1 00:00:00,640 --> 00:00:01,420 ‫Welcome back. 2 00:00:02,170 --> 00:00:13,780 ‫Let's now do the conversion exercise, you had a rotation matrix called R, sub Z, Y, X using the 3 00:00:13,780 --> 00:00:14,730 ‫oil or angles. 4 00:00:14,950 --> 00:00:16,630 ‫So that was our convention. 5 00:00:17,840 --> 00:00:29,180 ‫Or you can write it down like this are sabzi times are sub Y, times are sub X, but for simplicity 6 00:00:29,180 --> 00:00:37,010 ‫let's just call it are now each matrix here was a three by three rotation matrix. 7 00:00:37,780 --> 00:00:44,260 ‫So if you multiply them together in this sequence, then this would be a result. 8 00:00:45,020 --> 00:00:53,780 ‫So you can use this rotation matrix to take measurements in the body frame and convert them into the 9 00:00:53,780 --> 00:00:54,740 ‫inertia frame. 10 00:00:55,740 --> 00:01:05,250 ‫But careful rotation matrices were only good for translational velocities, not angular velocities for 11 00:01:05,250 --> 00:01:06,670 ‫the angle of velocities. 12 00:01:06,960 --> 00:01:15,690 ‫You had this special transfer matrix, if you remember, and the reason for that was because when we 13 00:01:15,690 --> 00:01:24,750 ‫dealt with the rotations, we had a specific convention and then we had to derive a specific transfer 14 00:01:24,750 --> 00:01:34,530 ‫matrix for that specific convention, Zywiec using the oil or angles, meaning that you first rotate 15 00:01:34,830 --> 00:01:41,670 ‫about the body from Z axis, then you rotate about the body frame, y axis. 16 00:01:42,270 --> 00:01:48,990 ‫So about the moving frame and then you rotate about the body frame or moving frame x axis. 17 00:01:50,180 --> 00:02:01,850 ‫So anyway, if you have your you the N.W., which are your translation of the losses in the body frame, 18 00:02:02,540 --> 00:02:11,210 ‫what will you get if you take this rotation matrix and you multiply it by this vector, what will you 19 00:02:11,210 --> 00:02:11,480 ‫get? 20 00:02:12,380 --> 00:02:20,740 ‫You will get X, dot, why dot and Z that in the inertia frame. 21 00:02:21,290 --> 00:02:29,750 ‫And now if you have your body frame, angular velocity P, Q and R, then what will you get? 22 00:02:29,750 --> 00:02:40,670 ‫If you take the transfer matrix and you multiply it by this vector, you will get five that theta that 23 00:02:41,420 --> 00:02:42,650 ‫and upside that. 24 00:02:43,680 --> 00:02:44,740 ‫It makes sense, right? 25 00:02:45,210 --> 00:02:53,070 ‫Translation of last in the body frame and then you get a translation of velocities in the initial frame. 26 00:02:53,790 --> 00:03:00,870 ‫And here you have the angular velocities in the body frame and then you get your angular velocities 27 00:03:01,410 --> 00:03:02,710 ‫in the initial frame. 28 00:03:03,660 --> 00:03:06,720 ‫We can also rewrite it in a more compact way. 29 00:03:07,810 --> 00:03:14,970 ‫This entire vector in the inertia frame equals this matrix times. 30 00:03:16,180 --> 00:03:25,650 ‫This vector, so we managed to incorporate the rotation matrix and the transfer matrix in one big matrix 31 00:03:25,930 --> 00:03:35,920 ‫and we called it Matrix J and you multiply it by this vector and that's how you convert between body 32 00:03:35,920 --> 00:03:43,480 ‫and inertial frames and then you just perform the same kind of integration for the states in the initial 33 00:03:43,480 --> 00:03:43,950 ‫frame. 34 00:03:45,080 --> 00:03:53,120 ‫So here I have used the oilor method, the easier but more imprecise method, and soon you will learn 35 00:03:53,120 --> 00:03:55,110 ‫the wrong Akutan method. 36 00:03:56,000 --> 00:03:57,050 ‫So that's how you do it. 37 00:03:57,560 --> 00:03:59,540 ‫And let's complete the schematics now. 38 00:04:00,410 --> 00:04:03,860 ‫So what you definitely need to have in your plan box. 39 00:04:04,860 --> 00:04:12,870 ‫Is this Jay Matrix that was a function of our matrix and transfer matrix? 40 00:04:14,080 --> 00:04:18,910 ‫In fact, to be more precise than Jay. 41 00:04:20,060 --> 00:04:27,560 ‫Was a function of R.A., but then the rotation matrix itself was a function of. 42 00:04:28,870 --> 00:04:36,700 ‫Fi, Seeta and BPCI, so we have here Fi, Seeta and Passi. 