14 – KALMAN QUIZ Gaussian Motion 01 RENDER V2

So, let’s step a step back and look at what we’ve achieved. We knew there was a measurement update, an emotion update, which is also called prediction. We know that the measurement update is implemented by multiplication which is the same as Bayes rule. The motion update is done by total probability or an addition. So, we tackle the more complicated case, this is actually the hop from mathematically, and we solve this, we give an exact expression. We even derived mathematically, and you were able to write a computer program that implements this step of the Kalman Filter. I don’t want to go into too much depth here. This is a really, really easy step. Let me write it down for you. Suppose you live in a world like this, this is your current best estimate of where you are and this is your uncertainty. Now, say you move to the right side, a certain distance, and that motion itself has its own set of uncertainty. Then you arrive at a prediction that adds the motion command to the mean, and it has an increased uncertainty over the initial uncertainty. Intuitively, this makes sense. If you move to the right by this distance in expectation you’re exactly where you wish to be, but you’ve lost information because your motion tends to lose information as manifest by this uncertainty over here. Now, the math for this is really, really easy. A new mean, is your old mean plus the motion often called u. So, if you move over 10 meters, this will be 10 meters and you knew sigma square is your old sigma square plus variance of the motion Gaussian. This is all you need to know, it’s just an addition, and I won’t prove it to you because it’s really trivial. But in summary, we have a Gaussian over here. We have a Gaussian for the motion with u as the mean and r square has its own motion uncertainty. The resulting Gaussian in the prediction step just adds these two things up, mu plus u and sigma square plus r square. Since it was so simple, let me quiz you, we have a Gaussian before the prediction step which mu equals 8 and sigma square equals 4. We then move to the right, a total of 10 with a motion uncertainty of 6. Now, describe to me the predicted Gaussian and give me the new mu and the new sigma square.

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