Sample Odometry Motion Model
Rejection Sampling
If we have a function $f$ we can sample random values from this function using the following method:
- Sample $x$ from a uniform distribution $[-b,b]$.
- Sample $y$ from $[0,\max f]$
- If $f(x)>y$ then keep the sample.
- Otherwise reject.
Sampling vs Odometry
- Sampling
- Determines a predicted new pose based on an initial pose $x_{t-1}$ and the motion $u_t$.
When sampled multiple times, a point cloud is gained, representing the probability distribution for possible end locations.
- Odometry
- Provides a closed form expression for generating the probability of being in a location. This is computing the posterior: $p(x_t\mid u_t,x_{t-1})$.
Map-Consistent Motion Model
We can rewrite the bayesian filter to include background knowledge of a map. The motion model becomes:
\[p(x_t\mid x_{t-1}, u_t, m)\]graph LR
p((Past)) --> xt-1
xt-1 --> xt --> xt+1
ut-1 --> xt-1 --> zt-1
ut --> xt --> zt
ut+1 --> xt+1 --> zt+1
zt-1 --- m
zt --- m
zt+1 --- m
xt+1 --> a((future))
Knowlege about features like walls, change the resulting probabilities.