How can I use the Runge-Kutta integration method for quaternions
Kinematics (https://arxiv.org/pdf/1711.02508.pdf)
Incremental rotation $\Delta \theta = \omega_n \Delta t$
$ Exp(\omega \Delta t) = q\{\omega \Delta t\} = e^{\omega \Delta t /2 } = \begin{bmatrix} cos(||\omega|| \Delta t/2)\\ \frac{\omega}{||\omega||} sin(||\omega|| \Delta t/2)\end{bmatrix}$
Difference equation $q_{n+1} = q \otimes q\{\omega \Delta t\}$
Differential equation $\frac{dq}{dt} =\dot{q} = \frac{1}{2} q \otimes \omega$
Runga-Kutte of 4th order
$x_{n+1} = x_n + \frac{\Delta t}{6} \big( k_1 + 2k_2 + 2k_3 + k_4 \big)$
$k_1 = f(t_n, x_n)$
$k_2 = f(t_n + \frac{1}{2} \Delta t,\; x_n + \frac{\Delta t}{2} k_1 )$
$k_3 = f(t_n + \frac{1}{2} \Delta t,\; x_n + \frac{\Delta t}{2} k_2 )$
$k_4 = f(t_n + \Delta t,\; x_n + \Delta t \cdot k_3 )$
How can i apply the differential equation $\dot{q}$ to the Runga-Kutta method?