1 00:00:00,730 --> 00:00:01,480 ‫Welcome back. 2 00:00:02,080 --> 00:00:07,720 ‫So this was our state's basic equation in a very compact form. 3 00:00:08,560 --> 00:00:16,300 ‫Now, of course, we have to dig deeper than that and express this equation in a less compact way in 4 00:00:16,300 --> 00:00:17,920 ‫order to be able to work with that. 5 00:00:18,400 --> 00:00:21,820 ‫However, that's just a matter of representation. 6 00:00:22,450 --> 00:00:27,460 ‫You just expand vectors and matrices and see what's inside. 7 00:00:28,030 --> 00:00:37,840 ‫So, for example, instead of writing this vector like this, we can write it like this where you have 8 00:00:38,200 --> 00:00:44,570 ‫each time derivative of the state equals and then something else. 9 00:00:45,370 --> 00:00:48,470 ‫But before we do that, we have to do something else. 10 00:00:49,300 --> 00:00:53,020 ‫You see this lambda B vector here? 11 00:00:53,530 --> 00:00:57,280 ‫It's a vector that contains the net force and moments. 12 00:00:58,240 --> 00:01:00,220 ‫So you can write it out like this. 13 00:01:00,220 --> 00:01:04,900 ‫You have your F net and then M-Net in the body frame and then transpose. 14 00:01:05,770 --> 00:01:12,430 ‫But these are the net forces and moments that in itself does not tell you much. 15 00:01:12,970 --> 00:01:17,740 ‫The net forces and moments are composed of individual forces. 16 00:01:17,740 --> 00:01:23,950 ‫In moments like this, you have your F one plus F two plus F three plus. 17 00:01:24,070 --> 00:01:32,170 ‫How many forces you have there equals your FNET and then you have your M one plus M two plus M three 18 00:01:32,170 --> 00:01:32,890 ‫plus. 19 00:01:32,890 --> 00:01:37,150 ‫How many moments you have there equals your net moment. 20 00:01:37,990 --> 00:01:43,000 ‫That's why we also call them the sum of the forces and the sum of the moments. 21 00:01:43,970 --> 00:01:52,630 ‫So one thing that we have to do before we express our Newton oilor formulation in the states space form, 22 00:01:53,270 --> 00:02:03,110 ‫we have to identify the individual forces and the individual moments that our drone is exposed to. 23 00:02:04,170 --> 00:02:13,290 ‫So this is your little drone here and you can be exposed to, for example, two F one, and maybe then 24 00:02:13,500 --> 00:02:16,740 ‫later to F to. 25 00:02:17,830 --> 00:02:26,890 ‫And then you can have moment one and then perhaps moment two, so all these forces in moments need to 26 00:02:26,890 --> 00:02:32,100 ‫be added together to get your net force and net moment. 27 00:02:32,590 --> 00:02:36,320 ‫But we have to know the individual forces in moments as well. 28 00:02:37,030 --> 00:02:39,340 ‫So what is our drone exposed to? 29 00:02:40,240 --> 00:02:47,200 ‫Well, first of all, as you can imagine, the drone is exposed to force of gravity, right? 30 00:02:48,210 --> 00:02:57,090 ‫So, again, that's your drone here, and since the drone is mass symmetric, then the equivalent force 31 00:02:57,090 --> 00:03:03,210 ‫of gravity vector acts through the center of mass of the drone. 32 00:03:03,270 --> 00:03:05,880 ‫So this is your force of gravity here. 33 00:03:06,900 --> 00:03:15,090 ‫And the center of mass of the drone is itself aligned with the origin of the body frame, so the origin 34 00:03:15,090 --> 00:03:19,820 ‫of the body frame is also here where you have the center of mass. 35 00:03:20,640 --> 00:03:22,500 ‫See that and dot. 36 00:03:23,520 --> 00:03:32,760 ‫Therefore, you can say that the first lamda and let's call it Lunda one vector B, we can say that 37 00:03:32,760 --> 00:03:40,630 ‫this is for gravity and lets for clarity also call it like this, Lunda G in the body frame. 