TABLE OF CONTENTS In Chapter 3, we represent orientation by matrices in SO(3), which can be parameterized using three parameters. In this appendix, we describe some of the most popular methods of doing so, including Euler angles and angles with respect to a fixed frame. We also describe how orientation can be described as rotation about an arbitrary axis and by quaternions.
Recall that the Euler angles ?, ?, ? in chapter 3 correspond to successive rotations about body Z-Y-Z axes, and that the corresponding rotation matrix is obtained as
| (E.1) |  |
in which s ? and c ? denote sin ? and cos ? respectively.
Consider now the problem of using Euler angles to define a chart on some open set U ? SO(3). It is easy to see that a single chart cannot cover all of SO(3). For example, if R 33 = 1, it must be the case that ? = 0, and the rotation matrix is given by
| (E.2) |  |
In this case, it is not possible to uniquely define ? and ?, since only their sum is represented in R. A similar case occurs when R 33 = ?1.
To define a chart using Euler angles, we begin by defining the open set
and defining the chart ? such that
For any R ? U, not both of R 13, R 23 are zero. Then the...
