26. Motion Model Probabiity I

Now we will practice implementing the motion model to determine P(location) for our Bayesian filter. We discussed the derivation of the model in **Recursive Structure
** and Implementation Details for Motion Model .

Recall that we derived the following recursive structure for the motion model:

and that we will implement this in the discretized form:

Let's consider again what the summation above is doing - calculating the probability that the vehicle is now at a given location, $x_t$ .

How is the summation doing that? It's looking at each prior location where the vehicle could have been, $x_{t-1}$ . Then the summation iterates over every possible prior location, $x_{t-1}^{(1)}…x_{t-1}^{(n)}$ . For each possible prior location in that list, $x_{t-1}^{(i)}$ , the summation yields the total probability that the vehicle really did start at that prior location and that it wound up at $x_t$ .

That now raises the question, how do we calculate the individual probability that the vehicle really did start at that prior location and that it wound up at $x_t$ , for each possible starting position $x_{t-1}$ ?

That's where each individual element of the summation contributes. The likelihood of starting at $x_{t-1}$ and arriving at $x_{t}$ is simply
$p(x_t|x_{t-1}) * p(x_{t-1})$ .

We can say the same thing, using different notation and incorporating all of our knowledge about the world, by writing: $p(x_t|x_{t-1}^{(i)}, u_t, m) * bel(x_{t-1}^{(i)})$

From the equation above we can see that our final position probability is the sum of n discretized motion model calculations, where each calculation is the product of the 'i'th transition probability, $p(x_t|x_{t-1}^{(i)}, u_t, m)$ ,
and 'i'th belief state, $bel(x_{t-1}^{(i)})$ . Let's try out a single, discreet calculation.

' i ' th Motion Model Probability:

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Show Solution

In the next concept, we will practice determining other values relevant to the motion model.