43 00:04:37,580 --> 00:04:44,300 ‫And the transfer matrix was a function of Phi and Theta. 44 00:04:45,320 --> 00:04:56,780 ‫So Phi and theater like this, that means that this box will need values. 45 00:04:58,130 --> 00:05:02,870 ‫There are five Seeta and Passi. 46 00:05:03,920 --> 00:05:11,480 ‫And of course, since you have your states in your body frame, you just take your state's. 47 00:05:12,650 --> 00:05:21,050 ‫And you put them through this Jay box and then what you will get is X dot. 48 00:05:22,040 --> 00:05:35,030 ‫Y and Z dot and then find that C to that and upside that, and then you put them through another integrated 49 00:05:35,030 --> 00:05:43,490 ‫box, which is now this one in purple, if you use oilor or you can also use around Akutan, which we 50 00:05:43,490 --> 00:05:44,390 ‫are going to do soon. 51 00:05:45,590 --> 00:05:50,230 ‫So either Oilor or Runga Kuta. 52 00:05:51,260 --> 00:06:02,660 ‫And of course what we'll get out of this box will be your X, Y, Z, and then Fifita and Passi. 53 00:06:03,580 --> 00:06:14,200 ‫So you're five feet and I can now go into your matrix and one final thing, your integrated box here 54 00:06:14,650 --> 00:06:24,850 ‫now will also need your states now so all your six inertial frame states will go into this second integrated 55 00:06:24,850 --> 00:06:25,300 ‫box. 56 00:06:26,450 --> 00:06:32,810 ‫And then, of course, together with the body frame states, your inertial frame states, they will 57 00:06:32,810 --> 00:06:38,800 ‫go into your controller and then the controller will compute your use. 58 00:06:39,740 --> 00:06:48,980 ‫And that's how the loop will operate throughout your maneuver or throughout following your trajectory. 59 00:06:50,150 --> 00:06:57,530 ‫So I would say that this is a pretty detailed schematics that tells you what's going on in the plan 60 00:06:57,530 --> 00:06:57,940 ‫box. 61 00:06:59,000 --> 00:07:05,040 ‫And so we only have two things to do before we finish with our plans. 62 00:07:05,630 --> 00:07:15,350 ‫I need to teach you Runga quota so that you would only rely on oilor and then I need to teach you and 63 00:07:15,350 --> 00:07:23,060 ‫show you what's inside this or megabucks, and then you will be completely done with the planned part. 64 00:07:23,510 --> 00:07:29,770 ‫And then we can move to the controller part where we're going to use two types of controllers. 65 00:07:30,740 --> 00:07:36,730 ‫One will be the more predictive control which you learned in the previous course in this core series. 66 00:07:37,460 --> 00:07:47,690 ‫And so this time the section on model predictive control will be easier and shorter, because now I'm 67 00:07:47,690 --> 00:07:53,720 ‫just going to take what I had taught you in the previous course, and I'm just going to brutally apply 68 00:07:53,720 --> 00:07:55,190 ‫it to this system. 69 00:07:56,060 --> 00:08:05,810 ‫And you see that there are very many similarities between the vehicle case that we had in terms of the 70 00:08:05,810 --> 00:08:15,680 ‫lateral control case and this case where we have a USV and then we want to make it follow a 3D trajectory 71 00:08:16,520 --> 00:08:19,280 ‫while controlling it states. 72 00:08:20,180 --> 00:08:27,620 ‫But in combination with the model predictive control, there will be another controller working with 73 00:08:27,620 --> 00:08:30,760 ‫it so they will be synchronized with each other. 74 00:08:31,580 --> 00:08:35,260 ‫So we're going to spend a little bit more time with the other controller. 75 00:08:36,170 --> 00:08:40,730 ‫But the more primitive controller, you already know what it is and how to apply it. 76 00:08:41,120 --> 00:08:45,100 ‫And now you're just going to see how to apply it to a different system. 77 00:08:46,130 --> 00:08:52,730 ‫Now, in the next video, I'm going to spend just a little bit of more time on the oilor integration 78 00:08:52,730 --> 00:08:58,820 ‫method, and then right away we're going to move into Runga Kutta method. 79 00:08:59,510 --> 00:09:03,080 ‫So thank you very much and see you in the next video.