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Quaternions and Rotation Sequences: A Review of Jack B. Kuipers' Book
Quaternions are a type of hyper-complex numbers that have four components: a real part and three imaginary parts. They were discovered by William Rowan Hamilton in 1843, who carved his famous equations on a stone bridge in Dublin. Quaternions have many applications in mathematics, physics, engineering, computer graphics, and robotics, especially for representing rotations in three-dimensional space.
In his book Quaternions and Rotation Sequences, Jack B. Kuipers introduces and defines the quaternion, gives a brief introduction to its properties and algebra, and shows its primary application: the quaternion rotation operator. The quaternion rotation operator competes with the conventional matrix rotation operator in a variety of rotation sequences, such as Euler angles, yaw-pitch-roll angles, and gimbal angles. Kuipers also discusses the advantages and disadvantages of using quaternions over matrices for rotation computations.
The book is divided into three parts: Part I covers the basics of quaternions and their algebraic operations; Part II explores the quaternion rotation operator and its applications in different rotation sequences; Part III extends the concepts of quaternions to other areas, such as spherical trigonometry, attitude determination, celestial mechanics, and computer graphics. The book is written in an accessible and engaging style, with many examples, exercises, figures, and historical notes. It is suitable for undergraduate and graduate students, as well as researchers and practitioners who want to learn more about quaternions and their applications.
The book can be downloaded as a PDF file from the following link: [^1^]. It is also available in hardcover and paperback editions from various online retailers.
Quaternions have many modern uses in various fields that require three-dimensional rotations. For example, in astronautics, quaternions are used to represent the orientation of spacecraft and satellites, and to control their attitude using thrusters or reaction wheels. Quaternions avoid the problem of gimbal lock that can occur when using Euler angles or rotation matrices. Quaternions also allow for smooth and continuous interpolation of orientations, which is useful for animation and simulation. [^2^] [^3^]
In computer graphics, quaternions are used to rotate objects and cameras in virtual scenes. Quaternions can also be used to create realistic movements and special effects for characters and objects in movies and video games. For example, quaternions can be used to implement skeletal animation, where a hierarchy of bones and joints is used to animate a character. Quaternions can also be used to implement inverse kinematics, where the position of a joint is calculated from the position of an end-effector, such as a hand or a foot. [^1^]
In robotics, quaternions are used to represent the orientation of robots and their parts, such as arms, legs, and grippers. Quaternions can also be used to plan and execute motions for robots, such as reaching, grasping, and manipulating objects. Quaternions can also be used to estimate the pose (position and orientation) of a robot or a camera from sensor data, such as images or inertial measurements. [^2^]
One surprising application of quaternions is to electric toothbrushes. Tooth brushing is a blind process: you cannot easily see what you are doing or gauge the results of brushing. Quaternions have been used in the design of a system that tracks the position of a toothbrush in the mouth relative to the userâs teeth. The system uses a wireless sensor attached to the toothbrush handle that measures its orientation using quaternions. The sensor communicates with a smartphone app that displays a 3D model of the userâs teeth and shows which areas have been brushed and which need more attention. The system aims to improve oral hygiene and prevent dental diseases. aa16f39245