1 00:00:00,560 --> 00:00:01,400 ‫Welcome back. 2 00:00:02,420 --> 00:00:12,670 ‫So how do we find IDOT, Jadot and Cadart remember that I, J and K are unit vectors. 3 00:00:13,400 --> 00:00:15,500 ‫They're magnetites don't change. 4 00:00:15,770 --> 00:00:17,720 ‫They are always one. 5 00:00:18,290 --> 00:00:26,750 ‫The only thing that changes is the direction of the eye J and K when the body frame rotates with respect 6 00:00:26,750 --> 00:00:29,340 ‫to the inertia frame like this. 7 00:00:30,260 --> 00:00:32,990 ‫So this is your inertial frame here in white. 8 00:00:33,470 --> 00:00:42,170 ‫In red you have your body frame and then yellow you have the unit vectors in the body frame and in purple. 9 00:00:42,320 --> 00:00:49,100 ‫This W b, it's the general rotation of this body frame. 10 00:00:49,850 --> 00:00:55,640 ‫And you can see that this general rotation doesn't happen about any particular axis here. 11 00:00:56,480 --> 00:01:05,210 ‫That means that all unit vectors change IDOT, Jadot and Cadart because it's a general rotation that 12 00:01:05,420 --> 00:01:11,960 ‫rotates all the body frame axis with respect to the inertia frame and caution. 13 00:01:12,710 --> 00:01:30,710 ‫Don't confuse DWB with W in the V B vector because remember this vector was U V and W transposed metres 14 00:01:30,710 --> 00:01:31,640 ‫per second. 15 00:01:32,510 --> 00:01:41,720 ‫This is the translational motion of the body frame or the drone in the body from Z direction. 16 00:01:42,420 --> 00:01:49,930 ‫However, this guy here, this is an angular velocity vector in the body frame. 17 00:01:50,720 --> 00:01:53,960 ‫In other words, we can write it down like this. 18 00:01:54,590 --> 00:02:03,860 ‫W in the body frame equals P, Q and R transpose radians per second. 19 00:02:04,670 --> 00:02:07,940 ‫So I will always add a B to this guy here. 20 00:02:08,850 --> 00:02:16,910 ‫OK, but now in order to find don't jadot and K Dot, let's zoom in a little bit. 21 00:02:17,600 --> 00:02:21,470 ‫Let's look at this body frame rotation more closely. 22 00:02:22,880 --> 00:02:27,380 ‫So this is our body frame here in red and in yellow. 23 00:02:27,500 --> 00:02:37,340 ‫You have the unit vectors I, J and K, and now we're going to rotate our body frame about this purple 24 00:02:37,340 --> 00:02:47,990 ‫axis and let's call this purple axis l that is just a random axis that we choose to rotate about, which 25 00:02:47,990 --> 00:02:53,630 ‫doesn't correspond neither with X, Y or Z. 26 00:02:54,410 --> 00:03:06,230 ‫And let's say that we rotate about this purple axis by an angle, the alpha radiance, which is the 27 00:03:06,230 --> 00:03:10,370 ‫differential angle and infinitesimally small angle. 28 00:03:11,060 --> 00:03:20,480 ‫We have rotated our body frame about this purple axis by an infinitesimally small angle. 29 00:03:21,410 --> 00:03:31,370 ‫Since this purple angle doesn't correspond neither to X, Y or Z axis, it means that if we rotate about 30 00:03:31,370 --> 00:03:44,630 ‫it, then there will be a change in the direction of I, J and K, and there will only be a change in 31 00:03:44,630 --> 00:03:47,090 ‫the direction, not in magnitude. 32 00:03:48,020 --> 00:03:51,200 ‫The magnitude of I will always be one. 33 00:03:52,040 --> 00:04:00,800 ‫And the same thing is true for J and four K, I, J and K there you need the vectors. 34 00:04:00,800 --> 00:04:09,350 ‫So they do have a direction, but they're magnitudes by definition are always one, and therefore in 35 00:04:09,350 --> 00:04:15,260 ‫our case only that the direction of these unit vectors will change. 