I am beginning to explore using probability in my robotics applications. My goal is to progress to full SLAM, but I am starting with a more simple Kalman Filter to work my way up.

I am using Extended Kalman Filter, with state as [X,Y,Theta]. I use control input [Distance, Vector], and I have an array of 76 laser ranges [Distance,Theta] as my measurement input.

I am having trouble knowing how to decide on the covariance to use in my Gaussian function. Because my measurements are uncertain (The laser is about 1cm accurate at < 1meter, but can be up to 5cm accurate at ranges higher) I do not know how to create the 'function' to estimate the probability of this. I know this function is supposed to 'linearize' to be used, but I'm not sure how to go about this.

I am reasonably confident on how to decide on the function for my state Gaussian, I am happy to use a plain old mean=0,variance=1 on this.. This should work no? I would appreciate some help from people understanding Kalman Filters, because I think I may be missing something.