1 00:00:00,490 --> 00:00:12,340 ‫But what will happen if I make my imaginary parts a zero like this and like this, let's see what's 2 00:00:12,340 --> 00:00:13,200 ‫going to happen. 3 00:00:13,960 --> 00:00:22,530 ‫And so you can see that even though the tracking is very slow, but there is no oscillation. 4 00:00:23,350 --> 00:00:25,060 ‫So this is the Z dimension. 5 00:00:25,780 --> 00:00:36,040 ‫You see, you don't have any oscillation, but you track the Z dimension very slowly y well because 6 00:00:36,370 --> 00:00:38,430 ‫you don't have any imaginary parts. 7 00:00:38,440 --> 00:00:46,080 ‫So you cannot have oscillation, but your real part is very small. 8 00:00:46,090 --> 00:00:48,040 ‫So it's minus zero point one. 9 00:00:48,850 --> 00:00:58,570 ‫And now let's go back to our complex polls and now let's make the real poll not minus zero point one, 10 00:00:58,960 --> 00:01:04,290 ‫but plus zero point zero one. 11 00:01:05,080 --> 00:01:10,990 ‫So a positive number, but 10 times smaller, zero point zero one. 12 00:01:11,800 --> 00:01:13,640 ‫And let's see what's going to happen. 13 00:01:14,680 --> 00:01:20,850 ‫So the drone oscillates in the Z dimension and this oscillation is growing. 14 00:01:21,820 --> 00:01:26,530 ‫You see this constellation is growing in the Z dimension. 15 00:01:27,590 --> 00:01:34,910 ‫And the reason for that is because you have complex polls here, imaginary parts give you a solution 16 00:01:35,120 --> 00:01:42,470 ‫and then you have a small, positive, real part that will increase your error. 17 00:01:43,500 --> 00:01:45,640 ‫And now I want to show you something else. 18 00:01:46,260 --> 00:01:51,400 ‫How about if I put zero here and zero here? 19 00:01:52,260 --> 00:02:02,790 ‫So now you can imagine that you think you would oscillate and your amplitude wouldn't change and, well, 20 00:02:02,940 --> 00:02:04,210 ‫that's the result. 21 00:02:04,530 --> 00:02:07,950 ‫However, as you can see, the amplitude does change. 22 00:02:08,850 --> 00:02:17,190 ‫So this is the Z dimension and you can see that the amplitude grows even though your real part is zero. 23 00:02:17,280 --> 00:02:18,840 ‫And why is it like that? 24 00:02:19,560 --> 00:02:26,700 ‫Well, the explanation that I can give you is the following, that theoretically it shouldn't grow, 25 00:02:27,600 --> 00:02:38,970 ‫because what happens is that your lump, the one equals zero plus I times zero point three and then 26 00:02:39,180 --> 00:02:46,800 ‫your lump, the two equals zero minus eight times zero point three. 27 00:02:47,730 --> 00:02:50,250 ‫That's for your Z dimension, right? 28 00:02:50,250 --> 00:02:52,300 ‫For your use of Z. 29 00:02:53,100 --> 00:03:00,630 ‫And so your K one value here would be minus zero point zero nine. 30 00:03:00,750 --> 00:03:07,980 ‫I know that because I looked it up and then K two would be zero. 31 00:03:08,850 --> 00:03:13,620 ‫So in your Z dimension differential equation, you OK? 32 00:03:13,650 --> 00:03:18,570 ‫One would have this value and your K two would have this value. 33 00:03:19,550 --> 00:03:26,840 ‫And so if you calculate your error as a function of time, then you will have an expression that only 34 00:03:26,840 --> 00:03:35,090 ‫consists of cosigns and signs and you will not have something like this where you have the oil and no 35 00:03:35,600 --> 00:03:41,870 ‫to the power of their real part of the polls times time. 36 00:03:42,440 --> 00:03:46,590 ‫You will not have that because this one will be zero. 37 00:03:46,610 --> 00:03:48,760 ‫So this entire thing will be won. 38 00:03:49,670 --> 00:03:52,750 ‫So the amplitude of the oscillation shouldn't grow. 39 00:03:53,660 --> 00:03:56,000 ‫This shouldn't grow, but it does. 40 00:03:56,840 --> 00:04:07,910 ‫Well, the reason for that is that these key values here, they will give you the right e double dot 41 00:04:08,510 --> 00:04:16,190 ‫in the Z dimension so that your error would simply oscillate, but so that its amplitude wouldn't change. 