Spacecraft Dynamics and Control E-Book

Anton H. de Ruiter

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 572

Veröffentlichungsjahr: 2012

Beschreibung

Provides the basics of spacecraft orbital dynamics plus attitude dynamics and control, using vectrix notation

Spacecraft Dynamics and Control: An Introduction presents the fundamentals of classical control in the context of spacecraft attitude control. This approach is particularly beneficial for the training of students in both of the subjects of classical control as well as its application to spacecraft attitude control. By using a physical system (a spacecraft) that the reader can visualize (rather than arbitrary transfer functions), it is easier to grasp the motivation for why topics in control theory are important, as well as the theory behind them.  The entire treatment of both orbital and attitude dynamics makes use of vectrix notation, which is a tool that allows the user to write down any vector equation of motion without consideration of a reference frame. This is particularly suited to the treatment of multiple reference frames. Vectrix notation also makes a very clear distinction between a physical vector and its coordinate representation in a reference frame. This is very important in spacecraft dynamics and control problems, where often multiple coordinate representations are used (in different reference frames) for the same physical vector.

• Provides an accessible, practical aid for teaching and self-study with a layout enabling a fundamental understanding of the subject
• Fills a gap in the existing literature by providing an analytical toolbox offering the reader a lasting, rigorous methodology for approaching vector mechanics, a key element vital to new graduates and practicing engineers alike
• Delivers an outstanding resource for aerospace engineering students, and all those involved in the technical aspects of design and engineering in the space sector
• Contains numerous illustrations to accompany the written text. Problems are included to apply and extend the material in each chapter

Essential reading for graduate level aerospace engineering students, aerospace professionals, researchers and engineers.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Kinematics

