...................................................................................................... 104

5.16 Results of constant torque simulation using USL with a constant rate

vector............................................................................................................. 108

5.17 Constant torque simulation using USL with a white noise vector. ...... 111

6.1 More Detailed CMG Based ACS Diagram........................................... 116

6.2 -65˚ roll maneuver simulation conducted with MP inverse ................. 118

6.3 -65˚ roll maneuver simulation conducted with SR inverse ................... 122

6.4 -65˚ roll maneuver simulation conducted with GSR inverse initiated at

the elliptic singularity.................................................................................... 125

6.5 CMG Based ACS without actuator ....................................................... 130

6.6 Ideal system attitude, CMG cluster’s angular momentum and torque

histories to complete -65˚ roll maneuver....................................................... 131

xv

6.7 -65˚ roll maneuver simulation with USL conducted in pre-planned fashion

....................................................................................................................... 133

6.8 -65˚ roll maneuver simulation using USL with null vector ................... 136

6.9 -65˚ roll maneuver simulation using USL with an arbitrary and constant

rate vector...................................................................................................... 140

6.10 -65˚ roll maneuver simulation using USL with an arbitrary and constant

rate vector initiated at the elliptic singularity................................................ 143

6.11 -65˚ roll maneuver simulation using USL with intelligently selected

constant rate vector........................................................................................ 147

6.12 Attitude hold maneuver with ideal. ...................................................... 150

6.13 Attitude hold maneuver with SR inverse. ............................................ 153

6.14 Attitude hold maneuver with USL conducted in pre-planned fashion. 156

6.15 Attitude hold maneuver with USL conducted with intelligently selected

constant rate vector........................................................................................ 160

A.1 Coordinate frames ................................................................................. 165

A.2 Internal Elliptic Singularity at δs = [-90º, 0º, 90º, 0º] ........................... 174

A.3 External Elliptic Singularity at δs = [-90º, 180º, -90º, 0º]..................... 177

A.4 Hyperbolic Escapable Singularity at δs = [90º, 180º, -90º, 0º] ............. 178

xvi

NOMENCLATURE

Text = sum of external disturbance torques

HS/C = total angular momentum of the spacecraft

IS/C = inertia matrix of the whole spacecraft

3 4,I I = 3, 4 dimensional identity matrices

J = system Jacobian matrix

, ih h = total angular momentum, angular momentum of the ith CMG

n = noise vector of unit magnitude

p = node number

q = blending constant used in unified singularity robust steering law

t = time

t∆ = temporal distance between nodes

β = pyramid skew angle, 54.73º

δ = vector of gimbal angles

iδ = gimbal angle of the ith gyro

τ = torque of the CMG cluster

Superscripts

T = transpose of the matrix

xvii

LIST OF ACRONYMS

ADCS Attitude Determination and Control System

RW Reaction Wheel

MW Momentum Wheel

CMG Control Moment Gyroscope

SGCMG Single Gimbal Control Moment Gyroscope

DGCMG Double Gimbal Control Moment Gyroscope

VSCMG Variable Speed Control Moment Gyroscope

PD Proportion Derivative

PID Proportion Derivative Integral

MP Moore Penrose Pseudo Inverse

SR Singularity Robust

GSR Generalized Singularity Robust

IG Inverse Gain

USL Unified Steering Logic

SSTL Surrey Satellite Technology Limited

BILSAT BILTEN Satellite

1

CHAPTER 1

INTRODUCTION

1.1 MOTIVATION

Due to their superior properties such as large torque amplification and

momentum storage; control moment gyro (CMG) based attitude control systems

have made it very attractive for space applications. In fact they have been used in a

number of large spacecrafts such as MIR, Skylab, and ISS1,2. The possibilities of

using them on a smaller spacecraft are being investigated. For example the

BILSAT∗ microsatellite (Fig. 1.1) with an Earth observation mission, that was

launched on Sept. 27, 2003, carries an experimental CMG payload suitable for

small satellites.

∗ The satellite was built by SSTL of UK, together with the engineers from

TUBITAK-BILTEN, a research institute of Turkey.

2

CMG rotors usually operate at a constant speed. Exchange in momentum is

realized by changing the spin axis orientation with respect to the spacecraft.

Torquing the gimbal results in a reaction torque which is orthogonal to both the

gimbal and spin axes. CMGs have the advantage of producing considerably larger

output torques than the input torque required to drive the gimbal suspension,

provided that spacecraft has low inertial angular rates initially3. This property is the

well-known torque amplification property.

Fig. 1.1 BILSAT satellite launched on Sept. 27, 2003

For a specified torque level, single gimbal control moment gyros (SGCMG)

based attitude control systems exhibit benefits in power requirements, agility,

weight, and size over their competitors such as reaction and momentum wheels2.

Their construction is much simpler than double gimbal control moment gyros.

3

Besides their many advantages, they have an undesirable characteristic that make

their use in attitude control a real challenge. During large slew maneuvers, they

may steer towards singular configurations which allow no torque capability along a

particular direction1. Although SGCMG clusters containing redundant actuators

may reduce the risk of being trapped in singular configurations, it may not be

avoided completely. For this reason, there is a need to develop a steering law that

will drive the system away from the singularities within the system hardware

limitations1-3.

1.2 BACKGROUND

In this section, short overview of the literature relevant to this research is

presented.

An early work that investigates the use of control moment gyros for

spacecraft attitude control is carried out by Jacket and Liska4. Using basic

geometry, Margulies and Aubrun3 discussed and established the fundamental

properties of such clusters. They investigated the momentum envelope for various

CMG configurations and identified the singular configurations. They also

presented the possibilities of escaping from singular configuration by null motion

for redundant systems. Bedrossian et. al.5, 6, recognizing the similarities between

robotic manipulators and CMG systems, utilized the singularity-robust (SR)

4

inverse technique developed by Nakamura and Hanafusa7 to obtain approximate

solution of gimbal rates allowing some torque error in the vicinity of singularity.

In addition they proposed to add null motion to the particular solution to avoid

singularities. The amount of null motion added is inversely proportional to the

distance from singularity. Oh and Vadali8 provided complete set of equations of

motion including the rotor transverse inertia as well as gimbal inertia terms. Using

the Liapunov’s approach, they formulated an alternative feedback control law that

employs gimbal acceleration steering instead of velocity steering. Krishnan and

Vadali9, again using Liapunov’s method developed an inverse free technique for

spacecraft control. Wie et al.10 by modifying the SR-inverse method introduced a

new logic that helps the spacecraft transit through internal singularities. Ford and

Hall11 developed new singularity avoidance law by modifying the SR-inverse

method using singular value decomposition. This resulted in smoother gimbal rates

without altering the output torque in the singular direction. Schaub and Junkins12

proposed to use null motion with Variable Speed CMG (VSCMG) to have one

more extra degree of freedom. They showed that a drastic reduction occurs in the

required reaction wheel power consumption as CMG singular state is approached.

The required torque proves to be small and achievable by existing CMG hardware.

Wie1 presents an overview of the existing steering laws, describes basic equations

for construction of the mathematical model of CMG based attitude control systems.

Lappas2 verified the advantages that CMGs can provide to small satellites by

constructing small CMG system. He also proposed new control logic for

5

compensation of the gimbal angle deviations due to external disturbances using

magnetic control.

Paradiso13 presented a directed search algorithm, which is capable of globally

avoiding singular states in a feed-forward steering law utilizing null motion at

discrete nodes. Vadali et al.14 developed a method for determining a family of

preferred initial gimbal angles that would not encounter singularities during a

maneuver. Vadali and Krishnan15 also worked on explicitly avoiding singularities

by parameterizing gimbal rates as polynomial functions of time and optimizing the

parameters with respect to a singularity avoidance objective function.

1.3 ORIGINAL CONTRIBUTIONS

Main contributions of this thesis are:

a. A new steering law for single gimbal control moment gyros is developed.

b. The law is used for the preplanned maneuvering of spacecrafts and it is shown

to be capable of avoiding internal singularities.

c. A number of approaches for the spontaneous steering with the new steering

algorithm are also developed. Their superiority to previously developed

approaches is demonstrated.

6

1.4 SCOPE OF THE THESIS

The organization of the chapters in this thesis is as follows. In Chapter 2, the

attitude determination and control system hardware is described briefly. Main

advantages of CMG among all attitude control devices are emphasized. Finally, a

mathematical model of CMG based attitude control system is presented.

In Chapter 3, the unique characteristics of SGCMG are addressed. After

principals of operation of SGCMGs are given, basic equations of the minimally

redundant SGCMG cluster in a pyramid configuration are presented. The section

that follows presents the analysis of the singularity phenomenon. Singular states

seen in pyramid configuration are classified according to the different

considerations for better understanding of the singularity problem.

Chapter 4 provides an overview of the steering laws together with their

derivations. Steering laws are classified according to their operational type. Then,

using the experience obtained in derivation of existing steering laws, minimization

problem is solved to meet different objectives in order to find new methods. The

section that follows presents the derivation of the new unified steering law (USL),

together with a discussion on its relation to other steering laws and ways that it may

be employed.

7

In Chapter 5, simulations that show the effectiveness of the USL and

comparisons with the available methods are performed. In order to understand the

capability of the steering laws in avoiding elliptic singularity on the way, constant

torque and corner maneuver simulations are carried out. Different approaches

proposed for the employment of USL steering law are tested and evaluated.

In Chapter 6, simulations are continued in order to evaluate the performance

of the steering methods in spacecraft model. For this purpose, the mathematical

model of the attitude control system described in Chapter 2 is developed using the

MATLAB and SIMULINK software. Roll and attitude hold maneuver simulations

are performed to compare the performances of the proposed methods and the

existing methods.

Chapter 7 presents the summary and conclusions of the dissertation research.

In Appendix-A.1, basic coordinate frames used are reviewed. In Appendix-

A.2, attitude representation and equations of motion are explained briefly. In

Appendix-A.3, first, null space dimension and null space basis vector calculation is

described. Then, examples of elliptic singularities are detailed to support the

material given in Chapter 3. Finally, in Appendix-A.4, simulated annealing hide

and seek algorithm used in determination of optimum gimbal angles for a given

momentum state is summarized.

8

CHAPTER 2

ATTITUDE DETERMINATION AND CONTROL

SYSTEMS OF SPACECRAFTS

2.1 INTRODUCTION

Attitude determination and control system (ADCS) is one of the most crucial

subsystems of the spacecraft. Main function of ADCS is to stabilize the spacecraft,

and steer it to a particular direction correctly despite the internal and external

disturbance torques acting over spacecraft. Simplified block diagram of ADCS is

given in Fig. 2.1.

Fig. 2.1 General Block Diagram of ADCS

10

2.2.1 Attitude Measurement Hardware (Sensors)

Attitude measurement hardware is used to determine the attitude of the

spacecraft with respect to a specified reference frame. The final product could be

sun angles in body axis frame, Euler angles, or quaternions of the satellite with

respect to a particular reference frame. Depending on the reference frame in

measuring the attitude, following sensors may be used for attitude determination:

i) earth sensors, (i.e., infrared earth sensors)

ii) sun sensors,

iii) star sensors,

iv) rate and rate integrating sensors, based on gyroscopic, laser

or other solid state principles, and

v) magnetometers.

The accuracy that can be achieved depends on the sensor type and the quality

of the instruments. For instance, the accuracies that can be achieved with the earth

sensors range from 0.02º to 0.5º, depending on the complexity of the hardware and

the processing algorithm. On the other hand, the accuracies of the sun sensors vary

between 0.001º to 3º. Note that, for earth orbiting satellites, the earth and sun

sensors are usually utilized together in order to obtain full three axis attitude of the

satellite. If higher attitude accuracies are desired, star sensors with accuracies

reaching to 0.0003º could be used2,21.

11

2.2.2 Attitude Control Hardware

Main function of the attitude control hardware is to supply required

translational and rotational acceleration to spacecraft to accomplish the desired

maneuver. Control forces and torques can be obtained through different sources.

According these sources, attitude control hardware could be categorized as follows:

1) Propulsion systems: Simply by expelling its propellant, they provide

control forces and torques acting on the spacecraft enabling changes in translational

and angular velocities. They are divided into three sub groups: cold gas; chemical;

and electrical.

2) Momentum exchange devices: They produce torque by modifying their

angular momentum vector. The momentum exchange devices do not require tanks

to store expendables. While the whole spacecraft system momentum remains

constant, momentum of the actuator is transferred to the spacecraft to reorient it

into a desired attitude.

All actuators consist of a spinning disc with an angular velocity ω, (and

corresponding angular momentum h=Idisc ω) .They might be grouped according to

the means of torque production:

12

i) Momentum Wheels (MW) and Reaction Wheels (RW): Both

momentum and reaction wheels produce torque by increasing or decreasing the

rotation speed of the wheel. Momentum wheels provide constant angular

momentum for gyroscopic stabilization. Orientation of the spin axis is fixed with

respect to the inertial space. Attitude control is achieved by varying the spin speed

of the wheel about some nominal value whereas reaction wheel is nominally at rest.

ii) Control Moment Gyros (CMG): Control torques are generated by

changing the direction of the momentum vector (=the direction of the axis of

spinning wheel). They will be described in details throughout this thesis.

3) Magnetic torque rods: They are composed of a magnetic core and a coil.

As the coil is energized, the torque rods produce a magnetic dipole moment.

Torque produced by magnetic torque rods is proportional to the magnetic field of

Earth. However, the magnitude of the torque realized is usually not sufficient for

rapid orientation. Thus, magnetic torque rods are usually used for active damping

in gravity gradient attitude-stabilized spacecraft and in order to desaturate

momentum exchange devices.

4) Solar torque controllers: Solar radiation pressure on reflecting surfaces

is used to create control forces and torques. Solar torque controller hardware

consists of two solar panels and flaps which are attached to the panels for creating

unbalanced moments.

13

2.3 ADVANTAGES OF CMG

As the space missions become more demanding, designing an ADCS with

higher performance to respond the needs of future gets more important. In this

vein, CMGs are being seen as more efficient actuators due to their superior

properties compared to other momentum exchange devices. Following positive

features can be summarized for CMGs:

1. Of all actuator types, CMGs offer the greatest torque amplification.

A small torque input to the CMG generates a large torque output to the

spacecraft.

2. High angular momentum capability of the CMGs leads to a highly

stable platform. Vibration isolation is not a big problem for CMG based

spacecrafts.

