Integrated random walk === 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. pic a.svg Accelerations : Typical realization of 100 random accelerations. 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

FIGURE 2 Plot of acceleration, velocity and position, as well as the estimated fit, with respect to time.