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Simultaneous localization and mapping

Computational navigational technique used by robots and autonomous vehicles

  • 10 min read
  • 5 chapters

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SLAM algorithms are tailored to the available resources and are not aimed at perfection but at operational compliance. Published approaches are employed in self-driving cars, unmanned aerial vehicles, autonomous underwater vehicles, planetary rovers, newer domestic robots and even inside the human body.

Mathematical description of the problem

Given a series of controls ut and sensor observations ot over discrete time steps t, the SLAM problem is to compute an estimate of the agent's state xt and a map of the environment mt. All quantities are usually probabilistic, so the objective is to compute

P(mt+1,xt+1|o1:t+1,u1:t)

Applying Bayes' rule gives a framework for sequentially updating the location posteriors, given a map and a transition function P(xt|xt−1),

P(xt|o1:t,u1:t,mt)=∑mt−1P(ot|xt,mt,u1:t)∑xt−1P(xt|xt−1)P(xt−1|mt,o1:t−1,u1:t)/Z

where Z is the normalization constant, which ensures all the probabilities sum up to 1.

Similarly the map can be updated sequentially by

P(mt|xt,o1:t,u1:t)=∑xt∑mtP(mt|xt,mt−1,ot,u1:t)P(mt−1,xt|o1:t−1,mt−1,u1:t)

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 an expectation–maximization algorithm.

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