1.1 Physical Vectors

1.2 Reference Frames and Physical Vector Coordinates

1.3 Rotation Matrices

1.4 Derivatives of Vectors

1.5 Velocity and Acceleration

1.6 More Rigorous Definition of Angular Velocity

Notes

References

Chapter 2: Rigid Body Dynamics

2.1 Dynamics of a Single Particle

2.2 Dynamics of a System of Particles

2.3 Rigid Body Dynamics

2.4 The Inertia Matrix

2.5 Kinetic Energy of a Rigid Body

Notes

References

Chapter 3: The Keplerian Two-Body Problem

3.1 Equations of Motion

3.2 Constants of the Motion

3.3 Shape of a Keplerian Orbit

3.4 Kepler’s Laws

3.5 Time of Flight

3.6 Orbital Elements

3.7 Orbital Elements given Position and Velocity

3.8 Position and Velocity given Orbital Elements

Notes

References

Chapter 4: Preliminary Orbit Determination

4.1 Orbit Determination from Three Position Vectors

4.2 Orbit Determination from Three Line-of-Sight Vectors

4.3 Orbit Determination from Two Position Vectors and Time (Lam- bert’s Problem)

Notes

References

Chapter 5: Orbital Maneuvers

5.1 Simple Impulsive Maneuvers

5.2 Coplanar Maneuvers

5.3 Plane Change Maneuvers

5.4 Combined Maneuvers

5.5 Rendezvous

Notes

Reference

Chapter 6: Interplanetary Trajectories

6.1 Sphere of Influence

6.2 Interplanetary Hohmann Transfers

6.3 Patched Conics

6.4 Planetary Flyby

6.5 Planetary Capture

Notes

References

Chapter 7: Orbital Perturbations

7.1 Special Perturbations

7.2 General Perturbations

7.3 Gravitational Perturbations due to a Non-Spherical Primary Body

7.4 Effect of J2 on the Orbital Elements

7.5 Special Types of Orbits

7.6 Small Impulse Form of the Gauss Variational Equations

7.7 Derivation of the Remaining Gauss Variational Equations

Notes

References

Chapter 8: Low Thrust Trajectory Analysis and Design

8.1 Problem Formulation

8.2 Coplanar Circle to Circle Transfers

8.3 Plane Change Maneuver

Notes

References

Chapter 9: Spacecraft Formation Flying

9.1 Mathematical Description

9.2 Relative Motion Solutions

9.3 Special Types of Relative Orbits

Notes

Reference

Chapter 10: The Restricted Three-Body Problem

10.1 Formulation

10.2 The Lagrangian Points

10.3 Stability of the Lagrangian Points

10.4 Jacobi’s Integral

Notes

References

Chapter 11: Introduction to Spacecraft Attitude Stabilization

11.1 Introduction to Control Systems

11.2 Overview of Attitude Representation and Kinematics

11.3 Overview of Spacecraft Attitude Dynamics

Chapter 12: Disturbance Torques on a Spacecraft

12.1 Magnetic Torque

12.2 Solar Radiation Pressure Torque

12.3 Aerodynamic Torque

12.4 Gravity-Gradient Torque

Notes

Reference

Chapter 13: Torque-Free Attitude Motion

13.1 Solution for an Axisymmetric Body

13.2 Physical Interpretation of the Motion

Notes

References

Chapter 14: Spin Stabilization

14.1 Stability

14.2 Spin Stability of Torque-Free Motion

14.3 Effect of Internal Energy Dissipation

Notes

References

Chapter 15: Dual-Spin Stabilization

15.1 Equations of Motion

15.2 Stability of Dual-Spin Torque-Free Motion

15.3 Effect of Internal Energy Dissipation

Notes

References

Chapter 16: Gravity-Gradient Stabilization

16.1 Equations of Motion

16.2 Stability Analysis

Notes

References

Chapter 17: Active Spacecraft Attitude Control

17.1 Attitude Control for a Nominally Inertially Fixed Spacecraft

17.2 Transfer Function Representation of a System

17.3 System Response to an Impulsive Input

17.4 Block Diagrams

17.5 The Feedback Control Problem

17.6 Typical Control Laws

17.7 Time-Domain Specifications

17.8 Factors that Modify the Transient Behavior

17.9 Steady-State Specifications and System Type

17.10 Effect of Disturbances

17.11 Actuator Limitations

Notes

References

Chapter 18: Routh’s Stability Criterion

18.1 Proportional-Derivative Control with Actuator Dynamics

18.2 Active Dual-Spin Stabilization

Notes

References

Chapter 19: The Root Locus

19.1 Rules for Constructing the Root Locus

19.2 PD Attitude Control with Actuator Dynamics - Revisited

19.3 Derivation of the Rules for Constructing the Root Locus

Notes

References

Chapter 20: Control Design by the Root Locus Method

20.1 Typical Types of Controllers

20.2 PID Design for Spacecraft Attitude Control

Notes

References

Chapter 21: Frequency Response

21.1 Frequency Response and Bode Plots

21.2 Low-Pass Filter Design

Notes

References

Chapter 22: Relative Stability

22.1 Polar Plots

22.2 Nyquist Stability Criterion

22.3 Stability Margins

Notes

References

Chapter 23: Control Design in the Frequency Domain

23.1 Feedback Control Problem - Revisited

23.2 Control Design

23.3 Example - PID Design for Spacecraft Attitude Control

Notes

References

Chapter 24: Nonlinear Spacecraft Attitude Control

24.1 State-Space Representation of the Spacecraft Attitude Equations

24.2 Stability Definitions

24.3 Stability Analysis

24.4 LaSalle’s Theorem

24.5 Spacecraft Attitude Control with Quaternion and Angular Rate Feedback

Notes

References

Chapter 25: Spacecraft Navigation

25.1 Review of Probability Theory

25.2 Batch Approaches for Spacecraft Attitude Estimation

25.3 The Kalman Filter

Notes

References

Chapter 26: Practical Spacecraft Attitude Control Design Issues

26.1 Attitude Sensors

26.2 Attitude Actuators

26.3 Control Law Implementation

26.4 Unmodeled Dynamics

Notes

References

Appendix A: Review of Complex Variables

A.1 Functions of a Complex Variable

A.2 Complex Valued Functions of a Real Variable

A.3 The Laplace Transform

A.4 Partial Fraction Expansions

A.5 Common Laplace Transforms

A.6 Example of Using Laplace Transforms to Solve a Linear Differential Equation

Appendix B: Numerical Simulation of Spacecraft Motion

B.1 First Order Ordinary Differential Equations

B.2 Formulation of Coupled Spacecraft Orbital and Attitude Motion Equations

Notes

Reference

Index

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Library of Congress Cataloging-in-Publication Data

De Ruiter, Anton H. J. Spacecraft dynamics and control : an introduction / Anton H.J. de Ruiter, Christopher Damaren, James R. Forbes. pages cm Includes bibliographical references and index. ISBN 978-1-118-34236-7 (hardback) 1. Space vehicles–Attitude control systems. 2. Space vehicles–Dynamics. I. Damaren, Christopher. II. Forbes, James R. III. Title. TL3260.D33 2013 629.4′1–dc23 2012033616

A catalogue record for this book is available from the British Library.