36 00:04:15,920 --> 00:04:25,040 ‫And so by rotating about this purple axis, this eye unit vector will go from here to here. 37 00:04:25,760 --> 00:04:34,630 ‫Let's call it I at T plus T, so that's a differential change in time. 38 00:04:35,390 --> 00:04:42,710 ‫So after a very small time, your eye unit vector will move from here to here. 39 00:04:43,730 --> 00:04:49,100 ‫And the same thing happens with the J unit vector and K unit vector. 40 00:04:50,030 --> 00:04:55,220 ‫They all rotate a little bit like this. 41 00:04:56,360 --> 00:04:59,600 ‫That means that this vector here. 42 00:05:00,110 --> 00:05:11,300 ‫Light blue is D I is the change of the eye vector right, first your eye vector was here, now it's 43 00:05:11,300 --> 00:05:11,800 ‫here. 44 00:05:12,020 --> 00:05:14,690 ‫Therefore this is the change. 45 00:05:15,770 --> 00:05:29,660 ‫You get it like this d I equals the eye unit vector at T plus the T minus the unit vector at TI. 46 00:05:30,500 --> 00:05:37,700 ‫And so when you do this vector subtraction then you will get your di the same thing, you will get your 47 00:05:38,150 --> 00:05:50,000 ‫D.J., which is the infinitesimal change of your vector, and here you will have your DKA, the change 48 00:05:50,000 --> 00:05:55,310 ‫of your K unit vector and all these changes are vector changes. 49 00:05:56,060 --> 00:06:02,720 ‫And now we want to relate our unit vector changes with our actual unit vectors. 50 00:06:03,740 --> 00:06:12,770 ‫And with this, the alpha radiance, the angle that we have rotated about this purple axis that we called 51 00:06:13,040 --> 00:06:22,470 ‫hat in so we can write D A equals D Alpha Cross. 52 00:06:23,180 --> 00:06:38,560 ‫I also d.j equals the Alpha Cross, J and K equals the Alpha Cross K y. 53 00:06:39,140 --> 00:06:40,850 ‫Well, it makes sense actually. 54 00:06:41,270 --> 00:06:49,490 ‫If you take for example, di equals the Alpha Cross I so only the component of the Alpha and that is 55 00:06:49,490 --> 00:07:00,830 ‫perpendicular to the unit vector ie is multiplied by EI and the result is D I and DS perpendicular to 56 00:07:00,830 --> 00:07:04,460 ‫both I and D Alpha. 57 00:07:15,570 --> 00:07:26,820 ‫Also, if you use your right hand rule and you first point your fingers towards the alpha and then towards 58 00:07:26,820 --> 00:07:39,060 ‫eye, then your thump points towards the eye and the same logic applies to Jay and D.J. and Kay and 59 00:07:39,060 --> 00:07:39,860 ‫Dick. 60 00:07:40,680 --> 00:07:42,840 ‫So you have these three equations. 61 00:07:43,230 --> 00:07:54,480 ‫And now what we are going to do, we're going to divide both sides by DETI, also here and here, the 62 00:07:54,480 --> 00:07:55,970 ‫differential time change. 63 00:07:56,760 --> 00:08:08,520 ‫Then this guy becomes I dot, this guy becomes Jadot and this guy becomes K dot, because that's what 64 00:08:08,520 --> 00:08:10,300 ‫this point above means. 65 00:08:10,350 --> 00:08:12,180 ‫It means time derivative. 66 00:08:12,180 --> 00:08:13,710 ‫It's just another notation. 67 00:08:14,720 --> 00:08:28,250 ‫And so since the Alpha was in Radiance, then the Alpha, the tea is in radians per second and so this 68 00:08:28,460 --> 00:08:39,250 ‫the Alpha deti is nothing else but our angular velocity vector in the body frame about this. 69 00:08:39,470 --> 00:08:46,430 ‫L had access, therefore I that equals w. 70 00:08:47,460 --> 00:09:04,710 ‫B vector cross i j dot equals W. b. vector cross J and K dot equals w. b vector cross. 71 00:09:05,340 --> 00:09:07,830 ‫OK, so there you go. 72 00:09:07,860 --> 00:09:12,960 ‫Now you know what your idot Jadot and K Dot are.