Quaternions and Rotation Sequences

Name: Quaternions and Rotation Sequences | A Primer with Applications to Orbits, Aerospace and Virtual Reality
Brand: Princeton University Press
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A Primer with Applications to Orbits, Aerospace and Virtual Reality

J. B. Kuipers(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 31. März 2020

400 Seiten

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978-0-691-21170-1 (ISBN)

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Inhalt

Cover Page
Half-title Page
Title Page
Copyright Page
Dedication
Contents
List of Figures
About This Book
Acknowledgements
1. Historical Matters
1.1 Introduction
1.2 Mathematical Systems
1.3 Complex Numbers
1.4 Polar Representation
1.5 Hyper-complex Numbers
2. Algebraic Preliminaries
2.1 Introduction
2.2 Complex Number Operations
2.2.1 Addition and Multiplication
2.2.2 Subtraction and Division
2.3 The Complex Conjugate
2.4 Coordinates
2.5 Rotations in the Plane
2.5.1 Frame Rotation - Points Fixed
2.5.2 Point Rotation - Frame Fixed
2.5.3 Equivalent Rotations
2.5.4 Matrix Notation
2.6 Review of Matrix Algebra
2.6.1 The Transpose
2.6.2 Addition and Subtraction
2.6.3 Multiplication by a Scalar
2.6.4 Product of Matrices
2.6.5 Rotation Matrices
2.7 The Determinant
2.7.1 Minors
2.7.2 Cofactors
2.7.3 Determinant of an n x n Matrix
2.8 The Cofactor Matrix
2.9 Adjoint Matrix
2.10 The Inverse Matrix- Method 1
2.11 The Inverse Matrix - Method 2
2.12 Rotation Operators Revisited
3. Rotations in 3-space
3.1 Introduction
3.2 Rotation Sequences in the Plane
3.3 Coordinates in R3
3.3.1 Successive Same-axis Rotations
3.3.2 Signs in Rotation Matrices
3.4 Rotation Sequences in R3
3.4.1 Some Rotation Geometry
3.4.2 Rotation Eigenvalues & Eigenvectors
3.5 The Fixed Axis of Rotation
3.5.1 Rotation Angle about the Fixed Axis
3.6 A Numerical Example
3.7 An Application - Tracking
3.7.1 A Simpler Rotation-Axis Algorithm
3.8 A Geometric Analysis
3.8.1 All Axes which take X into x2
3.8.2 All Axes which take Y into y2
3.8.3 Rotation Axis for both X into x2and Y into y2
3.9 Incremental Rotations in R3
3.10 Singularities in SO(3)
4. Rotation Sequences in R3
4.1 Introduction
4.2 Equivalent Rotations
4.2.1 New Rotation Symbol
4.2.2 A Word of Caution
4.2.3 Another Word of Caution
4.2.4 Equivalent Sequence Pairs
4.2.5 An Application
4.3 Euler Angles
4.4 The Aerospace Sequence
4.5 An Orbit Ephemeris Determined
4.5.1 Euler Angle-Axis Sequence for Orbits
4.5.2 The Orbit Ephemeris Sequence
4.6 Great Circle Navigation
5. Quaternion Algebra
5.1 Introduction
5.2 Quaternions Defined
5.3 Equality and Addition
5.4 Multiplication Defined
5.5 The Complex Conjugate
5.6 The Norm
5.7 Inverse of the Quaternion
5.8 Geometric Interpretations
5.8.1 Algebraic Considerations
5.8.2 Geometric Considerations
5.9 A Special Quaternion Product
5.10 Incremental Test Quaternion
5.11 Quaternion with Angle ? = p/6
5.12 Operator Algorithm
5.13 Operator action on v = kq
5.14 Quaternions to Matrices
5.15 Quaternion Rotation Operator
5.15.1 Lq(v) = qvq* is a Linear Operator
5.15.2 Operator Norm
5.15.3 Prove: Operator is a Rotation
5.16 Quaternion Operator Sequences
5.16.1 Rotation Examples
6. Quaternion Geometry
6.1 Introduction
6.2 Euler Construction
6.2.1 Geometric Construction
6.2.2 The Spherical Triangle
6.3 Quaternion Geometric Analysis
6.4 The Tracking Example Revisited
7. Algorithm Summary
7.1 The Quaternion Product
7.2 Quaternion Rotation Operator
7.3 Direction Cosines
7.4 Erame Bases to Rotation Matrix
7.5 Angle and Axis of Rotation
7.6 Euler Angles to Quaternion
7.7 Quaternion to Direction Cosines
7.8 Quaternion to Euler Angles
7.9 Direction Cosines to Quaternion
7.10 Rotation Operator Algebra
7.10.1 Sequence of Rotation Operators
