1 00:00:00,690 --> 00:00:10,940 ‫However, now we will make an important assumption, you see, this June cannot have a large fire angle 2 00:00:10,950 --> 00:00:13,860 ‫and it cannot have a large city angle. 3 00:00:14,130 --> 00:00:15,120 ‫And why is that? 4 00:00:15,990 --> 00:00:19,590 ‫Well, let's look at the way these side view. 5 00:00:20,610 --> 00:00:28,860 ‫Let's establish our coordinate frame here, the inertia from Z axis is up, the inertia frame Y axis 6 00:00:28,860 --> 00:00:29,670 ‫is this one. 7 00:00:30,740 --> 00:00:37,550 ‫And then the axis that points towards you, that comes out of the screen, that would be the inertia 8 00:00:37,550 --> 00:00:38,780 ‫from x axis. 9 00:00:39,720 --> 00:00:45,150 ‫And let's say that this here, it would be the body from the axis. 10 00:00:46,320 --> 00:00:56,610 ‫This here would be the body frame Y-axis and then the body frame X-axis would go straight out of the 11 00:00:56,610 --> 00:00:57,820 ‫screen towards you. 12 00:00:59,040 --> 00:01:05,340 ‫That means that this here is more or to you have more for here. 13 00:01:06,310 --> 00:01:16,150 ‫And actually, you would be looking at moral one from the side, which would be this one here, and 14 00:01:16,150 --> 00:01:23,920 ‫the more three would also be not seen because you're looking at the drone directly from the side view, 15 00:01:24,280 --> 00:01:26,470 ‫from the body frame perspective. 16 00:01:26,470 --> 00:01:28,870 ‫But it would be behind that drone. 17 00:01:30,000 --> 00:01:38,820 ‫So this here would be your model, too, which is this one here, then you're more four would be here, 18 00:01:39,660 --> 00:01:44,970 ‫more one would be here, which you cannot see because it's from the side. 19 00:01:45,780 --> 00:01:53,760 ‫And then behind the drone, you would have more three, which is from this perspective on the other 20 00:01:53,760 --> 00:01:54,660 ‫side of the drone. 21 00:01:55,830 --> 00:02:01,500 ‫But we can draw more one and more three separately. 22 00:02:02,590 --> 00:02:14,950 ‫So this is more one and then this is more three, and so we assume that we have a big fire angle like 23 00:02:14,950 --> 00:02:17,770 ‫this, this would be our fire angle. 24 00:02:18,790 --> 00:02:24,580 ‫And you know that this is a fire angle because you can see that your body frame X X-axis has the same 25 00:02:24,580 --> 00:02:27,580 ‫direction, like your initial frame x axis. 26 00:02:28,550 --> 00:02:37,760 ‫And so if you rotate about the body frame X-axis and then your body frame, X-axis has the same direction, 27 00:02:38,000 --> 00:02:43,220 ‫like the initial frame x axis, then that means that this has to be a fine angle here. 28 00:02:44,350 --> 00:02:46,060 ‫And it's a pretty big angle. 29 00:02:47,230 --> 00:02:58,630 ‫And so the thrust vectors, the tease that I'm going to draw here and here and here and also here. 30 00:02:59,820 --> 00:03:05,760 ‫They are produced by the rotor's and they are perpendicular to the rotors. 31 00:03:05,790 --> 00:03:14,160 ‫All right, so this is your model and your trust factor is perpendicular to it if you not decompose 32 00:03:14,340 --> 00:03:15,780 ‫this t vector. 33 00:03:16,930 --> 00:03:23,170 ‫Along the inertial y axis and the inertial z axis. 34 00:03:24,380 --> 00:03:35,180 ‫And let's call it T, Y, and then T, Z, then you see that the component in the inertia frame Z direction 35 00:03:35,360 --> 00:03:36,860 ‫is quite small. 36 00:03:37,940 --> 00:03:43,580 ‫And I will also do the same thing here for this murder and for this murder. 37 00:03:44,760 --> 00:03:52,380 ‫And for this mother as well, and so the bigger the fire angle you have here, the bigger this angle 38 00:03:52,380 --> 00:03:53,040 ‫is. 39 00:03:54,230 --> 00:04:01,190 ‫The smaller the thrust vectors vertical component is, it makes sense, right? 40 00:04:01,370 --> 00:04:08,240 ‫The bigger the inclination, the smaller the vertical component becomes, because the thrust vector 41 00:04:08,240 --> 00:04:11,170 ‫will become more and more horizontal. 42 00:04:12,050 --> 00:04:19,850 ‫And also you have a force of gravity in the negative inertia from the Z axis. 43 00:04:19,850 --> 00:04:20,150 ‫Right. 44 00:04:20,170 --> 00:04:26,140 ‫So this is the positive axis and down it would be a negative inertia from the axis. 45 00:04:26,420 --> 00:04:28,490 ‫So that would be a force of gravity. 46 00:04:29,610 --> 00:04:32,740 ‫That acts through the center of mass of the drone. 47 00:04:33,750 --> 00:04:45,750 ‫And so if the angle fi is not small, then the vertical components of all the four motors will not be 48 00:04:45,750 --> 00:04:46,350 ‫able. 49 00:04:47,310 --> 00:04:51,300 ‫To counteract the downward force of gravity. 50 00:04:52,280 --> 00:05:02,840 ‫And so your net vertical force will be downwards and you can computed like this, you have your four 51 00:05:03,200 --> 00:05:08,000 ‫vertical thrust forces four times. 52 00:05:10,080 --> 00:05:16,710 ‫They are our ports, so they are positive they're these ones here you have four of them, four times 53 00:05:16,710 --> 00:05:24,330 ‫two minus the force of gravity that the UAE is experiencing. 