7 – KALMAN Kalman Filter Design RENDER V2

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 of an object. We know that new location is the old location plus velocity to in this new this matrix. You have a 1 over here, and a 1 over here. The new velocity should just be the old velocity because the 0 over here, and a 1 over here. If you multiply this matrix with this vector, this is exactly what you’re getting. For the measurement, you only observe the first component of the place not the velocity, and that uses a vector or matrix like this. So, this matrix will be called F, and this H. The actual update equations for Kalman Filter are involved. I give them to you, but please don’t memorize them and I won’t prove them for you, even the prove is very involved. Every time I use images, look them up. There’s a prediction step where I take my best estimate x multiplied with the state transition matrix F, and I add whatever motion I know, u. That gives me my new x. I also have a covariance that characters my uncertainty. That is updated as follows, where T is the transpose. There’s also a measurement update step where we use the measurement z. We compare the measurement with our prediction where H is the measurement function that maps the state to measurements. We call this the error. The error is mapped into a matrix S which is obtained by projecting the system uncertainty into the measurement space using the measurement function protection, plus a matrix R, the character is a measurement noise. This is then mapped into a variable called K which is often called the common gain, where we invert the matrix S. Then finally, we actually update our estimate and our uncertainty using what ought to be the most cryptic equation that you’ve seen in a long time. Now, vote this down so that you have a complete definition, but this is something you should not memorize. If you really wish to understand, this math happens to be just a generalization of the math I gave you to higher dimensional spaces. But I would recommend just not to worry about this. There’s a set of linear algebraic equations that implement the Kalman Filter in higher dimensions.

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