Why is data association “given for free” here?

Real robots must solve **permutation** ambiguity when landmarks look alike. Fixing IDs keeps the Jacobian dimensions small and the failure modes (permutation errors) separate from the EKF linearization story.

Why can the map drift even when measurements keep arriving?

The EKF keeps only a **single Gaussian** peak. Nonlinear motion and measurement models plus repeated linearization can introduce **bias**; inconsistent covariances (underestimated uncertainty) make drift worse. This toy exposes the coupling but does not implement loop closure or batch nonlinear refinement.

What does the amber ellipse around a landmark mean?

It is a quick **2×2 marginal** visualization of the EKF covariance on that landmark’s **(x, y)** coordinates after projecting the full **9×9** matrix—useful for intuition, not a rigorous uncertainty polygon.

Why world-frame odometry noise instead of a bicycle model?

A planar bicycle or unicycle model is more realistic but adds bookkeeping. Shared **Δx, Δy, Δθ** noise keeps the motion update transparent so the lesson stays on **stacked Jacobians** and **map updates**.

EKF SLAM (toy)

This page implements a planar augmented-state EKF for a toy SLAM problem with three point landmarks whose coordinates sit in the same state vector as the robot pose (x, y, θ). A scripted ground-truth trajectory (smooth Lissajous-like motion) drives the robot while the filter receives noisy odometry increments in the world frame and range + bearing measurements to each landmark with known data association (each sensor reading is labeled with the correct landmark ID). The map is deliberately initialized wrong with a large positional covariance so you can watch the map–trajectory coupling characteristic of EKF SLAM: landmark rows of the covariance matrix become correlated with the robot as the vehicle moves and re-observes features. Updates use the standard linearized measurement Jacobian for polar observations about the current mean, a 2×2 innovation covariance with inversion in closed form, and the usual P ← (I − K H) P covariance propagation (teaching-grade numerics—symmetrized after each landmark pass). Compare the green truth robot and map with the red EKF pose and amber map estimates plus faint 2σ ellipses from the landmark marginal covariances.

Who it's for: Students who finished a Kalman filter homework and want a minimal, moving EKF SLAM picture before tackling graph-based SLAM, loop closure, or unknown correspondence.

Key terms

Extended Kalman filter
SLAM
Augmented state
Landmark
Range and bearing
Data association
Innovation covariance
Map–robot correlation

How it works

Augmented-state EKF on a planar toy: three landmarks in the state vector, noisy range & bearing updates, odometry prediction — compare EKF map and pose to ground truth.

Frequently asked questions

Why is data association “given for free” here?: Real robots must solve permutation ambiguity when landmarks look alike. Fixing IDs keeps the Jacobian dimensions small and the failure modes (permutation errors) separate from the EKF linearization story.
Why can the map drift even when measurements keep arriving?: The EKF keeps only a single Gaussian peak. Nonlinear motion and measurement models plus repeated linearization can introduce bias; inconsistent covariances (underestimated uncertainty) make drift worse. This toy exposes the coupling but does not implement loop closure or batch nonlinear refinement.
What does the amber ellipse around a landmark mean?: It is a quick 2×2 marginal visualization of the EKF covariance on that landmark’s (x, y) coordinates after projecting the full 9×9 matrix—useful for intuition, not a rigorous uncertainty polygon.
Why world-frame odometry noise instead of a bicycle model?: A planar bicycle or unicycle model is more realistic but adds bookkeeping. Shared Δx, Δy, Δθ noise keeps the motion update transparent so the lesson stays on stacked Jacobians and map updates.

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