1 00:00:00,590 --> 00:00:01,310 ‫Welcome back. 2 00:00:02,240 --> 00:00:10,420 ‫So what about this BPCI, our angle, how does it look like in this entire story here? 3 00:00:11,270 --> 00:00:14,960 ‫I have a reader on the axis here and this is your angle. 4 00:00:15,920 --> 00:00:25,160 ‫And even though the Passi are is provided by the planner and not by this feedback legalization controller, 5 00:00:25,700 --> 00:00:35,690 ‫it is still served to the NPC controller in the same way you have five data points from zero up until 6 00:00:35,690 --> 00:00:37,430 ‫zero point four seconds. 7 00:00:38,330 --> 00:00:40,040 ‫And these data points are equal. 8 00:00:41,050 --> 00:00:45,760 ‫So even though your BPCI are comes from the planner. 9 00:00:46,870 --> 00:00:52,750 ‫It is given to the NPC controller when the outer loop happens. 10 00:00:53,790 --> 00:01:03,120 ‫So this is your BPCI are one and then your side true, your true, your angle will adjust itself like 11 00:01:03,120 --> 00:01:03,540 ‫this. 12 00:01:04,680 --> 00:01:10,890 ‫And so when the next outer loop happens, then the planner will give another batch of information to 13 00:01:10,890 --> 00:01:12,660 ‫the NPC controller. 14 00:01:13,720 --> 00:01:22,060 ‫And the NPC will start working to produce a response like this, for example, and that would be your 15 00:01:22,100 --> 00:01:30,970 ‫BPCI are two and then a new outflux happens and you have your BPCI are three. 16 00:01:32,220 --> 00:01:41,430 ‫And the embassy will do its magic in zero point one second intervals, which would be the simple time 17 00:01:41,430 --> 00:01:42,070 ‫interval. 18 00:01:42,960 --> 00:01:49,940 ‫And so here you can see the drone in action trying to follow this trajectory. 19 00:01:50,460 --> 00:02:00,030 ‫And here you have the reference X dimension, then you have the reference Y dimension and then you have 20 00:02:00,030 --> 00:02:02,400 ‫the reference Z dimension. 21 00:02:03,060 --> 00:02:06,990 ‫You can see that the trajectory has some kind of initial height. 22 00:02:07,110 --> 00:02:13,410 ‫So even though your drone starts at the altitude of zero meters, then the trajectory starts at the 23 00:02:13,410 --> 00:02:14,860 ‫altitude of five meters. 24 00:02:15,120 --> 00:02:19,940 ‫So the drone needs to catch the altitude of the trajectory as well. 25 00:02:20,700 --> 00:02:23,730 ‫And then these are the angles here. 26 00:02:23,910 --> 00:02:31,710 ‫So sirtf the reference, your angle comes from the planner, but then fire off and then theatric. 27 00:02:31,980 --> 00:02:35,520 ‫They are calculated by the feedback lionization controller. 28 00:02:36,360 --> 00:02:40,370 ‫And so I want to zoom in on these angles here. 29 00:02:40,980 --> 00:02:49,060 ‫And so in blue you have the reference values and then in red you have the true values. 30 00:02:49,500 --> 00:02:54,470 ‫So essentially the red line is trying to catch the blue line. 31 00:02:54,990 --> 00:03:00,760 ‫In other words, your drone is trying to catch the blue reference values. 32 00:03:01,590 --> 00:03:07,590 ‫And so, for example, if you know, zoom in, let's take this section here. 33 00:03:08,370 --> 00:03:15,300 ‫So you see, this is what I have been talking about, the outer loop, the feedback mineralisation controller 34 00:03:15,690 --> 00:03:25,290 ‫gives you a reference value for zero point four seconds, which is a constant value here, a straight 35 00:03:25,290 --> 00:03:26,880 ‫line in this section. 36 00:03:27,660 --> 00:03:32,960 ‫But then the NPC controller is trying to catch up. 37 00:03:32,970 --> 00:03:43,890 ‫It's trying to reach this blue reference line in four iterations where one iteration lasts for zero 38 00:03:43,890 --> 00:03:45,240 ‫point one seconds. 39 00:03:46,230 --> 00:03:52,060 ‫So we can also try to zoom in this part here and here as well. 40 00:03:52,620 --> 00:04:01,900 ‫So this would be, for example, K plus one K plus two K plus three and then K plus four. 41 00:04:02,730 --> 00:04:08,030 ‫Now here I showed you that when you're outer loop finishes. 42 00:04:08,310 --> 00:04:10,640 ‫So this is your K plus four here. 43 00:04:10,950 --> 00:04:14,580 ‫Then from here you start with your K. 44 00:04:15,410 --> 00:04:24,020 ‫However, Python doesn't plot too values for the same time, so for zero point four seconds is just 45 00:04:24,020 --> 00:04:26,840 ‫going to plot this lower point. 