> ## Documentation Index
> Fetch the complete documentation index at: https://aegean.ai/llms.txt
> Use this file to discover all available pages before exploring further.
# SLAM: Simultaneous Localization and Mapping
> Simultaneous Localization and Mapping (SLAM) - building maps while tracking robot pose in unknown environments.
The **SLAM problem** asks: can a robot simultaneously build a map of an unknown environment and estimate its own trajectory within that map?
## Problem formulation
Given:
* The robot's controls $u_{1:t}$
* Observations of nearby features $z_{1:t}$
Estimate:
* The map $m$
* The path $x_{1:t}$
Formally, we want the posterior distribution
$$
p(x_{1:t}, m \mid z_{1:t}, u_{1:t}),
$$
which couples both localization and mapping. There are two principal formulations:
* **Full SLAM** estimates the entire path and the map:
$$
p(x_{1:t}, m \mid z_{1:t}, u_{1:t}).
$$
* **Online SLAM** estimates only the current pose and the map:
$$
p(x_t, m \mid z_{1:t}, u_{1:t}).
$$
Graphical models make the conditional dependencies explicit: the map, robot poses, controls, and observations are jointly dependent.
## Why SLAM is hard
1. **Coupled uncertainties**, errors in robot pose propagate into errors in the map, and vice versa.
2. **Data association**, observations must be correctly matched to landmarks. Wrong associations lead to catastrophic map corruption.
3. **Correlation of landmarks**, as Dissanayake et al. (2001) showed, in the limit all landmark estimates become fully correlated.
## Kalman filter for SLAM
The **Extended Kalman Filter (EKF)** is the classical solution (Smith & Cheesman, 1986). The state vector is:
$$
x_t = \begin{bmatrix}
x_t^r \\
m
\end{bmatrix},
$$
where $x_t^r$ is the robot pose and $m$ the landmark locations.
The update consists of:
* **Prediction**, propagate pose using the motion model:
$$
\hat{x}_t = f(x_{t-1}, u_t) + \epsilon_t.
$$
* **Correction**, incorporate the observation:
$$
K_t = \Sigma_t H_t^\top (H_t \Sigma_t H_t^\top + Q_t)^{-1},
$$
$$
x_t \leftarrow \hat{x}_t + K_t (z_t - h(\hat{x}_t)),
$$
$$
\Sigma_t \leftarrow (I - K_t H_t)\Sigma_t.
$$
The EKF maintains correlations between all landmarks through the covariance matrix.
### Properties of EKF-SLAM
* Complexity $O(n^2)$ in the number of landmarks
* Proven convergence for linear cases
* Diverges if nonlinearities are severe
* In the limit, landmarks become fully correlated
## Techniques for consistent maps
* **Scan matching**, align consecutive scans by maximizing their likelihood under a relative pose.
* **EKF-SLAM**, model the posterior as a joint Gaussian over robot pose and landmarks.
* **FastSLAM**, factorize the problem using Rao-Blackwellisation (Montemerlo et al., 2002).
* **Graph-SLAM / SEIFs**, represent constraints sparsely in graph or information form.
## Approximations and alternatives
* **Submaps** (Leonard et al., 1999; Bosse et al., 2002), partition environment into local maps.
* **Sparse links** (Lu & Milios, 1997; Guivant & Nebot, 2001), reduce correlations.
* **SEIF** (Sparse Extended Information Filter), exploit sparsity in the information matrix.
* **FastSLAM** (Montemerlo et al., 2002), factorize into a particle filter over robot trajectories and EKFs for landmarks.
## Summary
SLAM is fundamentally a **Bayesian estimation problem**. Classical EKF-SLAM maintains correlations but is quadratic in complexity. FastSLAM and Graph-SLAM exploit structure for efficiency. Data association and uncertainty coupling remain the central challenges.
***
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