1
00:00:00,420 --> 00:00:11,500
‫So let's define a vector now, big Gummo, superscript E equals big X, big Y and big Z.

2
00:00:11,970 --> 00:00:15,570
‫This is our position vector in the inertial frame.

3
00:00:15,720 --> 00:00:20,430
‫And that's why you have this letter E here, because you can also call it the earth frame.

4
00:00:20,880 --> 00:00:28,650
‫If we take the time derivative of this vector, this output, that means time derivative, the change

5
00:00:28,650 --> 00:00:30,650
‫of this vector with respect to time.

6
00:00:30,930 --> 00:00:37,450
‫If we do that, then we obtain the velocity vector in the initial frame.

7
00:00:37,830 --> 00:00:43,850
‫So as you can see, all the elements here have dots now, X, Y and Z dot.

8
00:00:44,340 --> 00:00:54,690
‫And now let's define a velocity vector in the body frame, which will be small X dot, small Y dot and

9
00:00:54,690 --> 00:00:56,610
‫small Z dot.

10
00:00:57,030 --> 00:01:05,610
‫However, like we have already mentioned in this course, we will use this notation small U, small

11
00:01:05,610 --> 00:01:11,740
‫V in small W, so that will be our velocity vector in the body frame.

12
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‫In other words, you ve and W on the body frame velocities in the body frame X, Y and Z direction respectively.

13
00:01:24,310 --> 00:01:32,880
‫And now we use the rotation matrix to make the connection between the body frame and the inertial frame

14
00:01:33,180 --> 00:01:34,140
‫velocities.

15
00:01:34,500 --> 00:01:42,240
‫If you have this information and you want to go here and obtain this information, then you need this

16
00:01:42,480 --> 00:01:48,450
‫rotation matrix, our sub body frame, Z, Y, X.

17
00:01:48,450 --> 00:01:55,890
‫So we are using the oilor angle approach where rotating about the moving body frame first about the

18
00:01:55,890 --> 00:01:57,960
‫Z axis, then Y and then X.

19
00:01:58,540 --> 00:02:06,690
‫However, if we have this information here and we want to obtain this information will then very simply

20
00:02:07,080 --> 00:02:11,640
‫we take this rotation matrix and we take the inverse of it.

21
00:02:12,060 --> 00:02:20,220
‫And remember, since rotation matrices are also normal matrices, then that means that you can just

22
00:02:20,220 --> 00:02:24,330
‫take the transpose of this matrix.

23
00:02:24,750 --> 00:02:30,450
‫And if you apply it to this vector, then you will obtain this vector here.

24
00:02:31,020 --> 00:02:35,660
‫By the way, I will also add an arrow here and here.

25
00:02:35,880 --> 00:02:39,070
‫So it just means that we're dealing with vectors.

26
00:02:39,090 --> 00:02:46,230
‫Sometimes I forget these arrows, but if you really want to be very correct with notation, then you

27
00:02:46,230 --> 00:02:53,180
‫should put these arrows on top of these letters in order to say that you're dealing with vectors.

28
00:02:53,610 --> 00:03:04,320
‫And so this entire thing here can also be represented like this gamma dot, superscript e vector.

29
00:03:04,740 --> 00:03:06,550
‫And this is what you are looking for.

30
00:03:07,020 --> 00:03:12,030
‫And let's suppose that you have the velocity vector in the body frame.

31
00:03:12,600 --> 00:03:20,730
‫And since we are using this convention here and we want to get this rotation matrix here, then you

32
00:03:20,730 --> 00:03:28,920
‫can get this rotation matrix by first putting a rotation matrix about the initial frame X here, then

33
00:03:29,460 --> 00:03:38,430
‫a rotation matrix about the initial frame Y here, and then a rotation matrix about the inertia frame,

34
00:03:38,700 --> 00:03:39,800
‫the Z here.

35
00:03:40,380 --> 00:03:47,730
‫Remember, this triple multiplication was for this convention originally.

36
00:03:48,270 --> 00:03:56,880
‫When you rotate about the inertia frames, however, it is also suitable for our case because we are

37
00:03:57,240 --> 00:04:04,810
‫rotating about the moving frame and our sequences opposite to the one that happens in the inertia frame.

38
00:04:05,220 --> 00:04:13,830
‫So if you rotate about the inertia frame axis here in this sequence, X, Y, Z, then if you rotate

39
00:04:13,830 --> 00:04:21,000
‫about the body frame Z, Y, X axis, then the orientation of the body frame will be the same.

