1 00:00:00,540 --> 00:00:10,350 ‫But now, if you look at this red line here, then it's not tilting at all, so how can the U.S. go 2 00:00:10,350 --> 00:00:11,670 ‫forward then? 3 00:00:12,510 --> 00:00:20,940 ‫How can it go forward if the red beam is not tilting down to provide a horizontal component in the forward 4 00:00:20,940 --> 00:00:21,570 ‫direction? 5 00:00:22,410 --> 00:00:31,210 ‫Well, remember that we did not model the drag force for the UVs body itself in this simulation. 6 00:00:31,260 --> 00:00:38,160 ‫There is no air resistance in the oves body from X, Y and Z direction. 7 00:00:38,940 --> 00:00:43,590 ‫That's not the reality, of course, but in this simulation, it's like that. 8 00:00:44,610 --> 00:00:51,690 ‫So essentially, the ovei receives one initial push in the forward direction and then it can just keep 9 00:00:51,690 --> 00:01:00,060 ‫going in the forward direction without providing any additional force in this horizontal direction to 10 00:01:00,060 --> 00:01:03,100 ‫have some kind of tilting of the red beam. 11 00:01:03,900 --> 00:01:07,100 ‫We have to add some air resistance. 12 00:01:07,920 --> 00:01:10,910 ‫Now, we didn't model the drag force itself. 13 00:01:10,920 --> 00:01:19,680 ‫However, I could still add some random values to introduce the drag and see if the red beam starts 14 00:01:19,680 --> 00:01:25,500 ‫tilting down in order to produce a horizontal thrust component. 15 00:01:25,890 --> 00:01:33,870 ‫So when we talked about the centripetal acceleration, then this one here was your body frame y axis. 16 00:01:34,770 --> 00:01:37,810 ‫But now we're talking about going forward. 17 00:01:38,040 --> 00:01:40,860 ‫So this is your body frame x axis. 18 00:01:41,880 --> 00:01:50,730 ‫The formulas for the drag are these ones F disp equals one 1/2. 19 00:01:51,640 --> 00:02:04,270 ‫Then you times the air density row times you squared times a sub you, so that's for the body frame 20 00:02:04,270 --> 00:02:12,550 ‫x axis, then this formula here with this sub V letter, that's for your body frame y axis. 21 00:02:13,530 --> 00:02:21,990 ‫And then when you have this letter W here, then that's your body from Z axis, we're now trying to 22 00:02:21,990 --> 00:02:30,810 ‫estimate the drag that the entirety of his body experiences this drag is not for the propeller's, this 23 00:02:30,810 --> 00:02:32,210 ‫is for the UVs body. 24 00:02:33,240 --> 00:02:43,920 ‫We will have three drak forces, F.T. Subcu, if these Sub V and F.T. Sub W, so if the wave travels 25 00:02:43,920 --> 00:02:51,960 ‫in the body frame X direction, then it experiences this drag here F sub D sub you. 26 00:02:52,830 --> 00:02:59,330 ‫If you go in the body frame y direction, then you experience this drag here. 27 00:02:59,370 --> 00:03:00,990 ‫If somebody says V. 28 00:03:01,890 --> 00:03:11,640 ‫And if you fly in the body, frame the direction, then that's what you experience of DFW, then this 29 00:03:11,640 --> 00:03:16,970 ‫valuable see somebody that's your drag coefficient. 30 00:03:17,550 --> 00:03:25,860 ‫It depends on the shape of the have a bigger C somebody means that your shape generates more air resistance. 31 00:03:26,340 --> 00:03:31,250 ‫A smaller see, somebody means that your shape is more aerodynamic. 32 00:03:31,920 --> 00:03:34,050 ‫You generate less air resistance. 33 00:03:34,710 --> 00:03:37,590 ‫And so we have three C sub D here. 34 00:03:38,460 --> 00:03:46,440 ‫So for example, if you're C, some DFW is bigger than the rest of the two. 35 00:03:47,490 --> 00:03:47,810 ‫Right. 