Quaternions and Rotation Sequences

A Primer with Applications to Orbits, Aerospace and Virtual Reality

By: J. B. Kuipers

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Paperback | 18 November 2002

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400 Pages

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25.4 x 19.0 x 2.54

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RRP $180.00

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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.

Industry Reviews

"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

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List of Figures	p. xv
About This Book	p. xix
Acknowledgements	p. xxi
Historical Matters	p. 3
Introduction	p. 3
Mathematical Systems	p. 4
Complex Numbers	p. 6
Polar Representation	p. 9
Hyper-complex Numbers	p. 11
Algebraic Preliminaries	p. 13
Introduction	p. 13
Complex Number Operations	p. 15
Addition and Multiplication	p. 15
Subtraction and Division	p. 17
The Complex Conjugate	p. 19
Coordinates	p. 21
Rotations in the Plane	p. 22
Frame Rotation - Points Fixed	p. 22
Point Rotation - Frame Fixed	p. 23
Equivalent Rotations	p. 25
Matrix Notation	p. 26
Review of Matrix Algebra	p. 27
The Transpose	p. 28
Addition and Subtraction	p. 28
Multiplication by a Scalar	p. 29
Product of Matrices	p. 29
Rotation Matrices	p. 31
The Determinant	p. 33
Minors	p. 34
Cofactors	p. 35
Determinant of an n x n Matrix	p. 35
The Cofactor Matrix	p. 36
Adjoint Matrix	p. 37
The Inverse Matrix - Method 1	p. 37
The Inverse Matrix - Method 2	p. 38
Rotation Operators Revisited	p. 39
Rotations in 3-space	p. 45
Introduction	p. 45
Rotation Sequences in the Plane	p. 45
Coordinates in R[superscript 3]	p. 47
Successive Same-axis Rotations	p. 50
Signs in Rotation Matrices	p. 51
Rotation Sequences in R[superscript 3]	p. 52
Some Rotation Geometry	p. 52
Rotation Eigenvalues & Eigenvectors	p. 54
The Fixed Axis of Rotation	p. 55
Rotation Angle about the Fixed Axis	p. 56
A Numerical Example	p. 57
An Application - Tracking	p. 59
A Simpler Rotation-Axis Algorithm	p. 65
A Geometric Analysis	p. 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
Introduction	p. 77
Equivalent Rotations	p. 77
New Rotation Symbol	p. 78
A Word of Caution	p. 79
Another Word of Caution	p. 80
Equivalent Sequence Pairs	p. 81
An Application	p. 81
Euler Angles	p. 83
The Aerospace Sequence	p. 84
An Orbit Ephemeris Determined	p. 86
Euler Angle-Axis Sequence for Orbits	p. 87
The Orbit Ephemeris Sequence	p. 89
Great Circle Navigation	p. 91
Quaternion Algebra	p. 103
Introduction	p. 103
Quaternions Defined	p. 104
Equality and Addition	p. 105
Multiplication Defined	p. 106
The Complex Conjugate	p. 110
The Norm	p. 111
Inverse of the Quaternion	p. 112
Geometric Interpretations	p. 113
Algebraic Considerations	p. 113
Geometric Considerations	p. 117
A Special Quaternion Product	p. 119
Incremental Test Quaternion	p. 120
Quaternion with Angle [theta] = [pi]/6	p. 123
Operator Algorithm	p. 124
Operator action on v = kq	p. 125
Quaternions to Matrices	p. 125
Quaternion Rotation Operator	p. 127
L[subscript q](v) = qvq is a Linear Operator	p. 127
Operator Norm	p. 128
Prove: Operator is a Rotation	p. 128
Quaternion Operator Sequences	p. 134
Rotation Examples	p. 136
Quaternion Geometry	p. 141
Introduction	p. 141
Euler Construction	p. 142
Geometric Construction	p. 143
The Spherical Triangle	p. 146
Quaternion Geometric Analysis	p. 147
The Tracking Example Revisited	p. 151
Algorithm Summary	p. 155
The Quaternion Product	p. 156
Quaternion Rotation Operator	p. 157
Direction Cosines	p. 158
Frame Bases to Rotation Matrix	p. 159
Angle and Axis of Rotation	p. 161
Euler Angles to Quaternion	p. 166
Quaternion to Direction Cosines	p. 167
Quaternion to Euler Angles	p. 168
Direction Cosines to Quaternion	p. 168
Rotation Operator Algebra	p. 169
Sequence of Rotation Operators	p. 169
Rotation of Vector Sets	p. 170
Mixing Matrices and Quaternions	p. 171
Quaternion Factors	p. 177
Introduction	p. 177
Factorization Criteria	p. 178