3. They provide power, mass and volume efficiency (Fig. 2.3)

compared to other actuators.

14

Fig. 2.3 SSTL actuator volume comparison2

4. Agility: Future missions requires an increased slew (=turning about

a fixed point or axis) maneuver rate capability, i.e., greater than current

standard 0.1-1º/s. Due to first two properties, superior slew rates and also

high precision tracking becomes possible with CMGs.

5. Use of CMGs instead of batteries to store energy in addition to the

attitude control is also being popular. It is called as integrated power attitude

control system16.

Table 2.12 demonstrates the superiority of CMGs over reaction wheels. With

considerably less mass, CMGs can produce the largest torque output and also they

can provide the highest average slew rate of 3º/s to a microsatellite. Furthermore,

on average CMG system is more efficient than RW system from the power point

view as seen in Table 2.1.

9

In this chapter, hardware items that are used in ADCS are introduced briefly.

Special attention is given to the control moment gyros. Their advantages over other

actuators are listed in section 2.3. In the section follows, a simple mathematical

model of CMG based attitude control system is presented.

2.2 ADCS HARDWARE

Hardware items in any spacecraft attitude determination and control system

can be grouped into two: instrumentation for the determination of the current

attitude of the satellite; and actuators for the production of forces and torques

necessary for steering the spacecraft.

Fig. 2.2 Attitude Determination and Control System Hardware

15

Besides these advantageous, their main drawback is the existence of

singularities. These are states in which, for a certain set of gimbal angles, the

CMG does not produce torque. Special techniques (called steering laws) should be

used to alleviate these undesired conditions.

Table 2.1 SSTL2 actuator comparison table demonstrating the potential of CMGs

Parameter CMG Microsat E. Microsat Minisat

Mass of S/C (kg) 50 50 350

Type of actuator 4 CMG 4 RW 4 RW

Mass of actuator system (kg) ~1 4 12.8

Power Av. (W) per actuator 0.75-4 0.8-3.5 3.3-14

Voltage (V) 5-12 12-16 24-32

Max. Angular Momentum (Nms) 1.1 0.36 4.2

Max. Torque Capability (mNm) 52.5 20 40

Average Slew Rate (°/s) 3 1.85 0.65

S/C Inertias (kg-m2) [2.5, 2.5, 2.5] [2.5, 2.5, 2.5] [40, 40, 40]

Min. time for 30° (s) 10 16.17 45.74

16

2.4 CMG BASED ATTITUDE CONTROL SYSTEMS

In this section, a simple mathematical model describing the attitude control

of a rigid spacecraft equipped with a cluster of redundant CMGs is presented (Fig.

2.4). The following analysis is adapted from Wie et al1,10. There are three main

parts to be considered. First the equations of motion describing the spacecraft

motion, second quaternion feedback controller for the kinematic update of the

spacecraft attitude, and finally the steering law part for the actuator. These are

presented below.

Fig. 2.4 CMG based attitude control system

Spacecraft dynamics

Total angular momentum of the spacecraft is expressed as the sum of

spacecraft main body angular momentum and the angular momentum of the CMG

cluster:

17

/ /= +S C S CH I ω h (2.1)

where HS/C is the total angular momentum of the system with respect to the

spacecraft’s body-fixed control axis; IS/C is the inertia matrix of the whole

spacecraft, ω is the spacecraft angular velocity vector; and h is the total CMG

angular momentum vector. The rotational equation of motion of such a spacecraft

may be described by:

+ × =S/C S/C extH ω H T (2.2)

where Text is the sum of external disturbance torques acting on the spacecraft

including the gravity gradient, solar pressure, and aerodynamic torques defined in

the same body-fixed control axes. Substituting Eq. (2.1) into the Eq. (2.2), we get:

( )/ /+ + × + =S C S C extI ω h ω I ω h T (2.3)

Rearranging this equation by introducing the internal control torque

generated by CMG cluster, u, we have:

/ /+ × = +S C S C extI ω ω I ω T u (2.4)

+ × = −h ω h u (2.5)

These equations together with the spacecraft kinematic differential equations

such as quaternions, or Euler angles constitute an attitude control system. The

18

spacecraft control input u will be generated by the controller to be realized by

CMG steering law. Then, the desired CMG momentum rate is selected as1:

= − × −h ω h u (2.6)

Quaternion feedback controller

Quaternion based feedback controller is employed. The quaternion error and

angular rate vectors are fed to the quaternion feedback controller to generate the

control torque vector u. Kinematic equations of motion used in this thesis are given

in A.2. A linear feedback controller of the following form is used1:

d= − −eu Kq Dω (2.7)

where [ ]1 2 3T

d e e eq q q=eq is the attitude quaternion error direction vector (See

Appendix-A.2.1). K and D are weighting matrices to be properly selected. The gain

selection1, 2 is made as:

/

/

kd

==

S C

S C

K ID I

(2.8)

where,

2

2

/ 2n

n

d

k

ζω

ω

=

= (2.9)

ζ is the damping ratio, and nω is the natural frequency.

19

Steering law

Since CMG steering law is one of the most crucial parts of any CMG based

ACS, Chapter 4 is completely devoted to this subject. Here only brief introduction

is given.

For a cluster of n SGCMG, the total angular momentum is the vector sum

of the individual momenta which are functions of CMG gimbal angles δ= (δ1,...,

δn):

=∑n

i ii=1

h h (δ ) (2.10)

The differential relationship between gimbal angles and the CMG angular

momentum vector h is obtained by taking time derivative of Eq. 2.10:

=h J(δ)δ (2.11)

where J is the 3 x n Jacobian matrix defined as:

i

j

hδ

∂∂= = ∂ ∂

hJ(δ)δ

(2.12)

Briefly, the task of the steering law is to determine the best gimbal angle

trajectories that are able realize the control torque u required by Eq. (2.8). By

means of gimbal rates that will realize the commanded h through Eq. (2.12) while

20

meeting various hardware constraints. Note that in CMG steering problem

formulation, both the gimbal inertias and the dynamics of the gimbal torquers are

ignored since their effects are very negligible.

21

CHAPTER 3

CHARACTERISTICS OF SINGLE GIMBAL CONTROL

MOMENT GYROSCOPES

3.1 INTRODUCTION

A Single Gimbal Control Moment Gyro (=SGCMG) is composed of a

flywheel spinning at a constant rate about an axis that is gimballed in one axis

orthogonal to the spin axis as shown in Fig. 3.1. The flywheel mounted on gimbal

frame produces constant magnitude angular momentum which is restricted to stay

in the plane of rotation. The gimbal motor on which the flywheel is mounted

changes the direction of the gimbal axis. Modification of the rotor axis orientation

produces the torque output that is normal to the both gimbal axis and spin axis.

If the wheel is gimballed in two axes instead of one axis, it is called as

Double Gimbal Control Moment Gyro (DGCMG; Fig. 3.2). Due to the extra

22

degree of freedom obtained by second gimbal, the rotor momentum of DGCMGs

can be oriented on a sphere along any direction. Consequently, the steering

problem is much simpler compared to SGCMGs. However, torque amplification

advantage is effectively lost in DGCMGs. They are usually appreciably heavier,

and their mechanical construction is much more complicated. In addition, they

consume more power.

Fig. 3.1 Single Gimbal CMG

Fig. 3.2 Double Gimbal CMGs

23

If the flywheel speed is allowed to be variable, the CMG is called a Variable

Speed CMG (VSCMG). It takes the advantageous of both a CMG and a reaction

wheel. Although speed variation provides extra degree of freedom as in the case of

DGCMG, they are also complicated to design12.

The type and also the number of CMGs that can be utilized in an attitude

control system is a trade off between performance, cost, reliability, difficulty level

of the mechanical implementation, and algorithm complexity. In this section, firstly

principal of operation of a single gimbal control moment gyros is given. Then,

problem formulation related to the minimally redundant SGCMG configuration is

presented. Singularity problem is defined with descriptive examples.

3.2 FORMALISM

A coordinate frame attached to a SGCMG is defined by the following

orthonormal basis vector as seen in Fig. 3.1:

, ,h g τe e e

where he : Unit vector along angular momentum

ge : Fixed unit vector along gimbal axis

τe : Unit vector along ×δ ge e (torque output direction)

24

The gimbal angle δ is measured with respect to the reference coordinate

frame with positive angular displacement defined by the gimbal axis direction.

Initial orientation of the gimbal fixed frame is defined as the reference frame,

, ,0 0 0h g τe e e .Thus, the expression for the unit vectors in any instant are given by:

cos sin

sin cos

δ δ

δ δ

= +

= − +

0 0h h τ

0 0τ h τ

e e e

e e e (3.1)

The expression of the angular momentum with respect to the reference

coordinate frame is:

h= hh e (3.2)

These equations can be generalized to the cluster of n identical single gimbal

CMGs. The total angular momentum of the cluster is the sum of all individual

momenta as described by Eq. (2.10).

Torquing the gimbal results in a precessional torque output along the

direction of τe . Since magnitude of the angular momentum of the SGCMG is a

constant, torque is produced only by the change of the direction of the momentum

vector. The output torque of a single gimbal CMG stationary in an inertial frame

may be given by:

= ×T δ h (3.3)

25

3.3 PYRAMID CONFIGURATION

As stated previously, since the momentum along the direction of spin axis is

constrained to lie in the plane of rotation, one CMG is not sufficient to provide the

three axis torque required to control the attitude of space vehicle. To provide full

attitude control at least three CMGs are needed. Situations exist, however, where

the CMG rotors are configured such that control authority can not be projected

along a certain direction (=singular direction). In order to reduce the effect of such

gimbal configurations, many redundant CMG array configurations have been

proposed in the past. The degree of redundancy is given by the difference between

the numbers of CMGs and the number of degrees of freedom to be controlled.

Fig. 3.3 Pyramid mounting arrangement of 4 CMGs

26

In this study, minimally redundant 4-SGCMG cluster in a typical pyramid

mounting arrangement (Fig. 3.3) is analyzed, since it is the most extensively

studied configuration among these redundant configurations. An extra degree of

control is provided by the 4th CMG as well. As seen from Fig. 3.3, gimbal axes of

each CMG are orthogonal to the faces of the pyramid. Each face is inclined with

pyramid skew angle of β=54.73º to the horizontal. This provides almost equal

momentum capability in all three axes1, 13, a nearly spherical momentum envelope

(Fig. 3.4). For pyramid configuration total angular momentum expression in Eq.

(2.10) takes the following form which may easily be derived from the geometry:

1 2 3 4

1 2 3 4

1 2 3 4

cos sin -cos cos sin coscos -cos sin -cos -cos sin

sin sin sin sin sin sin sin sino o o oh h h h

β δ δ β δ δδ β δ δ β δ

β δ β δ β δ β δ

-h = + + + (3.4)

It is assumed each CMG has equal and constant angular momentum of unit

magnitude (ho= 1 Nm.s). Total output torque for this system is given by the time

rate of change of total angular momentum vector:

τ = h = J(δ)δ (3.5)

where instantaneous Jacobian matrix is:

1 2 3 4

1 2 3 4

1 2 3 4

cos cos sin cos cos sinsin cos cos sin cos cos

sin cos sin cos sin cos sin cos

β δ δ β δ δδ β δ δ β δ

β δ β δ β δ β δ

− − ∂ = = − − ∂

hJ(δ)δ

(3.6)

27

Note that each column of the Jacobian matrix represents the output torque vector of

the respective CMG in the cluster. Minimum two norm solution of this problem

gives the Moore Penrose pseudo inverse:

1−

T T

MPδ = J(δ) J(δ)J(δ) τ (3.7)

For a given torque requirement, Eq. (3.6) represents a system of linear

simultaneous equations which should be solved instantaneously for current gimbal

rates. Most CMG steering laws as described in Chapter 4 are pseudoinverse based.

3.4 SINGULARITY AND SINGULARITY TYPES

3.4.1 Singularity Definition

Mathematically, Eq. (3.6) fails when J loses rank since inverse of JJT can not

be taken. Physical interpretation is that if all output torque vectors remain on the

same plane, no output torque can be produced along the direction normal to this

plane. If this is the case, the rank of the Jacobian matrix reduces to 2, indicating

that the system is in a singularity condition. The normal direction of this plane is

called as s, singularity direction. It corresponds to the direction of the eigenvector

corresponding to the minimum eigenvalue of the Jacobian matrix13, 18. Since the

output torque can not be generated along singular direction, three-axis

controllability is lost. The situation can be expressed as:

28

=TJ s 0 (3.8)

Eq. (3.5) represents a nonlinear vector valued mapping of the 4-dimensional

gimbal angle space (δ-space) into 3-dimensional angular momentum space3. The

corresponding derivative map described by Eq. (3.6) is a linear transformation from

the gimbal rate space δ•

-space into output torque space. Given any direction in

space, there exists 2n sets of gimbal angles for which the system is singular3.

Consequently, collection of all singular directions and singular states construct 24

dimensional singular surfaces in (δ -space) which are mapped piecewise onto

angular momentum space. Each point on a singularity surface represents a system

momentum state for which the available system torque in some direction is zero.

Resultant shape in the angular momentum space is created by the bounds of these

regions reflecting the capability of the CMG system (Fig. 3.4). The shape of outer

surface is called momentum envelope (boundary). Momentum envelope is nearly

spherical, roughly orthogonal shape, and it includes small dimples at the gimbal

configurations corresponding to the gimbal axes13, 18.

29

Fig. 3.4 Momentum Envelope for 4-CMG Pyramid Arrangement taken from

Reference 13.

One measure of singularity may be:

det( )m = TJ(δ)J(δ) (3.9)

3.4.2 Singularity Types:

In this section, all singularity types are summarized (Fig. 3.5) according to

number of criteria presented in the literature:

30

Fig. 3.5 Classification of all singularity types according to various criteria

I. Location in Momentum Envelope

According to the location of total angular momentum vector relative to the

momentum envelope, singular states can be categorized as external (surface) and

internal.

External singularities: Outer shape of the momentum envelope seen in the

fig.1 represents the external singularities. Almost 90 % of the momentum envelope

includes momentum states representing maximum angular momentum attainable by

the CMG cluster along any given direction. This type of singular states is also

called as saturation singularity. Since these surfaces represent the physical limits

of the system, it is impossible to escape from these singularities by managing CMG

31

redundancies. However, by the utilization of external torquers (such as gravity

gradient torques, jet firings) CMG system can be desaturated1,3,4,13.

