\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.

$$

\begin{equation}

\label{eq:gravt} F=G\frac{mM}{r^2}.

\end{equation}

$$

\[\ref{eq:gravt}\]

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