46 00:04:27,260 --> 00:04:32,000 ‫And then the next point that it will plot, it will be here. 47 00:04:32,150 --> 00:04:34,790 ‫So it will skip this one here. 48 00:04:35,300 --> 00:04:41,720 ‫So you will have a line from this point here up until this point here. 49 00:04:41,930 --> 00:04:42,520 ‫All right. 50 00:04:43,400 --> 00:04:45,750 ‫And that's what you see here. 51 00:04:45,800 --> 00:04:50,800 ‫So this is keep us four and then here you would have your back. 52 00:04:51,050 --> 00:04:52,340 ‫But Python doesn't plot. 53 00:04:52,370 --> 00:05:00,800 ‫It is just going to put K plus one here and you have a line here and then the NPCs trying to catch up. 54 00:05:01,790 --> 00:05:05,720 ‫By moving the drone angles closer to the reference values. 55 00:05:06,650 --> 00:05:13,340 ‫And then the true drone angles are represented with these red lines here. 56 00:05:14,430 --> 00:05:21,500 ‫And then, of course, we have other trajectories here as well, like, for example, this spiral trajectory 57 00:05:21,990 --> 00:05:26,830 ‫and again, the spiral trajectory has an initial height of five meters. 58 00:05:27,660 --> 00:05:29,840 ‫And so here you have the angles. 59 00:05:30,240 --> 00:05:34,620 ‫And if we zoom them in, then you can see the same thing. 60 00:05:34,800 --> 00:05:40,670 ‫You see, for example, let's take this area here and that's how it looks like. 61 00:05:41,340 --> 00:05:41,970 ‫And then. 62 00:05:42,940 --> 00:05:52,420 ‫The same thing here, and let's take this area so you see for each outer loop here, you have one, 63 00:05:52,900 --> 00:05:57,010 ‫two, three and four inner loops. 64 00:05:57,190 --> 00:05:57,850 ‫All right. 65 00:05:59,010 --> 00:06:01,420 ‫So that's the story with the angle. 66 00:06:02,430 --> 00:06:09,570 ‫In addition, the feedback in ization controller will also need something else from the planner. 67 00:06:10,110 --> 00:06:28,080 ‫It will need the time derivative of X or Y and Z are so X, dot R, then Y, dot R and then Z dot R. 68 00:06:28,950 --> 00:06:35,790 ‫And even that is not all because it will also need the second derivative of the reference value. 69 00:06:36,180 --> 00:06:51,180 ‫So X double dot R then Y double dot R and then Z double dot R and you will see Y when we start covering 70 00:06:51,480 --> 00:06:52,650 ‫this feedback. 71 00:06:52,660 --> 00:06:54,040 ‫Lionization controller. 72 00:06:54,930 --> 00:07:01,510 ‫Now this bpci r it served as a reference value for the NPC controller. 73 00:07:02,400 --> 00:07:08,000 ‫Now the feedback lionization controller will also need BPCI are. 74 00:07:08,850 --> 00:07:15,690 ‫However, the difference now is that here the PCR is not a reference value. 75 00:07:16,530 --> 00:07:24,570 ‫The feedback lionization controller will not need to find some kind of input to reach this BPCI our 76 00:07:24,840 --> 00:07:25,920 ‫reference value. 77 00:07:26,190 --> 00:07:28,440 ‫That's the job of the NPC controller. 78 00:07:29,440 --> 00:07:39,130 ‫However, in order to find you one by ah and then theater, you will have equation's there in which 79 00:07:39,130 --> 00:07:41,470 ‫you will also have BPCI. 80 00:07:41,470 --> 00:07:47,660 ‫Ah so the feedback plenary session controller doesn't need PCEHR as a reference value. 81 00:07:47,950 --> 00:07:57,010 ‫However, it does need it because it needs this value in some of its equations in order to find you 82 00:07:57,010 --> 00:07:58,930 ‫one firer and Sittar. 83 00:07:59,170 --> 00:08:06,880 ‫And again, once we start covering this part, you will see exactly how it works. 84 00:08:07,750 --> 00:08:19,100 ‫And in order to find X or Y or Z that are and also the double dot values, the procedure is very easy. 85 00:08:19,450 --> 00:08:28,360 ‫Let's, for example, take a spiral in order to achieve a spiral in three D, the equations for the 86 00:08:28,570 --> 00:08:36,040 ‫X Dimension, Y dimension and Z dimension are the following. 87 00:08:36,730 --> 00:08:41,420 ‫X equals the radius of the spiral. 88 00:08:42,160 --> 00:08:52,810 ‫So if you look at the spiral from the top, then that would be your radius here times cosine and then 89 00:08:52,810 --> 00:08:59,030 ‫you have two pi, then the frequency of the rotation times time. 90 00:08:59,710 --> 00:09:08,920 ‫So this F here, this is in Hertz, two pi is in radians and then time is in seconds. 