The null motion tests explained briefly in the following section reveal that

saturation singularities have elliptic characteristics. Since each individual CMG

angular momentum vector is positively projected along the singular direction, all

saturation singularities constitute complete class of 4H singularity type according

to the cutting plane technique (also xz plane view figure can be given here

(projection of the momentum envelope in Z-X plane) as seen in Fig. 3.6. Torque

can be generated along all directions from the tangent plane to the envelope and

inward normal to the envelope.

Fig. 3.6 Singular direction vs. projection of the momentum envelope in Z-X

plane18.

32

The deviation of the momentum envelope from the sphere occurs when the

momentum states corresponding to the singular direction is very closely aligned

with the one of the gimbal axis. Since CMG of this gimbal axis can not project its

angular momentum along singular direction, it should be nulled by CMG of the

opposite face. Therefore, projection of the angular momentum along the singular

direction has a sign pattern of (+, +, -, +), which is class of 2H singular states.

Therefore, over the momentum envelope, two different types of external

singularities can be observed. One is saturation singularity, properly on the

envelope, with all rotors aligned (4H), and the other is over the dimples including

singular states with having one rotor flipped with the singular direction (2H).

Internal singularities: Any singular state for which the total momentum

vector is inside the momentum envelope is called as so. All of the internal

singularities can be determined from the saturation singularity by reversing one or

more angular momentum vectors so that they are projected negatively on to the

singular direction. (When some angular momentum vectors are minimally

projected and the others are maximally projected, corresponding singular states are

all internal except the 2H singularities at the dimples). Therefore, internal

singularities include states with all even and odd number of combinations of

projections of individual angular momenta (2H & 0H) over singular directions

(except the 4H states).

33

II. Null Motion Characteristics

Null motions correspond to the homogenous gimbal rate solution of Eq. (3.6)

that produces zero torque with no change in the angular momentum state. Null

space basis vector computation is given in Appendix-A.3.1. Any singular point can

be classified according to whether it can be avoided or escaped by null motion or

not. If the rank of the Jacobian can be increased to full rank by given null

displacements, this singular state is escapable.

In order to understand null characteristic of the singular state, some simple

null tests can be carried out as described in References 3, 5 and 13. In all these

sources, tests are based on some sort of second order Taylor Series expansion of

the total CMG angular momentum about given singular point by allowing

infinitesimal null gimbal angle variations. The sign definiteness of the resultant

quadratic expression determines whether null motion is possible for the given state.

Following expression comes from the Taylor Series expansion3,5,6:

T s

i

: Variation in gimbal angle vector=diag(s h )

∂ ∂ =∂

Tδ P δ 0δ

P (3.10)

P is the projection matrix with diagonal elements representing the projections

of the singular angular momentum vector onto the singular direction. Expressing

the null motion as a linear combination of the null space basis vectors;

34

( )

1

: Scalar weighting factor: Null space basis vector (n-dimensional): number of CMGs in the cluster

n rank

ii

i

n

λ

λ

−

=

∂ = =∑J

i

i

δ n Nλ

n (3.11)

Substitution of the Eq. (3.11) into the Eq. (3.10) yields the desired quadratic

expression;

where

=

=

T

T

λ Qλ 0Q N PN

(3.12)

If Q is definite matrix, the singular state is called as elliptic. In the cases of

remaining alternatives (semi-definiteness or indefiniteness), the singular state is

called as hyperbolic. (These names come from the shape of the quadratic

expression obtained. It can be ellipsoid or hyperboloid, respectively.)

The definite Q matrix is an identifier of the characteristic of the impassible

singularity. The only possible solution of Eq. (3.10) for elliptic case is =λ 0

implying that no null motion is possible. Thus, escape by null motion from elliptic

type of singularity is not possible. There are two possibilities for Q to be positive

definite. First when all CMG angular momentum vectors have positive projections

onto the singular direction (4H) as in the case of saturation singularity. Second

when odd numbers of positive or negative projections (2H) onto the singular

direction can make Q definite. Most of these 2H singular states are internal and rest

are external corresponding to the dimples.

35

Hyperbolic singularities imply the possibility of escape. However, the

existence of null motion does not guarantee the escape from singularity all the time

since there are some degenerate null gimbal angle solutions. They are called as

degenerate since the rank of the Jacobian matrix (or the value of the singularity

measure, m) can not be increased although the singular state is altered. Therefore,

further tests should be done in order to understand whether the null motion can

reconfigure the system into nonsingular state or it can increase the singularity

measure.

III. Cutting Plane Technique

Cutting plane technique13,24 used in finding singularity surfaces can also be

used to classify singular states. In this technique, an arbitrary cutting plane is

chosen, and the individual momentum vectors are positioned normal to the

intersection of the momentum plane and the cutting plane. Since all the output

torques stay within the cutting plane, no torque can be generated along the normal

direction of this plane which is singular direction for this configuration. If the

momentum vectors are all positioned on the same side of the plane, singularity is

called as 4H, if one is reversed so that it is on the other side of the cutting plane,

corresponding singularity is called as 2H, and if two momentum vectors reversed,

this will determine 0H singularity. Then, this procedure can be repeated for

different cutting planes in order to extend singularity data to create three-

dimensional singularity surfaces. Note that momentum envelope in Fig. 3.4 is

obtained using this technique. Of course, this classification corresponds to the sign

36

pattern of the diagonal elements of the projection matrix, P, which represent the

individual angular momentum.

4H: Sign pattern of the diagonal elements of the P matrix is [+ + + +]. All

CMGs in the cluster reflect their maximum angular momentum projection

capability onto the singular direction which corresponds to the saturation

singularity as discussed previously.

2H: These are the most challenging singularity type, since their null

characteristics may not be clear directly. They are usually elliptic. Some of these

surfaces are external. As explained previously, these ones correspond to the

dimples of the momentum envelope (Fig. 3.7). [+ + + -], [+ + - +], [+ - + +], [- + +

+] are the projection sign patterns of the elliptic singularities.

Fig. 3.7 2H Singular Surfaces18

37

0H: All 0H states are completely inside the momentum envelope; therefore,

they are totally internal type of singularity. They occupy very small part of the total

angular momentum volume. They are usually hyperbolic and escapable via null

motion.

3.5 CONCLUSION

In order to develop an efficient steering law that avoids the singularity,

understanding of the momentum envelope structure is very important. In this

chapter, all types of singularities seen in momentum space of the 4-CMG in

pyramid arrangement are classified. Singular state examples for better

understanding of this classification are presented in Appendix-A.3. In these

examples, also null tests are carried out as explained in Reference 19.

38

CHAPTER 4

OVERVIEW OF THE STEERING LAWS

4. 1 INTRODUCTION

In this chapter, well known steering algorithms of the literature are

summarized. A new method to avoid the internal elliptic singularities is proposed.

In the following two chapters, performance of most of the existing and newly

proposed methods will be evaluated and compared through simulation studies.

An ideal steering law is expected to provide full avoidance of the singularity

while realizing the angular momentum commanded (or torque commanded).

Thus, the gimbal angles should be steered away from the internal singular states

using the system redundancy while satisfying the commanded torque, τ. If

singularity avoidance is not possible, it should at least transit through singularity

with minimum error in the required torque.

39

The solution of Eq. (3.5) can be divided into particular and homogenous

solution parts. While Eq. (3.7) gives the particular solution, there is also

homogenous solution that does not produce torque, hence called null motion:

null0 = J(δ)δ (4.1)

The Jacobian null vector being perpendicular to the rows of the Jacobian

matrix, may be calculated5,6,24 using the generalized cross or wedge product ∧ , or

derived using Jacobian cofactors:

[ ]31 2null

1 2 3

1 2 3 4

1

[ , , ]

, , , for =1,2,3

( 1) is the order 3 Jacobian cofactor for =1,...,4= det( ) is the order 3 Jacobian cofact

T

l l l l l

ii

i i

hh h

h h h

h h h h l

i

δ δ δ δ+

∂∂ ∂= ∧ ∧ =∂ ∂ ∂

=

∂ ∂ ∂ ∂ ∂= ∂ ∂ ∂ ∂ ∂ = −

1 2 3 4

i

δ C ,C ,C ,Cδ δ δ

h

hδ

C MM J

th

null

or minor

with column removedi i

m

=

=

J J

δ

(Appendix-A.3.1)

Since the particular and homogenous (null) solutions are orthogonal

complements to each other3,5,24, their linear combination is the general solution that

spans all possible CMG motions that satisfy the torque request.

particular homogenouscδ = δ + δ (4.2)

where, c is a real number.

40

4.2 CLASSIFICATION OF EXISTING STEERING LAWS

Moore-Penrose pseudo (MP) inverse steering law is the most basic form of

available steering laws. However, it almost always drives the cluster towards

singular configurations. For this reason, many other steering laws are developed,

achieving different levels of success. They utilize particular solution only, or a

combination of null solution with particular solution. Some emphasize quick

transition through an elliptic singularity, and some others strive for full avoidance.

These methods may be categorized as follows:

Fig. 4.1 Classification of Steering Laws

41

I. Instantaneous Methods

A. Particular Solution (Torque Producing Solution) Methods

i) Exact Solution Methods Based on Moore-Penrose Pseudo Inverse

If the Jacobian matrix is nonsingular, it has three linearly independent rows

vectors. The row space is spanned by these vectors, thus torque producing solution

can be written as follows24;

3

1th

1

where i Jacobian row vector

Plugging this equation into the constraint equation

solving for :

then replacing with the relatio

iiα

=

−

= =

=

=

=

∑ Tparticular i

Ti

T

T

δ R J(δ) α

R

J(δ)δ = τ :J(δ)J(δ) α τ

α

α J(δ)J(δ) τ

α n above in := Tparticularδ J(δ) α

1T T

particular MPδ δ = J(δ) J(δ)J(δ) τ−

= (4.3)

Most CMG steering laws computes torque-producing gimbal rates with some

variants of pseudoinverse. The pseudoinverse is the minimum two-norm vector

solution of Eq. (3.5). Therefore; it is the only torque producing method which

includes no null motion. The main drawback of this minimum norm property is the

high possibility of encountering singular states. For this reason other steering laws

are developed.

42

Weighted Pseudo Inverse: The generalized form of MP inverse steering law

can be posed as follows:

12

subject to

T

δmin δ Qδ

J(δ)δ = τ (4.4)

where Q is symmetric, positive definite weighting matrix.

( )

( )1

12

−

= ⇒

= ⇒ =

= −

T T

T -1 Tδ

-1 Tλ

-1 T

L = δ Q δ+ λ J(δ)δ - τ

L Q δ+ J(δ) λ = 0 δ = -Q J(δ) λ

L J(δ)δ - τ = 0 J(δ) -Q J(δ) λ τ

λ J(δ)Q J(δ) τ

1-1 T -1 Tδ = Q J(δ) J(δ)Q J(δ) τ−

(4.5)

If Q=I, then we have 1−

T Tδ = J(δ) J(δ)J(δ) τ which is MP method.

The weighted form is ineffective in passing or avoiding the singularity as in

the case of MP method24.

ii) Transition Methods

Minimum norm solution of Eq. (3.5) almost always drives the system to a

singular configuration. However; if one sacrifices the accuracy of the torque

generated, namely permitting small torque errors, it may be possible to transit

through all internal singularities. The feedback system may ideally compensate

43

these torque errors. The following methods are mainly developed using this line of

attack.

Singular Robust Inverse

Singularity Robust (SR) steering law is developed by Nakamura and

Hanafusa7 for robotic manipulators which are mechanical analogs to CMGs. It can

be derived by solving the following minimization problem 5,6,10,24:

12

+T Terr errδ

min δ Aδ τ Bτ (4.6)

where errτ = J(δ)δ - τ . A and B are positive, symmetric weighting matrices.

Solution of this problem yields the following expression:

( )( )

1−

=

⇒

+

Tδ

T T

T TSR

L J(δ) A J(δ)δ - τ + Bδ = 0

J(δ) AJ(δ) + B δ = J(δ) Bτ

δ = J τ = J(δ) AJ(δ) B J(δ) Aτ

(4.7)

If A and B are selected as diagonal matrices I3 and αI4, respectively; the SR

inverse is obtained:

1 1T T T Ta a− −

= + ≡ + SR 4 3J I J J J J I JJ (4.8)

where, α is the singularity avoidance parameter to be properly selected. It can be

shown that the matrix within brackets is never singular. Then, the required gimbal

rates are found from:

SR SRδ = J τ (4.9)

44

If α is taken zero, pseudoinverse is recovered. This parameter is usually

selected according to the singularity measure, m. For example 7:

2

0

0 for

.(1 / ) for cr

o cr

m ma

a m m m m

≥= − <

(4.10)

or5,24,

0 0max

max

0 for

for

for all other possibilities

crm ma aa am ma

≥= <

(4.11)

where m is computed from Eq. (3.8).

In the above expressions, α is negligible when the system is far away from

singularity, but it increases as the singularity is approached ensuring that a solution

for the gimbal rates always exists. However, as a singularity is approached, the

realized torques starts to differ from the desired torques. The main disadvantage

of this method is that internal elliptic singularities cannot be passed since SR

inverse use the same direction (torque producing particular solution direction) as

MP inverse. This property may be proved through singular value decomposition

of the SR-inverse 10,24.

Generalized Singular Robust Inverse Steering Law

SR inverse is modified by Wie10 by defining the weighting matrix A with

nonzero off diagonal elements instead of using a diagonal matrix in Eq. (4.7). In

45

this singularity robust (GSR) inverse, as singularity is approached, deliberate

deterministic dither (=excited or small oscillatory) signals of increasing amplitude

are used.

T T -1

G-SR [ ]0.01exp( 10 )m

λλ

= += −

δ J JJ E τ (4.12)

where; 3 2

3 1

2 1

1= 1 0

1

ε εε εε ε

>

E , and εi is the modulation function,

0.01sin(0.5 )i itε π φ= + , while iφ are 0, / 2π and, π , respectively.

The approach, however, does not avoid singularities and cannot generate the

desired torque around an elliptic singularity. Instead, it transits through a

singularity by generating small periodical disturbances. Although the generation

of small torque errors in unintended directions helps the transition, it creates

lengthy delays with large torque errors. Consequently, it may not be suitable for

precision tracking applications10.