42 00:04:17,060 --> 00:04:28,730 ‫But then this error, double dot needs to be converted to Phi R, Theta R and you won. 43 00:04:29,770 --> 00:04:37,780 ‫And not only this, of course, in the end, all your three erodable dots will need to be converted 44 00:04:38,230 --> 00:04:51,760 ‫into PHY are, Seeta are and you won and then you one will go to plant and then fire and Sittar, they 45 00:04:51,760 --> 00:04:57,540 ‫will go to the NPC controller and one happens in the NPC controller. 46 00:04:58,270 --> 00:05:02,520 ‫These values are your reference values, right. 47 00:05:03,490 --> 00:05:08,770 ‫But then the true PHY and Seeta values. 48 00:05:08,980 --> 00:05:12,310 ‫Right phy true and feet are true. 49 00:05:12,310 --> 00:05:14,460 ‫These are the values that belong to the drone. 50 00:05:15,070 --> 00:05:20,170 ‫These are the values that need to catch these phy are and Setara values. 51 00:05:21,070 --> 00:05:28,900 ‫They do track your fly are and Setara values, but they might not track them perfectly. 52 00:05:29,650 --> 00:05:34,690 ‫And because of this imperfection, your error here still grows. 53 00:05:35,410 --> 00:05:40,270 ‫So at least that's how I see it, because at the beginning I was surprised by it. 54 00:05:40,510 --> 00:05:42,190 ‫But then I thought about it. 55 00:05:42,760 --> 00:05:44,470 ‫And I think that that's the reason. 56 00:05:45,400 --> 00:05:55,390 ‫If I put here a very small negative, real number minus zero point zero one and then minus zero point 57 00:05:55,600 --> 00:06:02,590 ‫zero one, then you can see that your amplitude is not changing, at least it's not visible. 58 00:06:02,590 --> 00:06:04,980 ‫Maybe it's changing a little bit, but it's not visible. 59 00:06:05,950 --> 00:06:07,810 ‫You see, this is your Z dimension. 60 00:06:08,470 --> 00:06:14,260 ‫It's not very visible that your Z dimension is changing, maybe only a little bit. 61 00:06:15,210 --> 00:06:19,660 ‫Well, in fact, yes, it is changing a little bit here. 62 00:06:19,660 --> 00:06:22,720 ‫The distance is a little bit bigger than, for example, here. 63 00:06:23,680 --> 00:06:31,360 ‫But my point is that if you put a small, negative, real number here, then it might cancel out this 64 00:06:31,720 --> 00:06:35,890 ‫imperfection, this one here that we talked about. 65 00:06:36,640 --> 00:06:46,990 ‫And now let's make our Z dimension poles just real numbers, minus one and minus two, and let's make 66 00:06:46,990 --> 00:06:52,060 ‫our X dimension poles complex and let's see what's going to happen then. 67 00:06:53,010 --> 00:07:00,420 ‫You can see that now you have oscillation and then damping out, but in the X dimension. 68 00:07:01,420 --> 00:07:03,700 ‫You see now this is your X dimension. 69 00:07:04,570 --> 00:07:10,680 ‫You don't have that in the Y dimension and you don't have that in the Z dimension because now you have 70 00:07:10,690 --> 00:07:13,570 ‫strong, negative, real pulse. 71 00:07:14,550 --> 00:07:18,990 ‫And these are the angles here, the control inputs and the omega's. 72 00:07:19,950 --> 00:07:28,280 ‫And now let's do the same thing, but let's give complex pause to Y dimension. 73 00:07:29,190 --> 00:07:35,090 ‫So we're going to comment at the X dimension and we're going to give complex pause to the Y dimension. 74 00:07:35,610 --> 00:07:36,990 ‫So let's see what's going to happen. 75 00:07:37,860 --> 00:07:40,220 ‫Well, it seems to track it pretty well. 76 00:07:41,160 --> 00:07:46,620 ‫You do have oscillation in the Y dimension, but it damps out pretty quickly. 