54 00:05:25,370 --> 00:05:33,200 ‫And so if you're four times Tisza is less than your F g. 55 00:05:33,760 --> 00:05:36,210 ‫And we're talking about the magnitudes here. 56 00:05:37,280 --> 00:05:40,250 ‫Well, then your net vector will be downwards. 57 00:05:41,370 --> 00:05:48,690 ‫And then you will also have a component in the positive inertial y direction. 58 00:05:49,620 --> 00:06:00,660 ‫You have T.Y. here to here, here and also here, so it would be four times T.Y. like this. 59 00:06:01,860 --> 00:06:11,310 ‫And so that means that the drone will start sliding downwards and then the downward sliding direction 60 00:06:12,000 --> 00:06:13,140 ‫will be like this. 61 00:06:14,250 --> 00:06:18,330 ‫Your F net vector will be in this direction. 62 00:06:19,430 --> 00:06:23,630 ‫Four times, T.Y. will be in this direction. 63 00:06:24,800 --> 00:06:28,280 ‫And that would be your result in force. 64 00:06:28,310 --> 00:06:35,540 ‫OK, your let's say F, r and so your drone will start sliding down in this direction. 65 00:06:36,710 --> 00:06:44,540 ‫Of course, if you had some mega powerful propeller's, then they could produce such an awesome thrust 66 00:06:44,930 --> 00:06:54,980 ‫that would produce enough vertical component to counteract gravity even when and Seeta angles are not 67 00:06:54,980 --> 00:06:55,430 ‫small. 68 00:06:56,380 --> 00:06:58,960 ‫However, that's not the case with us. 69 00:06:59,940 --> 00:07:08,760 ‫Besides, in that case, the horizontal velocity of the UAE would be pretty large because of this horizontal 70 00:07:08,760 --> 00:07:10,470 ‫component of the thrust force. 71 00:07:11,380 --> 00:07:13,840 ‫And this is something that you might not want. 72 00:07:14,920 --> 00:07:25,210 ‫And so that means that in order to follow the trajectories, obviously there will be some PHY and Seeta 73 00:07:25,210 --> 00:07:29,740 ‫angles, however, they will be small, they have to be small. 74 00:07:31,090 --> 00:07:38,260 ‫Because only then the UAE can counteract the downward force of gravity and it won't start sliding down 75 00:07:38,260 --> 00:07:38,800 ‫like this. 76 00:07:40,060 --> 00:07:50,950 ‫And because of that, for control purposes, we can assume that fire and Seeta angles are zero radiance. 77 00:07:52,310 --> 00:07:59,870 ‫And this assumption won't affect the NPC controllers performance significantly, we are not going to 78 00:08:00,320 --> 00:08:08,900 ‫touch the transfer matrix in the planned model because we want the plan model to model the real system 79 00:08:08,900 --> 00:08:10,070 ‫more realistically. 80 00:08:11,270 --> 00:08:18,710 ‫However, if we can expect that the fire and see tangles will be small. 81 00:08:19,730 --> 00:08:28,460 ‫Then we can simplify the mall in the controller and we can use this simplified version in the controller, 82 00:08:29,330 --> 00:08:38,460 ‫and the way we simplify the model for the oveI is by assuming that PHI and Theta Angles both equals 83 00:08:38,460 --> 00:08:41,060 ‫zero inside the transfer matrix. 84 00:08:42,200 --> 00:08:49,480 ‫So if we now assume that this is our oves system and this is your feedback lionization controller, 85 00:08:49,490 --> 00:08:54,530 ‫so it goes here as well and your planner is here. 86 00:08:55,770 --> 00:09:06,960 ‫And then the planner also sends a reference sci angle here to the controller and then the feedback lionization 87 00:09:07,590 --> 00:09:12,060 ‫controller will send it's you one into the plan. 88 00:09:13,590 --> 00:09:23,190 ‫So what we want to know is that if in the planned box, in the transfer matrix, the fly and then the 89 00:09:23,730 --> 00:09:27,150 ‫Seeta angles, when they are not zero. 90 00:09:28,330 --> 00:09:38,050 ‫Then if we assume the fire and then the Seeta angles in the transfer matrix to be zero radians in the 91 00:09:38,050 --> 00:09:38,980 ‫controller. 92 00:09:40,100 --> 00:09:49,070 ‫Will this controller with this assumption still be able to control this plan without this assumption? 93 00:09:49,520 --> 00:09:55,550 ‫Well, will this simplified controller still control the plan? 94 00:09:56,420 --> 00:10:03,020 ‫Well, in which you did not have this assumption in the transfer matrix and since. 95 00:10:04,210 --> 00:10:12,010 ‫The U.S. needs need to maintain small fire, see tangles in order to follow a trajectory, then the 96 00:10:12,010 --> 00:10:14,070 ‫answer is yes, it can. 97 00:10:15,270 --> 00:10:18,930 ‫The controller with this assumption will simplify our lives. 98 00:10:19,930 --> 00:10:23,170 ‫But you will sacrifice very little in its performance. 99 00:10:24,270 --> 00:10:26,200 ‫And so this is an exercise for you. 100 00:10:26,220 --> 00:10:35,160 ‫Now, what will the transfer matrix equal to if your are and theta angles equals zero radiance? 101 00:10:36,130 --> 00:10:44,140 ‫So just make all five Seeta angles zero here in all the elements and see what you will get for your 102 00:10:44,140 --> 00:10:45,170 ‫transfer matrix. 103 00:10:46,120 --> 00:10:49,540 ‫Try it out yourself and then see you in the next video. 104 00:10:50,440 --> 00:10:51,250 ‫Thank you very much.