1 00:00:00,600 --> 00:00:01,380 ‫Welcome back. 2 00:00:01,950 --> 00:00:12,400 ‫So this is our general control structure here and now let's zoom in into this feedback lionization controller. 3 00:00:13,410 --> 00:00:15,630 ‫This is our magnifying glass here. 4 00:00:15,900 --> 00:00:19,980 ‫And now we're going to see what's inside this feedback penalization box. 5 00:00:20,880 --> 00:00:24,190 ‫So this is the feedback lionization controller block. 6 00:00:25,110 --> 00:00:30,930 ‫And here you have the reference values, X, R and X dot are. 7 00:00:31,740 --> 00:00:34,740 ‫Then here you have Y are. 8 00:00:35,010 --> 00:00:45,690 ‫And why dot r and then the Z R and then the Z dot are these are your reference values that come from 9 00:00:45,690 --> 00:00:46,370 ‫the planner. 10 00:00:47,220 --> 00:00:49,290 ‫So they are all positive here. 11 00:00:49,290 --> 00:00:50,880 ‫So I'm going to put Plus's here. 12 00:00:51,690 --> 00:01:01,110 ‫And then from the reference values you subtract the true X and the true X dot, then you also subtract 13 00:01:01,260 --> 00:01:13,380 ‫the true Y and the true Y dot and then you subtract the true Z and the true Z dot and as a result you 14 00:01:13,380 --> 00:01:25,590 ‫have your error X and then error dot X, then you have error Y and then error dot Y and then you have 15 00:01:25,590 --> 00:01:28,770 ‫error Z and error dot z. 16 00:01:29,620 --> 00:01:39,850 ‫And then you have this low key vector for the X values, for the Y values and for the Z values, and 17 00:01:39,850 --> 00:01:49,430 ‫so as a result, you will receive a control input, you X here, you Y and you Z. 18 00:01:50,350 --> 00:01:58,840 ‫And so, as you can already see in this controller, we are treating the X, Y and Z dimensions separately 19 00:01:59,770 --> 00:02:00,810 ‫from the planner. 20 00:02:01,060 --> 00:02:08,560 ‫You will get your X are an X dot are then why are and why that are. 21 00:02:08,560 --> 00:02:18,760 ‫And then also the Z, R and Z dot are these are your reference position states and their time derivatives. 22 00:02:19,710 --> 00:02:30,420 ‫And then you will get X and X dot and then also Y and Y dot and then the Z and Z dot, these are the 23 00:02:30,420 --> 00:02:31,610 ‫true values. 24 00:02:32,340 --> 00:02:37,790 ‫You get the X, Y, Z, and then you've W states from the planet. 25 00:02:38,460 --> 00:02:45,920 ‫And then additionally the U V, W states need to go through the rotation matrix. 26 00:02:45,930 --> 00:02:55,680 ‫So you have to multiply the rotation matrix by a vector that contains you, VW, a column vector, and 27 00:02:55,680 --> 00:03:00,430 ‫that's how you get X, Y and Z dot. 28 00:03:01,200 --> 00:03:03,990 ‫And so the error vectors look like this. 29 00:03:03,990 --> 00:03:15,870 ‫Then the error vector in the X dimension is ESOP X and E dot sup x or you can also write them down like 30 00:03:15,870 --> 00:03:28,290 ‫this X up R minus X and then here you will have X dot sub R minus X dot and the same thing applies to 31 00:03:28,290 --> 00:03:30,470 ‫the error vector in the Y dimension. 32 00:03:30,900 --> 00:03:36,120 ‫The first element in this vector is the actual error in the Y dimension. 33 00:03:36,450 --> 00:03:43,800 ‫And then the second element is the time derivative of that error, the time derivative of the error 34 00:03:43,890 --> 00:03:45,180 ‫in the Y dimension. 35 00:03:45,540 --> 00:03:48,050 ‫And you can also rewrite them like this, of course. 36 00:03:48,750 --> 00:03:52,520 ‫And then finally, this is for the error vector in the Z dimension. 37 00:03:53,340 --> 00:04:04,980 ‫And so then their control inputs are like this use up X equals the row vector of key values, times 38 00:04:04,980 --> 00:04:07,110 ‫the error vector in the X dimension. 39 00:04:07,350 --> 00:04:11,610 ‫And then of course you will get a scalar like this. 40 00:04:11,910 --> 00:04:19,800 ‫This will be your scalar k one times the error plus key two times the error dot. 41 00:04:20,790 --> 00:04:22,460 ‫Then you're you. 42 00:04:22,530 --> 00:04:30,630 ‫Why would be the same thing, it would look like this, and then this would be your scaler, of course, 43 00:04:31,540 --> 00:04:36,750 ‫and finally your use of Z would also be like that. 44 00:04:37,860 --> 00:04:38,600 ‫There you go. 45 00:04:38,820 --> 00:04:39,990 ‫It will be like that. 46 00:04:40,320 --> 00:04:43,130 ‫You will have a scalar value here. 47 00:04:44,100 --> 00:04:50,560 ‫Note that use up X of Y and then use Z. 48 00:04:51,030 --> 00:05:03,390 ‫They are not your phi are theta are and you one that the controller needs to produce for the NPC controller 49 00:05:03,390 --> 00:05:04,500 ‫and for the plant. 50 00:05:04,890 --> 00:05:08,280 ‫In fact they are intermediate control inputs. 51 00:05:08,820 --> 00:05:17,130 ‫These intermediate control inputs will be later converted into fly are theatergoer and you want control 52 00:05:17,130 --> 00:05:17,720 ‫input. 53 00:05:18,630 --> 00:05:22,660 ‫But first, let's understand this intermediate part better. 54 00:05:23,160 --> 00:05:28,250 ‫The question now becomes what the right key values are. 55 00:05:29,070 --> 00:05:31,130 ‫That's the question we need to answer. 56 00:05:31,440 --> 00:05:38,090 ‫And in order to answer that question, we need to talk a bit about differential equations. 57 00:05:38,910 --> 00:05:45,050 ‫So I'm going to leave the schematics B for now and then I'm going to come back to it later. 58 00:05:45,870 --> 00:05:50,050 ‫And so in the next video, we're going to talk about differential equations. 59 00:05:50,340 --> 00:05:51,210 ‫Thank you very much.