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Full Description
An authoritative and comprehensive foundation for professionals, educators, and students working in optical navigation
In Fundamentals of Spacecraft Optical Navigation, aerospace engineer John Christian delivers a rigorous and up-to-date discussion of optical navigation—the art of navigating spacecraft with camera images. The author combines the rich history of the fields of aerospace engineering, geometry, astronomy, planetary science, and computer vision with a robust treatment of the mathematics needed to solve contemporary problems.
Organized into ten chapters, the book provides the first comprehensive treatment of optical navigation. Readers will find:
A detailed account of the history of optical navigation, with plentiful excerpts and figures from original sources
A rigorous introduction to the mathematics of projective geometry
Thorough introductions to spacecraft dynamics, kinematics, and reference frames
Comprehensive explorations of star catalogs, astrometry, and planetary photometry
Overview of optical instrument hardware design
Practical discussion of celestial navigation and terrain relative navigation (TRN)
Perfect for graduate students interested in spacecraft guidance, navigation, and control, Fundamentals of Spacecraft Optical Navigation will also benefit aerospace faculty and aerospace professionals with a responsibility for designing, reviewing, operating, or working with optical navigation systems.
Contents
Foreword xi
Preface xiii
Acknowledgments xv
Acronyms xvii
About the Companion Website xix
1 Introduction 1
1.1 OpNav Pre-history 2
1.1.1 Maritime Celestial Navigation 2
1.1.2 Spacecraft OpNav before Sputnik 16
1.2 The Rise of Radio Navigation 18
1.3 OpNav in Crewed Spaceflight 19
1.4 OpNav in Robotic Spaceflight 20
1.4.1 Missions to the Outer Planets 22
1.4.2 Missions to Asteroids and Comets 25
1.5 Terrain Relative Navigation 28
2 Mathematical Foundations 31
2.1 Set Theory and Algebraic Structures 31
2.1.1 Sets 31
2.1.2 Number Systems 32
2.1.3 Numeral Systems 35
2.1.4 Set Operations 37
2.1.5 Groups, Rings, and Fields 40
2.2 Vector Spaces and Linear Algebra 43
2.2.1 Vector Spaces 43
2.2.2 Inner Products and Norms 44
2.2.3 Orthogonality and Linear Independence 45
2.2.4 Basis Vectors 46
2.2.5 Reference Systems and Reference Frames 47
2.2.6 Matrices 48
2.2.7 Linear Transformations and Group Actions 49
2.3 Change of Basis and Orthogonal Matrices 51
2.3.1 Representing a Vector with Different Bases 51
2.3.2 Attitude Transformations 54
2.3.3 Orthogonal Procrustes Problem 56
2.3.4 Weighted Orthogonal Procrustes Problem 60
2.4 Attitude Parameterizations 60
2.4.1 Euler's Rotation Theorem 60
2.4.2 Euler Angle and Euler Axis 61
2.4.3 Rotation Vectors 63
2.4.4 Attitude Quaternions 64
2.4.5 Rodrigues Parameters and Modified Rodrigues Parameters 68
2.5 Geometric Algebra 70
2.5.1 Grassmann Algebra 71
2.5.2 Clifford Algebra 75
2.6 Polynomials 78
2.6.1 Quadratics and Cubics 78
2.6.2 Polynomials in a Single Variable 79
2.6.3 Root Finding for Polynomials in Single Variable 82
2.6.4 Polynomials in a Single Variable as a Vector Space 85
2.6.5 Polynomials in Many Variables 86
2.6.6 Algebraic Varieties 88
2.7 Conics (A First Encounter) 88
2.7.1 Circles 90
2.7.2 Ellipses 92
2.7.3 Parabolas and Hyperbola 96
3 Projective Geometry 97
3.1 Perspective and the Pinhole Camera Model 98
3.1.1 Camera Obscura 98
3.1.2 The Image Plane 101
3.1.3 Pinhole Camera Model 103
3.2 Rules of Perspective Projection 104
