SLAM Tutorial¶

SLAM, simultaneous localization and mapping, is one of the defining problems of robotics because it forces two uncertainties to interact: the robot does not know exactly where it is, and it does not know the map perfectly either.

Learning goals¶

This chapter is meant to make the following ideas concrete:

why SLAM is harder than localization alone
how pose uncertainty and map uncertainty influence each other
how EKF-SLAM and graph-SLAM think about the problem differently
why loop closure is so powerful
what computational structure makes SLAM expensive in practice

Where SLAM is used¶

SLAM is central in robotics whenever a robot must operate in an environment that is not perfectly known in advance:

autonomous vehicles
drones
mobile service robots
warehouse and logistics robots
handheld mapping systems
AR and spatial computing systems

The joint estimation problem¶

SLAM estimates both trajectory and map:

\[ p(x_{1:t}, m \mid z_{1:t}, u_{1:t}) \]

That is why the problem is harder than localization. Errors in pose corrupt map estimates, and map errors feed back into later pose estimates.

The central conceptual jump is that pose and map cannot be treated as independent. Good SLAM methods succeed because they preserve and exploit that coupling.

EKF-SLAM: a probabilistic filter view¶

EKF-SLAM maintains a joint Gaussian over robot pose and landmarks. Prediction propagates the motion model and grows covariance. Measurement updates use the Kalman gain to reduce uncertainty when observations arrive.

The core update looks like:

\[ K = P H^T (H P H^T + R)^{-1} \]

followed by the usual mean and covariance correction. The key structural point is that pose-landmark cross-covariances are not incidental; they are the thing that lets a better pose estimate improve landmark estimates and vice versa.

Cost profile¶

EKF-SLAM is conceptually elegant, but it becomes expensive as the number of landmarks grows. Dense covariance structure pushes both memory and runtime upward.

At a high level:

memory can grow quadratically with the number of landmarks
updates can also become increasingly expensive because the covariance is large

That is one reason why graph-based formulations became so influential.

Graph SLAM compared to EKF-SLAM¶

Formulation	Main object	Strength	Main cost issue
EKF-SLAM	joint Gaussian belief	direct probabilistic interpretation	dense covariance growth
Graph SLAM	sparse optimization problem	strong global consistency and scalability	iterative nonlinear optimization

Graph SLAM: a sparse optimization view¶

Graph SLAM reframes the problem as sparse nonlinear least squares. Robot poses and landmarks become nodes, and measurements become constraints. The optimizer tries to minimize the residual energy across the whole graph.

That is why loop closure is so powerful in graph formulations: one good constraint can redistribute error across a long accumulated trajectory.

Cost profile¶

Graph-based methods are attractive because many SLAM problems are sparse:

each observation links only a small subset of variables
many constraints are local
sparse linear algebra scales better than dense covariance updates in many settings

That does not make graph SLAM free, but it often makes larger problems more tractable than dense filter formulations.

What to look at in the simulator¶

The most important things to watch are not just the final map shape, but the intermediate behavior:

pose errors distort the map
map errors distort future pose estimates
loop closure is valuable because it corrects both at once
drift accumulates before strong corrections arrive
landmark placement changes with trajectory quality

Why SLAM is such an important teaching topic¶

SLAM forces several robotics ideas into one place:

uncertainty propagation
data association
numerical optimization
linearization
memory scaling
delayed global correction

That makes it one of the best examples of why robotics is rarely just geometry or just probability or just control. It is usually all of them together.

Practical tradeoffs¶

Filter viewpoint¶

often conceptually closer to recursive estimation
good for understanding uncertainty propagation directly
can become expensive as the state grows

Graph viewpoint¶

often better for larger maps and longer trajectories
makes loop closure especially natural
depends heavily on optimization quality and data association

Complexity and memory summary¶

Formulation	Main representation	Typical strength	Main scaling concern
EKF-SLAM	joint Gaussian	clear probabilistic interpretation	dense covariance growth
Graph SLAM	sparse factor graph	strong global correction and scalability	nonlinear optimization cost

Try this¶

Experiment 1: Watch drift form¶

Start the demo and let the robot move.
Pay attention to how small pose errors accumulate into visible map distortion.

Experiment 2: Look for correction events¶

Focus on revisits or repeated observations.
Notice when the map and trajectory become more consistent again.
Ask what new information enabled that correction.

Experiment 3: Compare local consistency and global consistency¶

Watch the map before a strong correction event.
Then watch how the entire structure changes after new evidence arrives.
Notice that good global consistency may require revising earlier local beliefs.

Experiment 4: Think about cost¶

Imagine scaling the same problem to many more landmarks and poses.
Ask whether dense covariance or sparse graph structure seems more practical.
Use the simulator behavior as a prompt for algorithmic scaling intuition.

Common mistakes when studying SLAM¶

treating map building and localization as separate problems
ignoring cross-correlation structure
focusing only on final map appearance
assuming loop closure is a cosmetic improvement instead of a structural correction
forgetting that data association quality can dominate everything else

What this chapter is really teaching¶

SLAM is the point where uncertainty becomes geometric and geometry becomes probabilistic. If you understand why pose errors corrupt maps and why map corrections improve pose, you understand the heart of the SLAM problem.