B. Particular Solution Methods Armed with Null Motion

Since the homogenous solution produces no torque, it can be combined with

a pseudoinverse based method to avoid singular states on the trajectory of gimbal

angle set. Addition of proper amount of null motion to the torque producing

46

gimbal rates may allow following safer nonsingular trajectories (Eq. (4.2)).

However, it is not easy to decide on the proper magnitude.

i) Projection Matrix

Null vector may be obtained by projecting an arbitrary d vector into its null

space by using following relation:

homogenous

.T T 1

4δ I J( ) (J( )J( ) ) J( ) dρ δ δ δ δ− = − (4.13)

where, d is chosen by using a performance index P(δ), ( )T δd

δP∂

=∂

which is

referred to as gradient method. The weakness of this method is the need to

know singularity free gimbal angle trajectory beforehand in order to avoid

them. Selection of singularity measurement (m) and 1/m2 as a performance

index have been attempted by several authors2,3,5,24. Since the method is

instantaneously working, very abrupt changes in the determinant does not allow

addition of sufficient null motion to avoid internal singularities. Therefore; this

method is not studied in the simulation part.

ii) Inverse Gain

Other approach proposed is to use inverse gain (IG) method2,5,24:

homogenous c=.δ n (4.14)

47

6

6

, 1, 1

m mc

m m−

≥= <

(4.15)

Although this method is able to avoid all types of internal singularities, the

calculated null rates may become very high even though system is far away from

the singularity. In Chapter 5, IG method is combined with MP, SR and GSR

methods to see how these combinations affect the results.

C. Other Methods

Optimization based gimbal rate selection may be performed online by

optimizing a cost function. However, singularity avoidance is not guaranteed due

to use of gradient-based objective.

A transpose based method is proposed in Reference 9. Rather than inverse of

Jacobian matrix its transpose is used to develop a steering law. Since the pseudo

inverse calculation is not required, singularity avoidance is no longer a problem.

However, other performance criteria should be tested for this method.

48

II. Preplanned Methods

In real time applications, instantaneous methods may still take us to an

internal elliptic singularity. Torque errors may be intentionally produced to pass

through singularity rapidly by the steering logic itself as in the case of modified SR

method. During process, the maneuver will be delayed, and precision tracking can

not be realized. On the other hand if the spacecraft had an initial angular velocity

while approaching to the singularity, it will continue to roll until a torque command

in a different direction is requested by the feedback system. This may help the

spacecraft recover from singularity. Examples of this will be given in Chapter 5

and Chapter 6.

A. Preferred Initial Gimbal Angle Selection

If the angular momentum and torque envelopes are assumed to be known a

priori, a family of initial gimbal angles can be determined that avoids singularity all

along the path by back integration of the gyro torque equation from desired final

conditions14.

49

B. Global Steering Using Directed Search

Paradiso proposed a directed search that manages null motion about torque-

producing trajectories calculated with a singularity robust inverse as a feedforward

steering law. The search avoids or minimizes the effects of singular states13.

Defined cost and heuristic functions help search procedure in improving gimbal

trajectories. Note that, null motion is added at discretized gimbal positions along

the trajectory. However, this method can not be implemented on-line due to the

extensive computation requirements.

4.3 UNIFIED STEERING LAW

To avoid singularity, the gimbals shall be driven towards non-singular

configurations of a momentum state. To maneuver the spacecraft in a stable

fashion, however, torque required by the control system must also be realized,

which may be in conflict with the first objective. However, one may try to realize

these two goals simultaneously by posing a mixed minimization problem that tries

to go towards desired gimbal angles as well as generate the desired torque. Thus,

consider the following minimization problem:

err err err err12

+T T

δmin δ Qδ τ Rτ (4.16)

50

In this equation, err desiredδ δ - δ= , errτ = Jδ - τ , while Q and R are symmetric

positive definite weighting matrices. The above minimization problem may be

solved as:

( )

T Tdesired

12

then,

Thus;

= +

=

T Terr err err err

Tdesiredδ

T Tdesired

-1

USL

L δ Qδ τ Rτ

L Q (δ -δ ) - J(δ) R J(δ)δ - τ = 0

Q + J(δ) RJ(δ) δ = Qδ + J(δ) Rτ

δ = Q + J RJ Qδ + J Rτ

(4.17)

This steering logic blends the desired gimbal rates that will take the system to a

desired gimbal configuration and the required torque If 4q=Q I , and 3=R I , then,

Eq. (4.18) becomes;

T Tq q -1

USL 4 desiredδ = I + J J δ + J τ (4.18)

In this equation following may be observed:

I. If desired =δ 0 , then singularity robust steering law is recovered (Eq. (4.19))

II. If q is zero, MP inverse is recovered provided that J is a non-singular

square matrix: 1 1T T T T

MP

− − = = J J J J J JJ

51

However, for rectangular matrices (i.e. redundant CMG’s) the determinant of JTJ is

always zero and the inversion seen on the left hand side cannot be performed.

III. If q→∞ , then, desiredδ = δ . Thus, in this case the gimbals are driven at the

desired rate in an open loop fashion.

Selection of desired gimbal rate, desiredδ , and the blending coefficient, q, are

the key points in the utilization of this method. As to be shown below, the method

can be used as a preplanned method, or an instantaneous method, hence called the

Unified Steering Law (USL).

Preplanned steering

If CMG cluster’s angular momentum trajectory, to be followed during a

maneuver, is known beforehand, then gimbal angle solutions that have high

singularity measure, but accurately satisfying the momentum values at discrete

points in time (to be referred as nodes), may be calculated using an optimization

technique. Then, the system may be driven to go to the desired gimbal solutions at

these nodes by selecting the gimbal rates accordingly.

The main disadvantage of this method is the necessity to know the required

angular momentum history (or torque history) of the maneuver. Thus, maneuver

simulations are needed to find the desired gimbal angles and corresponding rates.

52

Therefore the approach may not be suitable when spontaneous response of the

spacecraft is needed.

On-line steering

For spontaneous response, an on-line steering law is necessary. Number of

methods has been developed to avoid or quickly transit through singularities. In IG

method for example, null rate is added with the expectation of avoiding internal

singularities. In the G-SR method, the system is disturbed by intentional

deterministic dither signals. In this way the system can pass through an internal

elliptic singularity. These approaches may also be applied to USL by injecting the

necessary disturbance to the system through the desired gimbal rate term, desiredδ , in

Eq. (4.18).

Since null vector is always perpendicular to the particular solution, it is a

good choice to be used as a disturbance. Another approach would be to use an

arbitrary vector for desiredδ . It may also be a dynamic vector with randomly

changing elements, imitating a white noise, especially effective when the cluster is

close to a singularity. These approaches are implemented and their relative

advantages are discussed in the next section.

53

CHAPTER 5

SIMULATION STUDY I

5.1 INTRODUCTION

Main goal of this chapter is to investigate extensively to understand the

performance of most widely used steering laws together with unified steering law

by carrying out some simulation studies using basic equations of a minimally

redundant CMG cluster presented in Chapter 3. Thus, constant torque and corner

maneuver simulations are performed with available methods (MP, SR, GSR) with

and without null motion addition. Here, it will be shown that nearly most of the

existing steering laws do not avoid elliptic singularities without addition of null

motion. In addition, the singularity avoidance or singularity transition properties of

the USL are evaluated through simulations.

54

5.2 DESCRIPTION OF THE SIMULATION WORK

Constant Torque Study

If a constant torque is required along a particular direction, the system either

encounters an internal singularity or reaches the external singularity. For example

starting from the initial gimbal angle configuration of [ ]0 ,0 ,0 ,0 T= ° ° ° °oδ (h=0),

requiring a constant torque [1.155,0,0]T=τ , and using a pseudoinverse steering

logic, the system encounters an internal elliptic singularity in about one second.

At this instant, the gimbal angles become [ ]90 ,0 ,90 ,0 T= − ° ° ° °oδ , and the angular

momentum is [ ]1.155,0,0 T=h . If a singularity is not encountered, the system

reaches an external singularity (in about 2.74 seconds), where [ ]3.15,0,0 T=h . At

an external singularity, momentum state has a unique solution for the gimbal angles

(i.e., [ ]90 ,180 ,90 ,0 T= − ° ° ° °δ ). This is a also called saturation type singularity

identified as un-escapable 6 as described in Chapter 3. Due to these properties of

the constant torque scenario described, is used in the simulations conducted below

to evaluate the proposed steering law and compare it with those available in the

literature 5,24.

Any method may be counted as successful if it can avoid or transit quickly

through all type of internal singularities until the momentum of the system reaches

to an external elliptic singularity while keeping the gimbal rates within certain

55

motor limits. In this study this limit is selected as ±π rad/s. Fig. 5.1 illustrates the

ideal torque and angular momentum profiles of the constant torque simulation

conducted.

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.1.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.1.b

Fig. 5.1 Ideal angular momentum and torque profiles

56

5.3 RESULTS OF THE EXISTING STEERING LAWS

5.3.1 MP Inverse Results

First a constant torque simulation study is conducted with MP inverse. As

expected system is trapped into the described elliptic singularity in one seconds. It

can be observed from Fig. 5.2, MP inverse leaves the system in a singular

configuration. In addition, more than half of the momentum capacity can not be

utilized. As seen from the gimbal rate plot (Fig. 5.2.e), at the elliptic singularity

instant, the gimbal rates go to infinity. As stated in previous chapter, the

eigenvector corresponding to the minimum eigenvalue of the system is the singular

direction. Since the desired torque command is collinear with the singular direction

which is [1; 0; 0] in this case, no torque can be generated in the requested direction.

Therefore, MP method can not always be utilized to produce torque.

The initial gimbal angle configuration has a considerable effect on the

solution. To demonstrate this, the simulation is started with a different initial

gimbal angle configuration, [ ]115 , 115 ,115 , 115 T= ° − ° ° − °oδ (h=0). Results of this

study are presented in Fig. 5.3. Although MP inverse is utilized, no internal

singularity is encountered during the maneuver and maneuver is completed ideally.

As suggested in reference 5, inverse gain (=IG) method is combined with MP

inverse. Results of this consideration are presented in Fig. 5.4. In this figure,

homogenous solution gimbal rates (null motion) are computed with IG Eq. (4.14)

57

and the particular solution gimbal rates computed with MP inverse are presented,

separately in Fig. 5.4.f and Fig. 5.4.g. From the results, it may be observed that

angular momentum and torque profiles (Fig. 5.4.a and Fig. 5.4.b) are realized with

no observable error, however, extremely high gimbal rates prevents this approach

from being a feasible steering law (Fig. 5.4.e).

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.2.a

58

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.2.b

0 0.5 1 1.5 2 2.50

0.5

1

1.5Singularity Measure

t (s)

m

5.2.c

59

0 0.5 1 1.5 2 2.5-100

-80

-60

-40

-20

0

20

40

60

80

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.2.d

0 0.5 1 1.5 2 2.5-30

-20

-10

0

10

20

30Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.2.e

Fig. 5.2 Constant torque simulation with MP inverse

60

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.3.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.3.b

61

0 0.5 1 1.5 2 2.50

0.5

1

1.5Singularity Measure

t (s)

m

5.3.c

0 0.5 1 1.5 2 2.5-200

-150

-100

-50

0

50

100

150

200

250

Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.3.d

62

0 0.5 1 1.5 2 2.5

-3

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.3.e

Fig. 5.3 Constant torque simulation with MP inverse with preferred initial gimbal

angle set.

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.4.a

63

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.4.b

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.4.c

64

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.4.d

0 0.5 1 1.5 2 2.5-10

-8

-6

-4

-2

0

2

4

6

8

10

12Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.4.e

65

0 0.5 1 1.5 2 2.5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Homogenous Gimbal Rate Solution

t (s)

Gdo

th(r

ad/s

)

g1dothg2dothg3dothg4doth

5.4.f

0 0.5 1 1.5 2 2.5-10

-8

-6

-4

-2

0

2

4

6

8

10

12Torque Producing Gimbal Rates

t (s)

Gdo

tp (r

ad/s

)

g1dotTg2dotTg3dotTg4dotT

5.4.g

Fig. 5.4 Constant torque simulation with MP inverse combined with IG method

Null rates from Inverse Gain

66

5.3.2 SR Inverse Results

Here α is selected as suggested by Bedrossian. [α0=0.1, αmax=0.2 and mcr=1]

(Eq. 4.9-11). The corresponding results are given in Fig. 5.5 that SR inverse

method can not avoid the internal elliptic singularity encountered at the first second

of the simulation. Although the gimbal rates can still be computed when the

Jacobian matrix is singular, realized torques is almost zero.

IG method is combined with SR inverse in order to see if SR inverse can

avoid singularity with the help of null motion addition. Results of this case are

presented in Fig. 5.6. From these results, it may be observed that angular

momentum and torque profiles (Fig. 5.6.a and Fig. 5.6.b) deviate from the desired

values around the region of elliptic singularity. Furthermore, although the

homogenous gimbal rates (Fig. 5.6.f) are essentially same with MP inverse results

(Fig. 5.4.f), the torque producing gimbal rates computed (Fig. 5.6.g) are lower than

the ones obtained by MP inverse (Fig. 5.4.g). Thus, this approach is much

successful than the SR inverse only approach, since in some cases it can transit

through internal singularities.

67

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.5.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.5.b

68

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

Singularity Measure

t (s)

m

5.5.c

0 0.5 1 1.5 2 2.5-100

-80

-60

-40

-20

0

20

40

60

80

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.5.d

69

0 0.5 1 1.5 2 2.5-1.5

-1

-0.5

0

0.5

1

1.5Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.5.e

Fig. 5.5 Constant torque simulation with SR inverse

0 0.5 1 1.5 2 2.5 3-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.6.a

70

0 0.5 1 1.5 2 2.5 3-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.6.b

0 0.5 1 1.5 2 2.5 30.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

5.6.c

71

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.6.d

0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.6.e

72

0 0.5 1 1.5 2 2.5 3-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Homogenous Gimbal Rate Solution

t (s)

Gdo

th(r

ad/s

)

g1dothg2dothg3dothg4doth

5.6.f

0 0.5 1 1.5 2 2.5 3-1.5

-1

-0.5

0

0.5

1

1.5Torque Producing Gimbal Rates

t (s)

Gdo

tp (r

ad/s

)

g1dotTg2dotTg3dotTg4dotT

5.6.g

Fig. 5.6 Constant torque simulation with SR inverse combined with IG method

Null rates from Inverse Gain

73

5.3.3 GSR Inverse

GSR inverse simulation results performed with the suggested parameter10

values given in Eq. (4.12) are shown in Fig. 5.7. Around elliptic singularity, the

singularity measure is almost zero for a long time with large torque errors; in

addition gimbal rates reached to 5 rad/s. Transition through elliptic singularity

caused about 1.5 second delay in the simulation.