77 00:07:47,790 --> 00:07:53,360 ‫And now let's give all palls complex pause. 78 00:07:53,410 --> 00:08:02,250 ‫OK, let's make sure that the X, Y and Z dimension, all of them have a negative real value and some 79 00:08:02,250 --> 00:08:05,080 ‫kind of imaginary value as well. 80 00:08:05,100 --> 00:08:06,480 ‫Let's see what's going to happen. 81 00:08:07,490 --> 00:08:16,100 ‫So the drone is starting to follow the trajectory and you can see that in the Z dimension, it has some 82 00:08:16,100 --> 00:08:17,300 ‫kind of oscillation. 83 00:08:18,440 --> 00:08:26,510 ‫But it's approaching the trajectory, so in the end, you have this kind of tracking, you have oscillation 84 00:08:26,510 --> 00:08:33,460 ‫in the X dimension, but it damps itself out thanks to negative real numbers. 85 00:08:34,310 --> 00:08:41,600 ‫You have the same thing in the Y Dimension Oscillation, but it disappears and you have the same thing 86 00:08:41,600 --> 00:08:48,980 ‫with the Z dimension where you have a little bit of oscillation and then it damps itself out thanks 87 00:08:48,980 --> 00:08:49,460 ‫to. 88 00:08:50,530 --> 00:09:01,300 ‫The negative real numbers here that are minus one and minus one, values are here quite big, so they 89 00:09:01,300 --> 00:09:04,030 ‫cancel out this oscillation here pretty quickly. 90 00:09:04,960 --> 00:09:12,520 ‫But now what happens if I put here minus 10 and also here minus 10? 91 00:09:12,730 --> 00:09:13,930 ‫What's going to happen? 92 00:09:14,350 --> 00:09:23,040 ‫The real number in the Z dimension, now it's minus 10 and now I'm going to run the code and look, 93 00:09:23,680 --> 00:09:25,670 ‫the code stops running. 94 00:09:26,230 --> 00:09:26,900 ‫So what is it? 95 00:09:26,900 --> 00:09:31,750 ‫Say you can't take a square root of a negative number. 96 00:09:32,320 --> 00:09:38,460 ‫The problem might be that the trajectory is too chaotic or it might have large discontinues jumps. 97 00:09:38,980 --> 00:09:42,370 ‫Try to make a smoother trajectory without discontinuous jumps. 98 00:09:43,000 --> 00:09:50,590 ‫Are the possible causes might be values for variables such as the sample time interval or horizon period 99 00:09:50,800 --> 00:09:53,950 ‫or other variables here, including poles. 100 00:09:54,490 --> 00:09:56,360 ‫So they might be problematic. 101 00:09:56,980 --> 00:10:01,370 ‫So in our case, the problem is Poles. 102 00:10:01,990 --> 00:10:04,420 ‫So what has happened? 103 00:10:05,140 --> 00:10:06,370 ‫Well, this has happened. 104 00:10:06,370 --> 00:10:11,440 ‫You cannot take a square root of a negative number if you remember. 105 00:10:11,830 --> 00:10:24,400 ‫Then in the end of the third chapter, you had to compute your omega one, Omega to Omega three and 106 00:10:24,400 --> 00:10:25,840 ‫Omega four. 107 00:10:26,810 --> 00:10:37,520 ‫And so you ended up with equations that had square root of something, a one for Megahed, two for Omega 108 00:10:37,520 --> 00:10:43,310 ‫three and four Megafaun, so that was the end of Chapter three. 109 00:10:44,150 --> 00:10:50,040 ‫So the program stops working when you have negative numbers under the square root here. 110 00:10:50,750 --> 00:10:52,450 ‫So what does it mean physically? 111 00:10:53,180 --> 00:11:02,540 ‫It means that the controllers tell you that in order to perform a maneuver at the given time, you need 112 00:11:02,540 --> 00:11:04,190 ‫to have control inputs. 113 00:11:04,880 --> 00:11:11,150 ‫You one, you to use three and you four. 114 00:11:12,200 --> 00:11:17,450 ‫You need to have certain values for them calculated by the controller. 