Localising in a corridor

A belief over cells

The rover is somewhere in a 1.16 m corridor, and it doesn't know where. Cut the corridor into $N = 23$ cells of 5 cm. The belief $b(j)$ is the probability that the rover is in cell $j$; it starts flat, $b(j) = 1/N$.

What the rangefinders say

Two rangefinders look sideways. Shelves stick 2 cm out of both walls, so a beam reads about 4 cm at a shelf and 6 cm at bare wall, with $\sigma = 1$ cm of noise. EXPECTED[j] holds the ranges $[E_{j,r}, E_{j,l}]$ (right, left) you'd read in cell $j$: that's the map. The likelihood of cell $j$ is how well it explains a reading $z = [z_r, z_l]$:

Update: Bayes' rule

Cells that explain the reading gain probability and the others lose it. One noisy reading fits many cells, so the belief has many peaks at first. Foundations: Bayes' rule derives this.

Predict: the rover moved

After each 5 cm drive the rover is probably one cell on, but the wheels aren't exact: KERNEL $= [0.05, 0.9, 0.05]$ holds the probabilities that it really moved 0, 1 or 2 cells.

The far wall stops the rover, so probability that would pass cell $N-1$ stays there. Predicting blurs the belief; updating sharpens it. Only the true cell keeps explaining reading after reading.

The program

The rover starts somewhere random, reads, then drives 14 cells, predicting and updating after each. The belief floats over the corridor and appears in the Images tab; the Plots tab shows the readings.

Your task

Write likelihood(z), update(belief, like) and predict(belief). Each returns an (N,) array. The grader checks them on fixed cases, then checks that your final belief peaks within one cell of the rover.