Particle Filter for Robot Localization

Unlike the Kalman filter — which represents belief as a single Gaussian ellipse — the particle filter represents the robot's belief as a cloud of $N$ random samples (particles). Each particle is a hypothesis: "maybe I am at pose $(x_i, y_i, \theta_i)$." This non-parametric approach handles multi-modal beliefs and non-Gaussian noise naturally.

Each particle $i$ carries a weight $w_i$ proportional to how likely it is given the latest sensor measurement $z$:

$w_i \propto p(z \mid x_i)$

After resampling, each particle is drawn with probability $w_i / \sum_j w_j$, so high-weight particles survive and low-weight ones are discarded.

The normalised mean pose estimate:

$\hat{x} = \sum_{i=1}^{N} w_i\, x_i$

1. Initialise: scatter $N$ particles uniformly across the map (global uncertainty) or near a known start.
2. Predict (motion update): move every particle by the motion command $u$, plus random noise drawn from the motion noise model.
3. Update (sensor update): compute weight $w_i = p(z \mid x_i)$ for each particle (how well does the sensor reading match if the robot were at $x_i$?).
4. Normalise weights: $w_i \leftarrow w_i / \sum_j w_j$.
5. Resample: draw $N$ new particles with replacement proportional to weights. Cloud collapses toward the true pose.

The Roomba vacuum uses a simplified particle filter (Monte Carlo Localisation) to know where it has cleaned. Google's self-driving car used 1 million particles at startup for "kidnapped robot" recovery. Particle filters also power face tracking in phone cameras.

1. If all particles have equal weight after the sensor update, what does this tell us about the quality of the sensor reading?
2. Why might a particle filter outperform an EKF in a corridor with two identical-looking junctions?