Why does the cyan cloud sometimes “collapse” to a few locations?

After resampling, many particles share the same ancestor state—that is normal when the posterior is sharp. If **σ_meas** is tiny, weights become peaked quickly; increasing measurement noise or motion noise spreads mass again.

Is the red ring the same as the MAP estimate?

No—it is the **posterior mean under the particle approximation**. With multimodal posteriors (common early on), the mean can sit between modes; the cloud itself is the richer object.

Why give the filter the true odometry delta?

This demo factors out **control-model error** so you can focus on **sensor geometry** and resampling behavior. A production stack would propagate **wheel ticks** or **IMU** deltas with their own bias and slip models.

Does systematic resampling guarantee diversity?

It reduces variance compared with multinomial resampling but can still copy the same high-weight particle many times if the posterior is extremely peaked—**jitter** or **roughening** (not drawn here) is used in practice after resampling.

Particle Filter Localization (MCL)

This page is a compact Monte Carlo localization (particle filter) sketch for a toy robot whose true trajectory is simulated for you while the filter only sees noisy range readings to fixed landmarks (orange markers) plus a noisy odometry increment each update. The motion model is deliberately simple: every filter step applies the same planar displacement the truth experienced between measurements, then adds isotropic Gaussian jitter independently to each particle so uncertainty grows under prediction. The measurement model treats each beacon range as Gaussian around the geometric distance with variance σ², so weights are proportional to a product of likelihoods (computed in log space with a max-subtraction trick for stability). When the effective sample size ESS = 1 / Σ w_i² falls below a user threshold times N, the ensemble is systematically resampled so duplicate particles do not silently starve diversity—watch the counter and the cyan cloud collapse or revive as you raise and lower σ_meas and σ_motion. A weighted mean (red ring) is shown for intuition; it is not a full Rao–Blackwellized map estimate.

Who it's for: Robotics or estimation students who have seen Bayes filters on slides and want a moving picture of predict → weight → resample before EKF or SLAM stacks.

Key terms

Particle filter
Monte Carlo localization
Importance sampling
Effective sample size
Systematic resampling
Odometry model
Range landmark
Sensor likelihood

How it works

Particle filter on a toy robot: odometry prediction and noisy range landmarks; ESS-triggered systematic resampling.

Frequently asked questions

Why does the cyan cloud sometimes “collapse” to a few locations?: After resampling, many particles share the same ancestor state—that is normal when the posterior is sharp. If σ_meas is tiny, weights become peaked quickly; increasing measurement noise or motion noise spreads mass again.
Is the red ring the same as the MAP estimate?: No—it is the posterior mean under the particle approximation. With multimodal posteriors (common early on), the mean can sit between modes; the cloud itself is the richer object.
Why give the filter the true odometry delta?: This demo factors out control-model error so you can focus on sensor geometry and resampling behavior. A production stack would propagate wheel ticks or IMU deltas with their own bias and slip models.
Does systematic resampling guarantee diversity?: It reduces variance compared with multinomial resampling but can still copy the same high-weight particle many times if the posterior is extremely peaked—jitter or roughening (not drawn here) is used in practice after resampling.

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Particle Filter Localization (MCL)

How it works

Frequently asked questions

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EKF SLAM (toy)

MPC Pendulum Swing-Up (MPPI)

A* / Dijkstra Pathfinding (Grid)

RRT Path Planner (grid)

Differential Drive Odometry

Holonomic 2D Hovercraft MPC (MPPI)

Particle Filter Localization (MCL)

How it works

Frequently asked questions