1 00:00:00,460 --> 00:00:01,630 ‫Welcome back. 2 00:00:01,810 --> 00:00:04,180 ‫So let's see our exercise. 3 00:00:04,390 --> 00:00:11,950 ‫The difference between the force of gravity and the control input force and torques is that the control 4 00:00:11,950 --> 00:00:16,570 ‫inputs are in the body frame now, not in the inertial frame. 5 00:00:17,140 --> 00:00:25,330 ‫And since we work in the body frame, we don't have to worry about rotation nor transfer matrices. 6 00:00:26,110 --> 00:00:30,250 ‫Therefore, this vector here looks like this. 7 00:00:31,180 --> 00:00:37,510 ‫That's how it looks like thrust acts in the body frame, the Z direction. 8 00:00:38,050 --> 00:00:40,630 ‫That is why it's the third element. 9 00:00:40,630 --> 00:00:44,560 ‫And then you to use three and four. 10 00:00:44,590 --> 00:00:51,460 ‫They are control moments that are the fourth, fifth and sixth element of this vector. 11 00:00:52,390 --> 00:00:53,420 ‫So that's it. 12 00:00:53,440 --> 00:00:56,260 ‫You have another force and moment vector. 13 00:00:57,360 --> 00:00:58,100 ‫Okay. 14 00:00:58,110 --> 00:01:01,830 ‫So do we have other significant forces or moments? 15 00:01:02,070 --> 00:01:08,040 ‫Well, in a matter of fact, we have to consider one more thing. 16 00:01:08,040 --> 00:01:12,120 ‫And that thing is called a gyroscopic effect. 17 00:01:12,270 --> 00:01:16,560 ‫And I think here we need to do a bit of explaining. 18 00:01:17,340 --> 00:01:26,790 ‫When I tell you gyroscopic effect, I want you to associate it with angular momentum change in direction. 19 00:01:27,450 --> 00:01:34,350 ‫So just remember this expression angular momentum, change in direction. 20 00:01:34,350 --> 00:01:38,760 ‫And angular momentum is a vector that has a direction. 21 00:01:39,090 --> 00:01:48,210 ‫And so when the direction of that angular momentum vector changes, then that's when you have your gyroscopic 22 00:01:48,210 --> 00:01:48,960 ‫effect. 23 00:01:49,740 --> 00:01:54,060 ‫First, let's use a classical mechanism to explain it. 24 00:01:55,040 --> 00:01:58,430 ‫Imagine that I have a rod from the original. 25 00:01:58,460 --> 00:02:00,470 ‫Oh, till the desk. 26 00:02:00,500 --> 00:02:04,280 ‫This desk rotates at omega radians per second. 27 00:02:04,490 --> 00:02:06,810 ‫In the positive y direction. 28 00:02:06,830 --> 00:02:12,080 ‫So your right hand thumb points in the positive y direction. 29 00:02:12,320 --> 00:02:21,050 ‫The entire structure is connected to 0.0 in such a way that it is free to rotate about the x axis. 30 00:02:21,080 --> 00:02:24,400 ‫The disc and the rod have a mass. 31 00:02:24,410 --> 00:02:28,760 ‫Their combined center of gravity is a shown in the image. 32 00:02:28,760 --> 00:02:36,470 ‫And so through that point you have the force of gravity of the entire structure which points in the 33 00:02:36,470 --> 00:02:37,910 ‫negative Z direction. 34 00:02:38,060 --> 00:02:49,490 ‫And so you have a moment ahm d from 0.0 until CG, thus you have a moment vector due to gravity about 35 00:02:49,490 --> 00:02:54,920 ‫the negative x axis its magnitude is f g times. 36 00:02:54,920 --> 00:02:55,430 ‫DX. 37 00:02:55,700 --> 00:03:04,310 ‫Now normally you would expect the entire structure to swing about the negative x axis, and that would 38 00:03:04,310 --> 00:03:07,670 ‫be the case if the disk didn't rotate. 39 00:03:07,670 --> 00:03:16,760 ‫However, the disk rotates about the positive y axis and this disk has a mass moment of inertia. 