ISBN: 9781118342367

To Janice, Thomas, Benjamin, Therese and Marie A.dR

To Yvonne, Gwen, and Georgia C.J.D

For Allison J.R.F

Preface

This book presents a fundamental introduction to spacecraft orbital and attitude dynamics as well as its control. There are several excellent books related to these subjects. It is not our intention to compete with these well-established texts. However, many of them assume relatively significant backgrounds on behalf of the reader, which can make them difficult to follow for the beginner. It is our hope that this book will fill that void, and that by studying this book, more advanced texts on the subject will become more accessible to the reader. This book is suitable for first courses in spacecraft dynamics and control at the upper undergraduate level or at the beginning graduate level. The book is naturally split between orbital mechanics, and spacecraft attitude dynamics and control. It could therefore be used for two one semester courses, one on each subject. It could also be used for self-study.

The primary objective of this book is to educate, and the structure of the book reflects this. This book could also be used by the professional looking to refresh some of the fundamentals. We have made this book as self-contained as possible. In each chapter we develop a subject at a fundamental level (perhaps drawing on results from previous chapters). As a result, the reader should not only understand what the key mathematical results are, but also how they were obtained and what their limitations are (if applicable). At the end of each chapter we provide a few recommended references should the reader have interest in exploring the subject further.

The assumed reader background is minimal. Junior undergraduate level mathematics and mechanics taught in standard engineering programs should be sufficient. While a background in classical control theory would help, it is not necessary to have it in order to be able to follow the treatment of spacecraft attitude control. The presentation in this book on spacecraft attitude control is completely self-contained, and it could in fact be used as a substitute for a complete first (undergraduate level) course in classical control. The reader without a control background will learn classical control theory motivated by a real system to be controlled, namely, a spacecraft (as opposed to some abstract transfer functions). The reader with a prior control background may gain new appreciation of the theory by seeing it presented in the context of an application.

In Chapters 1 and 2, we present the vector kinematics and rigid body dynamics required to be able to describe spacecraft motion. Chapters 3 to 10 contain the orbital mechanics component of this book. Topics include the two-body problem, preliminary orbit determination, orbital and interplanetary maneuvers, orbital perturbations, low-thrust trajectory design, spacecraft formation flying, and the restricted three-body problem. Chapter 11 presents a high level overview of both passive and active means of spacecraft attitude stabilization, and provides an introduction to control systems. Chapters 12 to 16 present aspects of spacecraft attitude dynamics (disturbance torques and a solution for torque-free motion), and more detailed treatments of passive means of spacecraft attitude stabilization. Chapters 17 to 23 present active means of spacecraft attitude control using classical control techniques. Chapters 24 and 25 present introductions to some more advanced topics, namely nonlinear spacecraft attitude control and spacecraft attitude determination. These chapters also provide a brief introduction to nonlinear control theory and state estimation. Chapter 26 presents an overview of practical issues that must be dealt with in designing a spacecraft attitude control system, namely different spacecraft attitude sensor and actuator types, digital control implementation issues and effects of unmodeled dynamics on spacecraft attitude control systems. Finally, Appendices A and B contain some background reference material.

After finishing this book, the reader should have a strong understanding of the fundamentals of spacecraft orbital and attitude dynamics and control, and should be aware of important practical issues that need to be accounted for in spacecraft attitude control design. The reader will be well-prepared for further study in the subject.

The first author would like to express his deep gratitude to the Department of Mechanical and Aerospace Engineering at Carleton University in Ottawa, Canada, for the opportunity to develop and teach courses in orbital mechanics and spacecraft dynamics and control. The notes developed for these courses were the starting point for much of this book.

The reader will notice that this book contains no exercises. This was a decision made in order to keep the page count down. However, the reader will find a full set of exercises accompanying the book, as well as other supplementary material on the book’s companion website: http://arrow.utias.utoronto.ca/damaren/book/.

