Fundamentals of Dynamics and Control of Space Systems

361 Pages·2012·7.66 MB·English

by  Dr. Krishna D Kumar

Fundamentals of Dynamics
and Control of Space
Systems
Krishna Dev Kumar
ii
Dr. Krishna. D. Kumar
Professor and Canada Research Chair in Space Systems
Department of Aerospace Engineering
Ryerson University
Toronto, Ontario
Canada M5B 2K3
Email: [email protected]
http://www.ryerson.ca/kdkumar
Copyright c2012 by Krishna Dev Kumar. All rights are reserved whether
(cid:13)
thewholeorpartofthematerialisconcerned,especiallytherightoftransla-
tion, reprinting, reuse of illustrations, recitation, broadcasting, reproduction
on microfilm or any other ways, and storage in data banks.
Cover page: Full view of the International Space Station as photographed
from the Space Shuttle Discovery during the STS-114 Return to Flight mis-
sion, following the undocking of the two spacecraft. (Courtesy of NASA)
Fundamentals of Dynamics
and Control of Space
Systems
Krishna Dev Kumar
Professor and Canada Research Chair in Space Systems
Department of Aerospace Engineering
Ryerson University
Toronto, Canada
iv
Preface
This book is an outcome of the lectures delivered by the author for engi-
neering students at Kyushu University, Japan, Korea Advanced Institute of
ScienceandTechnology,SouthKorea,andRyersonUniversity,Canada. The
text started when the author was at Kyushu University, Japan. The author
has been doing research on the subject of “Dynamics and Control of Space
Systems” for the last fifteen years. During the course of research and teach-
ing this subject, the author realized that there is an earnest need of a text
which may provide basic understanding of inherent complexity in modelling
space systems including environmental forces and torques. This text fulfils
this goal through a step by step approach that explains each concept in a
simple way with mathematical details and simple notations.
This textbook is written for engineers and students pursuing study or
researchonthe dynamicsandcontrolofspacesystems. The authorhas used
allcontentsofthistextbookinacorecourseonSpaceSystems(onesemester,
fourhoursaweek). Thistextassumesthatthereaderiswellacquaintedwith
elementary calculus and linear algebra. Enough care has been taken so that
the reader may attempt different types of problems. Each concept has been
discussed with sufficient mathematical background supported by examples.
A complete set of problems has been added to the end of every chapter. A
“Solution Manual” accompanying this book provides a solution to most
of the these exercise problems. In addition, a “Laboratory Manual” with
this textbook provides the practical aspects of designing space systems with
a focus on the construction of a Can-sized satellite.
During the courseofwriting Ihavereceivedseveralsuggestionsandcom-
mentsfrommycolleaguesandstudents. IespeciallythankmystudentsArad-
hana Choudhuri, Mike Alger, Geoffrey McVittie, Mike Tai, KamranShahid,
Antonio Mauro, and Tarunkumar Patel who have gone through previous
drafts of this book, and provided me with valuable suggestions.
Finally, I express my sincere appreciation to my parents, my wife, Annu,
and children, Diksha and Saurabh for their support and patience, without
which it would have been impossible to complete this book. This textbook
vi PREFACE
is dedicated to my parents and family for their inspiration and support.
In writing this textbook, I have tried my best to provide the best rele-
vant materials on the subject. I encourage and welcome all comments and
suggestions to make this book more relevant to the contents of your course
or researchstudy. Please address your comments to [email protected].
September 3, 2012 Krishna Kumar
Contents
Preface v
1 Introduction to Space Systems 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Space System . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Images of Space Systems . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Earth Orbiting Missions . . . . . . . . . . . . . . . . . 21
1.3.2 Small Satellites . . . . . . . . . . . . . . . . . . . . . . 23
1.3.3 Communication Satellites . . . . . . . . . . . . . . . . 24
1.3.4 Astronomy Satellites . . . . . . . . . . . . . . . . . . . 26
1.3.5 Interplanetary Space Systems . . . . . . . . . . . . . . 27
1.3.6 Manned Orbiting Platforms . . . . . . . . . . . . . . . 29
1.3.7 Space Robotics . . . . . . . . . . . . . . . . . . . . . . 32
1.3.8 Launch and Deployment Systems . . . . . . . . . . . . 36
1.3.9 Future Space Systems . . . . . . . . . . . . . . . . . . 37
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Kinematics, Momentum and Energy 39
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Point Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 Position, Velocity and Acceleration . . . . . . . . . . . 40