Transition Rotation Operators	p. 179
The Factorization M = TA	p. 180
Rotation Matrix A Specified	p. 180
Rotation Axes Orthogonal	p. 182
A Slight Generalization	p. 185
Three Principal-axis Factors	p. 186
Factorization: q = st = (s[subscript 0] + js[subscript 2])t	p. 189
Principal-axis Factor Specified	p. 190
Orthogonal Factors	p. 191
Euler Angle-Axis Factors	p. 192
Tracking Revisited	p. 194
Distinct Principal Axis Factorization	p. 197
Repeated Principal Axis Factorization	p. 200
Some Geometric Insight	p. 202
More Quaternion Applications	p. 205
Introduction	p. 205
The Aerospace Sequence	p. 205
The Rotation Angle	p. 207
The Rotation Axis	p. 208
Computing the Orbit Ephemeris	p. 209
The Orbit Euler Angle Sequence	p. 210
Orbit Ephemeris	p. 212
Great Circle Navigation	p. 216
Quaternion Method	p. 218
Reasons for the Seasons	p. 222
Sequence #1: Polygon - XSPAX	p. 223
Sequence #2: Trapezoid - APSBA	p. 224
Sequence #3: Triangle - XSBAX	p. 225
Summary of Rotation Angles	p. 226
Matrix Method on Sequence #1	p. 227
Matrix Method on Sequence #2	p. 229
Matrix Method on Sequence #3	p. 230
Seasonal Daylight Hours	p. 231
Spherical Trigonometry	p. 235
Introduction	p. 235
Spherical Triangles	p. 235
Closed-loop Rotation Sequences	p. 237
Rotation Matrix Analysis	p. 242
Right Spherical Triangle	p. 243
Isoceles Spherical Triangle	p. 244
Quaternion Analysis	p. 244
Right Spherical Triangle	p. 248
Isoceles Spherical Triangle	p. 248
Regular n-gons on the Sphere	p. 249
Area and Volume	p. 252
Quaternion Calculus for Kinematics and Dynamics	p. 257
Introduction	p. 257
Derivative of the Direction Cosine Matrix	p. 258
Body-Axes [characters not reproducible] Euler Angle Rates	p. 259
Perturbations in a Rotation Sequence	p. 260
Derivative of the Quaternion	p. 263
Derivative of the Conjugate	p. 264
Quaternion Operator Derivative	p. 265
Quaternion Perturbations	p. 268
Rotations in Phase Space	p. 277
Introduction	p. 277
Constituents in the ODE Set	p. 277
The Phase Plane	p. 280
Phase Plane Stable Node	p. 282
Phase Plane Saddle	p. 282
Phase Plane Stable Focus	p. 283
Some Preliminaries	p. 284
Linear Differential Equations	p. 285
Initial Conditions	p. 289
Partitions in R[superscript 3] Phase Space	p. 290
Space-filling Direction Field in R[superscript 3]	p. 293
Locus of all Real Eigenvectors	p. 295
Non Autonomous Systems	p. 296
Phase Space Rotation Sequences	p. 297
Rotation Sequence for State Vector	p. 297
Rotation Sequence for Velocity Vector	p. 299
A Quaternion Process	p. 303
Introduction	p. 303
Dipole Field Structure	p. 305
Electromagnetic Field Coupling	p. 305
Unit Z-axis Source Excitation	p. 307
Unit X-axis Source Excitation	p. 309
Source-to-Sensor Coupling	p. 311
Source-to-Sensor Distance	p. 314
Angular Degrees-of-Freedom	p. 315
Preliminary Analysis	p. 315
Closed-form Tracking Angle Computation	p. 316
Closed-Form Orientation Angle Computation	p. 317
Quaternion Processes	p. 318
Partial Closed-form Tracking Solution	p. 320
An Iterative Solution for Tracking	p. 323
Orientation Quaternion	p. 327
Position & Orientation	p. 329
Computer Graphics	p. 333
Introduction	p. 333
Canonical Transformations	p. 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 Coordinates	p. 337
An Object in R[superscript 2] Transformed	p. 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 Projections	p. 347
Perspective Projections	p. 347
Coordinate Frames	p. 348
Perspective--Simple Case	p. 350
Parallel Lines in Perspective	p. 351
Perspective in General	p. 352
Objects in Motion	p. 354
Incremental Translation Only	p. 355
Incremental Rotation Only	p. 356
Incremental Rotation Quaternion	p. 357
Incremental Rotation Matrix	p. 357
Aircraft Kinematics	p. 358
n-Body Simulation	p. 361
Further Reading and References	p. 365
Index	p. 367
Table of Contents provided by Syndetics. All Rights Reserved.

Quaternions and Rotation Sequences

A Primer with Applications to Orbits, Aerospace and Virtual Reality

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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