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← 02psps-01 Question 5 Help

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Showing Revision 1 created 10/24/2012 by Amara Bot.

  1. [Andy:] On the forums, we saw a lot of confusion about homeworks 2.5 and 2.6.
  2. Specifically, questions about what was going on with the linear algebra.
  3. I want to talk about some of that today, and I want to do it by comparing the 2D case --
  4. and that's the case we talked about in lecture--
  5. with the 4D case, which is what you're asked about on the homework.
  6. So in the 2-dimensional case, I want to first talk about this f matrix
  7. that Sebastian was calling the "state transition matrix."
  8. The idea behind this matrix was that we wanted to take some old beliefs,
  9. some old state, which in the 2-dimensional case was represented by x and ẋ
  10. where ẋ is our velocity and x is our position,
  11. and from that we want to extract our predict some new state,
  12. which was called x-prime and ẋ-prime.
  13. The question was what do we fill in here to get the proper values for x-prime and ẋ-prime.
  14. Let's think.
  15. What should our position be--our predicted position after some time has elapsed?
  16. Well, we want to include our old position, right?
  17. Lets first write out these formulas.
  18. We expect that x-prime will be composed of our old position
  19. plus whatever motion was occurring due to the velocity.
  20. That is going to be dt--the time elapsed--times our velocity.
  21. This is just velocity times time, which tells us how much our position has changed.
  22. Now, in matrix terms how do we express that?
  23. We're talking about x-prime, so that means we're going to think about this top row here.
  24. We want to keep x, which means a 1 goes here.
  25. We want to multiply ẋ by dt, so that means dt goes here.
  26. Just like that we figured out the first row of our F matrix.
  27. I'll label it here--F. Now what about the second row?
  28. Let's do a similar thing for ẋ prime to figure out where we should go in the second row.
  29. After our prediction, we said that we're just
  30. going to assume that the velocity hasn't changed.
  31. Velocity after equals velocity before.
  32. That means we don't want to have anything to do with this x, meaning a 0 goes here.
  33. We want everything to do with this ẋ--we want to keep this--so we put a 1 here.
  34. Okay. This kind of gives us some intuition for how this works in 2 dimensions.
  35. Let's see if we can generalize to 4.
  36. Now, again, we're going to some new state,
  37. and we're doing that by multiplying a state-transition matrix by some old belief.
  38. But now instead of x and ẋ, we also have y coordinates.
  39. So we have x, y, ẋ, and ẏ.
  40. Here we're going to, of course, get x, y, ẋ, ẏ, and all of those are prime,
  41. because they indicate after our prediction.
  42. Now, I'm not going to fill in this 4 x 4 matrix for you,
  43. but I think using similar reasoning to what we did in the 2-dimensional case,
  44. you can come up with what these formulas should be,
  45. and from that fill in this matrix appropriately,
  46. remembering that this entry corresponds to the first row,
  47. this entry the second, and so on. Good luck.