Introduction to particle filter Presented by: Amir Monemian Advisor : Dr Shiry

Why should you be interested in particle filters? great way to track the state of a dynamic system Bayesian model you can use particle filters to track your belief state

why should you use particle filters instead of Kalman filters? the main reason is that for a lot of large or high-dimensional problems, particle filters are tractable whereas Kalman filters are not. particle filters, which let you use the full, complex model, but just find an approximate solution instead.

discuss the formalization of the general problem Bayes Filtering used to discuss the method of using a predict/update cycle to estimate the state of a dynamical system from sensor measurements p(xt | do…t) p(zt | xt), p(xt | xt-1, ut-1) probability of xt given all the data we’ve seen so far X is the state variable Xt is the state variable at time t U is the inputs to your system perceptual model, z is the observations made by the sensors action model d just refers to inputs and observations together

What we are given is the inputs, the observations, the perceptual model, which is the probability that you’ll see a particular observation given that you’re in some state at time t, and the action model, which is the probability that you’ll end up in state xt at time t, assuming that you started in state xt-1 at time t-1, and input ut-1 to your system

the basic premise of particle filters The basic idea of particle filters is that any pdf can be represented as a set of samples (particles). the density of your samples in one area of the state space represents the probability of that region. This method can represent any arbitrary distribution, making it good for non-Gaussian, multi-modal pdfs. the key idea is that you find an approximate representation of a complex model (any arbitrary pdf) rather than an exact representation of a simplified mode (Gaussians).

how do you sample from your posterior? you had some belief from the last time step that you know how to update with your motion model. (prior belief q(x) ) sample from q(x), and then for each sample that you made, update it using what we will call an ‘importance weight’, based on the observations made.

the particle filter algorithm step by step To start the algorithm, we need the initial belief state, p(x0). This is just our initial guess of the pdf. For robot localization, if we have no idea, we can just scatter particles all over the map. then loop with three phases: prediction, update, and resample

the formalization of the steps calculating this equation from right to left

In the prediction step, we take each particle and add a random sample from the motion model. the update step , weight that is equal to the probability of observing the sensor measurements from that particle’s state. in the resample step, a new set of particles is chosen so that each particle survives in proportion to its weight.

Demo