# 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.

One possible way to tackle this is defining the acceleration as another state, and taking the derivative of the force, rather than the force itself, as the control input. The system would therefore be:

$\dot x = v$

$\dot v = a$

$\dot a = \frac{u}{m} + w_{proc}$

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.