ABSTRACT
Chapter 3 addresses rotational sequences and the quaternions. Section 3.1 introduces some commonly used frames in spacecraft attitude determination and control, which includes body-fixed frame, ECI frame, LVLH frame, SEZ frame, NED frame, ECEF frame, PQW frame, and RSW frame. Section 3.2 describes the rotational sequences and the corresponding mathematical representations. Section 3.3 discusses the transformation between coordinate systems, and some important transformations are used as examples. Section 3.4 provides a brief introduction of quaternion and its properties. This section is particularly important because almost all concepts and results described in this section will be repeatedly used in the rest of the book. Section 3.4.1 defines equality and addition of quaternions. Section 3.4.2 introduces quaternion multiplication and the identity quaternion. Section 3.4.3 discusses the complex conjugation, the norm, and the inverse of the quaternion. Section 3.4.4 describes vector rotation by a quaternion operator. Section 3.4.5 gives the matrix form of quaternion production. And lastly, section 3.4.6 derives the derivative of the quaternion.