2.2.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
viii CONTENTS
2.2.3 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3 Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3.1 Defining Attitude . . . . . . . . . . . . . . . . . . . . . 71
2.3.2 Position, Velocity and Acceleration . . . . . . . . . . . 77
2.3.3 Momentum . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.4 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4 Flexible Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4.1 Defining Deformation . . . . . . . . . . . . . . . . . . 95
2.4.2 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.5.1 Kinematics: Point Mass . . . . . . . . . . . . . . . . . 110
2.5.2 Kinematics: Rigid Body . . . . . . . . . . . . . . . . . 112
2.5.3 Kinematics: Flexible Body . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3 Forces and Torques 123
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2.1 Simplified GravitationalForce Model . . . . . . . . . . 125
3.2.2 Generalized Gravitational Force Model . . . . . . . . . 126
3.3 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . 128
3.3.1 Simplified Aerodynamic Force Model . . . . . . . . . . 128
3.3.2 Free-Molecular Aerodynamic Force Model . . . . . . . 131
3.4 Solar Radiation Pressure Force . . . . . . . . . . . . . . . . . 133
3.4.1 Solar Radiation Pressure Force Model I . . . . . . . . 133
3.4.2 Solar Radiation Pressure Force Model II . . . . . . . . 137
3.4.3 Earth shadow . . . . . . . . . . . . . . . . . . . . . . . 138
3.5 Magnetic Field Torque . . . . . . . . . . . . . . . . . . . . . . 139
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.7.1 Atmospheric Data . . . . . . . . . . . . . . . . . . . . 147
CONTENTS ix
3.7.2 International Geomagnetic Reference Field . . . . . . 151
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4 Dynamics I 159
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . 160
4.3 Kepler’s Laws of Orbit Motion . . . . . . . . . . . . . . . . . 162
4.4 Two-Body Motion . . . . . . . . . . . . . . . . . . . . . . . . 163
4.4.1 Angular Momentum Vector . . . . . . . . . . . . . . . 166
4.4.2 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.4.3 Eccentricity Vector . . . . . . . . . . . . . . . . . . . . 167
4.4.4 Solution of Relative Motion . . . . . . . . . . . . . . . 168
4.5 Conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5.1 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.5.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.6 Orbit Motion in Relation with Time . . . . . . . . . . . . . . 178
4.7 Describing Orbit Motion in Space . . . . . . . . . . . . . . . . 183
4.8 Relating Orbital Elements with Position and Velocity Vectors 186
4.8.1 Determination of Orbital Elements from Position and
Velocity Vectors . . . . . . . . . . . . . . . . . . . . . 186
4.9 Orbital Perturbations . . . . . . . . . . . . . . . . . . . . . . 190
4.9.1 Non-Keplerian Orbit Motion . . . . . . . . . . . . . . 190
4.9.2 Angular Momentum Vector . . . . . . . . . . . . . . . 191
4.9.3 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.9.4 Eccentricity Vector . . . . . . . . . . . . . . . . . . . . 192
4.9.5 Equations of Motion (Orbital Elements) . . . . . . . . 193
4.9.6 Earth’s Oblateness . . . . . . . . . . . . . . . . . . . . 201
4.9.7 Planetary Gravitational Perturbations . . . . . . . . . 211
4.9.8 Aerodynamic Drag . . . . . . . . . . . . . . . . . . . . 218
4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
x CONTENTS
5 Dynamics II 225
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.2 Euler Equations of Motion . . . . . . . . . . . . . . . . . . . . 225
5.3 Lagrange’s Equations of Motion . . . . . . . . . . . . . . . . . 226
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6 Mathematical Analysis and Simulation 263
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.2 Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . 264
6.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.5 Closed-form Solution . . . . . . . . . . . . . . . . . . . . . . . 270
6.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 282
6.7 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
7 Control System 301
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7.2 Linear Controllers . . . . . . . . . . . . . . . . . . . . . . . . 303
7.3 Nonlinear Controllers. . . . . . . . . . . . . . . . . . . . . . . 309
7.3.1 Fuzzy Logic Controllers . . . . . . . . . . . . . . . . . 309
7.3.2 Neural Networks Controllers . . . . . . . . . . . . . . 310
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
8 Formation Flying 323
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.2 Formation Classification . . . . . . . . . . . . . . . . . . . . . 325
8.3 Leader Satellite in Circular Orbit . . . . . . . . . . . . . . . . 326