Grid Localisation
This is where we model the world as a 2D or 3D array. We can then store the probability that we are in any one location in this array.
Piecewise Constant Representation
Markov localisation utilised a probability distribution across all states:
- Represented as $\text{Bel}(x_t)$ for each $x_t$.
We can model this discretely as a histogram (bins):
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If there are $N$ rooms, then we can model this as:
\[p(n)\forall n\in N\]
The sum over all bins should be:
\[\sum x_t=1\]We can then calculate like so:
- Start with a uniform distribution (if the location is unknown), or bounded Gaussian (if a location is known).
- The robot observes a feature:
- The Initial belief is proportional to the observation likelihood.
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The histogram then represents likely locations of the robot using:
\[P(z_t\mid x_t)\]
- The robot now moves:
- We convolve out current distribution with our motion model.
- This translates the histogram and smooths peaks.
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This is represented as:
\[\sum_{x_{t-1}}P(x_t\mid u_t,x_{t-1})Bel(x_{t-1})\]
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The final distribution is the product of the observation model and resulting distribution with the shifted motion model:
\[Bel(x_t)=\eta P(z_t\mid x_t)\sum_{x_{t-1}}P(x_t\mid u_t,x_{t-1})Bel(x_{t-1})\]$\eta$ is a normalisation factor that ensures $\sum x_t=1$.