7.10.2 Rotation of Vector Sets
7.10.3 Mixing Matrices and Quaternions
8. Quaternion Factors
8.1 Introduction
8.2 Factorization Criteria
8.3 Transition Rotation Operators
8.4 The Factorization M = TA
8.4.1 Rotation Matrix A Specified
8.4.2 Rotation Axes Orthogonal
8.4.3 A Slight Generalization
8.5 Three Principal-axis Factors
8.6 Factorization: q = st = (s0+js2)t
8.6.1 Principal-axis Factor Specified
8.6.2 Orthogonal Factors
8.7 Euler Angle-Axis Factors
8.7.1 Tracking Revisited
8.7.2 Distinct Principal Axis Factorization
8.7.3 Repeated Principal Axis Factorization
8.8 Some Geometric Insight
9. More Quaternion Applications
9.1 Introduction
9.2 The Aerospace Sequence
9.2.1 The Rotation Angle
9.2.2 The Rotation Axis
9.3 Computing the Orbit Ephemeris
9.3.1 The Orbit Euler Angle Sequence
9.3.2 Orbit Ephemeris
9.4 Great Circle Navigation
9.5 Quaternion Method
9.6 Reasons for the Seasons
9.6.1 Sequence # 1 : Polygon - XSPAX
9.6.2 Sequence #2: Trapezoid - APSBA
9.6.3 Sequence #3: Triangle - XSBAX
9.6.4 Summary of Rotation Angles
9.6.5 Matrix Method on Sequence # 1
9.6.6 Matrix Method on Sequence #2
9.6.7 Matrix Method on Sequence # 3
9.7 Seasonal Daylight Hours
10. Spherical Trigonometry
10.1 Introduction
10.2 Spherical Triangles
10.3 Closed-loop Rotation Sequences
10.4 Rotation Matrix Analysis
10.4.1 Right Spherical Triangle
10.4.2 Isoceles Spherical Triangle
10.5 Quaternion Analysis
10.5.1 Right Spherical Triangle
10.5.2 Isoceles Spherical Triangle
10.6 Regular n-gons on the Sphere
10.7 Area and Volume
11. Quaternion Calculus for Kinematics and Dynamics
11.1 Introduction
11.2 Derivative of the Direction Cosine Matrix
11.3 Body-Axes --& Euler Angle Rates
11.4 Perturbations in a Rotation Sequence
11.5 Derivative of the Quaternion
11.6 Derivative of the Conjugate
11.7 Quaternion Operator Derivative
11.8 Quaternion Perturbations
12. Rotations in Phase Space
12.1 Introduction
12.2 Constituents in the ODE Set
12.3 The Phase Plane
12.3.1 Phase Plane Stable Node
12.3.2 Phase Plane Saddle
12.3.3 Phase Plane Stable Focus
12.4 Some Preliminaries
12.5 Linear Differential Equations
12.6 Initial Conditions
12.7 Partitions in R3 Phase Space
12.8 Space-filling Direction Field in R3
12.9 Locus of all Real Eigenvectors
12.10 Non Autonomous Systems
12.11 Phase Space Rotation Sequences
12.11.1 Rotation Sequence for State Vector
12.11.2 Rotation Sequence for Velocity Vector
13. A Quaternion Process
13.1 Introduction
13.2 Dipole Field Structure
13.3 Electromagnetic Field Coupling
13.3.1 Unit Z-axis Source Excitation
13.3.2 Unit X-axis Source Excitation
13.4 Source-to-Sensor Coupling
13.5 Source-to-Sensor Distance
13.6 Angular Degrees-of-Freedom
13.6.1 Preliminary Analysis
13.6.2 Closed-form Tracking Angle Computation
13.6.3 Closed-Form Orientation Angle Computation
13.7 Quaternion Processes
13.8 Partial Closed-form Tracking Solution
13.9 An Iterative Solution for Tracking
13.10 Orientation Quaternion
13.11 Position & Orientation
14. Computer Graphics
14.1 Introduction
14.2 Canonical Transformations
14.3 Transformations in R2
14.3.1 Scale in R2
14.3.2 Translation in R2
14.3.3 Rotation in R2
14.4 Homogeneous Coordinates
14.5 An Object in R2 Transformed
14.6 Concatenation Order in R2
14.7 Transformations in R3
14.8 What about Quaternions?
14.9 Projections R3 --& R2
14.9.1 Parallel Projections
14.9.2 Perspective Projections
14.10 Coordinate Frames
14.10.1 Perspective - Simple Case
14.10.2 Parallel Lines in Perspective
14.10.3 Perspective in General
14.11 Objects in Motion
14.11.1 Incremental Translation Only
14.11.2 Incremental Rotation Only
14.11.3 Incremental Rotation Quaternion
14.11.4 Incremental Rotation Matrix
14.12 Aircraft Kinematics
14.13 n-Body Simulation
Further Reading and References
Index

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