1 00:00:00,570 --> 00:00:01,350 ‫Welcome back. 2 00:00:01,830 --> 00:00:08,010 ‫So I hope that you have tried the exercise and now I will show you how it is done. 3 00:00:08,640 --> 00:00:16,620 ‫So to derive a mathematical model for a system from the first principles of physics, you should start 4 00:00:16,620 --> 00:00:26,640 ‫from the Newton's second law, which is the net force vector equals the time derivative of the translational 5 00:00:26,820 --> 00:00:30,390 ‫momentum and times the velocity vector. 6 00:00:31,140 --> 00:00:37,710 ‫And so from calculus, you know that you first take the derivative of the mass with respect to time 7 00:00:38,190 --> 00:00:45,480 ‫and you leave the velocity vector alone and then you leave the mass alone and you take the time derivative 8 00:00:45,480 --> 00:00:57,150 ‫of the velocity vector like this, since the change of mass with respect to time equals the zero kilograms 9 00:00:57,150 --> 00:01:00,650 ‫per second, which means that the mass is constant. 10 00:01:00,660 --> 00:01:02,850 ‫It doesn't change with time. 11 00:01:03,780 --> 00:01:14,130 ‫It means that this term here becomes a zero and that means that F net equals mass times acceleration, 12 00:01:14,130 --> 00:01:15,680 ‫which is DVD t. 13 00:01:16,570 --> 00:01:25,410 ‫Next, we have to find the second time derivative of our position vector Gummo in the inertial frame, 14 00:01:25,950 --> 00:01:33,240 ‫which was X, Y and then Z transposed because it was a column vector. 15 00:01:34,080 --> 00:01:40,680 ‫And the reason for why we need to find the second time derivative of this position vector is because 16 00:01:40,920 --> 00:01:45,960 ‫Gunma doubled that in the inertia frame as a vector. 17 00:01:46,530 --> 00:01:51,270 ‫It is our divi deti. 18 00:01:51,600 --> 00:01:54,240 ‫All right, so let's do that. 19 00:01:54,780 --> 00:02:06,090 ‫We can rewrite Gummo in the inertia frame like this X then the unit vector ie plus why the unit vector 20 00:02:06,090 --> 00:02:20,250 ‫J and plus Z, the unit vector K and remember the unit vectors I, J and K are now along the inertial 21 00:02:20,250 --> 00:02:25,770 ‫frame axis, not the body frame axis, but the inertia frame axis. 22 00:02:26,640 --> 00:02:37,200 ‫And so gamma dot in the inertia frame equals x dot and then you have this unit vector and then you have 23 00:02:37,200 --> 00:02:43,320 ‫X and then the derivative of the unit vector, the same for Y. 24 00:02:43,590 --> 00:02:54,030 ‫So first of all you have Y daytime's J and then plus Y times Jadot and then plus the Z dot times the 25 00:02:54,030 --> 00:03:02,250 ‫unit vector K and then plus the Z times the time derivative of the unit vector K. 26 00:03:02,490 --> 00:03:16,050 ‫So like this since now we are in the initial frame, then that means that I dot j dot and k dot they 27 00:03:16,050 --> 00:03:18,000 ‫are all zero. 28 00:03:18,510 --> 00:03:19,020 ‫All right. 29 00:03:19,140 --> 00:03:20,780 ‫Zero units per second. 30 00:03:21,690 --> 00:03:29,370 ‫That means that this term here becomes zero, this term here becomes zero and also this term here becomes 31 00:03:29,400 --> 00:03:29,940 ‫zero. 32 00:03:30,870 --> 00:03:37,710 ‫And that means that your final gamma dot vector looks like this. 33 00:03:38,550 --> 00:03:42,710 ‫And now we'll take another time derivative of this vector here. 34 00:03:43,350 --> 00:03:50,550 ‫So gamma double dot in the initial frame equals these two terms. 35 00:03:50,550 --> 00:03:56,430 ‫First of all, you have X double dot times I plus X dot times. 