# Kalman filter using accelerometer and system dyanamical model

After a lot of reading about the Kalman filter, I still cannot understand how to apply it when I have accelerometer measurements. Let's say my system is simply a body of mass $m$ on a 1-D world, and that a force $F$ is applied as a control input. So, my system is described by $\ddot x = \frac{F}{m} + w_{proc}$, where $w_{proc}$ is the process random noise. Putting this as a first order system, we get

$\dot x = v$

$\dot v = \frac{F}{m} + w_{proc}$

If our measurement device gave us a position or velocity measurement, applying the Kalman filter would be straightforward. However, what we have is an accelerometer that gives us a measurement $a_{meas}$, affected by the random noise $w_{meas}$. The problem is that this is not a measurment of any state, but rather of the derivative of a state. We could say that $\dot v = a_{meas} + w_{meas}$. But this means we now have two equations defining $\dot v$. How should I proceed?

I could resort to a different method, such as moving horizon estimation, but I am curious as to how to solve it using the Kalman filter.

$\dot x = v$
$\dot v = a$
$\dot a = \frac{u}{m} + w_{proc}$
where $u = \dot F$ is the input to the system. Now we have the acceleration $a$ as a state, whose measurement equation is $a = a_{meas} + w_{meas}$, and the Kalman filter can be applied in a straightforward fashion.