The big idea: a swarm of guesses

Imagine scattering 1000 tiny paper boats on a map, each one a guess of where your robot might be. Then repeat:

1. Move every boat as the robot moved, plus a little random jitter (because odometry is noisy).
2. Score every boat by how well it explains the sensor reading. A boat in a spot where the sensor would read 3.1 m when it actually read 3.0 m gets a high score; one that would read 7 m gets almost zero.
3. Resample: throw away low-scoring boats and make copies of high-scoring ones.

After a few rounds the boats cluster around the true position. If two places explain the data equally well, you get two clusters, and the filter keeps both alive until new data breaks the tie. A Kalman filter would be forced to average them into one guess in between, which may be inside a wall.

This is a Monte Carlo approximation of the Bayes filter:

Each particle \(x^{(i)}\) is a hypothesis; each weight \(w^{(i)}\) is how much we believe it. More particles means a better approximation and more computation.