## 30.12.17

### SLAM Simultaneous Localization and Mapping

SLAM Simultaneous Localization and Mapping

In robotic mapping and navigationsimultaneous localization and mapping (SLAM) is the computational problem of constructing or updating a map of an unknown environment while simultaneously keeping track of an agent's location within it.

Given a series of sensor observations  over discrete time steps , the SLAM problem is to compute an estimate of the agent's location  and a map of the environment . All quantities are usually probabilistic, so the objective is to compute:
${\displaystyle P(m_{t},x_{t}|o_{1:t})}$
Applying Bayes' rule gives a framework for sequentially updating the location posteriors, given a map and a transition function ${\displaystyle P(x_{t}|x_{t-1})}$,
${\displaystyle P(x_{t}|o_{1:t},m_{t})=\sum _{m_{t-1}}P(o_{t}|x_{t},m_{t})\sum _{x_{t-1}}P(x_{t}|x_{t-1})P(x_{t-1}|m_{t},o_{1:t-1})/Z}$
Similarly the map can be updated sequentially by
${\displaystyle P(m_{t}|x_{t},o_{1:t})=\sum _{x_{t}}\sum _{m_{t}}P(m_{t}|x_{t},m_{t-1},o_{t})P(m_{t-1},x_{t}|o_{1:t-1},m_{t-1})}$
Like many inference problems, the solutions to inferring the two variables together can be found, to a local optimum solution, by alternating updates of the two beliefs in a form of EM algorithm.