1 00:00:00,590 --> 00:00:01,350 ‫Welcome back. 2 00:00:01,940 --> 00:00:10,630 ‫Let's now put our Newton oilor form into the state's equation for so that's your equation here. 3 00:00:11,120 --> 00:00:21,410 ‫And then remember that these three terms here, they gave you a net force and moment vector, but now 4 00:00:21,410 --> 00:00:25,460 ‫you're going to write it out in terms of these three terms. 5 00:00:26,270 --> 00:00:27,920 ‫One term is for gravity. 6 00:00:28,580 --> 00:00:31,130 ‫The other term is for gyroscopic effect. 7 00:00:31,430 --> 00:00:34,510 ‫And then the third term is for control inputs. 8 00:00:35,420 --> 00:00:43,560 ‫So the first order of business is that we're going to compute this, including the minus sign. 9 00:00:44,510 --> 00:00:55,280 ‫So this matrix here that you see in the body frame matrix, this is your velocity vector in the body 10 00:00:55,280 --> 00:00:55,760 ‫frame. 11 00:00:56,590 --> 00:00:59,210 ‫And don't forget about the minus sign here. 12 00:01:00,020 --> 00:01:06,890 ‫And if you're going to compute all that, then this is what you will get you will get this entire thing 13 00:01:06,890 --> 00:01:07,320 ‫here. 14 00:01:08,300 --> 00:01:22,730 ‫So when you multiply this first row with this velocity vector here with these six elements, then you 15 00:01:22,730 --> 00:01:25,150 ‫will get this thing here. 16 00:01:25,940 --> 00:01:33,920 ‫And when you do the same thing with the second row here and then you multiply that rule by these six 17 00:01:33,920 --> 00:01:34,950 ‫vector elements. 18 00:01:35,420 --> 00:01:38,080 ‫So it's a standard matrix vector multiplication. 19 00:01:38,750 --> 00:01:42,500 ‫Then you will get this thing here like this. 20 00:01:43,440 --> 00:01:49,650 ‫And so forward and you just do it for all the rows and then that's what you will get. 21 00:01:50,970 --> 00:02:02,670 ‫So now you take this entire thing here and you put it right here and then you add the other terms to 22 00:02:02,670 --> 00:02:02,910 ‫it. 23 00:02:03,840 --> 00:02:11,910 ‫And then you take this inverse matrix here, this entire thing, and you multiply it by whatever you 24 00:02:11,910 --> 00:02:12,780 ‫have here. 25 00:02:13,840 --> 00:02:17,770 ‫So all that here in purple is. 26 00:02:18,850 --> 00:02:29,380 ‫This so it's all here combined, and this is your mass and mass moment of inertia, matrix inverse. 27 00:02:30,070 --> 00:02:31,270 ‫So that's what it is. 28 00:02:31,900 --> 00:02:34,660 ‫And now you just have to multiply them together. 29 00:02:35,590 --> 00:02:45,360 ‫Note that this purple thing is actually six by one vector, because, for example, if you take this 30 00:02:45,370 --> 00:02:50,530 ‫first element here, this one, then this is just one row. 31 00:02:51,630 --> 00:02:58,230 ‫And you only have one column here, so if you had numbers for all these variables, you would just get 32 00:02:58,440 --> 00:03:00,010 ‫one scalar number here. 33 00:03:00,390 --> 00:03:07,170 ‫So this is actually a six by one vector and this is a six by six matrix. 34 00:03:08,160 --> 00:03:15,490 ‫So if you start multiplying them, then what will happen is that these one over Ms. 35 00:03:15,510 --> 00:03:23,870 ‫And one over mass moments of inertia, they will start appearing here in the denominator's. 36 00:03:25,290 --> 00:03:27,990 ‫So, for example, this would be for the first row. 37 00:03:29,280 --> 00:03:36,450 ‫That's because you're taking this first element here and you're applying it to the first row of this 38 00:03:36,450 --> 00:03:37,210 ‫vector. 39 00:03:37,890 --> 00:03:38,790 ‫It's like this. 40 00:03:39,820 --> 00:03:43,900 ‫Then the second element would be applied here. 41 00:03:44,380 --> 00:03:47,540 ‫However, since it's zero, then it doesn't matter. 42 00:03:47,570 --> 00:03:50,230 ‫Nothing will happen and it's better. 43 00:03:50,770 --> 00:03:57,280 ‫And because of the fact that this yellow matrix here is a diagonal matrix, then. 44 00:03:58,460 --> 00:04:05,030 ‫The first rule of this matrix only affects the first rule of this vector. 