In order to see the effect of null motion, this time IG method is combined

with GSR. Corresponding results are presented in Fig. 5.8. The maneuver is

completed on time with smaller torque errors around elliptic singularity. Although

gimbal rates are lower than the previous case, they are still quite large.

0 1 2 3 4-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.7.a

74

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.7.b

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

5.7.c

75

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

5.7.d

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-5

-2.5

0

2.5

5Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.7.e

Fig. 5.7 Constant torque simulation with GSR inverse

76

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.8.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.8.b

77

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Singularity Measure

t (s)

m

5.8.c

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.8.d

78

0 0.5 1 1.5 2 2.5-5

-2.5

0

2.5

5Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.8.e

0 0.5 1 1.5 2 2.5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Homogenous Gimbal Rate Solution

t (s)

Gdo

th(r

ad/s

)

g1dothg2dothg3dothg4doth

5.8.f

Null rates from Inverse Gain

79

0 0.5 1 1.5 2 2.5-5

-2.5

0

2.5

5Torque Producing Gimbal Rates

t (s)

Gdo

tp (r

ad/s

)

g1dotTg2dotTg3dotTg4dotT

5.8.g

Fig. 5.8 Constant torque simulation with GSR inverse combined with IG method

5.4 RESULTS OF USL

Pre-planned steering

The aim of the method is to drive the gimbal angles to a desired configuration

while satisfying torque requirements. For this purpose, first, the angular momentum

trajectory of the CMG cluster during a slew maneuver is obtained through

computer simulation. Then, desirable gimbal angles at the discrete instants (nodes)

of the angular momentum trajectory are calculated by sequentially solving the

following optimization problem:

80

1 max 1 max

max

( ) 1k

k k

k k k

m

k = ,.., pt t− −

=− ∆ ≤ ≤ + ∆

δ

h h δδ δ δ δ δ

(5.1)

The last equation imposes an additional constraint on the gimbal rates, while

finding the gimbal configuration of the next node. In this study the gimbal angle

configuration that realizes the required angular momentum is found using a

simulated annealing algorithm32. Instead of optimizing sequentially one node at a

time, all the nodal values of gimbal angles may be optimized together with an

objective of maximizing the lowest singularity measure.

A numerical study is conducted to examine the feasibility of this approach.

For this purpose eight nodes are used. Through optimization best gimbal angles for

each of these nodes are calculated, using a gimbal rate limit of ±2.5 rad/s (Table 1).

Then the constant torque simulation is conducted. During the simulation, at any

instant, desired gimbal rate is calculated by dividing the difference between the

gimbal angle of the next node and the current gimbal angle to the temporal distance

of the next node. To avoid excessive gimbal rates, the target node is switched to

the follow up node as the denominator becomes small. Thus;

( 1)

1,...,

k k t t k tk pk t t

ε ε− ∆ − < < ∆ −−=

=∆ −δ δδ (5.2)

The blending coefficient, q (Eq. (4.19)), and ε are both taken 0.005. The

simulation results are given in Fig. 9. From the results, it may be observed that the

81

torque requirement of [1.155,0,0]T is almost exactly satisfied. Throughout the

simulation maximum torque error was less than 1.5% (Fig. 5.9.a). The singularity

measure plot shows that the value is above 0.3, even in the neighborhood of the

expected location of elliptic singularity (i.e., 1st = ) (Fig. 5.9.b). Gimbal angles

and gimbal rates are also given. The gimbal rates are within ±3 rad/s which is quite

acceptable (Fig. 5.9.d). Table 5.1 also lists the achieved gimbal angles, which

shows that they are very close to the desired gimbal angles. When the simulation is

repeated with a ten times smaller blending coefficient it is observed that the

maximum toque error is also reduced ten times, while gimbal angles and gimbal

rates did not change at all.

Table 5.1 Nodal locations, desired gimbal angles, and gimbal angles attained with

USL steering law in constant torque simulation.

82

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.9.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.9.b

83

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.9.c

0 0.5 1 1.5 2 2.5

-3

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.9.d

84

0 0.5 1 1.5 2 2.5

-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

5.9.e

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.9.f

Fig. 5.9 Constant torque simulation results using optimized gimbal angles of Table 5.1

85

To verify the effectiveness of the steering, the corner maneuver described by

Bedrossian et al.5, is also tested. The maneuver again starts at the origin, h=0 with

zero gimbal angles. The torque profile of the maneuver is described below:

desired

1 [1,1,0] , t<0.83442

1 [1, 1,0] , t>0.8344 2

T

T

= −

τ (5.3)

The pseudoinverse steering logic completes the maneuver without

encountering any singularities. However, adding null motion with inverse gain

( nullρδ , Eq. (4.14)) to the SR inverse steering law drives the system to an internal

singularity 5. To ride the system towards the desired gimbal angles, four equally

spaced nodes are used. For each of these nodes desirable gimbal angles are again

found by solving optimization problem described in Eq. (5.1) using gimbal rate

limit of 2.5rad/s± . The desirable gimbal angles are listed in Table 5.2. Simulation

results with weight, q, being 0.005 are given in Fig. 5.10. From the results it may

be observed that not only the desired maneuver is completed without any difficulty,

but the torque and the angular momentum trajectories are very closely realized as

well. Singularity measure is high, gimbal angles are very close to the desired

values as before, and required gimbal rates are sufficiently low.

86

Table 5.2 Nodal locations and desired gimbal angles used in corner maneuver,

together with the corresponding singularity measures.

0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

1

Angular Momentum Trajectory

t (s)

h (N

m.s

) hxhyhz

5.10.a

Nodal

Locations (s)

Desired gimbal angles

(deg)

Singularity

Measure

0.0 [0, 0, 0, 0] 1.20

0.417 [-34.4, -12.0, 12.0, 34.4] 1.39

0.834 [-56.6, -37.5, 37.5, 56.7] 1.30

1.252 [-43.8, -52.6, 30.5, 78.0] 0.94

1.669 [-67.6, -59.7, 80.7, 126.8] 1.12

87

0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Torque Realized

t (s)

T (N

m)

TxTyTz

5.10.b

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Singularity Measure

t (s)

m

5.10.c

88

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.10.d

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-80

-60

-40

-20

0

20

40

60

80

100

120

Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.10.e

Fig. 5.10 Corner maneuver simulation results using optimized gimbal angles of Table

5.2

89

In contrast to the pre-planned methods, online methods that use MP-inverse

or SR-inverse usually drift away from desired configurations and hence are not

repeatable12, 13. To demonstrate the repeatability of USL, another simulation is

conducted. First a positive torque (0 < t < 2.45 s), then a negative torque (2.45 < t

< 4.9) of same magnitude is requested. The nodal values given in Table 5.1 are

employed to calculate desired gimbal rates for both forward and backward paths.

The results of the simulation with q = 0.0005 are presented in Fig. 5.11. Fig.

5.11.a gives the realized torque while Fig. 5.11.b gives the gimbal angle history of

the maneuver. The symmetry of the gimbal angle history with respect to the

switching point at (t = 2.45 s) shows that preplanned USL is repeatable. The small

difference between the right and left halves of the maneuver is mainly due to the ε

parameter used in calculating desiredδ (Eq. (5.2)).

0 1 2 3 4-0.5

0

0.5

1

1.5

2

2.5

3Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.11.a

90

0 1 2 3 4

-1

-0.5

0

0.5

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.11.b

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.11.c

91

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-3

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.11.d

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4gdes1gdes2gdes3gdes4

5.11.e

Fig. 5.11 Maneuver with desired gimbal angles given in Table 5.1 to demonstrate

repeatability.

92

Space may be viewed as a benign environment, with rather predictable

disturbance torques during a slew maneuver. Thus, the torque and angular

momentum profiles obtained from simulations shall not differ too much from those

that may be encountered during the actual maneuver. However, to consider

unpredictable occurrences, another simulation is conducted. For this purpose

constant torque maneuver is used. However, torque desired is changed during the

simulation by adding a sinusoidal part to the constant torque.

[ ]T T1.15 0 0 0.115 [1 1 1] sin(2 )t+desiredτ = (5.4)

The results of the simulation, conducted with 0.005q = , is given in Fig.

5.12. From the plots it may be observed that the realized torque follows the desired

torque quite closely with maximum error being less than 0.03 N-m (Fig. 5.12.a).

Other plots of the simulation show that the nodal values of the gimbal angles are

close to those given in Table 5.1 with maximum difference being less than six

degrees (Fig. 5.12.b). The gimbal rates are within 3rad/s± (Fig. 5.12.c). To

remove the small difference between the desired torque and realized torque,

simulation is repeated with a lower blending coefficient ( 0.0005q = ). The torque

history given in Fig. 5.12.d is quite smooth with torque errors being one tenth of

the previous case. However, there was no observable change in the gimbal angle

and gimbal rate histories. Other disturbances tested, containing sinusoids of higher

and lower frequencies, as well as amplitudes also gave similar results. In all cases

it is observed that the required torque profile is very closely satisfied. However, to

93

reduce the deviation from the desired gimbal angle history, the momentum history

of the desired maneuver shall not deviate too much form the planned maneuver.

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.12.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Torque Realized

t (s)

T (N

m)

TxTyTz

5.12.b

94

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.12.c

0 0.5 1 1.5 2 2.5

-3

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.12.d

95

0 0.5 1 1.5 2 2.5

-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

5.12.e

Fig. 5.12 Simulation results with dynamic torque input. The desired gimbal rates are

computed from the gimbal angles used in constant torque maneuver (Table 5.1).

These results show that USL steering logic may be used for the pre-planned

slew maneuver of spacecrafts. The logic responds with accurate torque profiles

even if the planned and required profiles are different, indicating that it may be

effectively used in feedback control.

96

On-line Steering

Steering with null motion

The constant torque simulation is repeated using null vector instead of

desired gimbal rate. In this study cofactors algorithm is used to find null vector

(Eq. (A.3)), while its magnitude is adjusted using inverse gain ( desired nullρ=δ δ ).

The system may be continuously fed by the null rate; however, to avoid too much

disturbance, proper selection of the blending coefficient is important. As before,

0.005q = is used. From the simulations (Fig. 5.13), it may be observed that the

realized torque trajectory matches desired trajectory very closely (Fig. 5.13.a) with

maximum torque error below 2%. The lowest singularity measure was 0.4, while

the gimbal rates did not exceed ±π rad/s. These results are much better than those

presented in the literature using SR inverse with null 5. Another simulation using

0.0005µ = was also conducted to see its effect on the torque history. Torque error

is reduced below 0.2% (Fig. 5.13.f).

97

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.13.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.13.b

98

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.13.c

0 0.5 1 1.5 2 2.5-4

-3

-2

-1

0

1

2

3

4Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.13.d

99

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.13.e

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.13.f

Fig. 5.13 Constant torque maneuver simulation results using USL with null vector.

100

Since, the value of the blending coefficient has an important bearing on the

quality of the results; different q values are also tested. For 0.0005q = , the error in

torque is reduced to 0.2% while, there was no observable deviation in gimbal angle

and gimbal rate histories.

The corner maneuver simulation is also carried out using null vector addition

as well. As reported in Reference 5, adding null vector with inverse gain to MP

inverse or SR inverse steering logics drives the cluster to a singular configuration.

Simulation results conducted using q=0.025 are given in Fig. 5.14. From the figure

it may be observed that the system approaches to a singular configuration around

the corner, however, recovers with some torque errors and corresponding error in

the momentum trajectory, with reasonable gimbal rates. Torque and momentum

errors encountered are much less than those obtained by Bedrossian et. al.5,24, using

SR inverse. In the same reference, although smaller error is observed in the

momentum when MP inverse with null addition is used, gimbal rates were

excessive reaching over 7 rad/s at the corner.

101

0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

1

Angular Momentum Trajectory

t (s)

h (N

m.s

)hxhyhz

5.14.a

0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Torque Realized

t (s)

T (N

m)

TxTyTz

5.14.b

102

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Singularity Measure

t (s)

m

5.14.c

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.14.d

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-50

0

50

100

150

Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

5.14.e

Fig. 5.14 Corner maneuver simulation using USL with null vector.

Steering with a constant vector

This time an arbitrarily selected constant desired rate vector,

Tdesired [0,1,0,0]=δ , is employed while conducting the constant torque maneuver.

The weight value is dynamically adjusted with respect to the singularity measure,

using the following formula:

e mq ηµ −= (5.5)

where, µ and η are positive constants. Thus, the coefficient is small when the

system is away from a singularity and increases as the system approaches to a

singularity, similar to the weight selection used in G-SR inverse (Eq. (4.12)).

104

Initially, µ and η are selected as 0.5 and 10, respectively. The simulation results

are given in Fig. 5.15. From these results it may be observed that, the singularity

region is passed, however, with torque errors. The results are comparable to the

GSR steering logic simulation results, given in Fig. 5.7. However, with USL, the

singularity measure was never zero and gimbal rates were within 2.5 rad/s± . Other

trials with different arbitrary constant rate vector directions were also successful.

Different values for µ and η are tested as well. In general increasing µ or

decreasing η resulted in lower gimbal rates. For example with 52e mq −= , gimbal

rates were reduced to the 1.7 rad/s± band (Fig. 5.15.f).

0 0.5 1 1.5 2 2.5 3 3.5-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.15.a

105

0 0.5 1 1.5 2 2.5 3 3.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.15.b

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

5.15.c

106

0 0.5 1 1.5 2 2.5 3 3.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.15.d

0 0.5 1 1.5 2 2.5 3 3.5-100

-50

0

50

100

150

Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

5.15.e

107

0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.15.f

Fig. 5.15 Constant torque simulation results using USL with an arbitrary and constant

rate vector.

