Page Author: [Shubodh Sai](mailto:[email protected]?Subject=The%20Notion%20Note-taking%20Project%20|%20SLAM%20Backend) and Udit Singh Parihar

Using this Notion page

• Read this to know how to use this page the right way.

Resources

• Resources

The Complete SLAM Problem

$$ P (X_{t}, M | U_{t-1}, Z_{t}) $$

The Pose Graph SLAM problem

$$ P (X_{t}| U_{t-1}) $$

• Motion Model:

  $$ U_{T} = [u_{0}, u_{1}, ..., u_{T}] $$

  $$ x_{t+1} = f(x_{t}, u_{t}) + w_{t} $$

  $$ w_{t} \: \sim \: \mathcal{N}(0, \Sigma_{t}) $$

  $$ x_{t+1} \: \sim \: \mathcal{N}(f(x_{t}, u_{t}), \Sigma_{t}) $$

• Pose Graph:

Pose Graph Representation

Odometry constraints:

$$ x_{i+1} \: \sim \: \mathcal{N}(f(x_{i}, u_{i}), \Sigma_{i}) $$

Loop closure constraints:

$$ x_{j} \: \sim \: \mathcal{N}(f(x_{i}, u_{ij}), \Lambda_{ij}) $$

<aside> 💡 Both of these are actually what makes it an optimization problem!

</aside>

The Optimal Set of Robot Poses:

$$ X^* = \underset{X}{argmax}\medspace \: P(X|U) $$