### MathJax

$$\large \left [ w, \mathbf{v}\right ] \\ \mathbf{v} = \left [ x, y, z \right ]$$

$\arctan \left ( \frac{0.5}{100.0} \right )$

The IMU is measuring both the gravitational acceleration, g, and the angular motion, $\omega$, around three axis.

The rotation around x, $\theta _{x}$, is a function of the changes in the angles of the y and z axis, $a _{y}$ and $a _{z}$ respectively, and the angular movement of the x axis, $\omega _{x}$. This can be generalized to the function for any axis, k, $\theta _{k}$ as:
$\large \theta _{k} = f( a _{k-1}, a _{k+1}, \omega _{k} )$
where the decrements and increments of k indicate the previous and next axis.

However, as z rotates there is no change in the orientation of the x and y axis. They remain horizontal so their measure of g remains constant. The expression for $\theta _{z}$ reduces to:
$\large \theta _{z} = f(\omega _{z} )$
which is rather frustrating.
$$\label{eq:gravt} F=G\frac{mM}{r^2}.$$

$\ref{eq:gravt}$

$\large[w, x, y, z]$