The equations for an accelerometer can be put in a state space form for analysis and simulation. The basic mass-spring-damper system equations
$$m\ddot{x}+b\dot{x}+kx=0$$ can be re-written as the two first-order equations,
$$\begin{matrix}
\dot{x}=\dot{x} \\
\ddot{x}=-\frac{b}{m}\dot{x}-\frac{k}{m}x+F
\end{matrix}$$
From these equations we can see that our two states are $x$ and $\dot{x}$. We then make the state space model from these equations and states
$$\begin{bmatrix}
\dot{x} \\ \ddot{x}
\end{bmatrix}=
\begin{bmatrix}
0&1\\
-\frac{k}{m}&-\frac{b}{m}
\end{bmatrix}
\begin{bmatrix}
x\\\dot{x}
\end{bmatrix}+
\begin{bmatrix}
0\\1
\end{bmatrix}
F$$
Our sensor model is then proportional to acceleration time a conversion factor $C$
$$y=C\ddot{x}=
\begin{bmatrix}
0&C
\end{bmatrix}
\begin{bmatrix}
\dot{x}\\\ddot{x}
\end{bmatrix}$$
This output can then be multiplied out into the state-space model above to get the output measurement model
$$y=\begin{bmatrix}
-\frac{k}{m}C & -\frac{b}{m}C
\end{bmatrix}
\begin{bmatrix}
\dot{x}\\\ddot{x}
\end{bmatrix}+
CF
$$
[[Piezoelectric Accelerometer State-Space Model]]
[[Capacitive Accelerometer State-Space Model]]
[[X-38 Certification]] – used 1st order accelerometer model
- candyAccelerometerMode
Backlinks:
[[Accelerometer]]
[[State-Space Model]]