Other constant rate vectors are also tested. Since the aim is to transit

through elliptic singularity at [ ]1.155,0,0 T=h , gimbal angles with high singularity

measures, but satisfying the momentum requirement at the elliptic singularity point

are found through optimization. Among the five different sets calculated the

following is selected: [ ]58 ,139 ,107 ,130 T− ° ° ° ° . Since the expected elliptic

singularity occurs in about one second, this vector is taken as the desired gimbal

rate (i.e, [ ]4 -0.257, 0.615, 0.474, 0.576 T= ×desiredδ ). This simulation was again

conducted with µ and η being 0.5 and 10, respectively. The realized torque

history plot given in Fig. 5.16.b, shows that the torque requirement is perfectly

108

satisfied. In addition lowest singularity (Fig. 5.16.c) was above 0.7, and gimbal

rates were within rad/sπ± (Fig. 5.16.d).

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.16.a

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.16.b

109

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Singularity Measure

t (s)

m

5.16.c

0 0.5 1 1.5 2 2.5

-3

-2

-1

0

1

2

3

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.16.d

110

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

150

200Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.16.e

Fig. 5.16 Results of constant torque simulation using USL with a constant rate vector.

Steering with white noise

In GSR inverse steering logic, a particular dither signal is used to transit

through singularity. Although it is possible to find such a signal for the USL as

well, it is decided to examine a more general case. For this purpose, a white noise

of unit magnitude is employed as the desired rate vector (i.e., desiredδ ). However,

the noise is applied only when singularity measure is below a critical value not only

to make sure that good tracking is realized but also stop noise input outside

singularity regions:

111

10

if

exp( ), else

1 ,

crm m

q m

q e

µ η γ

−

<

= − =

= =

desired

desired

δ n

δ 0

(5.6)

Results of the related simulation are presented in Fig. 5.17 with parameters

mcr, µ, η, and γ selected as 0.6, 0.5, 10, and 1 respectively. Plots given in Fig. 5.17

show that the elliptic singularity region is successfully passed, however, with

torque errors as well as some noise. Gimbal rates were within 2.5 rad/s± . Another

simulation using 0.05µ = was also conducted to see its effect on the torque

history. The history given in Fig. 5.17.f shows that the noise level is reduced.

However, gimbal rates are increased to 3.35 rad/s± band.

These results demonstrate that USL may be used as an online algorithm for

quick transition through internal singularities, with the possibility of avoiding them

as well.

0 1 2 3 4-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

5.17.a

112

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.17.b

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

5.17.c

113

0 0.5 1 1.5 2 2.5 3 3.5 4-3

-2

-1

0

1

2

3Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

5.17.d

0 0.5 1 1.5 2 2.5 3 3.5 4-100

-50

0

50

100

150

Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

5.17.e

114

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

Torque Realized

t (s)

T (N

m)

TxTyTz

5.17.f

Fig. 5.17 Constant torque simulation results using USL with a white noise vector.

115

CHAPTER 6

CMG BASED ATTITUDE CONTROL MODEL

SIMULATION

6.1 INTRODUCTION

Mathematical model described in section 2.4 for the attitude control of a

spacecraft equipped with CMGs is constructed in the MATLAB® and SIMULINK®

software. Using this model, performance evaluation studies of the unified steering

law presented in previous chapter, is extended to a spacecraft platform. Fig. 6.1

shows the block diagram of the spacecraft simulation together with the attitude

control system.

In steering logic block, required gimbal rates to generate the internal control

torque ucommanded is determined using a desired steering logic algorithm. Then,

these rates are fed forward to CMG Dynamics block, where the gimbals are driven.

Inside the CMG Dynamics block, the gimbal rates are integrated to obtain gimbal

angles instantaneously which are used to compute instantaneous angular

116

momentum, and torque of the CMG cluster. The torque and angular momentum

values give urealized to Eq. (2.5) which is fed into the spacecraft dynamics block.

Fig. 6.1 More Detailed CMG Based ACS Diagram constructed in SIMULINK®

environment.

6.2 SIMULATION PARAMETERS

In the following section, simulation results of -65˚ roll maneuver are given.

The values of the common parameters used in the simulations below are tabulated

in Table 6.12. Same model with same values of common parameters are used in

117

each simulation with different steering logic algorithm. Values of the parameters

used in each steering logic algorithm are given in the relevant simulation section.

Table 6.1 Simulation parameters

Parameter Value

RPYinitial: Initial attitude roll-pitch-yaw [0˚,0˚,0˚]

RPYcommanded: Attitude commanded [-65˚,0˚,0˚]

I: Inertia Matrix 2

10 0 00 10 0 kg.m0 0 10

Text: External disturbance torque Zero

allowableδ : Allowable gimbal rate 2 rad/s

h0: Angular momentum magnitude 1.0 N.m.s

oδ : Initial gimbal configuration [ ]70 ,0 ,70 ,0 T− ° ° ° °

Β : Pyramid skew angle 54.73˚

This initial gimbal angle configuration is purposely selected to examine the

performance of the particular steering logic when the system is very close to an

elliptic singularity at =δ [-90˚, 0˚, 90˚, 0˚].

In quaternion feedback controller block, gains are selected to produce ts

(=settling time) of 150s and ζ (= damping ratio) of 0.707 2. The natural frequency

of the system is found as nω =0.0283 rad/s from following relation:

0.05n ste ζω− = (6.1)

118

Thus, the gain coefficients are computed from Eq. (2.10) as k=0.0016 and d=0.04.

Note that gain selection mainly affects the response profile and time.

6.3 RESULTS OF EXISTING STEERING LAWS

6.3.1 MP Inverse Results

As expected from previous chapter results, requested roll maneuver from 0˚

to -65˚ can not be achieved using MP inverse since system is trapped into the

elliptic singularity at =δ [-90˚, 0˚, 90˚, 0˚]. It can be seen from Fig. 6.2, that the

cluster enters the singularity state after first 10 seconds, where gimbal rates try to

go to infinity at this singularity instant. Consequently, the spacecraft can not realize

the desired roll maneuver.

0 50 100 150 200 250 300-120

-100

-80

-60

-40

-20

0

20Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.2.a

119

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.2.b

0 50 100 150 200 250 300-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2Torque Realized

t (s)

T (N

m)

TxTyTz

6.2.c

120

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

Singularity Measure

t (s)

m

6.2.d

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.2.e

121

0 50 100 150 200 250 300-100

-80

-60

-40

-20

0

20

40

60

80

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.2.f

Fig. 6.2 -65˚ roll maneuver simulation conducted with MP inverse

6.3.2 SR Inverse Results

Same simulation is conducted with SR inverse steering law. The singularity

avoidance parameter, α, is selected as 0.01exp( 10 )mα = − . The results of this

simulation are presented in Fig. 6.3. It can be observed from these plots that

although the law can not avoid the internal elliptic singularity, the desired attitude

is still reached after about 300s. System stays locked in the singularity from about

t=20s to t=150s. However, after t=150s, spacecraft recovers from singularity due to

a change of satellite orientation causing a change in the direction of the torque

requirement and accomplishes its maneuver with about 130s delay.

122

0 100 200 300 400-80

-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.3.a

0 100 200 300 400-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.3.b

123

0 100 200 300 400-2

0

2

4

6

8 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.3.c

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Singularity Measure

t (s)

m

6.3.d

124

0 50 100 150 200 250 300 350 400-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.3.e

0 50 100 150 200 250 300 350 400-100

-80

-60

-40

-20

0

20

40

60

80

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.3.f

Fig. 6.3 -65˚ roll maneuver simulation conducted with SR inverse

125

6.3.3 GSR Inverse Results

Simulation for the same maneuver using GSR inverse logic is conducted

employing the same parameter values given in Eq. (4.12). However, initial gimbal

angle configuration is [ ]90 ,0 , 90 ,0 T= ° ° − ° °oδ to see if the system can escape from

singularity as demonstrated in the constant torque simulation studies of the

previous chapter. Results given in Fig. 6.4 illustrates that system stays in the

elliptic singularity region for 15 seconds. Then, it gets out of the singularity rapidly

due to intentionally produced dither signals. This simulation shows that GSR can

be used as an online steering logic for many applications other than precision

tracking application.

0 100 200 300 400-80

-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.4.a

126

0 100 200 300 400-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.4.b

0 100 200 300 400-2

-1

0

1

2

3

4

5

6

7 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.4.c

127

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7Singularity Measure

t (s)

m

6.4.d

0 50 100 150 200 250 300 350 400-0.04

-0.03

-0.02

-0.01

0

0.01

0.02Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.4.e

128

0 50 100 150 200 250 300 350 400-100

-80

-60

-40

-20

0

20

40

60

80

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.4.f

Fig. 6.4 -65˚ roll maneuver simulation conducted with GSR inverse initiated at the

elliptic singularity.

6.4 USL SIMULATIONS

The preliminary simulations with the USL method showed that Eq. (4.19)

needs to be updated for the feedback control system. In this vein, a new scale factor

is defined to adjust the magnitudes of the terms according to the desired control

torque magnitudes.

T Tq q κ -1

USL 4 desiredδ = I + J J δ + J τ (6.2)

where dynamic scale coefficient κ is:

129

T

( ) / ( )

: Constant

Crr norm normC

κ =

= desiredJ τ δ (6.3)

The constant C in the Eq. (6.3) is selected such that terms U1 and U2 have

comparable magnitudes with respect to each other.

6.4.1 Pre-Planned Steering

The aim of the method is to steer the gimbals to the desired configuration

while completing the desired maneuver on time. For this purpose, desired angular

momentum trajectory of the CMG cluster is found through a simulation with the

ACS model shown in Fig. 6.5. The model is actually the same model shown in Fig.

6.1 except that it does not have a steering block. Thus, it is assumed that actuator

respond precisely to generate the internal torque requirements to complete the

maneuver (ucommanded = urealized). The results of the simulation performed with this

model are given in Fig. 6.6. As before, desired gimbal angles at the discrete nodes

of the angular momentum trajectory (Fig. 6.6.b) are computed by solving the

optimization problem described in Eq. (5.1). For this purpose only three nodes,

spaced 100 seconds from one to another are selected.

130

Fig. 6.5 CMG Based ACS without actuator constructed in SIMULINK®

0 50 100 150 200 250 300-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.6.a

131

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.6.b

0 50 100 150 200 250 300-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.6.c

Fig. 6.6 Ideal system attitude, CMG cluster’s angular momentum and torque

histories to complete -65˚ roll maneuver

132

The best gimbal angles for each of these nodes are computed, using a gimbal rate

limit of 2 rad/s. Then, -65˚ roll maneuver simulation is conducted with the model

seen in Fig. 6.1. During the simulation, the desired gimbal rate at any instant is

computed by dividing the difference between the gimbal angle of the next node and

the current gimbal angle to the temporal distance of the next node as expressed in

Eq. (5.2). The blending coefficient, q, the dynamic scale constant, C, and ε are

taken as 0.005, 30, and 0.5, respectively. The simulation results are given in Fig.

6.7. Despite the small differences between the realized attitude profiles (Fig. 6.7.a)

and the ideal ones (Fig. 6.6.a), it is observed that required torque maneuver is

completed on time. Realized angular momentum history (Fig. 6.7.b) is very closer

to the desired one, although only three nodes are utilized in the procedure.

However, there is some difference in the torque profiles. Fig. 6.7.f illustrates the

gimbal angle history and the desired gimbal configuration on the nodes

simultaneously. It can be observed from this figure, gimbal angles are steered

precisely to the desired configurations at the nodes. Singularity measure

approaches to the zero in the initial phase of the simulation. Hopefully, singularity

is not a major problem for USL method when used in a preplanned fashion, since

gimbals are driven to the desired gimbal configuration of next node successfully

whether singular states or nonsingular states are transited between the nodes.

133

0 50 100 150 200 250 300-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.7.a

0 50 100 150 200 250 300-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.7.b

134

0 50 100 150 200 250 300-4

-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.7.c

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4Singularity Measure

t (s)

m

6.7.d

135

0 50 100 150 200 250 300

-0.1

-0.05

0

0.05

0.1

0.15Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.7.e

0 50 100 150 200 250 300-150

-100

-50

0

50

100Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4gdes1gdes2gdes3gdes4

6.7.f

Fig. 6.7 -65˚ roll maneuver simulation with USL conducted in pre-planned fashion

136

6.4.2 Online Steering

Steering with null motion

Simulations are repeated using null vector instead of desired gimbal rate as

done in constant torque simulations. The blending coefficient, q, and the dynamic

scale constant, C are taken as 0.005 and 50, respectively. From the simulation

results (Fig. 6.8), it may be observed that realized attitude, angular momentum and

realized torque histories are very close to desired profiles given in Fig. 6.6. The

lowest singularity measure during simulation is 0.05 which corresponds to the

initial phase of the simulation. Thus, it may be stated that addition of null motion

steers the system away from internal singularities in this particular maneuver.

0 50 100 150 200 250-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.8.a

137

0 50 100 150 200 250-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.8.b

0 50 100 150 200 250-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.8.c

138

0 50 100 150 200 2500

0.5

1

1.5

2

2.5Singularity Measure

t (s)

m

6.8.d

0 50 100 150 200 250

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.8.e

139

0 50 100 150 200 250-150

-100

-50

0

50

100

150

200

250Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.8.f

Fig. 6.8 -65˚ roll maneuver simulation using USL with null vector

Steering with a constant vector

In this case an arbitrarily selected constant desired rate vector,

Tdesired [0,1,0,0]=δ , is employed. The weight value of blending coefficient is

dynamically adjusted using 100.0005e mq −= . The dynamic scale constant used is C

= 100. From the simulation results (Fig. 6.9), it may be observed that the

singularity is avoided, however, with small torque errors in the region of

singularity. Fortunately, the feedback system compensates these errors, and system

completes the maneuver successfully.

140

0 50 100 150 200 250-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.9.a

0 50 100 150 200 250-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.9.b

141

0 50 100 150 200 250-4

-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m) Tx

TyTz

6.9.c

0 50 100 150 200 2500

0.5

1

1.5

2

2.5Singularity Measure

t (s)

m

6.9.d

142

0 50 100 150 200 250-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.9.e

0 50 100 150 200 250-150

-100

-50

0

50

100

150

200

250Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.9.f

Fig. 6.9 -65˚ roll maneuver simulation using USL with an arbitrary and constant rate

vector

143

The simulation is also started at an internal singularity state (i.e.

[ ]90 ,0 , 90 ,0 T= ° ° − ° °oδ ), to compare it with the results of the GSR simulations.

The weight values are selected with the same values as in the steering with null

case. The results given in Fig. 6.10 shows that the initial singular state is rapidly

escaped and the maneuver is completed successfully.

