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Quaternions and Rotation Sequences : A Primer with Applications to Orbits, Aerospace and Virtual Reality - J.B. Kuipers

Quaternions and Rotation Sequences

A Primer with Applications to Orbits, Aerospace and Virtual Reality

Paperback Published: 8th September 2002
ISBN: 9780691102986
Number Of Pages: 400

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Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations.

The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.

"This book will appeal to anyone with an interest in three-dimensional geometry. It is a competent and comprehensive survey... This book is unique in that it is probably the only modern book to treat quaternions seriously... A valuable asset."--Aeronautical Journal "[A] splendid book ... everything one could wish for in a primer. It is also beautifully set out with an attractive layout, clear diagrams, and wide margins with explanatory notes where appropriate. It must be strongly recommended to all students of physics, engineering or computer science."--Peter Rowlands, Contemporary Physics

List of Figuresp. xv
About This Bookp. xix
Acknowledgementsp. xxi
Historical Mattersp. 3
Introductionp. 3
Mathematical Systemsp. 4
Complex Numbersp. 6
Polar Representationp. 9
Hyper-complex Numbersp. 11
Algebraic Preliminariesp. 13
Introductionp. 13
Complex Number Operationsp. 15
Addition and Multiplicationp. 15
Subtraction and Divisionp. 17
The Complex Conjugatep. 19
Coordinatesp. 21
Rotations in the Planep. 22
Frame Rotation - Points Fixedp. 22
Point Rotation - Frame Fixedp. 23
Equivalent Rotationsp. 25
Matrix Notationp. 26
Review of Matrix Algebrap. 27
The Transposep. 28
Addition and Subtractionp. 28
Multiplication by a Scalarp. 29
Product of Matricesp. 29
Rotation Matricesp. 31
The Determinantp. 33
Minorsp. 34
Cofactorsp. 35
Determinant of an n x n Matrixp. 35
The Cofactor Matrixp. 36
Adjoint Matrixp. 37
The Inverse Matrix - Method 1p. 37
The Inverse Matrix - Method 2p. 38
Rotation Operators Revisitedp. 39
Rotations in 3-spacep. 45
Introductionp. 45
Rotation Sequences in the Planep. 45
Coordinates in R[superscript 3]p. 47
Successive Same-axis Rotationsp. 50
Signs in Rotation Matricesp. 51
Rotation Sequences in R[superscript 3]p. 52
Some Rotation Geometryp. 52
Rotation Eigenvalues & Eigenvectorsp. 54
The Fixed Axis of Rotationp. 55
Rotation Angle about the Fixed Axisp. 56
A Numerical Examplep. 57
An Application - Trackingp. 59
A Simpler Rotation-Axis Algorithmp. 65
A Geometric Analysisp. 67
X into x[subscript 2]p. 69
Y into y[subscript 2]p. 71
X into x[subscript 2] and Y into y[subscript 2]p. 71
Incremental Rotations in R[superscript 3]p. 73
Singularities in SO(3)p. 74
Rotation Sequences in R[superscript 3]p. 77
Introductionp. 77
Equivalent Rotationsp. 77
New Rotation Symbolp. 78
A Word of Cautionp. 79
Another Word of Cautionp. 80
Equivalent Sequence Pairsp. 81
An Applicationp. 81
Euler Anglesp. 83
The Aerospace Sequencep. 84
An Orbit Ephemeris Determinedp. 86
Euler Angle-Axis Sequence for Orbitsp. 87
The Orbit Ephemeris Sequencep. 89
Great Circle Navigationp. 91
Quaternion Algebrap. 103
Introductionp. 103
Quaternions Definedp. 104
Equality and Additionp. 105
Multiplication Definedp. 106
The Complex Conjugatep. 110
The Normp. 111
Inverse of the Quaternionp. 112
Geometric Interpretationsp. 113
Algebraic Considerationsp. 113
Geometric Considerationsp. 117
A Special Quaternion Productp. 119
Incremental Test Quaternionp. 120
Quaternion with Angle [theta] = [pi]/6p. 123
Operator Algorithmp. 124
Operator action on v = kqp. 125
Quaternions to Matricesp. 125
Quaternion Rotation Operatorp. 127
L[subscript q](v) = qvq is a Linear Operatorp. 127
Operator Normp. 128
Prove: Operator is a Rotationp. 128
Quaternion Operator Sequencesp. 134
Rotation Examplesp. 136
Quaternion Geometryp. 141
Introductionp. 141
Euler Constructionp. 142
Geometric Constructionp. 143
The Spherical Trianglep. 146
Quaternion Geometric Analysisp. 147
The Tracking Example Revisitedp. 151
Algorithm Summaryp. 155
The Quaternion Productp. 156
Quaternion Rotation Operatorp. 157
Direction Cosinesp. 158
Frame Bases to Rotation Matrixp. 159
Angle and Axis of Rotationp. 161
Euler Angles to Quaternionp. 166
Quaternion to Direction Cosinesp. 167
Quaternion to Euler Anglesp. 168
Direction Cosines to Quaternionp. 168
