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,

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

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

$\displaystyle \dot{R} = \omega \times R .$ (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

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

Steven M LaValle 2012-04-20