In a robot or spacecraft attitude representation, using quaternion is relevant. The kinematic equation is as below
$$ \begin{bmatrix} \dot{q}_0 \\ \dot{q} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} -q^\top \\ q_0\,I_3 + q^\times \end{bmatrix} \omega=H(q) \omega $$
is there any way to solve this equation based on $\omega$ and $q$? actually I want to integrate the quaternion, but the nonlinear property of the equation makes it hard and complicate.
edit: as @fibonatic proved in this post, $$ \omega = 4\,H(q)^\top \dot{\textbf{q}} $$ so we can write $$ \omega.dt= 4\,H(q)^\top .d\textbf{q} $$ but I can't integrate $4\,H(q)^\top$ respect to $\textbf{q}$, because $\textbf{q}$ is a vector and I don't know how integrate based on a vector.
explaining why I want to do above calculation:
in a robotic problem I have below kinematic and dynamic equations: $$ \begin{cases} \dot{\textbf{q}} = H\omega \\ \dot{\omega} = -J^{-1} \omega^\times J \omega + J^{-1} u = f(\omega) + J^{-1} u \end{cases} \qquad \qquad (1) $$ the input %u% is designed such that, $\omega$ and $\textbf{q}$ follow a desired values, $\omega_{d}$ and $\textbf{q}_{d}$ // in a complicated process, I have define new $\Omega$ $$ \Omega = \nu(\omega) $$
which $\nu$ is a nonlinear function of $\omega$. I rederive the kinematic and dynamic equations for new $\Omega$ $$ \begin{cases} \dot{\textbf{q}_{new}} = H(q_{new})\Omega \\ \dot{\Omega} = g(\omega) + J^{-1} u \end{cases} \qquad \qquad (2) $$ which $\nu(\omega)$ is a nonlinear function of $\omega$. as you see, I have new quaternion calculation which differs from real quaternion in $equation (1)$. if we want to $\textbf{q}_{new}$ to reach $\textbf{q}_{d}$, it create steady state error in real quaternion, $\textbf{q}$. therefore, I should change the desired value of new quaternion, $\textbf{q}_{new\ d}$ such than real quaternion,$\textbf{q}$ reaches desired value $\textbf{q}_{d}$.
to find the $\textbf{q}_{new\ d}$, I should calculate $\dot{\textbf{q}_{new}} = H(q_{new})\Omega$
I wish, you understand what I want to calculate.
q(0)
andw(t)
and you want to find q(t) ? Then the problem is one dimensional integration based on the variablet
. Is it enough to find the result q(t) numerically ? Then the problem is quite straightforward. $\endgroup$