Simultaneously localization and mapping (SLAM) is a chicken-and-egg problem.

Localization Estimate the path of the robot, i.e., a sequence of poses and locations,

\[x_{0:T} = {x_{0}, x_{1}, x_{2}, ..., x_{T}}\]

Mapping Build a map of the environment,

\[m\]

SLAM Simultaneously estimate the path of the robot and build a map of the environment given the robot’s controls

\[u_{1:T} = {u_{1}, u_{2}, ..., u_{T}}\]

and observations

\[z_{1;T} = {z_{1}, z_{2}, ..., z_{T}}\]

Probabilistic approaches

The real word is full of uncertainty, especially when we talk about robot perception and robot control. Graphical models are commonly used to represent uncertainty.

Graphical models

Full SLAM estimates the full robot’s path and the map:

\[p(x_{0:T}, m | z_{1:T}, u_{1:T})\]

Online SLAM seeks to recover only the most recent pose:

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

It means marginalizing out the previous poses (recursively):

\[p(x_{t}, m | z_{1:t}, u_{1:t}) = \int_{x_0} ... \int_{x_{t-1}} p(x_{0:t}, m | z_{1:t}, u_{1:t}) dx_{t-1} ... dx_{0}\]

Motion Model

How does the robot move?

\[p(x_{t} | x_{t}-1, u_{t})\]

Example: Standard Odometry model

• Robot moves $ (x, y, \theta) $ -> $ (x’, y’, \theta’) $

• Odometry information $ u = (\delta_{rot1}, \delta_{trans}, \delta_{rot2}) $

Observation Model

How do I interpret the observations?

\[p(z_{t} | x_{t})\]

Example: Standard Observation Model

Side notes

Why SLAM is HARD?

• Both the map and robot path are unknown, and estimations of map and robot path are correlated
• Observations to map association is also unknown. Wrong data association could cause catastrophical consequences

Taxonomy of the SLAM problem

• Volumetric vs. feature-based SLAM
• Topologic vs. geometric maps
• Known vs. unknown correspondence
• Static vs. dynamic environments
• Small vs. large uncertainty
• Active vs. passive SLAM
• Any-time and any-space SLAM
• Single-robot vs. multi-robot SLAM

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