38 00:03:41,550 --> 00:03:51,870 ‫So G stands for gravity so we can rewrite this lambda G in the body frame in this way here, here this 39 00:03:52,200 --> 00:04:00,450 ‫f g in the body frame is a three by one force vector that has three elements. 40 00:04:01,510 --> 00:04:06,170 ‫So it has effects F, Y, and then F, Z. 41 00:04:06,880 --> 00:04:14,920 ‫And remember, this is in the body frame and now it's important to note that there is no moment of gravity 42 00:04:15,460 --> 00:04:23,650 ‫because the equivalent gravity force vector goes through the center of mass of the wave and the origin 43 00:04:23,650 --> 00:04:27,370 ‫of the body frame is where the center of mass is. 44 00:04:28,270 --> 00:04:32,940 ‫If the body frame origin was somewhere else, like, for example here. 45 00:04:33,190 --> 00:04:42,700 ‫And let's just imagine that this is our body from x axis and then that would be our body frame y axis 46 00:04:43,150 --> 00:04:46,090 ‫and then body frame Z axis. 47 00:04:47,020 --> 00:04:53,470 ‫Then you would have a distance here between the origin of the body frame and the center of mass of the 48 00:04:53,470 --> 00:04:54,010 ‫drone. 49 00:04:54,820 --> 00:05:02,650 ‫And then with respect to this purple body frame, your force of gravity would create a moment. 50 00:05:03,460 --> 00:05:04,920 ‫But we don't want that. 51 00:05:05,140 --> 00:05:08,230 ‫So we do it in a smart way. 52 00:05:08,470 --> 00:05:15,970 ‫And we put the body frame in the best place where its origin matches the center of mass. 53 00:05:16,970 --> 00:05:24,440 ‫So we put the origin here in the center of the drone, that means that no movement is created in our 54 00:05:24,440 --> 00:05:32,240 ‫body frame and what you have is a zero three by one vector. 55 00:05:33,150 --> 00:05:43,520 ‫As your gravity moment elements, so this zero three by one vector looks like this zero zero and zero, 56 00:05:44,310 --> 00:05:45,480 ‫that's how it looks like. 57 00:05:46,520 --> 00:05:52,730 ‫By the way, what do I mean when I say equivalent gravity force vector? 58 00:05:53,630 --> 00:05:56,750 ‫Well, gravity is actually a distributed force. 59 00:05:57,810 --> 00:06:05,700 ‫Imagine that you have a square plate here, then force of gravity acts on every single point on this 60 00:06:05,700 --> 00:06:06,310 ‫object. 61 00:06:07,230 --> 00:06:17,970 ‫Let's say that the mass of the plate is 50 kilograms and let's say that the area of the plate is five 62 00:06:17,970 --> 00:06:19,260 ‫square metres. 63 00:06:20,310 --> 00:06:27,170 ‫You know that the force of gravity equals F G equals mass times G. 64 00:06:28,230 --> 00:06:37,860 ‫Well, that's approximate G to be 10 meters per second squared, even though in reality it's nine point 65 00:06:38,040 --> 00:06:41,480 ‫eight one meters per second squared. 66 00:06:42,360 --> 00:06:46,530 ‫But just to make it easier, let's just say that it's 10 meters per second squared. 67 00:06:47,650 --> 00:06:53,360 ‫Then that means that f some G equals mass times G. 68 00:06:53,450 --> 00:07:01,750 ‫So 50 times 10 equals five hundred Newtons. 69 00:07:02,730 --> 00:07:07,830 ‫In reality, though, this is not a point force, it's a distributed force. 70 00:07:09,020 --> 00:07:20,210 ‫So it would be F g divided by the area of the plate, and that would be five hundred divided by five, 71 00:07:20,810 --> 00:07:27,080 ‫and then that would equal 100 Newton per square meter. 72 00:07:27,410 --> 00:07:29,200 ‫That's how it is in reality. 73 00:07:29,420 --> 00:07:37,790 ‫However, since we assume that the plate is homogeneous, which means uniform in terms of mass distribution. 