36 00:03:47,940 --> 00:04:00,640 ‫If this is bigger than this and also bigger than this, then it means that the upper side of your drone. 37 00:04:01,650 --> 00:04:10,580 ‫This one here, it generates more air resistance in terms of shape than the size of the other. 38 00:04:10,630 --> 00:04:13,790 ‫You like this one here, then you roll. 39 00:04:13,800 --> 00:04:23,490 ‫That's your air density roll equals one point two to five kilograms per cubic meter. 40 00:04:24,270 --> 00:04:34,440 ‫And so your you variable, the variable and w variable, they are your translational velocities in the 41 00:04:34,440 --> 00:04:39,730 ‫body frame, X, Y and Z direction, respectively. 42 00:04:40,590 --> 00:04:49,220 ‫And finally, a sub you as a V and a sub W, they are the reverse cross-sectional areas. 43 00:04:50,230 --> 00:04:57,160 ‫So a subcu and ASV, they would be something like this, it would be like your side, cross-sectional 44 00:04:57,160 --> 00:05:05,050 ‫area and then a W would be something like this, it would be your top cross-sectional area. 45 00:05:05,800 --> 00:05:13,630 ‫From the formulas, you see that the drag forces depend on the velocities squared in the respective 46 00:05:13,630 --> 00:05:17,620 ‫direction, not velocity, but velocity squared. 47 00:05:18,490 --> 00:05:27,070 ‫So you see that as your wave speeds up your drag in the opposite direction, but along the same dimension 48 00:05:27,580 --> 00:05:28,700 ‫also goes up. 49 00:05:29,410 --> 00:05:38,350 ‫So the faster you go in the body frame X direction, the greater your you velocity here because you 50 00:05:38,350 --> 00:05:41,050 ‫is your velocity in the body from X direction. 51 00:05:41,830 --> 00:05:50,320 ‫Now just to specify that in reality the drone doesn't really move in the body frame X direction, it 52 00:05:50,320 --> 00:05:52,240 ‫moves, let's say, horizontally. 53 00:05:53,200 --> 00:06:01,570 ‫Or maybe if the drone also goes a little bit up, then it moves in this direction in your body frame 54 00:06:01,930 --> 00:06:06,550 ‫x axis would still be pointing in this direction. 55 00:06:07,570 --> 00:06:13,210 ‫Then that means that your your velocity would actually be in this direction. 56 00:06:14,140 --> 00:06:23,020 ‫So the faster you go in the body frame x axis, the bigger your U velocity is and the bigger your drag 57 00:06:23,020 --> 00:06:27,040 ‫force in the body frame X direction is. 58 00:06:27,940 --> 00:06:30,610 ‫And of course, that drag would be in the opposite direction. 59 00:06:30,620 --> 00:06:42,730 ‫So the body frame x axis drag force would be in this direction aifs up the sub you I don't know the 60 00:06:42,730 --> 00:06:48,550 ‫exact up D and then the area values for this drone. 61 00:06:48,970 --> 00:06:56,070 ‫However, I will choose some random variables that kind of make sense to me just to introduce the drag. 62 00:06:56,890 --> 00:07:02,950 ‫Remember, I'm not really trying to model the drag forces here in this course. 63 00:07:03,850 --> 00:07:13,180 ‫I'm just trying to introduce some kind of drag force that depends on the drones velocities because a 64 00:07:13,180 --> 00:07:16,000 ‫drag force does depend on drones velocities. 65 00:07:16,810 --> 00:07:25,660 ‫And I want to introduce some kind of drag force here just to see if the red beam of the OV starts tilting 66 00:07:25,660 --> 00:07:26,320 ‫forward. 67 00:07:27,040 --> 00:07:29,950 ‫So the drag force part here, it's not about precision. 68 00:07:29,950 --> 00:07:32,770 ‫I'm not trying to exactly model the drag force. 