1 00:00:00,750 --> 00:00:01,500 ‫Welcome back. 2 00:00:01,980 --> 00:00:08,970 ‫So I hope that you've tried this little thought, exercise yourself and let's look at it now. 3 00:00:09,540 --> 00:00:15,600 ‫If you're PHI and Theta are zeros, then this element here will be zero. 4 00:00:16,200 --> 00:00:25,890 ‫Cosine zero will be one size zero over cosine zero will be zero because you would have zero divided 5 00:00:25,890 --> 00:00:26,480 ‫by one. 6 00:00:27,150 --> 00:00:31,980 ‫And now in this term you would have one times zero. 7 00:00:32,550 --> 00:00:36,340 ‫So zero minus sine zero is also zero. 8 00:00:36,930 --> 00:00:41,330 ‫And here you will have coastline zero over cosine theta. 9 00:00:41,490 --> 00:00:44,520 ‫So one over one so that it will be one. 10 00:00:45,030 --> 00:00:53,900 ‫And so you will have a one here, here and here and the rest of the elements will be zeroes. 11 00:00:54,300 --> 00:01:01,050 ‫So with this assumption, your transfer matrix becomes an identity matrix. 12 00:01:01,470 --> 00:01:10,800 ‫And that is very good, because if the transfer matrix is the thing that connects your P, Q, R and 13 00:01:10,800 --> 00:01:18,450 ‫find our Theta and apply that, then if you have an identity matrix here, then that means that you're 14 00:01:18,600 --> 00:01:26,750 ‫Phi, that equals P Seton, that equals Q and sign, that equals R. 15 00:01:27,300 --> 00:01:37,920 ‫And so when we reach the section in which I teach you the NPC controller, you will see that this result 16 00:01:37,920 --> 00:01:44,420 ‫here will simplify our NPC math greatly here. 17 00:01:44,430 --> 00:01:47,250 ‫I want to point out something here. 18 00:01:47,250 --> 00:01:55,680 ‫With this transfer function, I literally assume that my sine Phi and Tangent Theta are zeros. 19 00:01:56,280 --> 00:02:03,900 ‫This is a little bit different from a small angle approximation where I would say that my Phi's approximately 20 00:02:03,900 --> 00:02:07,540 ‫zero and my seat is approximately zero. 21 00:02:08,160 --> 00:02:18,210 ‫In that case, sine Phi and Tangent Theta could be approximated like this Phi and Theta. 22 00:02:18,840 --> 00:02:24,600 ‫This kind of approximation is more precise than what I'm doing here. 23 00:02:25,080 --> 00:02:35,250 ‫However, if I do it in this way, then I will not be able to express my transfer matrix as an identity 24 00:02:35,250 --> 00:02:35,750 ‫matrix. 25 00:02:36,210 --> 00:02:40,600 ‫And then these relationships here, they would not happen. 26 00:02:41,160 --> 00:02:50,010 ‫However, if a drone operates close to its hovering position, which means that the tilting angles, 27 00:02:50,370 --> 00:02:57,720 ‫PHI and Theta are very close to zero all the time, then if my velocity and acceleration requirements 28 00:02:57,720 --> 00:03:05,360 ‫are small, then this rough approximation for the controller will still get the job done. 29 00:03:06,000 --> 00:03:10,490 ‫I will lose in precision, but I will gain in simplicity. 30 00:03:11,070 --> 00:03:18,810 ‫In other words, in the plans where this assumption is not made, if you apply a controller to it, 31 00:03:19,410 --> 00:03:27,210 ‫where this assumption is made, then the drone will still be able to follow the trajectory in a very 32 00:03:27,210 --> 00:03:27,890 ‫smooth way. 33 00:03:28,500 --> 00:03:35,340 ‫It's important to understand is that this simplification that we have made, this is for the controller. 34 00:03:35,880 --> 00:03:38,540 ‫I'm not doing this simplification in the plant. 35 00:03:39,000 --> 00:03:41,660 ‫That's not the case in the plant. 