# Spacecraft Dynamics and Control. An Introduction

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1 Brochure More information from Spacecraft Dynamics and Control. An Introduction Description: 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. Contents: Preface xvii 1 Kinematics Physical Vectors Reference Frames and Physical Vector Coordinates Rotation Matrices Derivatives of Vectors Velocity and Acceleration More Rigorous Definition of Angular Velocity References 45 2 Rigid Body Dynamics Dynamics of a Single Particle Dynamics of a System of Particles Rigid Body Dynamics 52

2 2.4 The Inertia Matrix Kinetic Energy of a Rigid Body References 63 3 The Keplerian Two-Body Problem Equations of Motion Constants of the Motion Shape of a Keplerian Orbit Kepler s Laws Time of Flight Orbital Elements Orbital Elements given Position and Velocity Position and Velocity given Orbital Elements References 98 4 Preliminary Orbit Determination Orbit Determination from Three Position Vectors Orbit Determination from Three Line-of-Sight Vectors Orbit Determination from Two Position Vectors and Time (Lambert s Problem) References Orbital Maneuvers Simple Impulsive Maneuvers Coplanar Maneuvers Plane Change Maneuvers Combined Maneuvers Rendezvous Reference Interplanetary Trajectories Sphere of Influence 129

3 6.2 Interplanetary Hohmann Transfers Patched Conics Planetary Flyby Planetary Capture References Orbital Perturbations Special Perturbations Cowell s Method General Perturbations Gravitational Perturbations due to a Non-Spherical Primary Body Effect of J2 on the Orbital Elements Special Types of Orbits Small Impulse Form of the Gauss Variational Equations Derivation of the Remaining Gauss Variational Equations References Low Thrust Trajectory Analysis and Design Problem Formulation Coplanar Circle to Circle Transfers Plane Change Maneuver References Spacecraft Formation Flying Mathematical Description Relative Motion Solutions Special Types of Relative Orbits Reference The Restricted Three-Body Problem Formulation The Lagrangian Points 212

4 10.3 Stability of the Lagrangian Points Jacobi s Integral References Introduction to Spacecraft Attitude Stabilization Introduction to Control Systems Overview of Attitude Representation and Kinematics Overview of Spacecraft Attitude Dynamics Disturbance Torques on a Spacecraft Magnetic Torque Solar Radiation Pressure Torque Aerodynamic Torque Gravity-Gradient Torque Reference Torque-Free Attitude Motion Solution for an Axisymmetric Body Physical Interpretation of the Motion References Spin Stabilization Stability Spin Stability of Torque-Free Motion Effect of Internal Energy Dissipation References Dual-Spin Stabilization Equations of Motion Stability of Dual-Spin Torque-Free Motion Effect of Internal Energy Dissipation

5 References Gravity-Gradient Stabilization Equations of Motion Stability Analysis References Active Spacecraft Attitude Control Attitude Control for a Nominally Inertially Fixed Spacecraft Transfer Function Representation of a System System Response to an Impulsive Input Block Diagrams The Feedback Control Problem Typical Control Laws Time-Domain Specifications Factors that Modify the Transient Behavior Steady-State Specifications and System Type 311 JWST251-FM JWST251-De-Ruiter Printer: Yet to Come November 2, :18 Trim: 244mm 168mm viii Contents 2.4 The Inertia Matrix A Parallel Axis Theorem A Rotational Transformation Theorem Principal Axes Kinetic Energy of a Rigid Body References 63

6 3 The Keplerian Two-Body Problem Equations of Motion Constants of the Motion Orbital Angular Momentum Orbital Energy The Eccentricity Vector Shape of a Keplerian Orbit Perifocal Coordinate System Kepler s Laws Time of Flight Circular Orbits Elliptical Orbits Parabolic Orbits Hyperbolic Orbits Orbital Elements Heliocentric-Ecliptic Coordinate System Geocentric-Equatorial Coordinate System Orbital Elements given Position and Velocity Position and Velocity given Orbital Elements 94 98

7 References 98 4 Preliminary Orbit Determination Orbit Determination from Three Position Vectors Orbit Determination from Three Line-of-Sight Vectors Orbit Determination from Two Position Vectors and Time (Lambert s Problem) The Lagrangian Coefficients References Orbital Maneuvers Simple Impulsive Maneuvers Coplanar Maneuvers Hohmann Transfers Bi-Elliptic Transfers Plane Change Maneuvers 123 FOR SCREEN VIEWING IN DART ONLY JWST251-FM JWST251-De-Ruiter Printer: Yet to Come November 2, :18 Trim: 244mm 168mm Contents ix 5.4 Combined Maneuvers Rendezvous Reference Interplanetary Trajectories Sphere of Influence 129

8 6.2 Interplanetary Hohmann Transfers Patched Conics Departure Hyperbola Arrival Hyperbola Planetary Flyby Planetary Capture References Orbital Perturbations Special Perturbations Cowell s Method Encke s Method General Perturbations Gravitational Perturbations due to a Non-Spherical Primary Body The Perturbative Force Per Unit Mass Due to J Effect of J 2 on the Orbital Elements Special Types of Orbits Sun-Synchronous Orbits Molniya Orbits 169

