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.

Accelerations — FIGURE 1 Typical realization of 100 random accelerations.

To do this, we conduct a numerical experiment. We repeat the simulation times and then plot the RMS value of the 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

—
— $t^{0.5}$
—
— $t^{1.5}$

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