4-6. Matrices and Transformation of State

These are the common filter equations in all their beautiful complexity. But what do they all mean.? Before we jump into all the matrix math behind these equations, I’d like to just do some basic visual cleanup. And to do that, I want to make three modifications to these equations. Let’s just focus on this … Read more

For the rest of this course, we are going to focus on building a working knowledge of matrix math, which is also known as linear algebra. But what do I mean by working knowledge? As Sebastian has repeatedly said, it is not important to memorize equations, but as a self-driving car engineer, it will be … Read more

So, when you design a Kalman Filter, you need effectively two things. For the state, you need a state-transition function, and that’s usually a matrix. So, we’re now in the world of linear algebra. For the measurements, you need a measurement function. So, let me give you those for our example of the 1D motion … Read more

When you put all this together, you find that all these possibilities on the Gaussian over here link to a Gaussian that’s just like this. This is a really interesting two-dimensional Gaussian which you should really think about. Clearly, if it was to project this Gaussian uncertainty into the space of possible locations, I can’t … Read more

And just like before, it’ll be over here with velocity of two initial position of one, we find ourselves in three. And again, this model assumes that in the absence of more knowledge, diversity shouldn’t really change.

Let me consider a velocity of two, which means this is our starting point. And let me ask you, where you would expect among those choices to be the most plausible prediction to be?

The answer is right over here. Why? If our current starting point is the point over here, for which we know the location is one, and the velocity is one, and we predict one time step in the future. Then for that prediction, we know the location will be two, and the velocity might be … Read more

In Kalma Filter then, we’re going to build a two-dimensional estimate. One for the location and one for the velocity denoted x. But it also can be 0, it can be negative, or it can be positive. If initially, I know my location but not my velocity, then I represented Gaussian as elongated around the … Read more

This is a lot of math and I get it. I can tell you I don’t myself remember all these formulas. I have to look them up. I remember in early days with Udacity, I made a course on Robotics where I couldn’t remember how to invert matrices so I go with it. I stole … Read more

To explain how this works, I have to talk about high-dimensional Gaussians. These are often called Multivariate Gaussians. The mean is now a vector with one element for each of the dimensions. The variance square is replaced by what’s called a covariance and it’s a matrix with D rows and D columns if the dimensionality … Read more