3.3 An Axiomatic Perspective 110
3.3.1 The Axioms of Euclidean Geometry 111
3.3.2 The Axioms of Projective Geometry 111
3.3.3 Duality 112
3.3.4 Perspectivities and Projectivities 113
3.3.5 Desargues's Theorem 115
3.3.6 Harmonic Sets 116
3.3.7 Fundamental Theorem of Projective Geometry 117
3.4 An Algebraic Perspective 120
3.4.1 Projective Space, P n 120
3.4.2 Homogeneous Coordinates 121
3.4.3 Points, Lines, and Planes 123
3.4.4 Conics 138
3.4.5 Quadrics 150
3.5 Invariants 155
3.5.1 Cross-ratio in P 1 156
3.5.2 Cross-ratio in P 2 159
3.5.3 View Invariants 162
3.6 Two-dimensional Transformations 163
3.6.1 Euclidean Group 166
3.6.2 Similarity Group 168
3.6.3 Affine Group 170
3.6.4 Projective Linear Group 171
4 Time, Reference Frames, and Orbits 181
4.1 Time and Angle 181
4.1.1 Subdivision of the Day and Circle 181
4.1.2 Angle Measurements 182
4.1.3 SI Second, International Atomic Time, and Terrestrial Time 186
4.1.4 Julian Dates 187
4.2 Equinoxes and Solstices 188
4.3 Celestial Reference Frames 192
4.3.1 Coordinate Systems on the Celestial Sphere 192
4.3.2 Dynamically-defined Reference Systems 195
4.3.3 Kinematically-defined Reference Systems 197
4.4 Days, Calendars, and Civil Time 198
4.4.1 Day, Month, and Year 198
4.4.2 Universal Time 203
4.4.3 Civil Time 205
4.5 Two-body Orbital Mechanics 205
4.5.1 Equations of Motion 209
4.5.2 Conservation of Energy 211
4.5.3 Conservation of Angular Momentum 212
4.5.4 The Two-body Trajectory Equation 214
4.5.5 Orbital Period 217
4.5.6 Perifocal Frame and Orbital Elements 217
4.5.7 Orbital Hodograph 219
4.5.8 An Algebraic View of Orbits 221
4.6 Dissemination of Celestial Geometry 227
4.6.1 Historical Astronomical Tables and Ephemerides 228
4.6.2 Modern Astronomical Tables and Ephemerides 230
5 Astrometry and Star Catalogs 233
5.1 The Propagation of Light 234
5.1.1 Maxwell's Equations 234
5.1.2 The Wave-like Behavior of Light 235
5.1.3 Wavefronts and Rays 239
5.1.4 Refraction, Snell's Law, and Fermat's Principle 239
5.2 Asterisms and Constellations 244
5.3 Classical Star Catalogs 245
5.3.1 Babylonian Star Catalogs 245
5.3.2 Classical Greek Star Catalogs 246
5.3.3 Visualizations of the Celestial Sphere in Classical Antiquity 249
5.3.4 Other Catalogs in Antiquity 253
5.3.5 European Star Catalogs and Celestial Cartography 255
5.3.6 Star Catalogs in the 20th Century 258
5.4 Modern Astrometry and Star Catalogs 259
5.4.1 Modern Astrometric Models 259
5.4.2 USNO Catalogs 268
5.4.3 Catalogs from Space-based Scanning Astrometry 270
5.5 Stochastic Catalogs 276
5.5.1 Uniform Sampling on the Sphere 277
5.5.2 Distribution of Inter-star Angles on the Sphere 278
5.6 Theory of Relativity 279
5.6.1 Principle of Relativity 280
5.6.2 Special Theory of Relativity 281
5.6.3 Stellar Aberration 287
5.6.4 General Theory of Relativity 291
5.6.5 Practical Computation of Gravitational Light Deflection 300
6 Radiometry and Photometry 303
6.1 Electromagnetic Spectrum 304
6.2 Photons and Quantum Electrodynamics 305
6.3 Radiometric Units of Measure 308
6.3.1 Power, Flux Density, and Irradiance 308
6.3.2 Solid Angles 312
6.3.3 Intensity and Radiance 316
6.4 Blackbody Radiation 318
6.5 Apparent Magnitude 322
6.6 Photometric Systems 324
6.7 Transmittance and Optical Depth 325
6.8 Single-scattering Phase Function 328
6.8.1 Rayleigh Phase Function 329
6.8.2 Henyey-Greenstein Phase Function 330
6.9 Reflectance Models 332
6.9.1 Reflectance and Photometric Functions 332
6.9.2 Conservation of Energy 333
6.9.3 Helmholtz Reciprocity 334
6.9.4 Fresnel Equations (Specular Reflection) 335