36 00:03:56,430 --> 00:04:03,300 ‫I dot the same thing for the Y dimension and for the Z dimension. 37 00:04:04,170 --> 00:04:11,970 ‫Again, this term here becomes zero, then this term here becomes zero and this term here becomes zero 38 00:04:12,000 --> 00:04:14,700 ‫y because we are in the inertia frame. 39 00:04:14,970 --> 00:04:22,230 ‫So that means that the axis of an inertial frame, they are fixed. 40 00:04:22,350 --> 00:04:22,800 ‫All right. 41 00:04:22,800 --> 00:04:25,860 ‫They don't rotate, they don't move anywhere. 42 00:04:25,860 --> 00:04:26,880 ‫They are fixed. 43 00:04:27,600 --> 00:04:36,210 ‫And if they are fixed, then that means that these unit vectors here, they cannot move. 44 00:04:36,450 --> 00:04:39,260 ‫So their time derivatives must be zero. 45 00:04:39,930 --> 00:04:47,400 ‫And so you can rewrite this vector like this gunman double dot in the initial frame equals. 46 00:04:47,700 --> 00:04:57,360 ‫And then you have here X double dot, y double dot and then Z double dot like this. 47 00:04:58,020 --> 00:04:59,450 ‫And then of course E. 48 00:04:59,840 --> 00:05:10,100 ‫Your D.V. deti, all right, and that also means that in the inertia frame, the equations of motion 49 00:05:10,610 --> 00:05:11,660 ‫look like this. 50 00:05:12,500 --> 00:05:22,060 ‫This is the net force vector and it equals mass times, X, Y and Z double dot. 51 00:05:23,150 --> 00:05:31,730 ‫And now we need to express our F net vector in terms of the individual relevant forces that the drone 52 00:05:31,730 --> 00:05:32,900 ‫is exposed to. 53 00:05:33,380 --> 00:05:38,360 ‫If you remember, then we have three force moment vectors for that. 54 00:05:39,050 --> 00:05:48,950 ‫We had Londa in the body frame for the gravity, then also lambda in the body frame for the gyroscopic 55 00:05:48,950 --> 00:05:55,580 ‫effect and lambda in the body frame for the inputs, control inputs. 56 00:05:55,580 --> 00:05:57,740 ‫You want you to use three and you four. 57 00:05:58,710 --> 00:06:07,350 ‫So that was the gravity here in the body frame and you had this inverse rotation matrix here that was 58 00:06:07,380 --> 00:06:15,210 ‫Lunda in the body frame for the gyroscopic effect and that was Lunda in the body frame for the control 59 00:06:15,210 --> 00:06:15,880 ‫inputs. 60 00:06:16,800 --> 00:06:24,390 ‫In our case, though, we only need the first three elements from these vectors so we can forget about 61 00:06:24,390 --> 00:06:33,390 ‫this one and then we can forget about this one and then also this one, because we only need the force 62 00:06:33,390 --> 00:06:37,320 ‫elements and we can neglect the moment elements. 63 00:06:37,980 --> 00:06:44,400 ‫And in addition, our force vectors have to be in the initial frame, not in the body frame. 64 00:06:44,550 --> 00:06:48,810 ‫Here they are in the body frame, but we need them in the initial frame. 65 00:06:49,530 --> 00:07:01,520 ‫So this one here, that was our force of gravity, but in the body frame then we had no gyroscopic force. 66 00:07:01,530 --> 00:07:03,960 ‫We only had gyroscopic moments. 67 00:07:04,140 --> 00:07:09,230 ‫That gyroscopic effect had to do with moments, not with force. 68 00:07:09,510 --> 00:07:12,360 ‫So that's even irrelevant here. 69 00:07:12,360 --> 00:07:18,210 ‫So we don't have to worry about this entire vector at all, not even in the initial frame. 70 00:07:18,960 --> 00:07:26,880 ‫But you could say that this is your gyroscopic force in the body frame, which is a zero zero zero. 71 00:07:27,030 --> 00:07:28,200 ‫So it's a zero vector. 