Differential rotations

To express the change in the moment of momentum in detail, the concept of a differential rotation is needed. In the plane, it is straightforward to define $\omega = {\dot \theta}$; however, for $SO(3)$, it is more complicated. One choice is to define derivatives with respect to yaw-pitch-roll variables, but this leads to distortions and singularities, which are problematic for the Newton-Euler formulation. Instead, a differential rotation is defined as shown in Figure 13.11. Let $v$ denote a unit vector in ${\mathbb{R}}^3$, and let $\theta$ denote a rotation that is analogous to the 2D case. Let $\omega$ denote the angular velocity vector,

$$\omega = v \frac{d\theta}{dt} .$$ (13.83)

This provides a natural expression for angular velocity.^13.9 The change in a rotation matrix $R$ with respect to time is

$$\dot{R} = \omega \times R .$$ (13.84)

This relationship can be used to derive expressions that relate $\omega$ to yaw-pitch-roll angles or quaternions. For example, using the yaw-pitch-roll matrix (3.42) the conversion from $\omega$ to the change yaw, pitch, and roll angles is

$$\begin{pmatrix}\dot{\gamma} \\ \dot{\beta} \\ \dot{\alpha} \\ \en... ...end{pmatrix} \begin{pmatrix}\omega_1 \\ \omega_2 \\ \omega_3 \\ \end{pmatrix} .$$ (13.85)