45 00:04:06,080 --> 00:04:14,090 ‫The second diagonal element affects only the second row of this vector. 46 00:04:15,140 --> 00:04:18,680 ‫The third element in this diagonal. 47 00:04:19,690 --> 00:04:27,040 ‫Matrix only affects the third row of this vector and so forth. 48 00:04:27,880 --> 00:04:31,120 ‫This would only affect this row. 49 00:04:32,130 --> 00:04:44,100 ‫This one would only affect this row here and finally, this one will only affect this row here, and 50 00:04:44,100 --> 00:04:49,620 ‫it's like that only because this is a diagonal matrix and all other elements here are zeroes. 51 00:04:50,700 --> 00:04:55,470 ‫And that's why you have mass here in the denominator because of this. 52 00:04:56,610 --> 00:05:05,220 ‫And you'll have mass in the denominator in the second row of this vector because of this in the third 53 00:05:05,220 --> 00:05:05,610 ‫row. 54 00:05:05,790 --> 00:05:08,420 ‫It's thanks to this element here. 55 00:05:09,240 --> 00:05:17,070 ‫Now, in the fourth element in this diagonal, we have the mass movement of inertia about the body frame 56 00:05:17,070 --> 00:05:17,940 ‫x axis. 57 00:05:18,920 --> 00:05:22,980 ‫So this thing will be thanks to this element here. 58 00:05:24,470 --> 00:05:31,910 ‫Then this element here will be applied to the fifth row, and that's mass moment of inertia, but the 59 00:05:31,910 --> 00:05:41,200 ‫y axis body frame y axis and finally, this element is for the sixth row. 60 00:05:42,230 --> 00:05:51,680 ‫And of course, when your purple vector is in this form, then this disappears because it's either the 61 00:05:51,680 --> 00:05:58,100 ‫yellow matrix times the purple vector without these red denominators. 62 00:05:59,160 --> 00:06:05,880 ‫Or when you have these denominators already, then that means that you have already multiplied the yellow 63 00:06:05,880 --> 00:06:12,420 ‫matrix by the purple vector and so you can make quite a lot of simplifications here. 64 00:06:13,120 --> 00:06:22,610 ‫You can cancel out these masses and these masses and these masses also here and here as well. 65 00:06:23,610 --> 00:06:25,560 ‫And the rest you cannot cancel. 66 00:06:26,580 --> 00:06:29,160 ‫And so that would be your first role. 67 00:06:29,190 --> 00:06:37,680 ‫You take this, you dot equals and then this first rule here, that's this. 68 00:06:38,960 --> 00:06:40,280 ‫That's your second role? 69 00:06:41,450 --> 00:06:45,190 ‫Not that I have gotten rid of the masses because I was able to cancel them. 70 00:06:46,500 --> 00:06:48,670 ‫This would be your third row here. 71 00:06:50,160 --> 00:06:58,770 ‫This would be your fourth row note, would I have done here I have factor out queue are here and here. 72 00:06:59,790 --> 00:07:02,220 ‫So I was able to write it down like this. 73 00:07:02,700 --> 00:07:13,590 ‫I said y y minus I sub Z that you have here and then divide it by the mass movement of inertia about 74 00:07:13,590 --> 00:07:14,610 ‫the x axis. 75 00:07:15,610 --> 00:07:24,520 ‫That will be your fifth row and again, what I have done here, I have factored out PR in order to be 76 00:07:24,520 --> 00:07:25,980 ‫able to write it down like this. 77 00:07:26,960 --> 00:07:37,980 ‫And that would be your sixth row again, PKU was factored out right over here and there you go. 78 00:07:38,630 --> 00:07:49,490 ‫You finally have your these mathematical model in the state space form, in the body frame. 79 00:07:50,510 --> 00:07:55,490 ‫You took your equations of motion that came from the laws of physics. 80 00:07:56,750 --> 00:08:07,010 ‫You adjusted it to a rotating body frame and then you converted it into a state space form, which is 81 00:08:07,010 --> 00:08:09,480 ‫more convenient for control problems. 82 00:08:10,610 --> 00:08:19,460 ‫Remember the equations of motion in the inertia frame, in the body frame and in the state's space form? 83 00:08:20,060 --> 00:08:26,580 ‫They all mathematically describe the same thing, but simply in a different way, in a different form. 84 00:08:27,350 --> 00:08:36,050 ‫And so depending on what you want to do, one form is more convenient than another form for that specific 85 00:08:36,050 --> 00:08:36,620 ‫purpose. 