1
00:00:00,520 --> 00:00:01,370
‫Welcome back.

2
00:00:02,080 --> 00:00:08,350
‫So we have just covered an enormous section in how to derive the kinematics and dynamics, fundamental

3
00:00:08,350 --> 00:00:16,210
‫equations we learned about rotation and transfer matrices in order to connect and be able to transfer

4
00:00:16,210 --> 00:00:19,870
‫information between the inertial and body frames.

5
00:00:20,410 --> 00:00:29,560
‫And then I showed you how the Newton oilor formulation in the body frame for a six degree of freedom

6
00:00:29,560 --> 00:00:31,480
‫system is achieved.

7
00:00:32,080 --> 00:00:36,820
‫It was a big section, however, it was also a necessary one.

8
00:00:37,570 --> 00:00:44,020
‫Without that foundation, you would not fully understand the Oves plan model.

9
00:00:44,950 --> 00:00:52,480
‫And so now I am going to derive a specific plan model for our USV.

10
00:00:52,780 --> 00:00:59,740
‫Using the results from the previous section, you will see that you have already done the hard part.

11
00:01:00,610 --> 00:01:07,180
‫Applying stuff from the previous section to our USV will be very easy.

12
00:01:08,170 --> 00:01:09,610
‫So let's get onto it.

13
00:01:10,960 --> 00:01:20,530
‫In the end of the last section, we had a Newton oilor formulation that looked like this, and then

14
00:01:20,860 --> 00:01:31,420
‫we explored an alternative form for it, which looks like this, this equation is the exact same equation

15
00:01:32,110 --> 00:01:34,030
‫like this first one.

16
00:01:34,390 --> 00:01:37,860
‫The only thing is that it is written in a different form.

17
00:01:38,800 --> 00:01:50,290
‫So this matrix M in the B frame, it was this matrix here, this entire one, then this vector here

18
00:01:50,830 --> 00:01:52,540
‫would be this one.

19
00:01:53,700 --> 00:01:58,560
‫Then this vector here would be this entire thing here.

20
00:01:59,870 --> 00:02:07,550
‫And the biggest difference was the second term that appeared because the body frame rotated with respect

21
00:02:07,550 --> 00:02:08,750
‫to the inertia frame.

22
00:02:09,750 --> 00:02:20,280
‫So we took this term here and we converted it into this term here and now we are going to take this

23
00:02:20,280 --> 00:02:23,610
‫alternative form and work with that.

24
00:02:24,680 --> 00:02:33,010
‫Now, in principle, our goal is the same that we had when we worked with the vehicle in case of a vehicle,

25
00:02:33,290 --> 00:02:36,320
‫we went from the equations of motion.

26
00:02:37,780 --> 00:02:39,670
‫That looked like this.

27
00:02:40,800 --> 00:02:45,660
‫To the states based representation that looked like this in red.

28
00:02:46,750 --> 00:02:53,920
‫If you remember then the A, B, C, D, E and F.

29
00:02:55,000 --> 00:03:03,850
‫Constance were composed of many variables related to the vehicle, and since all those variables were

30
00:03:03,850 --> 00:03:17,140
‫themselves constant, including X Dot and I in the B frame, then D, A, B, C, D, E, F variables

31
00:03:17,140 --> 00:03:19,210
‫were also fixed values.

32
00:03:19,600 --> 00:03:21,490
‫And that's why we call them constants.

33
00:03:22,470 --> 00:03:30,540
‫Not that they don't have to be constant, they are constant in our case, thanks to assuming that our.

34
00:03:31,830 --> 00:03:40,090
‫X double dot or our longitudinal acceleration was zero meters per second squared.

35
00:03:40,320 --> 00:03:49,410
‫In other words, our longitudinal velocity was constant and that's why our A, B, C, D, e, f variables

36
00:03:49,560 --> 00:03:50,550
‫were constants.

37
00:03:51,550 --> 00:03:58,810
‫We learned that state's basic equations are first order differential equations because you only have

38
00:03:58,810 --> 00:04:07,600
‫there and input Delta states, why dad and BPCI that.

39
00:04:08,670 --> 00:04:17,070
‫And the first time derivative of the states, why double dot inside, double dot y double that can be

40
00:04:17,070 --> 00:04:22,580
‫rewritten in this form here and BPCI double dot in this form here.

41
00:04:23,460 --> 00:04:32,220
‫And when you take that first order differential equation and you express the time derivative of the

42
00:04:32,220 --> 00:04:39,450
‫states as a function of states themselves and the inputs like this.

43
00:04:40,520 --> 00:04:53,150
‫Why double date as a function of why that side dot and Delta equals something and then double that as

44
00:04:53,150 --> 00:04:55,550
‫a function of why dot?

