hard

SLAM

Build a map while localizing within it — occupancy grids, pose uncertainty, and loop closure.

Localization assumes you already have a map. But what if you don’t? SLAM — Simultaneous Localization And Mapping — has the robot build a map of an unknown environment while tracking its own pose inside that same, half-built map. It is the technology behind self-mapping vacuums, warehouse robots, and indoor drones.

Below, a robot drives a loop through a room, firing range beams at each pose. Cells a beam passes through are carved as free; the cell it strikes is marked occupied — together they fill an occupancy grid. The violet ellipse is the robot’s growing pose uncertainty. Drive it around and watch the loop closure at the end snap the drift away.

odometry drift = 1.0×

Speed

pose 1/64

true pose estimate ± uncertainty beam hit occupied cell

uncertainty 0.5 cells

Each pose the robot fires range beams: cells a beam passes through are carved as free, the cell it strikes is marked occupied, and the grid sharpens into a map. Meanwhile the estimated pose drifts from the truth as odometry error builds (violet ellipse grows). When the robot returns to a place it has seen before, a loop closure snaps the estimate back and shrinks the uncertainty — that is the heart of SLAM.

The chicken-and-egg problem

To build a good map you need to know where you are; to know where you are you need a map. SLAM does both at once, estimating the joint posterior over the whole path and the map together:

p\big(x_{1:t},\, m \mid z_{1:t},\, u_{1:t}\big).

Solving for both couples them: every new observation refines the map and the pose estimate. There is no fixed reference, so errors in one feed into the other.

Occupancy grids

A common map is the occupancy grid: the world is diced into cells, each holding the probability $p(m_i)$ that it is occupied. Beams update cells with a sensor model and are accumulated in log-odds, which makes repeated evidence a simple sum:

\ell_i \;\leftarrow\; \ell_i + \log\frac{p(m_i \mid z_t)}{1 - p(m_i \mid z_t)}.

Free, occupied, and unknown cells emerge as many noisy scans average out. The map is clean only if the poses that placed those scans were accurate.

Drift and loop closure

Pose comes from odometry — integrating wheel motion — but small errors compound, so the estimate drifts ever further from the truth (the ellipse swells). The fix is loop closure: when the robot recognizes a place it has already mapped, it adds a constraint tying “here” back to “there.” That single link lets the whole trajectory be re-optimized, snapping accumulated error out of both the path and the map.

Modern systems treat this as a graph: nodes are poses, edges are motion and loop-closure constraints, and a least-squares solver finds the configuration that best satisfies them all — graph-based (or pose-graph) SLAM. Filtering approaches like EKF-SLAM instead track poses and landmarks in one growing covariance matrix.

Takeaways

SLAM estimates the path and the map jointly — neither is known in advance.
Occupancy grids store per-cell occupancy probabilities, updated in log-odds from range scans.
Odometry drift accumulates; loop closure recognizes revisited places and corrects the whole trajectory, the defining trick of SLAM.

References

Thrun, Burgard & Fox, Probabilistic Robotics — Ch. 9 (occupancy grids), Ch. 10–13 (EKF and graph SLAM).
Grisetti, Kümmerle, Stachniss & Burgard, A Tutorial on Graph-Based SLAM — pose graphs and optimization.