40
00:04:21,150 --> 00:04:27,240
‫And therefore you can use this multiplication for this case and for our case.

41
00:04:27,630 --> 00:04:33,870
‫And then you multiplied by the velocity vector in the body frame and you get your velocities in the

42
00:04:33,870 --> 00:04:34,670
‫inertia frame.

43
00:04:35,190 --> 00:04:45,000
‫And if you want to go back, then you simply take this matrix multiplication and you take the inverse

44
00:04:45,000 --> 00:04:46,830
‫of it or transpose.

45
00:04:47,340 --> 00:04:53,280
‫And from linear algebra, you know that you can also write it down like this.

46
00:04:53,550 --> 00:04:59,310
‫So if you have this triple multiplication here and you take the inverse of it, then if you want to

47
00:04:59,310 --> 00:04:59,670
‫open.

48
00:04:59,740 --> 00:05:06,340
‫These parentheses, then the order of these matrices will change, meaning that I can write it down

49
00:05:06,340 --> 00:05:18,940
‫like this are sub X inverse times or sub Y inverse times are sub Z inverse times.

50
00:05:19,600 --> 00:05:22,750
‫Our government dot in the inertia frame.

51
00:05:23,290 --> 00:05:25,510
‫So that comes just from linear algebra.

52
00:05:26,020 --> 00:05:27,720
‫And one final remark.

53
00:05:28,390 --> 00:05:38,200
‫Sometimes I can write this B letter or this E letter over here because I've seen both annotations and

54
00:05:38,200 --> 00:05:42,420
‫sometimes I use one notation, sometimes I use another notation.

55
00:05:42,670 --> 00:05:50,200
‫There is no difference whatsoever whether I put this B here or here, it doesn't matter.

56
00:05:50,740 --> 00:05:58,240
‫It's just to let you know, are we measuring the velocities in the body frame or are we measuring the

57
00:05:58,240 --> 00:06:01,020
‫velocities in the inertia frame?

58
00:06:01,480 --> 00:06:03,220
‫So don't get confused by that.

59
00:06:03,490 --> 00:06:12,070
‫If sometimes I write it like this, the same thing that if sometimes I don't put an arrow here, but

60
00:06:12,070 --> 00:06:15,980
‫from the context, you know, that it's a vector, then it's the same thing.

61
00:06:16,750 --> 00:06:27,120
‫So now you can clearly see why you need these rotation matrices you needed in order to get the velocities

62
00:06:27,190 --> 00:06:34,630
‫of the drone in the inertial frame, because initially you have them in the body frame, either from

63
00:06:34,630 --> 00:06:43,060
‫calculations or from sensors, and then you use the rotation matrices to have them in the initial frame

64
00:06:43,570 --> 00:06:52,090
‫and you need them in the inertia frame in order to compute your position in the inertia frame, something

65
00:06:52,090 --> 00:06:56,500
‫like this that we did in the previous course in this series.

66
00:06:56,860 --> 00:07:05,050
‫But more on that later in the next video, we will find out that we also need to connect the angular

67
00:07:05,050 --> 00:07:15,790
‫velocities in the body frame p, q, r with how the oilor angles change with respect time, fi, dot,

68
00:07:16,240 --> 00:07:18,370
‫theta dot inside dot.

69
00:07:18,700 --> 00:07:26,290
‫And in order to make this connection, we need to derive something called a transfer matrix.

70
00:07:26,740 --> 00:07:30,520
‫And in the next video we will derive that transfer matrix.

71
00:07:31,060 --> 00:07:39,730
‫And then you will also see that there is a difference whether you choose this convention or this oilor

72
00:07:39,730 --> 00:07:41,290
‫angle approach convention.

73
00:07:41,590 --> 00:07:48,850
‫Even though the rotation matrices for both of these cases are the same, the transfer matrix will be

74
00:07:48,850 --> 00:07:49,600
‫different.

75
00:07:50,050 --> 00:07:56,850
‫And so we will derive the transfer matrix for our oilor approach case.

76
00:07:57,370 --> 00:08:04,780
‫And once you understand the method of how to derive this transfer matrix, then you will see, and I

77
00:08:04,780 --> 00:08:13,000
‫will explicitly show you that the transfer matrix for this convention would be way different than what

78
00:08:13,000 --> 00:08:15,480
‫we will get for this convention here.

79
00:08:16,120 --> 00:08:19,300
‫So thank you very much and see you in the next video.