# A brief description (by chapter) of the book QUATERNIONS & ROTATION SEQUENCES

## Chapter 01 - Historical Matters

The quaternion is presented as an element in the set of all numbers. The history of the quaternion as a hyper-complex extension of the familiar complex number of rank 2 is discussed. Those readers with strong or sufficient mathematical background may wish to peruse the first three chapters rather lightly.

## Chapter 02 - Algebraic Preliminaries

A brief review of some basic algebraic operations, required in later chapters, is presented. Simple but fundamental rotations in the plane, R2 are introduced. Those with a background in these matters may go directly to Chapter 4 or Chapter 5.

## Chapter 03 - Rotations in 3-space

Rotations in the plane are extended and generalized to define rotations in R3. Various notations are adopted, and sometimes used interchangeably in order to emphasize or specifically focus on certain important attributes of rotation operators. Algorithms are developed for relating independent coordinate frames in R3.

## Chapter 04 - Rotation Sequences in R3

Rotation sequences are defined and developed. Their underlying ideas are extended in some applications in later chapters. The important notions of a closed-loop rotation sequence, and equivalent sequences, fundamental when developing mathematical models for certain applications in kinematics and dynamics, are introduced. Applications of the Aerospace and other Euler angle rotation sequences are considered in some detail in this and later chapters.

## Chapter 05 - Quaternion Algebra

The algebraic properties of the quaternion are defined. The use of the quaternion in rotation operators is developed in detail, and the geometric interpretation of the quaternion-based rotation operator is discussed. This rotation operator is the primary application of the quaternion in applied mathematics. \newpage

## Chapter 06 - Quaternion Geometry

The composite quaternion axis and angle for a two-rotation quaternion operator sequence are determined. When applied to the familiar tracking example, the expected result which was achieved algebraically is confirmed geometrically.

## Chapter 07 - Algorithm Summary

Here we summarize the algorithms which relate the alternative matrix and quaternion rotation operators. Included are direction cosines, Euler angles, quaternion operators, rotation matrices, eigenvalues, and eigenvectors. This gives a more comprehensive view of these matters when addressing applications.

## Chapter 08 - Quaternion Factors

Given composite rotations in either matrix or quaternion form, the factorization of rotation matrices and quaternion rotation operators is investigated. This is motivated by possible real application requirements and/or their constraints.

## Chapter 09 - More Quaternion Applications

Aerospace and other alternative Euler angle sequences are related, using rotation matrices and quaternion operators. Great circle navigation and Orbit ephemeris determination are studied. In this same context, some simple celestial earth-sun models, which explain and model the seasons of the year, are developed.

## Chapter 10 - Spherical Trigonometry

The development of familiar (some not so familiar) formulas in spherical trigonometry are obtained, using rotation sequences of both rotation matrices and quaternion rotation operators. Interesting exercises provide some comparative insight into these analytical alternatives.

## Chapter 11 - Calculus for Kinematics/Dynamics

The time-derivative of the direction-cosine rotation matrix and of the quaternion and the quaternion rotation operator are derived. The propagation of errors through rotation sequences is introduced and developed. Examples are studied. Body axes angular rates are related to their resulting Euler angle angular rates. The perturbation method is introduced as an alternative for deriving various angular rate relationships. Orbit ephemeris and orbit parameter sensitivities are developed and applied in an example, of some current interest, in order to demonstrate the analytical techniques.

## Chapter 12 - Rotations in Phase Space

A new comprehensive view into the solution space for Ordinary Differential Equations is presented. After some preliminary review of the conventional and familiar phase plane techniques, we develop in some detail the R3 geometry of the solution space, within which reside the solutions of all 2nd order ODE's: Linear, Non-linear, Autonomous and Non-Autonomous!

## Chapter 13 - A Quaternion Process

A mathematical model for an electromagnetic six degree-of-freedom transducer, first conceived, defined, developed and patented by the author, is presented in detail. This new technology is used in several USAF Aerospace applications, as well as in a variety of commercial applications. Its design strategy and mathematical model are developed, both in terms of rotation matrices and quaternion rotation operators.

## Chapter 14 - Computer Graphics

The current interest in computer generated images begs for efficient rotation operators or incremental rotation operators. Perspectives on these matters are presented using the quaternion rotation operator. After a review of the canonical computer graphics operators in homogeneous coordinates the design of an application in Virtual Reality is considered. The six degree-of-freedom transducer, which was discussed in Chapter 13, is implicitly required and integrated in this design.

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