115 00:11:18,170 --> 00:11:28,040 ‫But then the model that we have derived tells you that there is no combination of Omega one, omega 116 00:11:28,040 --> 00:11:37,460 ‫two, omega three and Omega four that the drone can produce to achieve the required. 117 00:11:37,850 --> 00:11:42,410 ‫You won you to use three and you four. 118 00:11:43,370 --> 00:11:47,760 ‫That's why you have negative numbers under the square root. 119 00:11:48,260 --> 00:11:57,770 ‫It means that there is no real solution for the Omega's that usually happens when your trajectory changes 120 00:11:57,770 --> 00:12:07,880 ‫very abruptly or if you give the controller very strong pulls, forcing the ovei to do two sharp maneuvers 121 00:12:07,880 --> 00:12:09,250 ‫that it cannot handle. 122 00:12:10,040 --> 00:12:17,780 ‫So your trajectory must be smooth enough and the polls need to stay within a certain area in the Laplanche 123 00:12:17,780 --> 00:12:18,400 ‫domain. 124 00:12:19,010 --> 00:12:21,950 ‫So this is your plus domain. 125 00:12:23,110 --> 00:12:31,480 ‫And you need to make sure that your polls don't go very far away from this origin, because even if 126 00:12:31,480 --> 00:12:40,690 ‫they are negative but they are very far away from this origin point, then you might force your drone 127 00:12:40,690 --> 00:12:48,790 ‫to perform very aggressive maneuvers and then your controller will tell you that, OK, I need this. 128 00:12:48,790 --> 00:12:52,120 ‫You want this, you to do this, you three and this you four. 129 00:12:52,510 --> 00:12:59,530 ‫But then the model will tell you that, sorry, man, I cannot give you the omega one and two and three 130 00:12:59,530 --> 00:13:07,270 ‫and four that will allow you to achieve your required. 131 00:13:07,270 --> 00:13:10,350 ‫You want you to use three and you four. 132 00:13:11,260 --> 00:13:22,810 ‫So the conclusion is, have your trajectories be smooth, don't have big poles, be in the negative 133 00:13:22,810 --> 00:13:28,330 ‫region if you want your errors to approach zero. 134 00:13:29,480 --> 00:13:37,850 ‫And if you don't want oscillation, then be on the negative side and on this real number line, don't 135 00:13:37,850 --> 00:13:40,410 ‫go into the imaginary area. 136 00:13:41,120 --> 00:13:41,540 ‫Right. 137 00:13:42,780 --> 00:13:51,240 ‫OK, so you want your error and error not to go to zero for that, you chose the appropriate lump the 138 00:13:51,240 --> 00:13:57,150 ‫one and lump the two, you computed the right key one and key to values. 139 00:13:57,450 --> 00:14:08,130 ‫And that gave you the error double dot that will then make error and error dot approach zero. 140 00:14:08,430 --> 00:14:09,770 ‫But this error doubled. 141 00:14:09,780 --> 00:14:12,000 ‫That is just a required value. 142 00:14:12,390 --> 00:14:19,410 ‫In order to achieve this error double dot, you need to have the right you one you to use three and 143 00:14:19,410 --> 00:14:21,510 ‫you four control inputs. 144 00:14:22,140 --> 00:14:34,440 ‫And so that means that you need to find a connection between error dot and then you one by R and FITA 145 00:14:34,440 --> 00:14:41,460 ‫are, which are the control inputs of this feedback lionization controller. 146 00:14:41,970 --> 00:14:51,660 ‫And then you one will go straight to the plant and then fire and theatergoer along with your cirE will 147 00:14:51,660 --> 00:14:54,210 ‫go into your NPC controller. 148 00:14:54,540 --> 00:15:01,410 ‫That will give you the remaining you two, three and four control inputs. 149 00:15:01,840 --> 00:15:09,870 ‫And so in the next video, we're going to learn how to make a connection between your required E double 150 00:15:09,870 --> 00:15:14,370 ‫dot and then you one fire. 151 00:15:14,640 --> 00:15:17,100 ‫And Sittar, thank you very much.