1 00:00:00,630 --> 00:00:04,350 ‫OK, but now let's go back to our Akutan method. 2 00:00:05,630 --> 00:00:14,300 ‫You have your first order, Odie's, here, or ordinary differential equations, and let's just assume 3 00:00:14,300 --> 00:00:16,550 ‫that we know our current status. 4 00:00:17,640 --> 00:00:20,760 ‫This is our current state vector in the body frame. 5 00:00:21,880 --> 00:00:28,990 ‫And these are our current state vectors in the initial frame, and now we're going to use the wrong 6 00:00:28,990 --> 00:00:36,220 ‫Akutan method to find these state vectors, but at K plus one. 7 00:00:37,230 --> 00:00:39,510 ‫And to do that, we follow the steps. 8 00:00:40,600 --> 00:00:51,430 ‫We first compute the first slope's for our state vectors, this, this and this, so we need to get 9 00:00:51,850 --> 00:01:01,480 ‫the time derivative of this state vector, which will be the dot at K, then we need to get government 10 00:01:01,630 --> 00:01:03,850 ‫dot at K. 11 00:01:04,810 --> 00:01:09,850 ‫And then we need to get see Dot at Kay. 12 00:01:10,870 --> 00:01:17,700 ‫That's what we need to get and of course, this is in the body frame and it's very easy to get them, 13 00:01:18,850 --> 00:01:24,220 ‫it's very easy because you have all your state values here at Kay. 14 00:01:25,300 --> 00:01:34,600 ‫You take all the state values and you plug them into these equations, you plug them into the body frame 15 00:01:34,600 --> 00:01:42,790 ‫equations and on the right side, you're going to plug them into the rotation and transfer matrices. 16 00:01:43,770 --> 00:01:50,250 ‫And also into these vectors as well, you, VW and P, Q, R. 17 00:01:51,240 --> 00:01:55,410 ‫So you have everything you need to compute all the 12 derivatives here. 18 00:01:56,420 --> 00:02:00,470 ‫And just like that, you have all these time derivatives. 19 00:02:01,530 --> 00:02:07,740 ‫Now, that's the same thing like it was for the oilor approach, but now that's where the difference 20 00:02:07,740 --> 00:02:08,380 ‫comes in. 21 00:02:09,420 --> 00:02:19,050 ‫Now what I will do, I will take the Oilor method and I will compute a new state, but not at time T 22 00:02:19,050 --> 00:02:20,470 ‫plus T. 23 00:02:20,540 --> 00:02:29,610 ‫S instead, I'm going to compute new states at time T plus T. 24 00:02:29,640 --> 00:02:31,290 ‫S over to. 25 00:02:32,470 --> 00:02:39,190 ‫So here I will make a graph for one state as an example, but the same logic applies to other states 26 00:02:39,190 --> 00:02:39,670 ‫as well. 27 00:02:40,900 --> 00:02:49,690 ‫So I have the slope here that I computed at time equals T. So the slope is for this point here. 28 00:02:50,750 --> 00:02:58,280 ‫And then I project it until T plus takes over to to hear. 29 00:02:59,320 --> 00:03:04,030 ‫So this is the first slope and piece of cake. 30 00:03:04,510 --> 00:03:08,950 ‫So it's this red line here, the red straight line. 31 00:03:09,920 --> 00:03:19,790 ‫And so when I projected till this point, then it's the same thing like using the oil or method to compute 32 00:03:19,790 --> 00:03:24,620 ‫a new state, but until this time here. 33 00:03:25,730 --> 00:03:28,610 ‫So I'm going to cold this point a little bit differently. 34 00:03:29,590 --> 00:03:31,540 ‫I'm going to say that this point is. 35 00:03:32,510 --> 00:03:36,890 ‫P, sub K plus one. 36 00:03:38,010 --> 00:03:43,680 ‫But then here in the index, I'm going to put T plus T. 37 00:03:43,710 --> 00:03:56,250 ‫S over to so that you would know that this point is computed at this time period here at T plus T is 38 00:03:56,250 --> 00:04:06,750 ‫over to and I'm going to put one here, which means that I computed this point using the first slope. 39 00:04:08,040 --> 00:04:10,560 ‫So mathematically, it will look like this. 40 00:04:11,770 --> 00:04:20,710 ‫So this point pops up, Kate, plus one at T plus two or two using the first slope. 