45
00:04:56,620 --> 00:05:01,750
‫Cidade and Delta equals something else.

46
00:05:03,010 --> 00:05:05,290
‫Then you have a state based equation.

47
00:05:06,660 --> 00:05:19,140
‫In general variables, it would look like this if you assign X one to Y dot, X to topside that and

48
00:05:19,140 --> 00:05:21,750
‫then you to Delta.

49
00:05:23,050 --> 00:05:26,440
‫Then this is how it would look like like this.

50
00:05:27,590 --> 00:05:37,340
‫So even though you have a double time derivative here, then don't confuse that thing, because remember

51
00:05:37,340 --> 00:05:41,890
‫that other states, they already have one time derivative.

52
00:05:42,230 --> 00:05:49,580
‫So when you have a double time derivative here, then the difference is only one derivation with respect

53
00:05:49,580 --> 00:05:55,580
‫to time, and that's why it's a first order differential equation.

54
00:05:56,590 --> 00:06:04,690
‫If instead of why, that you would simply have why and then here you had why double that, then it would

55
00:06:04,690 --> 00:06:11,320
‫not be a first order differential equation, then it would be a second order differential equation,

56
00:06:11,320 --> 00:06:18,750
‫especially if you had another term, like, for example, why that here?

57
00:06:18,790 --> 00:06:23,080
‫So you would have y y dot and then y double dot.

58
00:06:23,770 --> 00:06:30,100
‫That wouldn't be the first order differential equation and therefore it wouldn't be a state based equation.

59
00:06:30,250 --> 00:06:38,290
‫A state's basic equation is a first order differential equation where you have the states, the inputs

60
00:06:38,740 --> 00:06:46,690
‫and then the time derivatives of the states and you express the time derivatives of the states as a

61
00:06:46,690 --> 00:06:50,050
‫function of the states and the inputs.

62
00:06:51,690 --> 00:06:56,310
‫And now we want to have the same thing in our USV case.

63
00:06:57,450 --> 00:07:07,050
‫This is our Newton oilor formulation, the alternative form of ID Lambda B that has three net force

64
00:07:07,050 --> 00:07:19,790
‫elements in it affects F, Y and Z and it has three net moment elements in it and X and Y and then Mzee.

65
00:07:20,340 --> 00:07:29,190
‫And remember, these are all net values, meaning that each element here is a summation of forces in

66
00:07:29,190 --> 00:07:33,960
‫their respective dimensions or moments in their respective dimensions.

67
00:07:34,800 --> 00:07:38,220
‫Then this is our state vector.

68
00:07:39,180 --> 00:07:41,580
‫In this state, Victor has six elements.

69
00:07:41,880 --> 00:07:45,870
‫Three for translational velocity in the body frame.

70
00:07:46,850 --> 00:07:58,610
‫You've N.W. meters per second and three, four angular velocities in the body frame, so P, Q and R.

71
00:07:59,640 --> 00:08:09,480
‫Radians per second, this vector looks like this, and then the time derivative of our states is this

72
00:08:09,480 --> 00:08:10,590
‫vector here.

73
00:08:11,730 --> 00:08:13,110
‫That's how it looks like.

74
00:08:14,070 --> 00:08:20,790
‫The first three elements here are translational acceleration in the body frame and then the last three

75
00:08:21,000 --> 00:08:24,170
‫are the angular acceleration in the body frame.

76
00:08:24,930 --> 00:08:28,740
‫And so what we want to do is essentially the same thing.

77
00:08:29,130 --> 00:08:36,920
‫We want to express the state time derivatives in terms of their states and inputs.

78
00:08:37,890 --> 00:08:46,650
‫So in other words, we want to put our Newton oilor equations of motion in the state's space form.

79
00:08:47,550 --> 00:08:54,440
‫So the data vector B equals everything else.

80
00:08:55,560 --> 00:09:04,260
‫And so you have an exercise now you have this equation here, this alternative Newton oilor form.

81
00:09:05,290 --> 00:09:13,430
‫And so your task now is to take this entire form and you have to put it in the same space form.

82
00:09:14,260 --> 00:09:16,620
‫Now, don't bother with specifics.

83
00:09:16,990 --> 00:09:22,960
‫This is a one minute exercise and you can do it in two or three operations.

84
00:09:23,950 --> 00:09:28,110
‫I just want you to do it before I show it to you in the next video.

85
00:09:28,630 --> 00:09:30,130
‫So give it a try.

86
00:09:30,340 --> 00:09:31,280
‫Thank you very much.

87
00:09:31,300 --> 00:09:32,800
‫And I see you in the next video.