40 00:03:16,880 --> 00:03:27,290 ‫Therefore you have an angular momentum vector h equals I times omega about the positive y axis. 41 00:03:27,440 --> 00:03:35,600 ‫And now something interesting happens when you introduce a torque or moment into the system like we 42 00:03:35,600 --> 00:03:38,510 ‫did with the moment due to gravity. 43 00:03:38,720 --> 00:03:48,650 ‫The angular momentum vector h starts chasing that moment vector due to gravity, the h vector will try 44 00:03:48,650 --> 00:03:52,160 ‫to align itself with the moment vector. 45 00:03:52,460 --> 00:03:58,820 ‫As a result, the entire structure starts rotating about the positive z axis. 46 00:03:59,030 --> 00:04:04,580 ‫Let's say that it rotates by the alpha radians. 47 00:04:04,580 --> 00:04:10,370 ‫So the alpha is the differential angle, the instantaneous change of alpha. 48 00:04:10,670 --> 00:04:19,820 ‫And so there is a differential change in the H vector in the direction of M minus X, the moment about 49 00:04:19,820 --> 00:04:26,120 ‫the negative x axis due to gravity and that change is d h how do you get that? 50 00:04:26,120 --> 00:04:34,370 ‫D You take the de alpha vector that points in the positive Z direction because that rotation is about 51 00:04:34,370 --> 00:04:39,560 ‫positive Z axis and then you cross it with the h vector. 52 00:04:39,560 --> 00:04:43,100 ‫So the h equals the alpha cross. 53 00:04:43,100 --> 00:04:52,550 ‫H right hand rule shows you that the cross product gives you a direction in the negative x direction. 54 00:04:52,820 --> 00:05:04,490 ‫If you divide both sides by the t, you will have d h over d t equals the alpha over t cross h or simply 55 00:05:04,760 --> 00:05:08,420 ‫h dot equals alpha dot cross h. 56 00:05:08,420 --> 00:05:14,210 ‫But remember the h vector is also a torque vector. 57 00:05:14,360 --> 00:05:23,420 ‫And so when you have rotating elements like for example, rotors in the drone, then you have to account 58 00:05:23,510 --> 00:05:35,030 ‫for that h dot and now we know that in our this case h dot that points in the negative x direction it 59 00:05:35,030 --> 00:05:43,490 ‫equals alpha dot in the positive z direction, cross the h vector in the positive y direction. 60 00:05:43,730 --> 00:05:52,820 ‫What happens next, of course, is that the moment vector will also rotate by the alpha radians about 61 00:05:52,820 --> 00:05:54,230 ‫the positive z axis. 62 00:05:54,470 --> 00:06:00,410 ‫And so the h vector tries to catch the moment vector, but it never catches it. 63 00:06:00,440 --> 00:06:05,120 ‫The entire structure keeps rotating about the positive z axis. 64 00:06:05,330 --> 00:06:08,390 ‫I can show it to you in a small experiment. 65 00:06:08,390 --> 00:06:10,550 ‫I have a small gyroscope here. 66 00:06:10,850 --> 00:06:12,350 ‫You can see the disk. 67 00:06:12,650 --> 00:06:13,730 ‫It rotates. 68 00:06:13,910 --> 00:06:19,790 ‫If I hang it as shown in the video, then we pretty much have our disk case. 69 00:06:19,790 --> 00:06:27,320 ‫When the gyro is not rotating, then it wants to rotate about an axis that points into the wall. 70 00:06:27,440 --> 00:06:28,820 ‫That's because of gravity. 71 00:06:28,820 --> 00:06:32,810 ‫Of course it will not stay in the horizontal position. 