Rotation Operator Algebrap. 169
Sequence of Rotation Operatorsp. 169
Rotation of Vector Setsp. 170
Mixing Matrices and Quaternionsp. 171
Quaternion Factorsp. 177
Introductionp. 177
Factorization Criteriap. 178
Transition Rotation Operatorsp. 179
The Factorization M = TAp. 180
Rotation Matrix A Specifiedp. 180
Rotation Axes Orthogonalp. 182
A Slight Generalizationp. 185
Three Principal-axis Factorsp. 186
Factorization: q = st = (s[subscript 0] + js[subscript 2])tp. 189
Principal-axis Factor Specifiedp. 190
Orthogonal Factorsp. 191
Euler Angle-Axis Factorsp. 192
Tracking Revisitedp. 194
Distinct Principal Axis Factorizationp. 197
Repeated Principal Axis Factorizationp. 200
Some Geometric Insightp. 202
More Quaternion Applicationsp. 205
Introductionp. 205
The Aerospace Sequencep. 205
The Rotation Anglep. 207
The Rotation Axisp. 208
Computing the Orbit Ephemerisp. 209
The Orbit Euler Angle Sequencep. 210
Orbit Ephemerisp. 212
Great Circle Navigationp. 216
Quaternion Methodp. 218
Reasons for the Seasonsp. 222
Sequence #1: Polygon - XSPAXp. 223
Sequence #2: Trapezoid - APSBAp. 224
Sequence #3: Triangle - XSBAXp. 225
Summary of Rotation Anglesp. 226
Matrix Method on Sequence #1p. 227
Matrix Method on Sequence #2p. 229
Matrix Method on Sequence #3p. 230
Seasonal Daylight Hoursp. 231
Spherical Trigonometryp. 235
Introductionp. 235
Spherical Trianglesp. 235
Closed-loop Rotation Sequencesp. 237
Rotation Matrix Analysisp. 242
Right Spherical Trianglep. 243
Isoceles Spherical Trianglep. 244
Quaternion Analysisp. 244
Right Spherical Trianglep. 248
Isoceles Spherical Trianglep. 248
Regular n-gons on the Spherep. 249
Area and Volumep. 252
Quaternion Calculus for Kinematics and Dynamicsp. 257
Introductionp. 257
Derivative of the Direction Cosine Matrixp. 258
Body-Axes [characters not reproducible] Euler Angle Ratesp. 259
Perturbations in a Rotation Sequencep. 260
Derivative of the Quaternionp. 263
Derivative of the Conjugatep. 264
Quaternion Operator Derivativep. 265
Quaternion Perturbationsp. 268
Rotations in Phase Spacep. 277
Introductionp. 277
Constituents in the ODE Setp. 277
The Phase Planep. 280
Phase Plane Stable Nodep. 282
Phase Plane Saddlep. 282
Phase Plane Stable Focusp. 283
Some Preliminariesp. 284
Linear Differential Equationsp. 285
Initial Conditionsp. 289
Partitions in R[superscript 3] Phase Spacep. 290
Space-filling Direction Field in R[superscript 3]p. 293
Locus of all Real Eigenvectorsp. 295
Non Autonomous Systemsp. 296
Phase Space Rotation Sequencesp. 297
Rotation Sequence for State Vectorp. 297
Rotation Sequence for Velocity Vectorp. 299
A Quaternion Processp. 303
Introductionp. 303
Dipole Field Structurep. 305
Electromagnetic Field Couplingp. 305
Unit Z-axis Source Excitationp. 307
Unit X-axis Source Excitationp. 309
Source-to-Sensor Couplingp. 311
Source-to-Sensor Distancep. 314
Angular Degrees-of-Freedomp. 315
Preliminary Analysisp. 315
Closed-form Tracking Angle Computationp. 316
Closed-Form Orientation Angle Computationp. 317
Quaternion Processesp. 318
Partial Closed-form Tracking Solutionp. 320
An Iterative Solution for Trackingp. 323
Orientation Quaternionp. 327
Position & Orientationp. 329
Computer Graphicsp. 333
Introductionp. 333
Canonical Transformationsp. 334
Transformations in R[superscript 2]p. 334
Scale in R[superscript 2]p. 334
Translation in R[superscript 2]p. 335
Rotation in R[superscript 2]p. 336
Homogeneous Coordinatesp. 337
An Object in R[superscript 2] Transformedp. 338
Concatenation Order in R[superscript 2]p. 339
Transformations in R[superscript 3]p. 341
What about Quaternions?p. 345
Projections R[superscript 3] [right arrow] R[superscript 2]p. 346
Parallel Projectionsp. 347
Perspective Projectionsp. 347
Coordinate Framesp. 348
Perspective--Simple Casep. 350
Parallel Lines in Perspectivep. 351
Perspective in Generalp. 352
Objects in Motionp. 354
Incremental Translation Onlyp. 355
Incremental Rotation Onlyp. 356
Incremental Rotation Quaternionp. 357
Incremental Rotation Matrixp. 357
Aircraft Kinematicsp. 358
n-Body Simulationp. 361
Further Reading and Referencesp. 365
Indexp. 367
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691102986
ISBN-10: 0691102988
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 400
Published: 8th September 2002
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 25.4 x 20.07  x 2.54
Weight (kg): 0.79