I have the following problem: given two Input Vectors $x = \begin{pmatrix}x\\y\\z\\v_x\\v_y\\v_z\\q_1\\q_2\\q_3\\q_4\end{pmatrix}$, $u = \begin{pmatrix}a_x\\a_y\\a_z\\w_x\\w_y\\w_z\end{pmatrix}$ time $t$ and gravity $g$ calculate $\dot{x}$. These vectors contain values recorded from an IMU.
for $\dot{x}$ I simply considered the motions of equation: $\dot{x} = f(x,u) = \begin{pmatrix}a_x*t+v_x\\a_y*t+v_y\\a_z*t+v_z\\a_x\\a_y+g\\a_z\\\dot{q_1}\\\dot{q_2}\\\dot{q_3}\\\dot{q_4}\end{pmatrix}$. Any $\dot{q_i}$ is calculated using a given Formula which is okay so far
Is my IMU-Model for $x$ and $\dot{x}$ correct? I need to calculate the Jacobian $A=\frac{\partial \dot{x}}{\partial x} $.
I dont really get to calculate $A$. it has to be $A \in \mathbb{R}^{10x10}$. But i always fail at calculating it, especially with respect to the quaternions.
My First 6 Lines look like that:
$\left[ {\begin{array}{cc} 0 & 0 & 0 & a_x & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & a_y & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & a_z & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ 0 & 0 & 0 & 0 & 0 & 0 & ? & ? & ? & ?\\ \end{array} } \right] $
Thanks for the help!
:EDIT:
As i just found out, any derivative $\frac{\partial \dot{q}}{\partial q}=0$. So i guess its:
$A=\left[ {\begin{array}{cc} 0 & 0 & 0 & a_x & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & a_y & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & a_z & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} } \right]$ ?