1 00:00:00,740 --> 00:00:01,490 ‫Welcome back. 2 00:00:02,210 --> 00:00:10,070 ‫Let's now find our Omega's in terms of use, you have this equation here and you can rewrite it like 3 00:00:10,070 --> 00:00:10,570 ‫this. 4 00:00:11,450 --> 00:00:18,350 ‫You can write these Omega squared here and then on the other side of the equation, sine you will have 5 00:00:18,740 --> 00:00:23,750 ‫you won, but now I'm going to divide it by the trust factor. 6 00:00:23,750 --> 00:00:27,950 ‫So this trust factor goes to the other side of the equation. 7 00:00:27,950 --> 00:00:34,040 ‫Sine so divided by C sub T, the second row would be like this. 8 00:00:34,970 --> 00:00:42,050 ‫Note that this cty times L will go to the other side of the equation sine where the denominator is just 9 00:00:42,050 --> 00:00:44,550 ‫like in the previous case only. 10 00:00:44,570 --> 00:00:46,510 ‫Now you also have this l here. 11 00:00:47,540 --> 00:00:54,050 ‫And since we don't have Omega one squared and omega three squared here, we can put zeroes here. 12 00:00:55,120 --> 00:01:02,560 ‫You can write the third wrote in a similar way, only now you have zeros for Omega two squared and Omega 13 00:01:02,560 --> 00:01:08,370 ‫four squared and then your seat times is here in the denominator. 14 00:01:09,340 --> 00:01:12,630 ‫And finally, your fourth row would be like this. 15 00:01:12,640 --> 00:01:16,440 ‫And now this torque factor would be on the other side of the equation. 16 00:01:16,450 --> 00:01:18,820 ‫Sign here in the denominator. 17 00:01:20,100 --> 00:01:26,970 ‫Next, you can rewrite this entire system in the vector matrix form, it will look like this. 18 00:01:27,480 --> 00:01:36,930 ‫And if you rewrite this vector matrix form, if you multiply this matrix by this vector, then you will 19 00:01:36,930 --> 00:01:40,470 ‫get the exact same form like it it is here. 20 00:01:41,660 --> 00:01:42,840 ‫It makes sense, right? 21 00:01:43,100 --> 00:01:49,640 ‫So here in the first row, you have all the positive signs, all the plus signs, and therefore you 22 00:01:49,640 --> 00:01:56,390 ‫have all ones here and then in the vector, you have the Omega squared values. 23 00:01:57,480 --> 00:02:04,260 ‫And then wherever you have a zero here, then that means you also have a zero here in The Matrix and 24 00:02:04,260 --> 00:02:10,680 ‫then whenever you have a negative sign, that means that here in The Matrix, you will have minus one, 25 00:02:11,610 --> 00:02:19,350 ‫because, for example, if you multiply minus one by omegle, one squared, then you will have this 26 00:02:19,350 --> 00:02:21,940 ‫term here like this. 27 00:02:22,860 --> 00:02:30,450 ‫So let's call this matrix as let's call this vector. 28 00:02:31,470 --> 00:02:44,670 ‫Omega squared vector, and let's call this vector, you vector and let's put an asterisk here just to 29 00:02:44,670 --> 00:02:51,390 ‫let you know that this vector contains these concerns in the denominator. 30 00:02:51,720 --> 00:02:58,320 ‫Not only you want you to use three and four, but also it has things in the denominator. 31 00:02:58,320 --> 00:03:00,870 ‫And that's why I'm putting an asterisk here. 32 00:03:01,880 --> 00:03:15,290 ‫So what you have then is as times omega squared, which is a vector, equals you vector asterisk. 33 00:03:16,530 --> 00:03:23,040 ‫To find this omega squared vector, you have to put an inverse of the matrix here. 34 00:03:24,030 --> 00:03:29,130 ‫And here, this will become your identity matrix then. 35 00:03:30,300 --> 00:03:36,210 ‫And so your omega squared vector equals the inverse matrix. 36 00:03:37,220 --> 00:03:39,890 ‫Times this, you asterisk, Victor. 37 00:03:40,980 --> 00:03:44,940 ‫And now you need to have this inverse matrix, which equals. 38 00:03:46,170 --> 00:03:50,340 ‫This this is your inverse matrix. 39 00:03:51,410 --> 00:03:59,630 ‫I used an inverse matrix online calculator, there are a lot of them online, so you just Google inverse 40 00:03:59,630 --> 00:04:02,840 ‫matrix calculator and then you take the first one there. 41 00:04:03,290 --> 00:04:08,540 ‫You just put this matrix in and it will automatically give you this inverse matrix. 42 00:04:09,570 --> 00:04:18,870 ‫So your omega squared vector will then equal this inverse matrix times this, your vector asterisk, 43 00:04:19,410 --> 00:04:21,640 ‫which now is in this form. 44 00:04:22,710 --> 00:04:27,400 ‫So if you write it all out, then this is what you will get in green. 