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.

FIGURE 1 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.