91 00:09:09,520 --> 00:09:19,700 ‫So the units would be like this, two pi would be radians and let's say radians per one round. 92 00:09:20,020 --> 00:09:24,340 ‫So two pi radians is one full circle. 93 00:09:24,640 --> 00:09:29,880 ‫Right then frequency would be rounds per second. 94 00:09:30,340 --> 00:09:40,120 ‫So cycles per second, which is Hertz so round per second and then time would be second. 95 00:09:41,160 --> 00:09:48,000 ‫So in the end, of course, you will cancel out the rounds and then the seconds, and that means that 96 00:09:48,420 --> 00:09:54,780 ‫inside this cosine parenthesis you will have radiance as you should. 97 00:09:55,700 --> 00:10:04,370 ‫However, the larger your F, the larger your frequency, the larger your rounds per second. 98 00:10:05,340 --> 00:10:15,540 ‫The faster you rotate, so here you can see the spiral and then you can see it's three dimensions here. 99 00:10:16,110 --> 00:10:20,280 ‫So the next dimension, why dimension and then the Z dimension. 100 00:10:20,940 --> 00:10:22,920 ‫And here you can see the X dimension. 101 00:10:22,930 --> 00:10:28,080 ‫It has an amplitude three two pi times zero point five hertz times. 102 00:10:29,190 --> 00:10:36,500 ‫So you can see that in 10 seconds it is able to do five rotations. 103 00:10:36,990 --> 00:10:38,790 ‫So this is third rotation. 104 00:10:39,720 --> 00:10:41,420 ‫This is the fourth rotation. 105 00:10:42,030 --> 00:10:44,480 ‫And then this is the fifth rotation. 106 00:10:45,360 --> 00:10:53,610 ‫But what happens if I increase this frequency, the zero point five, so I have a code here and then 107 00:10:53,970 --> 00:10:59,130 ‫I'm going to increase the frequency for the X dimension instead of zero point five hertz. 108 00:10:59,130 --> 00:11:05,220 ‫I'm going to put one hertz and I'm going to do the same thing for the Y dimension. 109 00:11:05,640 --> 00:11:09,170 ‫So I'm going to replace zero point five with one. 110 00:11:09,990 --> 00:11:18,260 ‫And now you can see that in 10 seconds it will make more rotations, 10 to be exact. 111 00:11:18,330 --> 00:11:26,100 ‫So you see in 10 seconds it is able to make more rotations because you have increased your frequency 112 00:11:26,520 --> 00:11:28,630 ‫in your equations here. 113 00:11:28,650 --> 00:11:31,570 ‫Now it's one hertz here and then here. 114 00:11:32,550 --> 00:11:37,560 ‫So this is the equation for the examination, for the Y dimension. 115 00:11:37,740 --> 00:11:46,470 ‫The equation is R times sine to pi frequency times time. 116 00:11:47,190 --> 00:11:51,540 ‫And then for the Z dimension, we want to have a straight line function. 117 00:11:51,840 --> 00:12:01,950 ‫So you will have the initial height of the trajectory plus the final height of the trajectory, minus 118 00:12:02,280 --> 00:12:14,700 ‫the initial height of the trajectory divided by the entire time that it takes you to perform the maneuver 119 00:12:15,360 --> 00:12:18,590 ‫and then you multiply that by time. 120 00:12:19,410 --> 00:12:25,740 ‫So you can see that here you have the Z dimension and you have a straight line function here. 121 00:12:25,860 --> 00:12:31,200 ‫However, the difference here is that here I didn't have an initial height. 122 00:12:31,950 --> 00:12:36,810 ‫I assumed that my initial height starts from zero meters. 123 00:12:37,170 --> 00:12:43,050 ‫However, in the drone case, we will have some kind of initial height. 124 00:12:43,710 --> 00:12:53,550 ‫And so again, these are the equations for the spiral for X or Y, R and R, and now to find their time 125 00:12:53,550 --> 00:13:02,460 ‫derivatives, you just take the derivatives of these equations with respect to time and then you will 126 00:13:02,460 --> 00:13:05,910 ‫also take the second derivative of those equations. 127 00:13:06,880 --> 00:13:09,990 ‫And so I will leave it to you as an exercise. 128 00:13:10,530 --> 00:13:17,970 ‫Try to find the derivatives of these equations the first and then the second derivatives, and then 129 00:13:17,970 --> 00:13:20,280 ‫you will see the solutions in the next video. 130 00:13:20,670 --> 00:13:21,570 ‫Thank you very much.