1 00:00:00,780 --> 00:00:07,700 ‫Let's now derive the four control actions with which we can control the use of his position and orientation, 2 00:00:07,830 --> 00:00:14,700 ‫the first control action is thrust and its unit, of course, is Newton. 3 00:00:15,030 --> 00:00:18,970 ‫And let's call thrust you want. 4 00:00:19,470 --> 00:00:23,970 ‫So our first control action will be you one, which is thrust. 5 00:00:24,270 --> 00:00:28,440 ‫In other words, thrust is one of our system inputs. 6 00:00:28,680 --> 00:00:31,800 ‫And for now, let's assume that there is no gravity. 7 00:00:32,040 --> 00:00:38,760 ‫We only have thrust that is generated from the four propellers, from the four motors. 8 00:00:38,760 --> 00:00:46,320 ‫And by the way, we abbreviate more with M so M one and two and three and then four are just more. 9 00:00:46,320 --> 00:00:48,420 ‫One more, two more, three and more four. 10 00:00:48,750 --> 00:00:56,760 ‫If you look at the universe here, then you can see the four propellers rotating at the same rate. 11 00:00:56,910 --> 00:01:08,310 ‫Even though M1 and M3 rotate counterclockwise and M2 and M4 rotate clockwise, all of them generate 12 00:01:08,550 --> 00:01:11,420 ‫the same thrust force upwards. 13 00:01:12,030 --> 00:01:22,680 ‫And so if all of them rotate a sigma age radians per second and you add a small amount of rotation that 14 00:01:22,680 --> 00:01:32,280 ‫we call Delta A. which is also in radians per second, and you apply this Delta A equally to all of 15 00:01:32,280 --> 00:01:39,210 ‫them, to all of the propellers, then all propellers generate the same amount of more thrust in the 16 00:01:39,210 --> 00:01:42,030 ‫body frame the Z direction. 17 00:01:42,360 --> 00:01:52,170 ‫If you add them all together, you will have a global thrust, which is you one and that you one generates 18 00:01:52,710 --> 00:01:59,860 ‫this acceleration, small z double dot, that is acceleration in the body frame Z direction. 19 00:02:00,450 --> 00:02:11,490 ‫Now if the body frame Z is aligned with the inertial frame Z axis, in other words, if the V is not 20 00:02:11,490 --> 00:02:17,940 ‫tilted, then the acceleration points upward in the big Z direction as well. 21 00:02:18,120 --> 00:02:27,240 ‫And if Delta A is equal in negative for all propellers, then that means that you're you one still points 22 00:02:27,450 --> 00:02:31,620 ‫in the body frame Z direction, but it is smaller. 23 00:02:32,070 --> 00:02:40,260 ‫And of course, if we assume gravity, then obviously to take off the ground, you're you one must be 24 00:02:40,260 --> 00:02:42,210 ‫greater than the force of gravity. 25 00:02:42,330 --> 00:02:42,700 ‫Right. 26 00:02:42,930 --> 00:02:52,830 ‫So if you have your force of gravity downwards and then you're you one upwards, then you one as a force 27 00:02:52,830 --> 00:02:55,200 ‫needs to be bigger than force of gravity. 28 00:02:55,440 --> 00:03:00,180 ‫And of course, we are talking about the magnitudes because the direction of the new one is upwards 29 00:03:00,180 --> 00:03:03,300 ‫and the direction of the force of gravity is downwards. 30 00:03:03,300 --> 00:03:08,070 ‫But the magnitude of you want needs to be bigger than the magnitude of the force of gravity. 31 00:03:08,250 --> 00:03:09,860 ‫Then the drone will take off. 32 00:03:09,870 --> 00:03:11,090 ‫Otherwise it won't. 33 00:03:11,610 --> 00:03:17,240 ‫And so we have our view one, which is our first control action. 34 00:03:17,670 --> 00:03:20,910 ‫This is what the controller will find for the drone. 35 00:03:21,270 --> 00:03:30,690 ‫And we have acceleration generated from you, one which is small Z, double dot this one and they are 36 00:03:30,690 --> 00:03:32,790 ‫related with mass. 37 00:03:33,330 --> 00:03:39,090 ‫So you one equals mass times, small Z double dot. 38 00:03:39,270 --> 00:03:46,410 ‫Remember, mass is resistance to linear acceleration and this is assuming that there is no force of 39 00:03:46,410 --> 00:03:46,980 ‫gravity. 40 00:03:47,190 --> 00:03:50,490 ‫You one is the only force that is applied to the drone.