Anton H.J. de Ruiter Christopher J. Damaren James R. Forbes

1

Kinematics

Spacecraft are free bodies, possessing both translational and rotational motion. The translational component is the subject of orbital dynamics, the rotational component is the subject of attitude dynamics. It will be seen that the two classes of motion are essentially uncoupled, and can be treated separately.

To be able to study the motion of a spacecraft mathematically, we need a framework for describing it. For this purpose, we need to have a solid understanding of vectors and reference frames, and the associated calculus.

1.1 Physical Vectors

A physical vector is a three-dimensional quantity that possesses a magnitude and a direction. A physical vector will be denoted as , for example. It can be represented graphically by an arrow. Vector addition is defined head-to-tail as shown in Figure 1.1. Multiplication of a vector by a scalar a scales the magnitude by |a|. If a is positive, the direction is unchanged, and if a is negative, the direction is reversed. It is also useful to define a zero-vector denoted by , which has magnitude 0, but no specified direction.

Figure 1.1 Physical vector addition

Under these definitions, physical vectors satisfy the following rules for addition:

and the following rules for scalar multiplication:

It is very important to note that the concept of a physical vector isindependentof a coordinate system.

1.1.1 Scalar Product

Given vectors and , the scalar (or dot) product between the two vectors is defined as

where is the small angle between the two vectors, as shown in Figure 1.2. By this definition, the scalar product is commutative, that is

As demonstrated in Figure 1.2, the scalar product is just the projection of onto multiplied by . Projections are additive, as shown in Figure 1.3, therefore, the scalar product is also distributive, that is

(1.1)

The following properties are also readily verified from the definition

(1.2)

(1.3)

(1.4)

(1.5)

Figure 1.2 Scalar product geometry

Figure 1.3 Distributivity of scalar product

1.1.2 Vector Cross Product

Given vectors and , the cross-product is defined as a vector , denoted by with magnitude

with a direction perpendicular to both and , chosen according to the right-hand rule, as shown in Figure 1.4. Note that is again the small angle between the two vectors.

Figure 1.4 Vector cross product

From the definition of the cross-product, it is clear that changing the order simply reverses the direction of the cross-product, that is

Now, as shown in Figure 1.5, the vector can be decomposed into two mutually perpendicular vectors , where is perpendicular to , and is parallel to . These components are given by

which is the projection of onto the direction of , and

Since (see Figure 1.5), and is perpendicular to , . Since lies in the plane defined by and , and points to the same side of as , has the same direction as . Therefore,

(1.6)

Now, we are in a position to show a distributive property of the cross-product. Consider three vectors , and . First of all, note that

Therefore, we have

Now, the vectors , and all are perpendicular to . Therefore,

Since the vectors , and are all perpendicular to , the cross-products , and are all simply the vectors , and rotated by 90 about the vector , and then scaled by the factor , as shown in Figure 1.6. What this shows is that

and therefore by (1.6),

(1.7)

which is the distributive property we wanted to show. Finally, the following results are also readily derived from the definition:

(1.8)

(1.9)

Figure 1.5 Parallel and perpendicular vector components

Figure 1.6 Distributivity of vector cross product

1.1.3 Other Useful Vector Identities

Some other useful vector identities are:

Note that the definitions of scalar- and cross-product and all of the associated properties and identities above areindependentof a coordinate system.

1.2 Reference Frames and Physical Vector Coordinates

Up to this point, we have only considered physical vectors, without any mention of a frame of reference. For computational purposes we need to introduce the concept of a reference frame. Reference frames are also needed to describe the orientation of an object, and are needed for the formulation of kinematics and dynamics.

To define a reference frame, say reference frame “1” (which we will label ), it is customary to identify three mutually perpendicular unit length (length of one) physical vectors, labeled as , and respectively. The notation used here corresponds to the usual x-y-z axes defined for a Cartesian three-dimensional coordinate system. These three vectors then define the reference frame. The unit vectors are chosen according to the right-handed rule, as shown in Figure 1.7. Under the right-handed rule, the unit vectors satisfy

Since they are perpendicular, they also satisfy

(1.10)

Now, since the three unit vectors form a basis for physical three-dimensional space, any physical vector can be written as a linear combination of the unit vectors, that is

(1.11)

where

(1.12)

is a column matrix containing the coordinates of the physical vector in reference frame , and

(1.13)

is a column matrix containing the unit physical vectors defining reference frame . We shall refer to as a (that is, a matrix of physical vectors).