6.9.5 Lambertian Model (Diffuse Reflection) 341
6.9.6 Lommel-Seeliger Model 343
6.9.7 Chandrasekhar Model 347
6.9.8 Hapke Model 351
6.9.9 Lunar-Lambert Model 357
6.10 Reflectance Models for Rough Planetary Surfaces 358
6.10.1 Modeling Rough Surfaces 359
6.10.2 Geometric Attenuation Factor 360
6.10.3 BRDFs for Rough Surfaces 373
6.10.4 Torrance-Sparrow Model 374
6.10.5 Oren-Nayar Model 376
6.11 Reflectance Model Comparisons 377
6.12 Resolved Photometry 378
6.12.1 Terrain Models 379
6.12.2 Ray Tracing 384
6.13 Unresolved (Disk-integrated) Photometry 391
6.13.1 Spherical Celestial Bodies 392
6.13.2 Disk-integrated Albedos 398
6.13.3 Apparent Magnitude of a Celestial Body 402
6.13.4 Lightcurve Analysis 406
7 Camera Hardware and Models 407
7.1 Overview of Camera Systems 407
7.1.1 Camera Obscura (Redux) 407
7.1.2 Framing Cameras 411
7.1.3 Scanning Cameras 413
7.2 Light Baffles 413
7.3 Optical Assembly 419
7.3.1 Mirrors (Reflecting Elements) 419
7.3.2 Lenses (Two-surface Refracting Elements) 425
7.3.3 Systems of Mirrors and Lenses 437
7.3.4 Stops and Pupils 439
7.3.5 Seidel Aberrations 441
7.3.6 Point Spread Function 444
7.3.7 Optical Distortion Model 448
7.4 Image Sensors 451
7.4.1 Digital Images 451
7.4.2 Photodetectors 462
7.4.3 Focal Plane Arrays 470
7.4.4 Dynamic Vision Sensors (Event Cameras) 483
7.5 Camera and Optical Instrument Design 485
8 Navigating with Stars 489
8.1 Modeling Stars in Digital Images 490
8.2 Star Detection and Centroiding 493
8.2.1 Star Detection 493
8.2.2 Idealized Star Centroiding 496
8.2.3 Practical Centroiding Methods 500
8.3 Attitude Determination 501
8.3.1 Calibrated vs. Uncalibrated Attitude Determination 501
8.3.2 Calibrated Attitude Determination: Wahba's Problem 502
8.3.3 Uncalibrated Attitude Determination 507
8.3.4 Attitude Covariance 516
8.4 Star Identification 519
8.4.1 A Framework for Automated Star Identification 520
8.4.2 Asterism Invariants 520
8.4.3 Asterism Catalog Curation 534
8.4.4 Database Management and Query 537
8.4.5 Systematic Construction of Image Asterisms 543
8.4.6 Asterism Validation 544
8.5 Velocity Estimation from Stellar Aberration 544
8.5.1 Measurement Model 545
8.5.2 Velocity-based Orbit Determination 548
9 Celestial Navigation 551
9.1 Global Shape of Self-gravitating Bodies 551
9.1.1 Nonrotating Bodies 552
9.1.2 Slowly Rotating Bodies 553
9.1.3 Arbitrarily Rotating Bodies 556
9.1.4 Spherical Harmonics 565
9.2 Images of Ellipsoidal Celestial Bodies 569
9.2.1 Analytic Horizon Projection 570
9.2.2 Analytic Terminator Projection 574
9.2.3 Photocenter Offset for Unresolved Objects 577
9.3 Horizon-based Position Estimation 580
9.3.1 Limb Scanning 581
9.3.2 Limb Localization 583
9.3.3 Non-iterative Horizon-based OpNav 590
9.4 Horizon-based Attitude Determination 601
9.5 Triangulation 603
9.5.1 Trigonometric Solutions 606
9.5.2 Triangulation with Noisy Measurements 610
9.5.3 Celestial Triangulation 613
9.6 Navigation Filters 619
9.6.1 State Selection and Dynamical Model 620
9.6.2 State and Covariance Propagation 623
9.6.3 State and Covariance Update 628
9.6.4 Measurement Models 631
10 Terrain Relative Navigation 637
10.1 Landmarks 637
10.1.1 Craters 638
10.1.2 Features 646
10.2 Map-Free TRN 648
10.2.1 Visual Odometry Measurements 649
10.2.2 Filter Processing of Visual Odometry Measurements 653
10.3 Map-based TRN 655
10.3.1 Monocular Position Estimation 655
10.3.2 Monocular Pose Estimation: Perspective-n-point Problem 657
10.3.3 Filter Processing of Landmark Observations 660
References 667
Index 709