72 00:06:33,080 --> 00:06:37,850 ‫However, I will now rotate the gyro disk very fast. 73 00:06:37,880 --> 00:06:46,760 ‫You can see from my right hand thumb that the disk rotation and thus the angular momentum vector point 74 00:06:46,760 --> 00:06:47,780 ‫towards you. 75 00:06:48,050 --> 00:06:53,660 ‫And now when I make the gyro disk rotate and when I let go. 76 00:06:54,010 --> 00:07:02,980 ‫Then as expected, the entire thing rotates about the positive z axis, which is your alpha dot. 77 00:07:03,130 --> 00:07:10,510 ‫So the h vector is chasing the moment vector that is caused by gravity, but it never catches it. 78 00:07:10,750 --> 00:07:19,690 ‫And the change of h vector with respect to time h dot which points in the direction of the moment due 79 00:07:19,690 --> 00:07:24,130 ‫to gravity and which itself is a moment too. 80 00:07:24,310 --> 00:07:25,840 ‫You can get it like this. 81 00:07:26,050 --> 00:07:29,240 ‫H dot equals alpha dot cross. 82 00:07:29,260 --> 00:07:32,090 ‫H But let's do another experiment. 83 00:07:32,110 --> 00:07:36,190 ‫I will now rotate the gyro disk in the opposite direction. 84 00:07:36,490 --> 00:07:41,320 ‫Now the angular momentum vector will point away from you. 85 00:07:41,350 --> 00:07:42,850 ‫Watch what happens. 86 00:07:42,880 --> 00:07:45,880 ‫The moment vector due to gravity hasn't changed. 87 00:07:45,910 --> 00:07:53,860 ‫It still points in the direction of the wall initially, but the h vector that points in the opposite 88 00:07:53,860 --> 00:07:55,750 ‫direction away from you. 89 00:07:55,960 --> 00:07:59,110 ‫It starts chasing the moment vector. 90 00:07:59,320 --> 00:08:04,420 ‫As a result, the entire structure rotates in the opposite direction. 91 00:08:04,780 --> 00:08:08,110 ‫Alpha dot points in the negative z direction. 92 00:08:08,410 --> 00:08:14,980 ‫You can see in the image that the h vector points in the negative y direction. 93 00:08:15,220 --> 00:08:22,600 ‫M minus x and h dot still have the same direction, but the entire structure rotates about the negative 94 00:08:22,600 --> 00:08:23,470 ‫z axis. 95 00:08:23,650 --> 00:08:29,320 ‫Thus alpha dot also points down in the negative z direction. 96 00:08:29,530 --> 00:08:33,070 ‫To get h dot, you apply the same formula. 97 00:08:33,370 --> 00:08:35,770 ‫H dot equals alpha dot cross. 98 00:08:35,770 --> 00:08:40,390 ‫H The right hand rule will give you the same direction. 99 00:08:40,630 --> 00:08:47,770 ‫What I want to emphasize is that physically you first need to introduce a torque into the system like 100 00:08:47,770 --> 00:08:54,970 ‫this M minus X, and then the angular momentum vector H will start chasing it. 101 00:08:55,000 --> 00:08:57,220 ‫For example, consider this case. 102 00:08:57,250 --> 00:09:01,300 ‫You have this disk and the counterweight on the other side. 103 00:09:01,390 --> 00:09:04,960 ‫So C g is at 0.0. 104 00:09:05,200 --> 00:09:07,900 ‫The gravity does not make the system tilt. 105 00:09:08,080 --> 00:09:10,390 ‫There is no moment due to gravity. 106 00:09:10,540 --> 00:09:17,500 ‫You have the disk rotation and thus the h vector in the positive y direction. 107 00:09:17,830 --> 00:09:25,480 ‫If you now apply a talk about the negative z axis, let's call it M minus Z. 108 00:09:25,750 --> 00:09:34,600 ‫Maybe because you apply forces as shown in the image, then the H vector will start chasing the m minus 109 00:09:34,600 --> 00:09:35,560 ‫z vector. 110 00:09:35,680 --> 00:09:45,280 ‫Then your alpha dot is about the negative x axis, but still h dot equals alpha dot cross h. 111 00:09:45,490 --> 00:09:54,190 ‫So always think in terms of h vector chasing the applied moment vector, then it's pretty intuitive. 112 00:09:54,400 --> 00:09:55,510 ‫Thank you very much.