0 50 100 150 200 250-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.10.a

144

0 50 100 150 200 250

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.10.b

0 50 100 150 200 250-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.10.c

145

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

Singularity Measure

t (s)

m

6.10.d

0 50 100 150 200 250-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.10.e

146

0 50 100 150 200 250-250

-200

-150

-100

-50

0

50

100

150

200

250Gimbal Angles

t (s)

G (d

eg) g1

g2g3g4

6.10.f

Fig. 6.10 -65˚ roll maneuver simulation using USL with an arbitrary and constant rate

vector initiated at the elliptic singularity.

Since the aim is to transit through the elliptic singularity at [ ]1.155,0,0 T=h ,

gimbal angles with high singularity measures, but satisfying the momentum

requirement at the elliptic singularity point are found through optimization as done

in the constant torque simulations. Among the four different sets calculated the one

with the highest singularity measure (m=2.07) is selected: [ ]160 , 82 ,120 , 12 T° − ° ° − ° .

The dynamic scale constant value is changed to C = 500 in order to increase the

effect of the desiredδ term. It can be seen from the results (Fig. 6.11) that internal

elliptic singularity is fully avoided and spacecraft completes the maneuver

successfully on time.

147

0 50 100 150 200 250-70

-60

-50

-40

-30

-20

-10

0

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.11.a

0 50 100 150 200 250-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.11.b

148

0 50 100 150 200 250-2

0

2

4

6

8

10 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.11.c

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

Singularity Measure

t (s)

m

6.11.d

149

0 50 100 150 200 250-100

-50

0

50

100

150Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.11.e

0 50 100 150 200 250-100

-50

0

50

100

150Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.11.f

Fig. 6.11 -65˚ roll maneuver simulation using USL with intelligently selected

constant rate vector

150

6.5 ATTITUDE HOLD SIMULATIONS

In this section, a hypothetical cyclic disturbance torque, Text, given in Eq.

(6.4) is assumed to be acting on the spacecraft with a constant orbital rate of n =

0.0011 rad/s.

[ ]0.0025 sin(2 ) 0 0 Tnt=extT (6.4)

Despite of the disturbance acting about one orbital period (~5400 s), the

spacecraft is commanded to maintain its initial attitude of RPYinitial = [0˚,0˚,0˚] all

the time. In order to see the ideal attitude, angular momentum and realized torque

histories, ACS model shown in Fig. 6.5 is utilized. The results of this simulation

performed with this model are given in Fig. 6.12.

0 1000 2000 3000 4000 5000 6000-0.06

-0.04

-0.02

0

0.02

0.04

0.06Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.12.a

151

0 1000 2000 3000 4000 5000 6000-0.5

0

0.5

1

1.5

2

2.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.12.b

0 1000 2000 3000 4000 5000 6000-5

-2.5

0

2.5

5 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.12.c

Fig. 6.12 Attitude hold maneuver with ideal system

152

Described attitude hold maneuver is simulated using different steering

methods. Initial gimbal configuration used in all simulations is [ ]0 ,0 ,0 ,0 T= ° ° ° °oδ .

Closeness between the results of these simulations and the results of the ideal

system given in Fig. 6.12 reflects the performance of the steering logic. First,

simulation using MP inverse is conducted. However, gimbal angles are trapped in a

singularity in the initial phase, and the system is not being able to recover from this

point on. Thus, the spacecraft can not maintain its attitude with MP inverse steering

logic.

The simulations are repeated with SR inverse. The results of SR inverse are

given in Fig. 6.13. From the results it may be observed that the spacecraft maintain

its attitude although there is a high attitude error with the instant of singularity.

After the singularity gimbal angle configuration changes substantially and does not

return to the original set even though the disturbance is cyclic. Hence, the gimbal

configuration is not repeatable. At the first encounter with singularity the gimbal

rates also increase substantially. The simulation is repeated with GSR inverse. The

results obtained were quite similar to those obtained with SR. Thus, corresponding

results are not repeated here.

153

0 1000 2000 3000 4000 5000 6000-2

0

2

4

6

8

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.13.a

0 1000 2000 3000 4000 5000 6000-0.5

0

0.5

1

1.5

2

2.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.13.b

154

0 1000 2000 3000 4000 5000 6000-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Torque Realized

t (s)

T (N

m) Tx

TyTz

6.13.c

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

6.13.d

155

0 1000 2000 3000 4000 5000 6000-1

-0.5

0

0.5

1

1.5

2Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.13.e

0 1000 2000 3000 4000 5000 6000-100

-50

0

50

100

150Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.13.f

Fig. 6.13 Attitude hold maneuver with SR inverse.

156

The simulations are also conducted using USL. First ideal gimbal angle is

decided to be initial configuration [ ]0 ,0 ,0 ,0 T= ° ° ° °oδ . Then system is driven to

this set throughout the maneuver. Since the excitation is periodic, the gimbal rate

is calculated by dividing the distance from the desired gimbal angle to the time

remaining to the next half period. The simulation results are given in Fig. 6.14.

From the results it may be observed that a singularity is encountered, but

successfully passed through. From the singularity measure plot (Fig. 6.14.d), it can

be seen that the singularity is encountered in every half period. This is due to the

fact that the gimbal configuration is repeatable even after encountering a

singularity.

0 1000 2000 3000 4000 5000 6000-2

0

2

4

6

8

10Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.14.a

157

0 1000 2000 3000 4000 5000 6000-0.5

0

0.5

1

1.5

2

2.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.14.b

0 1000 2000 3000 4000 5000 6000-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Torque Realized

t (s)

T (N

m)

TxTyTz

6.14.c

158

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

6.14.d

0 1000 2000 3000 4000 5000 6000-1

-0.5

0

0.5

1

1.5Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.14.e

159

0 1000 2000 3000 4000 5000 6000-100

-50

0

50

100

150Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.14.f

Fig. 6.14 Attitude hold maneuver with USL conducted in pre-planned fashion.

This may be considered disadvantage since there is a singularity on the path.

However, as before, the desired gimbal set may be intelligently selected to avoid

singularities all together. An example of this is given in Fig. 6.15. Here, the desired

gimbal configuration is selected to be the best gimbal angles at the possible

location of singularity encounter on the way (i.e. [ ]160 , 82 ,120 , 12 T° − ° ° − ° ). This

approach is similar to the intelligently selected gimbal rate used for -65˚ roll

maneuver (Fig. 6.11). It may be observed from these results that the singularity is

never encountered, and the time histories are identical to the ideal case simulations.

The gimbal histories repeat themselves after the first half period. Thus, in the first

half period gimbals are to taken to a preferred configuration by the intelligently

selected constant gimbal rate.

160

0 1000 2000 3000 4000 5000 6000-0.06

-0.04

-0.02

0

0.02

0.04

0.06Spacecraft Attitude Profile

t (s)

Rol

l-Pitc

h-Ya

w

(deg

.)

RollPitchYaw

6.15.a

0 1000 2000 3000 4000 5000 6000-0.5

0

0.5

1

1.5

2

2.5Angular Momentum Trajectory

t (s)

h (N

m.s

)

hxhyhz

6.15.b

161

0 1000 2000 3000 4000 5000 6000-5

-2.5

0

2.5

5 x 10-3 Torque Realized

t (s)

T (N

m)

TxTyTz

6.15.c

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Singularity Measure

t (s)

m

6.15.d

162

0 1000 2000 3000 4000 5000 6000-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02Gimbal Rates

t (s)

Gdo

t (ra

d/s)

g1dotg2dotg3dotg4dot

6.15.e

0 1000 2000 3000 4000 5000 6000-100

-50

0

50

100

150Gimbal Angles

t (s)

G (d

eg)

g1g2g3g4

6.15.f

Fig. 6.15 Attitude hold maneuver with USL conducted with intelligently selected

constant rate vector

163

CHAPTER 7

CONCLUSION

A new singularity robust steering law for a cluster of redundant single gimbal

CMGs is presented. The steering law combines desired gimbal rates with torque

requirements in a weighted fashion.

The law may be used to avoid internal singularities by planning the

maneuvers in advance. For this purpose, using the angular momentum history,

gimbal angles with large singularity measures are found. These gimbal angles are

employed to calculate desired gimbal angle rates, and shown that the system goes

to desired gimbal angles. The repeatability of the approach is demonstrated. It is

also shown that the law can generate the desired torque profile even if it is different

than the planned torque profile. However, if the maneuver’s momentum trajectory

is close to the trajectory used in planning, then the gimbal angle history will also be

close to the planned trajectory, avoiding getting trapped in singular configurations.

Thus, the law may be used together with a feedback system, generating the desired

torques.

164

The USL is also used online. Number of approaches are tested: To use null

vector for desired gimbal angle rates, to use an arbitrary constant vector, to select

the direction of the constant vector intelligently such that the system is driven away

from a singular configuration, and finally to use pure white noise. All approaches

were successful to either completely avoid or rapidly transit through singularities.

It is concluded that the unified steering law developed in this thesis may

effectively be used to steer clusters of SGCMGs. Simulations with USL methods

reflect the ability of this method. The applications of these ideas to a feedback-

controlled spacecraft are also presented. The results confirm the success of the

method. Simulations with USL method when employed in preplanned fashion

imply the applicability of the method to the precision tracking missions. The

simulation results were also satisfactory when the method is used online in the

spacecraft attitude control model.

In conclusion, the USL steering law may successfully replace all the

steering approaches presented in the literature.

165

APPENDIX

A.1 BASIC COORDINATE SYSTEMS

Different reference frames are used for different spacecraft tasks. Coordinate

frames used in this thesis to represent the attitude of a spacecraft is illustrated in

Fig. A.1:

Fig. A.1 Coordinate frames31

166

A.1.1 The Inertial Reference Frame

This frame XIYIZI, whose direction in space is fixed relative to the solar

system has its centre at the center of mass of the Earth. The first axis XI is aligned

with the vernal equinox direction, which intersects the celestial sphere at a point

named the first point of the Aries, γ, or the vernal point. The ZI axis is the axis of

rotation of the Earth in a positive direction, which intersects the celestial sphere at

the celestial pole. The third axis YI completes an orthogonal right-handed system.

XIYI plane of this frame is the equatorial plane of the Earth.

A.1.2 The Body Frame

The attitude of a three-dimensional body is most conveniently defined with a

set of axes fixed to the body. This set of axes is generally a triad of orthogonal

coordinates. The origin of this frame XBYBZB is attached to the spacecraft center

of mass. This frame is used to determine the satellite’s relative orientation with

respect to other reference frames.

A.1.3 The Orbit Reference Frame

Origin of this frame XRYRZR moves with the center of mass of the satellite in

the orbit. The ZR axis points toward the center of mass of the Earth. The XR axis is

defined in the plane of the orbit, in the direction of the velocity of the spacecraft

167

and perpendicular to the ZR axis. The YR axis is normal to the local plane of the

orbit, and completes a three-axis orthogonal system.

A.2 ATTITUDE REPRESENTATION AND EQUATIONS OF

MOTION

A.2.1 Attitude Representation & Kinematic Equations of Motion

The satellite’s attitude with respect to any reference frame can be defined by

the Euler angles, by a direction cosine matrix [A], or by its quaternion vector q.

Each representation can be parameterized in terms of each other.

The Euler angles are defined as the sequence of rotations about three

orthogonal axes. (φ (roll angle), θ (pitch angle), and ψ (yaw angle)). Geometrical

significance of Euler angles is more obvious than other representations.

The direction cosine matrix can be expressed in terms of the Euler angles.

According to the different order of rotation of the axes of the moving body with

respect to the reference frame, there may be as many as 12 direction cosine

matrices (since 12 possible sequences of rotation exist also for Euler angles)

expressed in trigonometric functions of the Euler angles.

168

In this thesis, since it is one of the mostly preferred sets in flight control

systems, order of axes transformation to transform from inertial frame to body

fixed frame is chosen as ψ→φ→ θ (with axes order of rotation 3→1→2). [Aψ] is

the first angular rotation about the ZB axis to be performed. The next rotation will

be about the new XBI axis by an angle φ and so on. Finally;

ψφθ θ φ ψ[A ] = [A ][A ][A ] (A.1)

Performing the matrix multiplication in Eq. (A.1) yields:

312

c c s s s c c s s s s cc s c c s

s c c s s s s c s c c c

θ ψ θ φ ψ θ ψ θ φ ψ θ φφ ψ φ ψ φ

θ ψ θ φ ψ φ ψ θ φ ψ θ φ

− + − = = − + −

ψφθ[A ] [A ] (A.2)

where c and s denote cos and sin, respectively. Eq. (A.2) is a direction cosine

matrix expressed in the Euler angles, and it shows the rotation of the body axes

relative to the reference axis frame. Knowledge of direction cosine matrix elements

is equivalent to knowing the attitude of the spacecraft relative to reference frame in

which the transformation matrix [A] is defined. In general, for a rotating body,

elements of this matrix change with time. It is shown in reference 29 that:

ddt

=[A] [Ω][A] (A.3)

with

3 2

3 1

2 1

00

0

ω ωω ωω ω

− = − −

[Ω] (A.4)

where 1 2 3, ,ω ω ω are angular velocities about the body coordinate axes. Numerical

integration of Eq. (A.3) requires knowledge of initial conditions [A(0)]. However,

169

this integration is more time-consuming. The main drawback of using this

representation and Euler angle representation is in computational complexity. Since

highly nonlinear trigonometric relations involved in computation, angles may

become undefined for some rotations. On the other hand the integration of the

quaternion vector is much more efficient, and thus much more popular.

Four-parameter set of quaternion representation provides easy numeric

computation. They do not require computation of trigonometric relationships in the

kinematic equations of motion. They have no singularities as in the case of the

Euler angles. Therefore, they are preferred mainly to represent the time evolution

of the attitude parameters in this thesis.

Their definition is a consequence of the properties of the direction cosine

matrix [A]. It is shown in reference that any attitude transformation in space by

consecutive rotations about the three orthogonal unit vectors of the coordinate

system can be achieved by a single rotation about a fixed axis qd. If the rotation

angle is called q4 then we obtain quaternion expression:

4 1 2 3 4q q q q q= + + + = + dq i j k q (A.5)

As with the direction cosine matrices, if the angular velocity of vector ω of

the body frame is known with respect to another reference frame, a differential

vector equation for q can be written as:

12

=q Qq (A.6)

170

where,

3 2 1

3 1 2

2 1 3

1 2 3

00

00

ω ω ωω ω ωω ω ωω ω ω

− − = − − − −

Q (A.7)

In order to express the difference between the current attitude, q, and the

commanded attitude, qc, quaternion error is defined as follows1:

1 4 3 2 1 1

2 3 4 1 2 2

3 2 1 4 3 3

4 1 2 3 4 4

e c c c c

e c c c c

e c c c c

e c c c c

q q q q q qq q q q q qq q q q q qq q q q q q

− − − − = − −

(A.8)

where, [ ]1 2 3 4T

e e e eq q q q=eq is the attitude quaternion error vector.