86 00:08:37,220 --> 00:08:38,510 ‫For control problems. 87 00:08:38,870 --> 00:08:43,520 ‫The state space form is better and it is not hard to see why. 88 00:08:44,510 --> 00:08:52,040 ‫In control problems, you constantly need to keep track of your states and the states based equations 89 00:08:52,040 --> 00:08:56,390 ‫make it quite easy, just like you learned in the previous course. 90 00:08:56,390 --> 00:09:03,350 ‫In this course series, you first define your initial states, which are the states when you start counting 91 00:09:03,350 --> 00:09:07,460 ‫time or when you start your maneuver. 92 00:09:08,420 --> 00:09:11,870 ‫And then you compute the new states like this. 93 00:09:13,320 --> 00:09:25,680 ‫You can compute the new states at K plus one using your state values now at K, the time derivative 94 00:09:25,920 --> 00:09:28,410 ‫of your state at K. 95 00:09:29,520 --> 00:09:31,980 ‫Times your sample time interval. 96 00:09:33,000 --> 00:09:43,170 ‫And divided by n your state time derivative at K, you get it from your state's base equations. 97 00:09:44,340 --> 00:09:55,860 ‫You define your sample time interval, and in our case, it was zero point one seconds and then N was 98 00:09:55,860 --> 00:10:03,000 ‫the number defining how many new state computations you do in one sample time interval. 99 00:10:03,930 --> 00:10:08,520 ‫So, for example, if An equals five, then. 100 00:10:09,700 --> 00:10:22,450 ‫If this is your tits, but you say three point three here and three point four here, then we compute 101 00:10:22,450 --> 00:10:30,420 ‫a new state every zero point zero two seconds, which is this interval here. 102 00:10:31,330 --> 00:10:41,410 ‫So we would do it at three point thirty two, then three point thirty four, three point thirty six, 103 00:10:41,410 --> 00:10:46,120 ‫three point thirty eight until you get to three point four. 104 00:10:46,930 --> 00:10:55,900 ‫So in each sub interval, you would compute a new state that increases the precision of your new state's. 105 00:10:56,970 --> 00:11:05,880 ‫Because like I said in the previous course, in this core series, this method here, it's not 100 percent 106 00:11:05,880 --> 00:11:06,510 ‫perfect. 107 00:11:06,870 --> 00:11:08,820 ‫So it gives you some kind of error. 108 00:11:09,210 --> 00:11:18,450 ‫So if you make these new state computations with a smaller interval, then you can decrease the error 109 00:11:18,480 --> 00:11:21,120 ‫and increase the precision of your state. 110 00:11:22,060 --> 00:11:29,320 ‫And of course, you don't only compute a new state for you variable, but you do the same thing for 111 00:11:29,320 --> 00:11:34,450 ‫V, W, P, Q and R. 112 00:11:35,600 --> 00:11:44,270 ‫So remember what they are, the first three states are the translational velocities of the drone in 113 00:11:44,270 --> 00:11:53,720 ‫the body frame and the other three states are the angular velocities of the drone in the body frame 114 00:11:53,720 --> 00:11:54,260 ‫as well. 115 00:11:55,460 --> 00:12:04,730 ‫So these states base equations here in white, this is what you have been after pretty much throughout 116 00:12:04,730 --> 00:12:09,020 ‫the entire course, and finally you have it. 117 00:12:10,250 --> 00:12:17,220 ‫However, the story's not over yet because these equations are only in the body frame. 118 00:12:17,690 --> 00:12:24,200 ‫However, we also have to work in the inertia frame because the reference values that we get. 119 00:12:25,210 --> 00:12:33,160 ‫Or in other words, the reference values that the controller gets are in the inertia frame, so therefore 120 00:12:33,760 --> 00:12:41,050 ‫we have to do something else in order to be able to make the plants suitable for the controller. 121 00:12:41,830 --> 00:12:44,200 ‫And I'm going to do that in the next video. 122 00:12:44,410 --> 00:12:45,370 ‫Thank you very much.