I have some issues with understanding the Particle Filter for navigation through a known map. So, consider a situation where I want to write a Particle filter to navigate through a maze or a map that is known beforehand. I also know where is the target (exit from the maze/map). So, basically the idea is quite simple, since the map is known I can easily know where I should move on next.
The fields on the map have randomly assigned integers to them, so I can use them as observations. The only unknown thing is basically the random start point of the robot.
So, generally I have the following algorithm, which is a quite classic Bootstrap Filter I guess:
1.Init Sample N samples from p(x0) Assign all the weights value of 1/N 2.Importance Sampling For 1 to N perform sampling xt from P(xt|xt-1) For 1 to N assign weight based on P(yt|xt) 3.Resample with replacement based on weights
The only thing that I don't understand is how should the P(xt|xt-1) works in this case. I assume no noise, in this case, to make it easier. I mean, since the map is known beforehand and the target is known too, this means that there should be only one desired xt for each xt-1, but I am not sure how to capture that. Will be glad for some guidance.
EDIT: So basically the idea is that the observations(yt) are as I said some numbers that are on the floor in the maze (each field has only 1 number), but the numbers can be duplicated between multiple fields. Xt is in this case position of the robot as it's randomly generated and unknown to the robot.