[ ]1 2 3 4T

c c c c cq q q q=q is the quaternion command vector

[ ]1 2 3T

d e e eq q q=eq is the attitude quaternion error direction vector which

is used in gain computations in Eq. (2.7).

The expression above is directly utilized in quaternion feedback controller part of

the spacecraft model constructed.

A.2.2 Equations of Motion

In this section, the equations of motion for a general rigid body are

summarized. These equations describe the time dependence of the spacecraft body

angular rate. Following well-known Euler equation relates time derivative of the

angular momentum vector, dH/dt, to the external torque:

171

/S C = − ×I I IB ext B BI ω τ ω Jω (A.9)

where; [ ]Tx y zω ω ω=I

Bω : Inertially referenced body angular rate vector

/S CI : Moment of inertia matrix of the spacecraft

extτ : External torque vector including active control torques produced by

thrusters and magnetometers, and environmental disturbances

A.3 NULL MOTION

A.3.1 Null Space Dimension and Null Space Basis Vector Calculation

As discussed in Chapter 3 and Chapter 4, homogenous solution of the Eq.

(3.5) are described by the null space of the Jacobian matrix. The dimension of the

null-space can be computed as:

( ) ( )NullDim n rank= −J J (A.10)

Considering pyramid configuration with four gimbals (n=4), consequently, when J

is nonsingular, dimension of the null space is 1. On the other hand, when J is

singular, dimension of the null space is 2. In either case, null space basis vector can

be determined using generalized vector cross product.

Null space basis vector calculation:

For nonsingular Jacobians, this method can be used directly. The null rate is

found from following expression:

172

1 2 3 431 2 dete e e ehh h ∂∂ ∂ = ∧ ∧ = ∂ ∂ ∂ 1 2 3 4

nJ J J Jδ δ δ

(A.11)

where,

[ ]

1 2 3

1 2 3 4

1 2 3 4

th

[ , , ]

, , , for =1,2,3

, , , : 4-dimensional gimbal angle space

: with i column removed

T

l l l l l

T

h h h

h h h h l

e e e e

δ δ δ δ

=

∂ ∂ ∂ ∂ ∂= ∂ ∂ ∂ ∂ ∂

i

h

hδ

J J

In addition, it can also be expressed as 3x3 Jacobian minors:

[ ]1

th

null

( 1) is the order 3 Jacobian cofactor = det( ) is the order 3 Jacobian cofactor minor

with column removed

i

i i

m

+

=

= −

=

=

1 2 3 4

i i

i i

n C ,C ,C ,C

C MM J

J J

δ

(A.12)

However, when the Jacobian is singular, there will be only two linearly

independent rows ( also two l.i. columns). This implies that all of the cofactors are

going to the zero, as singularity measure approaches to zero. In order to determine

null space basis vectors of singular configuration following procedure is proposed:

Step 1: Determine the two linearly independent row vectors of Jacobian.

Replace one of the dependent rows with one of the following vectors to obtain

Jacobian matrix with three linearly independent row vectors:

[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], and [0, 0, 0, 1]

173

Step 2: Using Eq. (A.11) or Eq. (A.12), calculate the first null vector for new

Jacobian matrix.

Step 3: Since dimension of the null space for singular configuration is 2, we

need to find one more null space basis vector. Second vector can be found from

repeating previous steps with a different row vector.

Step 4: Check if the found null vectors are linearly independent or not. If

they are not linearly independent, repeat step 3 by selecting different row vector.

It can be checked that whether the found vectors satisfy the following relation

coming from the solution homogeneity:

=Jn 0 (A.13)

A.3.2 Examples of Elliptic Singularities

i) Internal Elliptic Singularity Example

δs = [-90º, 0º, 90º, 0º] corresponds to the one of the worst singularity cases

(Fig. A.2).Individual angular momenta and system total momentum is computed

from Eq. (3.5) as:

0.5774 1 0.5774 1 1.15490 0 0 0 0

0.8164 0 0.8164 0 0δ

− + + + = −

∑4

i ii=1

h = h ( ) =

174

Fig. A.2 Internal Elliptic Singularity at δs = [-90º, 0º, 90º, 0º]

Rank of Jacobian matrix is 2. Eigenvalues corresponding to the JJT matrix

are 0, 1.331, and 2.669. Eigenvector corresponding to the minimum eigenvalue

which is zero is [1, 0, 0]. This direction is the singular direction along which no

torque can be produced.

If one looks for the projections of the individual angular momentum vectors

along the singular direction, [+, -, +, -] sign pattern is observed. Therefore, this

singular state corresponds to the 2H singularity according to the cutting plane

classification.

175

Singularity measure, m is calculated with MATLAB as 8.8893x10-33.

In order to perform null test described in Chapter 3, first null space basis

vector should be computed. The Jacobian matrix of this configuration is:

0 0 0 01 -0.5774 1 0.57740 0.8164 0 0.8164

J =

Replacing the first zero row of the Jacobian matrix with [1, 0, 0, 0] and [0, 1,

0, 0], respectively, and carrying out the generalized vector cross product yields the

following null space basis matrix:

0 0.81640.8164 0

[ ]0.9429 0.8164

0.8164 0

− = = − −

1 2N n n

Corresponding diagonal P matrix and Q matrix are computed using Eq. (3.8)

and Eq. (3.9):

0.5774 0 0 00 1 0 0

( )0 0 0.5774 00 0 0 1

0.5134 0.44450

0.4445 0.7697

T sidiag s h

− = =

= = >

T

P

Q N PN

Since Q matrix is positive definite, this gimbal set represents an example of

an elliptic type of singularity.

176

ii) External Elliptic Singularity Example

δs = [-90º, 180º, 90º, 0º] is the gimbal set that lies on the momentum

envelope (Fig. A.3). Therefore, it is an example of saturation singularity.

Singularity direction is same as previous example. Rank of the Jacobian is 2.

Singularity measure, m is calculated with MATLAB as 2.8883x10-32. Individual

angular momenta and system total momentum and projection matrix P are

computed as follows:

0.5774 1 0.5774 1 3.15490 0 0 0 0

0.8164 0 0.8164 0 0

0.5774 0 0 00 1 0 0

( )0 0 0.5774 00 0 0 1

T sidiag s h

δ + + + = − = =

∑4

i ii=1

h = h ( ) =

P

Sign pattern of diagonal elements of P matrix [+, +, +, +] indicates that all

CMGs in the cluster projected their maximum angular momentum capability along

the singular direction. Therefore, this is 4H external singularity. Angular

momentum of 3.1549 represents the maximum of the system that can be realized

along the x-direction.

177

Fig. A.3 External Elliptic Singularity at δs = [-90º, 180º, -90º, 0º]

Null test performed yielded the following N and Q matrices:

1 1.15490 1

[ ]1 00 1

1.1549 0.66680

0.6668 1.3850

− − = =

= = >

1 2

T

N n n

Q N PN

As confirmed by null test this configuration is also elliptic.

178

ii) Hyperbolic (Escapable) Singularity Example

δs = [90º, 180º, -90º, 0º] is an example (Fig. A.4) of also 0H singularity,

since sign pattern of the projection of individual angular momenta along the

singular direction (again s=[1, 0, 0]) is [-, +, -, +]. Total angular momentum of the

cluster is zero vector. Rank of the Jacobian is also 2.

Fig. A.4 Hyperbolic Escapable Singularity at δs = [90º, 180º, -90º, 0º]

Null test performed yielded the following N and Q matrices:

179

1 1.15490 1

[ ]1 00 1

1.1549 0.66680.6668 1.2300

− = =

= =

1 2

T

N n n

Q N PN

Since Q is indefinite, null motion is possible. However, further tests should

be done in order to be sure of escape by null motion, since possibility of null

motion does not assure the escape from singularity. However, further tests based on

Taylor series expansion shows that escape by null motion is possible for this

case3,5,6,24.

A.4 SIMULATED ANNEALING HIDE & SEEK ALGORITHM

This method is used in determination of gimbal angles in momentum

approach. Here a brief description taken completely from Reference 33 is

presented.

The Simulated annealing method

Simulated Annealing is an optimization method that simulates the physical

annealing process of finding the low energy states of a solid at a particular

180

temperature. For a given temperature T, the probability of a system to be in state r

may be found from the Boltzmann distribution ))(exp(Tk

E

B

r , where )(rE is the

energy of the configuration, Bk is the Boltzmann constant. To simulate the

annealing process, Metropolis criteria may be used. For this purpose, the change in

the energy of a system with the move of an atom is calculated (∆E). If the move

lowers the energy of the system, it is accepted (∆E ≤ 0). If the energy of the system

is increased by the move (i.e., ∆E > 0) then, it is accepted with probability of

)/exp()( TkEEP B∆−=∆ . In the simulated annealing optimization method, the

cost function replaces the energy of the system, and the optimization parameters

represent the atoms. This idea is initially used by Kirkpatrick et al. to solve

discrete combinatorial optimization problems. The technique is later extended to

the optimization of functions of continuous variables.

In simulated annealing the success of the algorithm to find the global

optimum by fewest number of function evaluations is closely related to the method

used in selecting the next candidate point. For this purpose, various methods are

proposed. For example, Vanderbilt et al., proposes to select the next iteration point

from a normal distribution. Then it is multiplied by the step size to find the next

test value. Thus, a random walk with a fixed maximum step size is used. This

maximum step size for each variable is updated after predetermined number of

trials. The new set of step sizes is selected proportional to the inverse of the

Hessian calculated. Corona et al. on the other hand displaces one variable at a time.

181

Siarry et al. also selects the next test point using random walk. However, only

randomly selected subset of optimization parameters, are displaced at each trial.

The step size is kept constant for a predetermined number of iterations. Others use

pure random walk, where both the search direction and step size are taken from

uncorrelated uniform distributions.

Another important aspect of simulated annealing is the cooling scheme used.

Various schemes are also proposed in the literature. The most common approach is

to cool by multiplying the current temperature by a fixed factor after predetermined

number of trials or records. Hajek gives the necessary and sufficient conditions for

a deterministic cooling schedule so that convergence to a global optimum is

guaranteed. Siarry et al. propose to change the temperature after predetermined

number of steps. The reduction in temperature is large, if in the previous stage too

many records are found. Otherwise, it is lowered by a small amount. Thus, two

separate factors for fast cooling and slow cooling are used. Others propose an

adaptive cooling schedule. The cooling is carried out whenever a new record is

found. The temperature selected depends on the distance of the record to the

estimated global optimum. Thus, the temperature is small if the current record is

close to the estimated global optimum. Otherwise, it is large.

182

The Hide and Seek Algorithm

Consider the following optimization problem:

Sf

∈ωω)(max

(A.14)

where, S is a compact body in dR . Thus, the objective is to find S∈ω such that

)()( ** ωω fff ≥≡ for all S∈ω .

First the Metropolis criterion shall be given:

),0.1min(),( /))()(( τυωτβωυ ffe −= (A.15)

Then the algorithm proceeds as follow:

Step 0: Choose a starting point 0ω in the interior of S, and a high enough

starting temperature 0τ , and set 0=k .

Step 1: Choose search direction kθ on the surface of a unit sphere with uniform

distribution.

Step 2: Choose kλ from the uniform distribution such that

):( SR kkkk ∈+∈=Λ θω λλ . Set kkkk θωυ λ+=+1 .

Step 3: Choose kV ( 10 ≤≤ kV ) from a uniform distribution. Determine the next

search point 1+kω from,

[ ][ ]

∈∈

=+

+++ 1),,( if

),(,0 if

1

111

kkTkk

kkTkkk V

Vυωωυωυ

ωββ

(A.16)

Step 4: Update temperature, if )( 1+kf ω is greater than all previous function

values. Otherwise go to Step 1.

183

The new temperature is calculated using:

[ ] )(/)(2 21

* dff pk −−⋅= χτ ω (A.17)

Since, maximum value, *f cannot be known in advance, its estimate f ,

instead of *f is employed. The estimate is calculated using the following heuristic

estimator:

1)1(

ˆ2/21

1 −−−

+= −dpffff (A.18)

where, 1f and 2f are the two largest order statistics respectively, and the

parameter p corresponds to the probability that the real maximum *f , is larger than

its estimate f . The algorithm stops when the improvement is smaller than some

specified value.

Estimating the maximum for temperature updates

From the above discussion it may be seen that one important aspect of the

Hide-and-Seek algorithm is to update the temperature whenever a new record is

found. This update is based on the estimate of the global optimum. Consequently,

better estimates shall ease the convergence to the global optimum. The estimator

proposed is based on the current and the previous record. If the current record is

too far away from the previous one, then the heuristic estimator gives unreasonably

high values. Consequently, the temperature suddenly increases. The opposite may

also happen and cause the temperature to drop prematurely. In either case the

184

algorithm may not converge to the true maximum. However, in many

maximization problems, one may reasonably guess an upper bound for the

optimum. This upper bound may also be used instead of the value calculated by

the heuristic estimator. Another possibility may be to use the heuristic estimator

with an upper bound (and/or lower bound). If the estimated value is above this

bound, the upper bound may be used to calculate the current temperature.

Constrained Optimization in Simulated Annealing

Simulated annealing algorithm is fundamentally an unconstrained

optimization algorithm. However, it is also used for constrained optimization

problems. The usual approach is to discard the iteration point if the constraints are

violated, and select another point in the same fashion. This approach is quite easy

to apply, if the constrains are simple upper and lower bounds on the optimization

parameters. At a given iteration point, nonlinear constraints may also be evaluated,

and the next trial point may be rejected if the constraints are violated. If these

constraints are equality constraints, then the use of rejection method for nonlinear

constraints may be very inefficient. Such nonlinear constraints may also be

handled by augmenting them to the cost function using penalty coefficients. In this

study constraints are augmented to the cost using penalty coefficients.

185

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