9 7.6 Small Impulse Form of the Gauss Variational Equations Derivation of the Remaining Gauss Variational Equations References Low Thrust Trajectory Analysis and Design Problem Formulation Coplanar Circle to Circle Transfers Plane Change Maneuver References Spacecraft Formation Flying Mathematical Description Relative Motion Solutions Out-of-Plane Motion In-Plane Motion 195 FOR SCREEN VIEWING IN DART ONLY JWST251-FM JWST251-De-Ruiter Printer: Yet to Come November 2, :18 Trim: 244mm 168mm x Contents Alternative Description for In-Plane Relative Motion Further Examination of In-Plane Motion Out-of-Plane Motion - Revisited Special Types of Relative Orbits 203

10 9.3.1 Along-Track Orbits Projected Elliptical Orbits Projected Circular Orbits Reference The Restricted Three-Body Problem Formulation Equations of Motion The Lagrangian Points Case (i) Case (ii) Stability of the Lagrangian Points Comments Jacobi s Integral Hill s Curves Comments on Figure References Introduction to Spacecraft Attitude Stabilization Introduction to Control Systems 220

11 Open-loop versus Closed-loop Typical Feedback Control Structure Overview of Attitude Representation and Kinematics Overview of Spacecraft Attitude Dynamics Properties of the Inertia Matrix - A Summary Disturbance Torques on a Spacecraft Magnetic Torque Solar Radiation Pressure Torque Aerodynamic Torque Gravity-Gradient Torque Reference Torque-Free Attitude Motion Solution for an Axisymmetric Body Physical Interpretation of the Motion References 245 FOR SCREEN VIEWING IN DART ONLY JWST251-FM JWST251-De-Ruiter Printer: Yet to Come November 2, :18 Trim: 244mm 168mm Contents xi 14 Spin Stabilization Stability Spin Stability of Torque-Free Motion Effect of Internal Energy Dissipation 252

12 Energy Sink Hypothesis Major Axis Rule References Dual-Spin Stabilization Equations of Motion Stability of Dual-Spin Torque-Free Motion Effect of Internal Energy Dissipation References Gravity-Gradient Stabilization Equations of Motion Stability Analysis Pitch Motion Roll-Yaw Motion Combined Pitch and Roll/Yaw References Active Spacecraft Attitude Control Attitude Control for a Nominally Inertially Fixed Spacecraft Transfer Function Representation of a System System Response to an Impulsive Input Block Diagrams 284

13 17.5 The Feedback Control Problem Typical Control Laws Time-Domain Specifications Factors that Modify the Transient Behavior Steady-State Specifications and System Type Effect of Disturbances Actuator Limitations References Routh s Stability Criterion Proportional-Derivative Control with Actuator Dynamics Active Dual-Spin Stabilization References The Root Locus Rules for Constructing the Root Locus PD Attitude Control with Actuator Dynamics - Revisited Derivation of the Rules for Constructing the Root Locus References Control Design by the Root Locus Method Typical Types of Controllers PID Design for Spacecraft Attitude Control References Frequency Response Frequency Response and Bode Plots Low-Pass Filter Design References Relative Stability Polar Plots 387

14 22.2 Nyquist Stability Criterion Stability Margins References Control Design in the Frequency Domain Feedback Control Problem - Revisited Control Design Example - PID Design for Spacecraft Attitude Control References Nonlinear Spacecraft Attitude Control State-Space Representation of the Spacecraft Attitude Equations Stability Definitions Stability Analysis LaSalle s Theorem Spacecraft Attitude Control with Quaternion and Angular Rate Feedback References Spacecraft Navigation Review of Probability Theory Batch Approaches for Spacecraft Attitude Estimation The Kalman Filter References Practical Spacecraft Attitude Control Design Issues Attitude Sensors Attitude Actuators Control Law Implementation Unmodeled Dynamics References

15 Appendix A: Review of Complex Variables 541 Appendix B: Numerical Simulation of Spacecraft Motion Reference 561 Index 563 Ordering: Order Online - Order by Fax - using the form below Order by Post - print the order form below and send to Research and Markets, Guinness Centre, Taylors Lane, Dublin 8, Ireland.

16 Page 1 of 2 Fax Order Form To place an order via fax simply print this form, fill in the information below and fax the completed form to (from USA) or (from Rest of World). If you have any questions please visit Order Information Please verify that the product information is correct. Product Name: Web Address: Office Code: Spacecraft Dynamics and Control. An Introduction SC Product Format Please select the product format and quantity you require: Hard Copy (Hard Back): Quantity USD USD 28 Shipping/Handling * Shipping/Handling is only charged once per order. Contact Information Please enter all the information below in BLOCK CAPITALS Title: Mr Mrs Dr Miss Ms Prof First Name: Last Name: Address: * Job Title: Organisation: Address: City: Postal / Zip Code: Country: Phone Number: Fax Number: * Please refrain from using free accounts when ordering (e.g. Yahoo, Hotmail, AOL)

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