This function computes the differential quaternion of measured rates after an amount of time. \\
Let $ \ra{} $ be the measured rates, then $\norm {\ra {}}$ represents the absolute value of rates. Therefore,
\begin{equation}
\Delta \alpha = \norm {\ra {}} \multiplication \Delta t
\end{equation}
is the rotational angle. The (normalized) axis of the rotation is then
\begin{equation}
\vect v = \frac{\ra{}}{\norm {\ra {}}}.
\end{equation}
The construction of a quaternion from an axis and an angle is
\begin{equation}
\quat{} = \begin{pmatrix}
\cos \tfrac \alpha 2 \\
\vect v \sin \tfrac \alpha 2
\end{pmatrix},
\end{equation}
so that the resulting quaternion of measured rates becomes
\begin{equation}
\quat{} = \begin{pmatrix}
\cos \tfrac{\norm {\ra {}} \multiplication \Delta t} 2 \\
 \frac{\ra{}}{\norm {\ra {}}} \sin \tfrac{\norm {\ra {}} \multiplication \Delta t} 2
\end{pmatrix}.
\end{equation}
\inHfile{FLOAT\_QUAT\_DIFFERENTIAL(q\_out, w, dt)}{pprz\_algebra\_float}