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

Princeton: Princeton University Press, 1999. — 391 p.Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the 19th century - a feat he celebrated by carving the founding equations into a stone bridge - mathematicians and engineers have been fascinated by these mathematical objects. They are used in applications as various as describing the geometry of space-time, 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.List of Figures.About This BookAcknowledgements.Historical Matters.Mathematical Systems.
Complex Numbers.
Polar Representation.
Hyper-complex Numbers.Algebraic Preliminaries.Complex Number Operations.
The Complex Conjugate.
Coordinates.
Rotations in the Plane.
Review of Matrix Algebra.
The Determinant.
The Cofactor Matrix.
Adjoint Matrix.
The Inverse Matrix - Method 1, 2.
Rotation Operators Revisited.Rotations in 3-space.Rotation Sequences in the Plane.
Coordinates in R³.
Rotation Sequences in R³.
The Fixed Axis of Rotation.
A Numerical Example.
An Application - Tracking.
A Geometric Analysis.
Incremental Rotations in R³.
Singularities in SO(3).Rotation Sequences in R³.Equivalent Rotations.
Euler Angles.
The Aerospace Sequence.
An Orbit Ephemeris Determined.
Great Circle Navigation.Quaternion Algebra.Quaternions Defined
Equality and Addition.
Multiplication Defined.
The Complex Conjugate.
The Norm.
Inverse of the Quaternion.
Geometric Interpretations.
A Special Quaternion Product.
Incremental Test Quaternion.
Quaternion with Angle θ = pi/6
Operator Algorithm.
Operator action on v = kq.
Quaternions to Matrices.
Quaternion Rotation Operator.
Quaternion Operator Sequences.Quaternion Geometry.Euler Construction.
Quaternion Geometric Analysis.
The Tracking Example Revisited.Algorithm Summary.
The Quaternion Product.
Quaternion Rotation Operator.
Direction Cosines.
Frame Bases to Rotation Matrix.
Angle and Axis of Rotation.
Euler Angles to Quaternion.
Quaternion to Direction Cosines.
Quaternion to Euler Angles.
Direction Cosines to Quaternion.
Rotation Operator Algebra.Quaternion Factors.Factorization Criteria.
Transition Rotation Operators.
The Factorization M=TA
Three Principal-axis Factors.
Factorization: g = st = (s_o +js₂)t.
Euler Angle-Axis Factors.
Some Geometric Insight.More Quaternion Applications.The Aerospace Sequence.
Computing the Orbit Ephemeris.
Great Circle Navigation.
Quaternion Method.
Reasons for the Seasons.
Seasonal Daylight Hours.Spherical Trigonometry.Spherical Triangles.
Closed-loop Rotation Sequences.
Rotation Matrix Analysis.
Quaternion Analysis.
Regular n-gons on the Sphere.
Area and Volume.Quaternion Calculus for Kinematics and Dynamics.Derivative of the Direction Cosine Matrix.
Body-Axes —> Euler Angle Rates.
Perturbations in a Rotation Sequence.
Derivative of the Quaternion.
Derivative of the Conjugate.
Quaternion Operator Derivative.
Quaternion Perturbations.Rotations in Phase Space.Constituents in the ODE Set.
The Phase Plane.
Some Preliminaries.
Linear Differential Equations.
Initial Conditions.
Partitions in R³ Phase Space.
Space-filling Direction Field in R³.
Locus of all Real Eigenvectors.
Non Autonomous Systems.
Phase Space Rotation Sequences.A Quaternion Process.Dipole Field Structure.
Electromagnetic Field Coupling.
Source-to-Sensor Coupling.
Source-to-Sensor Distance.
Angular Degrees-of-Freedom.
Quaternion Processes.
Partial Closed-form Tracking Solution.
An Iterative Solution for Tracking.
Orientation Quaternion.
Position & Orientation.Computer Graphics.Canonical Transformations.
Transformations in R².
Homogeneous Coordinates.
An Object in R² Transformed
Concatenation Order in R²?
Transformations in R³.
What about Quaternions?
Projections R³ > R².
Coordinate Frames.
Objects in Motion.
Aircraft Kinematics.
n-Body Simulation.Further Reading and References