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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:

$A=\left[ {\begin{array}{cc} 0 & 0 & 0 & a_x & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & a_y & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & a_z & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} } \right]$ ?

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A robot Jacobian is

The time derivative of the kinematics equations [...] which relates the joint rates to the linear and angular velocity of the end-effector.

I'll emphasize that it

relates the joint rates to the linear and angular velocity of the end-effector.

You give no information about a physical configuration, joints, or where the IMU is located. You give no information about what you're trying to do with the Jacobian.

You ask if your IMU model is correct, but then say, "any $\dot{q}_i$ is calculated using a given Formula which is okay so far." What is the formula? If it's working, then what is the question?

For the linear speed terms, you're failing to take into account the linear speed caused by rotation -

$$ v = \omega \times r \\ $$

It's also not clear what your terms are. You have $a_N$ - are these accelerometer readings? I ask because you have $\dot{v}_y = a_y + g$. This only works if $g$ is exactly colinear with the y-axis. Is this the case? No, unless your IMU is always perfectly level. Otherwise, you're going to need to apply a rotation matrix somewhere - either to rotate gravity to the local frame, or to rotate the local measurements (IMU measurements) to the global frame.

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