1 00:00:00,420 --> 00:00:01,230 ‫Welcome back. 2 00:00:01,770 --> 00:00:09,900 ‫You see, in engineering, it's very important to know in advance what kind of product you're designing, 3 00:00:10,350 --> 00:00:14,070 ‫you need to know very well what you want your product to do. 4 00:00:14,640 --> 00:00:20,140 ‫The requirements of your product should be specified as much as possible. 5 00:00:20,730 --> 00:00:21,330 ‫Why? 6 00:00:21,960 --> 00:00:29,100 ‫Well, first of all, if your clients tell you that they want X and you give them Y, then that would 7 00:00:29,100 --> 00:00:30,740 ‫make them pretty unhappy. 8 00:00:31,380 --> 00:00:33,540 ‫And we don't want them to be unhappy. 9 00:00:33,550 --> 00:00:33,930 ‫Right. 10 00:00:34,560 --> 00:00:43,500 ‫But another reason why you want to know exactly what your product will be doing is because then you 11 00:00:43,500 --> 00:00:48,400 ‫will be able to know where you can simplify your mathematics. 12 00:00:48,990 --> 00:00:52,440 ‫In other words, where you can make certain approximations. 13 00:00:52,920 --> 00:00:59,880 ‫If we want our drone to perform some kind of super complicated acrobatic maneuvers, well, then your 14 00:00:59,880 --> 00:01:01,790 ‫math will probably get harder. 15 00:01:02,340 --> 00:01:08,400 ‫But there are a lot of applications for drones that do not require high speed. 16 00:01:08,760 --> 00:01:13,000 ‫Nowadays, wind turbines are becoming pretty popular. 17 00:01:13,560 --> 00:01:22,660 ‫And so what you might want to do, you might want to send a drone there to perform an inspection. 18 00:01:23,250 --> 00:01:29,770 ‫Maybe this drone has some cameras and then you can inspect for cracks in the structure. 19 00:01:30,360 --> 00:01:37,950 ‫I know that they do it with airplanes as well, since airplanes are big structures and they constantly 20 00:01:37,950 --> 00:01:46,590 ‫must be inspected for cracks in the structure, then I know that they are using drones for that as well, 21 00:01:47,250 --> 00:01:56,620 ‫where a drone would fly close to an aircraft with cameras and then inspect for any possible problems. 22 00:01:57,030 --> 00:02:02,580 ‫So if there is some kind of crack here in the material, in the structure, they would know about it. 23 00:02:03,150 --> 00:02:13,350 ‫So in both cases, they would fly around this object and then inspect for some kind of problems in the 24 00:02:13,350 --> 00:02:14,010 ‫structure. 25 00:02:14,520 --> 00:02:19,500 ‫Obviously, in that case, the drone should fly pretty slowly. 26 00:02:20,100 --> 00:02:28,890 ‫The drones cannot fly fast because if it starts flying fast, then it won't be able to follow this trajectory. 27 00:02:29,460 --> 00:02:33,000 ‫It will just overshoot if it flies very fast. 28 00:02:33,420 --> 00:02:34,770 ‫And then there is no point. 29 00:02:35,310 --> 00:02:41,550 ‫The goal is to inspect the aircraft and find possible problems in the structure. 30 00:02:42,120 --> 00:02:50,880 ‫The drone should take its time and fly around the aircraft or the wind turbine and then inspect all 31 00:02:50,880 --> 00:02:51,770 ‫the surface area. 32 00:02:52,380 --> 00:02:59,820 ‫That means that we already know that the trajectories that we will give to our drone, they will not 33 00:02:59,820 --> 00:03:02,700 ‫require the drone to fly fast. 34 00:03:03,210 --> 00:03:12,840 ‫In other words, if we plot our X dimension as a function of time, Y dimension as a function of time, 35 00:03:13,470 --> 00:03:22,130 ‫and then a Z dimension as a function of time, we will not have some kind of crazy trajectories there. 36 00:03:22,680 --> 00:03:29,640 ‫We will have smooth trajectories like this and these smooth trajectories. 37 00:03:30,240 --> 00:03:40,180 ‫They have small slopes, which means that your X dot, y dot and Z, that will be small, so your velocities 38 00:03:40,230 --> 00:03:40,980 ‫will be small. 39 00:03:41,610 --> 00:03:45,860 ‫Now how does a drone generate velocities? 40 00:03:46,440 --> 00:03:54,070 ‫You know that it has four thrust vectors generated by those four propellers. 41 00:03:54,690 --> 00:04:01,350 ‫And so if you want to have some kind of velocity, then you have to tilt your drone, either like this 42 00:04:01,860 --> 00:04:03,100 ‫or like that. 43 00:04:03,570 --> 00:04:08,250 ‫Furthermore, don't forget that we are dealing with a drone. 