74 00:07:38,060 --> 00:07:48,200 ‫So you have constant density, constant kilograms per meter cubed, meaning that this part of the plate 75 00:07:48,350 --> 00:07:55,670 ‫is not heavier than this part of the plate, but the mass distribution is equal in all the locations 76 00:07:55,670 --> 00:07:56,480 ‫of the plate. 77 00:07:57,510 --> 00:08:06,120 ‫And this plate has all sides with equal length, then, for example, this side here has length L and 78 00:08:06,120 --> 00:08:12,930 ‫also this side here has length L and the same thing here L and then this one here as well. 79 00:08:12,930 --> 00:08:22,650 ‫L Then for modeling purposes, you take the total F G, which is five hundred Newtons, and you say 80 00:08:22,650 --> 00:08:31,260 ‫that if this force vector acts through the center of mass, which in our case is the middle of the plate, 81 00:08:31,680 --> 00:08:41,010 ‫let's say right here, then if this plate is a rigid body, then it has no difference whether you depict 82 00:08:41,010 --> 00:08:51,480 ‫this case in terms of distributed force of gravity or equivalent force of gravity f g that goes through 83 00:08:51,480 --> 00:08:54,000 ‫the center of mass of this plate. 84 00:08:54,870 --> 00:09:01,100 ‫It makes no difference in terms of what kind of behavior your mathematical model will predict. 85 00:09:01,740 --> 00:09:03,140 ‫No difference whatsoever. 86 00:09:04,110 --> 00:09:11,910 ‫It's just easier to model things with one force going through one specific point, like in this case 87 00:09:11,910 --> 00:09:16,500 ‫here s.m, but it's an equivalent situation. 88 00:09:16,800 --> 00:09:24,630 ‫Hence I'm using the word equivalent force vector going through the center of mass in the case of this 89 00:09:24,630 --> 00:09:32,730 ‫drone and also in the case of this plate, careful, though, these two representations are true for 90 00:09:32,730 --> 00:09:33,510 ‫rigid bodies. 91 00:09:33,510 --> 00:09:41,940 ‫Only if you remove this rigid body assumption or constraint however you want to call it, you are allowing 92 00:09:41,940 --> 00:09:51,450 ‫this plate to bend and it will bend differently when it is subjected to a distributers force compared 93 00:09:51,450 --> 00:09:53,010 ‫to a point force. 94 00:09:53,760 --> 00:09:56,910 ‫But that's more for structural engineers to worry about. 95 00:09:58,030 --> 00:10:00,220 ‫So I just wanted to clarify this point. 96 00:10:01,030 --> 00:10:08,620 ‫If your plate is rigid, then this representation and this representation, they're equivalent. 97 00:10:09,550 --> 00:10:12,180 ‫They will behave in the same way. 98 00:10:12,700 --> 00:10:16,690 ‫However, if they can bend, then they will bend differently. 99 00:10:16,930 --> 00:10:20,560 ‫But in our course, we don't worry about bending. 100 00:10:21,770 --> 00:10:24,470 ‫So that's your lamda gravity here. 101 00:10:25,490 --> 00:10:34,850 ‫So the question is, once the drones force of gravity, well, you know, that force of gravity equals 102 00:10:34,850 --> 00:10:36,240 ‫M times G, right? 103 00:10:37,610 --> 00:10:39,440 ‫So here is your exercise. 104 00:10:40,040 --> 00:10:50,210 ‫How would lamda G in the body frame look like as a six by one vector? 105 00:10:50,840 --> 00:10:53,730 ‫Again, avoid any lengthy calculations? 106 00:10:54,170 --> 00:10:56,790 ‫It is a one minute exercise and that's it. 107 00:10:57,230 --> 00:11:01,140 ‫Try it and I will show you the solution in the next video. 108 00:11:01,820 --> 00:11:09,170 ‫So essentially, if you know that F G equals M times G, right. 109 00:11:09,170 --> 00:11:18,140 ‫And you know that the force of gravity points down in the negative Z direction, then how would you 110 00:11:19,070 --> 00:11:24,660 ‫rewrite this vector as a six by one vector? 111 00:11:25,040 --> 00:11:26,330 ‫How would it look like? 112 00:11:27,020 --> 00:11:28,390 ‫See you in the next video.