3.1 Introduction

As discussed in Chapter 2, the direction cosine matrix (DCM) is the tool that transforms vectors from one coordinate frame to another. The elements of this matrix are the direction cosines of the principal axes of one frame to the other. However to construct the DCM from the direction cosines is not only unattractive, but also inconvenient. One reason is that the orthonormality property is lost and with it the length and orientation of the transformed vector. To preserve this property one can use the rotation matrix to construct the DCM. That is, to find the angle of rotation and the axis around which the moving frame has rotated. Two very closely related methods fall under this category: the rotation vector and the quaternion. Another elegant approach is to use the Euler angles. This approach is based on using a sequence of simple rotations about independent axes rather than one single rotation; the DCM is then the product of simple rotation matrices. It will be shown that rotations about the principal ( x, y, z) axes take very simple and standard formats. All we need is to know the amount of rotation associated with each axis to form these DCMs.

This chapter has two main objectives. The first is to introduce the quaternion and show how to use it to construct the DCM. We will describe the means for converting from one form of transformation to another, e.g. converting from Euler angles to quaternions. With...