45 00:04:28,500 --> 00:04:33,900 ‫And now what you want to do, you want to get rid of these squared powers here. 46 00:04:34,740 --> 00:04:42,540 ‫And in order to do that, you just take the square root of these equations or these equations to the 47 00:04:42,540 --> 00:04:48,370 ‫power of one half, which is the same thing like taking the square root of it. 48 00:04:49,140 --> 00:04:51,110 ‫So this one as well. 49 00:04:52,020 --> 00:04:53,240 ‫This one as well. 50 00:04:54,180 --> 00:04:55,790 ‫And also this one. 51 00:04:57,000 --> 00:04:57,980 ‫And there you go. 52 00:04:58,050 --> 00:05:02,670 ‫Now you have your omega one, omega two, omega three and Omega four. 53 00:05:03,480 --> 00:05:10,230 ‫All of them are in terms of you want you to Q3 and Q4. 54 00:05:11,210 --> 00:05:17,990 ‫So the controller will give you the control inputs and you can right away compute the Omega's for the 55 00:05:17,990 --> 00:05:19,020 ‫drone motors. 56 00:05:19,880 --> 00:05:30,050 ‫And then once you have all your for us, you will just some them all up like we have seen before. 57 00:05:31,150 --> 00:05:41,440 ‫It will be like this omega one, minus omega two, plus omega three and minus omega four because Omega 58 00:05:41,440 --> 00:05:46,420 ‫two and Omega four had negative rotation's, according to the right hand rule. 59 00:05:47,460 --> 00:05:49,660 ‫And that's it, you've got everything you need. 60 00:05:50,580 --> 00:05:56,480 ‫Not that the numbers here cannot be negative because this is square root, right? 61 00:05:57,300 --> 00:06:08,360 ‫A to the power of one over half equals square root of a and so you cannot have a negative A in this 62 00:06:08,550 --> 00:06:11,520 ‫square root if you're dealing with real numbers. 63 00:06:12,520 --> 00:06:22,330 ‫You could if you allowed complex numbers, however, you cannot give a complex number to an angular 64 00:06:22,630 --> 00:06:33,310 ‫rotor velocity, the complex number was, for example, A plus eight times B, where A was the real 65 00:06:33,310 --> 00:06:37,510 ‫number and then items B was the imaginary part. 66 00:06:37,930 --> 00:06:44,680 ‫And then this I that was square root minus one. 67 00:06:45,640 --> 00:06:49,720 ‫You can allow these kind of things, for example, in electronics. 68 00:06:50,240 --> 00:06:56,740 ‫However, not in this case, you cannot give something like that to Omega's. 69 00:06:58,000 --> 00:07:02,250 ‫And if you have a negative number here, then the simulation would fail. 70 00:07:03,150 --> 00:07:11,670 ‫And that can happen if your sample time interval, your teeth becomes too big or too small, then this 71 00:07:11,670 --> 00:07:12,720 ‫thing happens. 72 00:07:13,790 --> 00:07:22,730 ‫In the code, your sample time interval equals zero point one seconds, and with this value, it works 73 00:07:22,730 --> 00:07:23,170 ‫well. 74 00:07:23,930 --> 00:07:29,840 ‫And also this negative value problem can arise if the trajectory is very extreme. 75 00:07:30,870 --> 00:07:38,520 ‫Meaning that the trajectory has sudden big turns, the controller that we are building for our drone 76 00:07:38,520 --> 00:07:44,880 ‫is suitable for smooth 3D trajectories and we have several of them in the code. 77 00:07:46,110 --> 00:07:52,890 ‫And that's it, you've got your plan muscle, you now know exactly what's going on there. 78 00:07:53,640 --> 00:07:56,520 ‫You have your state base equations. 79 00:07:57,660 --> 00:08:02,250 ‫You can integrate these states using the oil or Akutan method. 80 00:08:03,300 --> 00:08:11,280 ‫You can move between the inertial and body frames using the rotation and transfer matrices. 81 00:08:12,340 --> 00:08:18,760 ‫And then integrate your inertial frame states as well, using the oilor and wrong Akutan method. 82 00:08:19,850 --> 00:08:27,890 ‫And now you can also convert the control inputs into the more angular velocities. 83 00:08:28,960 --> 00:08:38,020 ‫We are now done with the planned box and starting from the next section, we will start unpacking the 84 00:08:38,020 --> 00:08:46,390 ‫control box to make sure that our drone is able to follow the trajectories that are given to it. 85 00:08:47,490 --> 00:08:50,760 ‫Thank you very much and see you in the next section.