36 00:03:41,670 --> 00:03:46,020 ‫I want my mathematical model to be as realistic as possible. 37 00:03:46,560 --> 00:03:56,760 ‫So I keep my transfer matrix inside the plant in its original form that contains all its PHI and theta 38 00:03:56,760 --> 00:03:58,580 ‫angles in the plant. 39 00:03:58,590 --> 00:04:03,210 ‫I'm not simplifying anything, but in the controller I do. 40 00:04:03,810 --> 00:04:11,610 ‫And then in the controller I will be able to get this identity matrix in this relationship and in the 41 00:04:11,610 --> 00:04:21,300 ‫controller I will replace my p q r variables with fi that theatergoer and that not in the plans but 42 00:04:21,300 --> 00:04:22,380 ‫in the controller. 43 00:04:22,860 --> 00:04:31,380 ‫And then in the simulation you will see that the plant, without the simplifications and without the 44 00:04:31,380 --> 00:04:36,770 ‫substitutions, can be stabilized with this controller. 45 00:04:37,410 --> 00:04:46,290 ‫That does include this simplification and the substitutions provided that the velocity and acceleration 46 00:04:46,290 --> 00:04:54,240 ‫requirements for the drone are small and the drone mostly operates close to its hovering position, 47 00:04:54,870 --> 00:04:59,820 ‫meaning that there is tilting with very small fire theta angles. 48 00:05:00,390 --> 00:05:10,530 ‫And now you see why it is convenient to choose as this convention rotation about the moving from Axis 49 00:05:10,650 --> 00:05:19,020 ‫first Z, then Y than X, even though we have to derive one additional matrix, which is the transfer 50 00:05:19,020 --> 00:05:22,450 ‫matrix, but now you see the added value of it. 51 00:05:22,920 --> 00:05:32,820 ‫So for translational velocities, you VW that we measured in meters per second here, we do use our 52 00:05:32,820 --> 00:05:38,490 ‫rotation matrix and then we get this vector in the inertial frame. 53 00:05:39,090 --> 00:05:41,340 ‫That's translational velocity. 54 00:05:41,910 --> 00:05:49,920 ‫But with angular velocities, what you measure with radians per second, this convention will give us 55 00:05:50,250 --> 00:06:01,170 ‫a very convenient transfer matrix that relates my oilor angle time derivatives with my angular velocity 56 00:06:01,170 --> 00:06:02,240 ‫in the body frame. 57 00:06:02,850 --> 00:06:10,980 ‫And when your velocity and acceleration requirements are small, then it will allow you to use this 58 00:06:10,980 --> 00:06:11,730 ‫assumption. 59 00:06:12,360 --> 00:06:20,130 ‫If you had chosen another convention about moving from Axis, then you would still have two angles. 60 00:06:20,670 --> 00:06:28,410 ‫But maybe you would have PHY and Passi or Seeta and BPCI and BPCI was your motion. 61 00:06:28,410 --> 00:06:28,760 ‫Right? 62 00:06:29,340 --> 00:06:31,140 ‫And so you could not have said that. 63 00:06:31,590 --> 00:06:34,280 ‫OK, let's assume BPCI to be zero. 64 00:06:34,890 --> 00:06:36,600 ‫That would have been impossible. 65 00:06:37,140 --> 00:06:45,330 ‫I mean, if you have to fly in a spiral then obviously your sci angle will vary a lot. 66 00:06:45,690 --> 00:06:55,290 ‫And also if you had chosen your R, X, Y, Z convention, the rotation about the fixed axis, then 67 00:06:55,290 --> 00:07:02,720 ‫you would have had three angles here and then this result would have been impossible as well. 68 00:07:03,360 --> 00:07:09,710 ‫And starting from the next video, we will actually derive the transfer matrix mathematically. 69 00:07:10,260 --> 00:07:13,410 ‫Thank you very much and I'll see you in the next video.