44 00:04:08,760 --> 00:04:16,880 ‫We are not dealing with a car or other objects that experience some kind of significant friction force. 45 00:04:17,400 --> 00:04:22,480 ‫Let's imagine that this is our drone and it is hovering right now. 46 00:04:22,920 --> 00:04:30,040 ‫So you want this drone to go there and you want your velocity in that direction to be more or less constant. 47 00:04:30,480 --> 00:04:37,710 ‫So what you will do, you will slightly tilt your drone so it will have some kind of horizontal component. 48 00:04:38,250 --> 00:04:43,160 ‫And because of that, the drone will start moving in this direction. 49 00:04:43,680 --> 00:04:49,590 ‫But what will happen if you keep your drone tilted as a function of time? 50 00:04:50,130 --> 00:04:54,570 ‫Well, then you will keep having this horizontal component. 51 00:04:55,080 --> 00:04:59,400 ‫And you know, that force equals mass times the. 52 00:04:59,450 --> 00:05:06,320 ‫Celebration, so for as long as you keep your drone tilted, you will have a component in this direction 53 00:05:06,980 --> 00:05:09,110 ‫that will cause acceleration. 54 00:05:09,560 --> 00:05:15,650 ‫And so if you just keep this drone tilted like that, then the drone will start moving in this direction 55 00:05:15,650 --> 00:05:16,850 ‫faster and faster. 56 00:05:17,420 --> 00:05:18,790 ‫But we didn't want that. 57 00:05:18,980 --> 00:05:25,900 ‫We wanted stable, smooth, not so fast velocity with small accelerations. 58 00:05:26,300 --> 00:05:34,670 ‫So in order to do that, the drone can tilt for a short period of time, get its desired velocity in 59 00:05:34,670 --> 00:05:38,550 ‫this direction, and then get back to its hovering position. 60 00:05:39,090 --> 00:05:48,590 ‫However, even though there is no tilt anymore, your drone will still go in this direction because 61 00:05:48,590 --> 00:05:55,320 ‫what forces with the drone experience at that time, you don't have friction with the ground? 62 00:05:55,940 --> 00:06:03,470 ‫Well, there is force of gravity, of course, but that's perpendicular to this motion, so it won't 63 00:06:03,470 --> 00:06:04,000 ‫affect it. 64 00:06:04,580 --> 00:06:08,390 ‫And yes, you have a very small drag force here. 65 00:06:08,960 --> 00:06:14,210 ‫But if our velocities are very small, then our air resistance will be very small. 66 00:06:14,780 --> 00:06:17,330 ‫And so you will need some time to slow down. 67 00:06:17,870 --> 00:06:23,620 ‫And then let's say that now you want the drone to go in this direction. 68 00:06:24,110 --> 00:06:26,560 ‫So you would give it a small tilt. 69 00:06:27,080 --> 00:06:34,010 ‫There will be a small horizontal component in this direction and the drone will start flying in this 70 00:06:34,010 --> 00:06:34,580 ‫direction. 71 00:06:35,090 --> 00:06:39,940 ‫And then very shortly you will go back to the hovering position. 72 00:06:40,550 --> 00:06:44,740 ‫Then these two horizontal components will, of course, disappear. 73 00:06:45,380 --> 00:06:54,470 ‫But since you're a drag force will be very small in a forward velocity, will not decrease significantly 74 00:06:54,470 --> 00:06:55,150 ‫right away. 75 00:06:55,700 --> 00:07:05,690 ‫And this kind of behavior allows us to assume, at least in our mathematics, that our fi and theta 76 00:07:06,290 --> 00:07:15,620 ‫are zero radians, because when your drone flies close to a hovering position, then these two angles 77 00:07:16,190 --> 00:07:20,470 ‫indeed are very close to zero during most of the time. 78 00:07:20,960 --> 00:07:29,120 ‫But remember, we are only able to make this assumption because we are not designing a controller for 79 00:07:29,120 --> 00:07:32,090 ‫some kind of racing or acrobatics drone. 80 00:07:32,660 --> 00:07:40,340 ‫We are designing a controller for a drone that would perform these kind of functions like structure 81 00:07:40,340 --> 00:07:41,110 ‫inspection. 82 00:07:41,630 --> 00:07:51,920 ‫And so now if we are able to say that, then if you look at this transfer matrix, then tell me what 83 00:07:52,070 --> 00:08:00,640 ‫will this transfer matrix become if I substitute PHI and Theta Angles here with zeros? 84 00:08:01,100 --> 00:08:05,570 ‫So let it be your exercise and let's look at it in the next video.