41 00:04:21,680 --> 00:04:33,830 ‫It's this point here, you get it if you take your present, state your piece up cake and then plus 42 00:04:34,820 --> 00:04:43,340 ‫and then here I'm going to put the first derivative p dot subquery, which is this slope here and then 43 00:04:43,460 --> 00:04:45,410 ‫times T. 44 00:04:45,410 --> 00:04:49,290 ‫S over to it makes sense, right? 45 00:04:49,880 --> 00:04:59,660 ‫You have your slope here that you computed at time equals t where your present states are that slope 46 00:04:59,660 --> 00:05:03,980 ‫at this point and then you projected it till T plus T. 47 00:05:03,980 --> 00:05:13,160 ‫S over to and you reached here and mathematically you computed with this formula your present state 48 00:05:13,610 --> 00:05:16,880 ‫plus the derivative of your state. 49 00:05:16,880 --> 00:05:27,050 ‫With respect to time times, it's over two, because this interval here is your Ts over to and that's 50 00:05:27,050 --> 00:05:28,040 ‫where you put it here. 51 00:05:29,150 --> 00:05:36,980 ‫And then you do the same thing for all the other states, and here they are, you see this is your first 52 00:05:36,980 --> 00:05:37,580 ‫slope. 53 00:05:38,530 --> 00:05:48,190 ‫And this is your new state, but at T plus two or two, and it was computed using the first slope and 54 00:05:48,190 --> 00:05:52,120 ‫you will see in a moment why I use this notation. 55 00:05:53,420 --> 00:06:04,250 ‫And then what you do, you take all these new states that you have at T plus PTS over to using the first 56 00:06:04,250 --> 00:06:07,730 ‫slope, you take all these new states. 57 00:06:08,900 --> 00:06:18,890 ‫And you're going to put them here, so whenever you see a state variable, here we are CU or Phi Theta 58 00:06:19,310 --> 00:06:24,450 ‫PSI, you replace them all with these new states here. 59 00:06:25,160 --> 00:06:30,380 ‫So all these new states, they will go into these equations here. 60 00:06:31,520 --> 00:06:35,210 ‫And when you do that, you get a new set of slope's. 61 00:06:36,190 --> 00:06:39,460 ‫You will have another 12 slope's here. 62 00:06:40,560 --> 00:06:45,930 ‫Another 12 time derivatives and these will be your second slope's. 63 00:06:47,190 --> 00:06:51,540 ‫And so we need to somehow name our second slope's. 64 00:06:52,500 --> 00:06:54,360 ‫And I'm going to name them like this. 65 00:06:55,450 --> 00:06:58,990 ‫P dot, sub K plus one. 66 00:07:00,130 --> 00:07:02,770 ‫At T plus T. 67 00:07:02,800 --> 00:07:13,450 ‫S over two, so that's how I'm going to name them and I've put T plastics over to here to indicate that 68 00:07:13,450 --> 00:07:18,060 ‫this derivative was computed at time plus T. 69 00:07:18,060 --> 00:07:18,910 ‫S over to. 70 00:07:19,920 --> 00:07:20,940 ‫In other words. 71 00:07:22,070 --> 00:07:33,570 ‫We used the state values at T plus is over to that, we computed using the first slope, we use these 72 00:07:33,570 --> 00:07:37,810 ‫state values to compute our second slope. 73 00:07:38,870 --> 00:07:43,340 ‫And by the way, these notations, they are my own notations. 74 00:07:43,790 --> 00:07:48,740 ‫I'm not saying that they're some kind of official notation somewhere in books. 75 00:07:49,070 --> 00:07:54,220 ‫It's just I'm trying to give you the meaning of Kouta. 76 00:07:54,230 --> 00:08:02,990 ‫And right now I'm just using this kind of notation to differentiate between slope's and state values. 77 00:08:04,210 --> 00:08:11,080 ‫So that you would know which slopes I'm talking about and so that you would know whether your new state 78 00:08:11,080 --> 00:08:17,110 ‫values were computed using the slope one or slope two or slope three. 79 00:08:18,310 --> 00:08:24,670 ‫And so in the next video, we're going to see what we are going to do with our second slope, we're 80 00:08:24,670 --> 00:08:33,280 ‫going to need that second slope to compute new state values and then our third slope. 81 00:08:34,090 --> 00:08:35,980 ‫But more on that in the next video.