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title: "Integrated random walk"
date: "2015-10-06"
---
The uncertainty in estimating the position using a noisy accelerometer increases with time.
Let us model the acceleration as a uniformly random real number $a(t) = \operatorname{rand}(-1, 1)$. Suppose you have a random walk where $v(t) = \sum_{i=1}^t a(i)$. This is the velocity. Then, we would like to investigate $x(t) = \sum_{i=0}^t v(i)$, which is the position.
To do this, we conduct a numerical experiment. We repeat the simulation $n$ times and then plot the RMS value of the $n$ trials at each time step.
By inspection, we see that $v(t)\approx c_1 t^{0.5}$ and $x(t) \approx c_2 t^{1.5}$ where $c_1$ and $c_2$ are constants.
Number of trials: 100
Mean: 0
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FIGURE 2 Plot of acceleration, velocity and position, as well as the estimated fit, with respect to time.
