1 00:00:00,550 --> 00:00:01,360 ‫Welcome back. 2 00:00:02,110 --> 00:00:08,260 ‫Let's now substitute our fire and see tangles with zero radiance and see what we get. 3 00:00:09,070 --> 00:00:17,400 ‫So if you have zero radiance here and zero radiance here, you know that you're sign the Zero and Tenjin 4 00:00:17,410 --> 00:00:20,730 ‫zero, they will give you a zero element. 5 00:00:20,800 --> 00:00:22,630 ‫So this entire thing will be zero. 6 00:00:23,620 --> 00:00:31,010 ‫Cosine zero will give you one, but then Tenjin Zero will give you zero. 7 00:00:31,960 --> 00:00:37,710 ‫So you will have one here, but you will have zero here. 8 00:00:38,080 --> 00:00:41,690 ‫So one time zero, it will give you zero. 9 00:00:42,580 --> 00:00:47,790 ‫And here, of course, this was zero and then this was zero. 10 00:00:47,800 --> 00:00:51,720 ‫So you would have zero times zero, which would be zero. 11 00:00:52,300 --> 00:00:58,510 ‫And then here five zero and cosine zero is one. 12 00:00:59,230 --> 00:01:05,550 ‫Here you would have zero and minus sine zero will be zero. 13 00:01:06,370 --> 00:01:18,850 ‫If I make this five zero and then this theta zero, then this numerator will be zero and then this denominator 14 00:01:18,850 --> 00:01:20,230 ‫will be one. 15 00:01:20,650 --> 00:01:31,790 ‫So zero divided by one will give you zero and then five equals zero and theta equals zero. 16 00:01:32,410 --> 00:01:40,200 ‫So the numerator will be one because cosine zero is one and the same thing for the denominator. 17 00:01:40,510 --> 00:01:47,940 ‫So you will have one over one, which means that this entire thing, this entire element will be one. 18 00:01:48,850 --> 00:01:55,660 ‫And so that means that with this small angle assumption, your transfer matrix will become an identity 19 00:01:55,660 --> 00:01:56,230 ‫matrix. 20 00:01:57,190 --> 00:02:00,370 ‫And now something very interesting happens. 21 00:02:01,450 --> 00:02:11,200 ‫This relation that you have here that relates the body frame, angular velocity's with the inertia frame, 22 00:02:11,200 --> 00:02:17,310 ‫angular velocity is this a relationship will become like this. 23 00:02:18,220 --> 00:02:24,410 ‫So you multiply an identity matrix by the body frame, angular velocities. 24 00:02:25,420 --> 00:02:34,780 ‫And so if you write this equation out, then you will find that your Phi Dot, which is here, equals 25 00:02:34,960 --> 00:02:35,680 ‫your P. 26 00:02:36,830 --> 00:02:39,920 ‫Your seat that will equal your Q. 27 00:02:41,130 --> 00:02:52,620 ‫And your side, that will equal or safai, that equals this one theater that equals this one and that 28 00:02:52,620 --> 00:02:54,360 ‫equals this one. 29 00:02:55,550 --> 00:03:08,060 ‫And then it is also logical that your fi double dot equals p dot theater, double dot equals your cute 30 00:03:08,060 --> 00:03:15,480 ‫dot and then side double dot equals your R dot. 31 00:03:16,400 --> 00:03:23,240 ‫And so then you can take these three states based equations here for the attitude. 32 00:03:24,620 --> 00:03:29,750 ‫And you can replace all your PS Qs, R. 33 00:03:29,750 --> 00:03:39,500 ‫S P dots, Q dots and R dots with these values here, and if you do that, then this is what you will 34 00:03:39,500 --> 00:03:39,980 ‫get. 35 00:03:40,880 --> 00:03:49,430 ‫You see, instead of P Q hours, you will have five dots, Seeta dots and side dots, and then instead 36 00:03:49,430 --> 00:03:56,750 ‫of P does Q dots and or dots you will have five double dots, theta double doesn't side double dots. 37 00:03:57,850 --> 00:04:03,730 ‫So just to be clear, these equations here are used only in the NPC controller. 38 00:04:05,030 --> 00:04:13,040 ‫In the plant, you still use these original equations alongside with the transfer matrix where you did 39 00:04:13,040 --> 00:04:21,290 ‫not have that PHI and Theta equals zero radians assumption, where you did not have that small angle 40 00:04:21,290 --> 00:04:22,370 ‫approximation. 41 00:04:23,390 --> 00:04:31,130 ‫So this entire form, these three equations here and then this transfer matrix relationship that would 42 00:04:31,130 --> 00:04:33,410 ‫be in the plant still. 43 00:04:34,490 --> 00:04:43,910 ‫But because in general, the financial tangles will be small throughout the entire maneuver, then because 44 00:04:43,910 --> 00:04:52,520 ‫of this small angle approximation in the controller, in the NPC controller, we will use these equations. 45 00:04:53,690 --> 00:05:02,600 ‫And the performance of the controller because of this assumption will not change significantly because, 46 00:05:03,140 --> 00:05:09,950 ‫see, tangles throughout the maneuver will be small as they should be, because if they are not, then 47 00:05:10,280 --> 00:05:14,120 ‫the U.S. simply won't be able to fly. 48 00:05:15,380 --> 00:05:22,940 ‫Because the propellers of the oveI are not that powerful, that they would be able to produce enough 49 00:05:22,940 --> 00:05:32,390 ‫vertical components thrust in order to counteract the force of gravity when you're fine and FITA angles 50 00:05:32,390 --> 00:05:34,640 ‫are large. 51 00:05:35,720 --> 00:05:44,030 ‫And also, if we think about it logically, then the feedback lionization controller shouldn't really 52 00:05:44,030 --> 00:05:54,770 ‫give big theta reference angles to the NPC because if you're fired, see tangles are large, then you 53 00:05:54,770 --> 00:05:58,280 ‫will have a very huge component in the horizontal direction. 54 00:05:59,090 --> 00:06:05,810 ‫And then if you have some kind of trajectory that you want your you have to follow. 55 00:06:06,950 --> 00:06:12,830 ‫Then with a large horizontal velocity, it will be difficult. 56 00:06:12,830 --> 00:06:16,010 ‫You know, it might drift from the trajectory. 57 00:06:16,940 --> 00:06:19,610 ‫Whenever there is a sharper turn, for example.