69 00:07:33,580 --> 00:07:37,180 ‫I just want to introduce some air resistance in my model. 70 00:07:38,050 --> 00:07:47,170 ‫And so my drag forces here, they will update every zero point one second because I will have new U, 71 00:07:47,560 --> 00:07:50,860 ‫V and W velocities every loop. 72 00:07:51,670 --> 00:07:59,590 ‫And so these are my drones states, base equations in the body frame that you thought v the W dot, 73 00:07:59,890 --> 00:08:01,990 ‫dot, dot and our dot. 74 00:08:02,800 --> 00:08:08,170 ‫These are these equations here you see in the code they're here. 75 00:08:08,950 --> 00:08:17,770 ‫And so I just take these drag forces and I include them in my body frame, state based equations like 76 00:08:17,770 --> 00:08:28,480 ‫this minus the drag force in the you direction, minus the drag force in the V direction and minus the 77 00:08:28,480 --> 00:08:31,300 ‫drag force in the W direction. 78 00:08:32,200 --> 00:08:37,930 ‫Those drag forces should decrease your you dot vw dot. 79 00:08:38,710 --> 00:08:45,160 ‫Remember that you dot VW dot, they are your accelerations in those directions. 80 00:08:45,850 --> 00:08:50,890 ‫And so they have to become smaller because after all, drag is air resistance. 81 00:08:51,190 --> 00:08:58,810 ‫If you walk on the ground and all of a sudden the wind blows in your face, then your forward acceleration 82 00:08:58,810 --> 00:08:59,680 ‫should go down. 83 00:09:00,460 --> 00:09:05,500 ‫And that is why I put a minus here, here and here. 84 00:09:06,400 --> 00:09:11,650 ‫And of course, I have to divide the drag forces by the oves mass. 85 00:09:11,650 --> 00:09:16,270 ‫So here, then here and also here. 86 00:09:17,230 --> 00:09:20,650 ‫And that's because I need to have an acceleration term here. 87 00:09:21,430 --> 00:09:26,890 ‫I have to go from Newton's to meters per second squared. 88 00:09:27,760 --> 00:09:33,910 ‫And if you remember then other forces here were also divided by the oves mass. 89 00:09:34,930 --> 00:09:45,070 ‫And so I want to choose if I include the Drac forces in my simulation or not, so I have an if statement 90 00:09:45,070 --> 00:09:53,850 ‫here and this if statement, it says this if drag's, which equals one, then these are your drak forces. 91 00:09:54,760 --> 00:09:57,420 ‫You see, these are your drak forces here. 92 00:09:57,550 --> 00:10:01,800 ‫The formulas here they are the same like here. 93 00:10:02,500 --> 00:10:09,790 ‫But then if my drag's switch equals a zero, then make my drak forces zero. 94 00:10:10,510 --> 00:10:13,470 ‫And then these terms here they become zero. 95 00:10:14,350 --> 00:10:22,300 ‫And then if it's something else then you switch variable must be either zero or one in the unit function. 96 00:10:23,080 --> 00:10:32,560 ‫And so if we go back to our init function, then you can assign a value for this Drac switch here. 97 00:10:33,280 --> 00:10:38,320 ‫So if you put here zero, then there will be no drag force in your simulation. 98 00:10:38,920 --> 00:10:45,900 ‫But if you put here one, then you will include some kind of air resistance in your simulation. 99 00:10:46,660 --> 00:10:55,060 ‫And here you have the values for the Drac coefficients, their random values and the same thing here. 100 00:10:55,060 --> 00:11:00,400 ‫These are your cross-sectional areas in the you, the N.W. direction. 101 00:11:01,270 --> 00:11:03,310 ‫And again, these are not the real values. 102 00:11:03,490 --> 00:11:08,050 ‫I simply put some values that felt logical to me. 103 00:11:08,980 --> 00:11:11,830 ‫And then this is your air density here.