72 00:07:28,980 --> 00:07:30,540 ‫And then this one here. 73 00:07:30,750 --> 00:07:37,860 ‫Well, that was your input vector in the body frame, this one here. 74 00:07:38,640 --> 00:07:45,120 ‫But in the initial frame, those vectors, those force vectors here, they would look like this. 75 00:07:46,080 --> 00:07:49,230 ‫The force of gravity in the initial frame looks like this. 76 00:07:50,290 --> 00:07:57,010 ‫The gyroscopic force in the initial frame is like this, so zero zero zero, meaning that it doesn't 77 00:07:57,010 --> 00:07:57,640 ‫exist. 78 00:07:58,480 --> 00:08:05,230 ‫And now remember that this force vector here is in the body frame. 79 00:08:05,230 --> 00:08:10,020 ‫This is zero zero and you one that's in the body frame. 80 00:08:10,090 --> 00:08:14,260 ‫So if you want to put it in the initial frame, well, then. 81 00:08:15,210 --> 00:08:23,700 ‫You have to take your rotation matrix and multiplied by this vector here, which is your force vector 82 00:08:23,700 --> 00:08:25,330 ‫in the body frame. 83 00:08:25,350 --> 00:08:33,180 ‫So if you multiply the rotation matrix by this body frame, input force vector, then you will get the 84 00:08:33,180 --> 00:08:35,410 ‫input force vector in the inertia frame. 85 00:08:36,390 --> 00:08:43,400 ‫So this element here, you one that was in the body frame Z direction. 86 00:08:44,400 --> 00:08:47,770 ‫So I guess you need the rotation matrix after all. 87 00:08:48,270 --> 00:08:52,370 ‫And so this is your rotation matrix here. 88 00:08:52,680 --> 00:08:59,190 ‫And if you multiply this rotation matrix by this vector zero zero, you one. 89 00:09:00,210 --> 00:09:10,920 ‫Then you can extract this element here for the X dimension, this element here for the Y dimension and 90 00:09:10,920 --> 00:09:15,640 ‫this element here for the Z dimension, it makes sense, right? 91 00:09:15,900 --> 00:09:26,730 ‫So if you take this vector here, F input and in the inertial frame and then you say that this is your 92 00:09:26,730 --> 00:09:37,110 ‫rotation matrix times this vector zero zero, you one, then remember this rotation matrix is a three 93 00:09:37,110 --> 00:09:40,220 ‫by three matrix with those huge elements there. 94 00:09:41,130 --> 00:09:48,990 ‫And since the first two elements here are zeros, that means that when you multiply this rotation matrix 95 00:09:48,990 --> 00:09:51,600 ‫by this vector, then. 96 00:09:52,840 --> 00:09:59,650 ‫This one gets multiplied by this one for the X dimension. 97 00:10:00,620 --> 00:10:10,640 ‫Then this one gets multiplied by this one for the Y dimension and this one gets multiplied by this one 98 00:10:11,000 --> 00:10:16,530 ‫for the Z dimension, and that's why you extract these elements here. 99 00:10:17,360 --> 00:10:22,300 ‫So in the end, the input force vector in the initial frame looks like this. 100 00:10:22,520 --> 00:10:32,520 ‫You see, these are the elements from this rotation matrix and then you multiply them by you one. 101 00:10:33,410 --> 00:10:35,870 ‫And now let's put it all together. 102 00:10:36,740 --> 00:10:48,560 ‫Mass Times, Gunma, double dot in the inertia frame equals and then your net force vector in the inertia 103 00:10:48,560 --> 00:10:55,940 ‫frame and then you rewrite this net force vector in terms of the three individual force vectors like 104 00:10:55,940 --> 00:10:56,420 ‫this. 105 00:10:57,230 --> 00:11:05,810 ‫Or you can rewrite it like this where you show what those individual force vectors are and then instead 106 00:11:05,810 --> 00:11:14,930 ‫of Gummo double that, you rewrite it like this equals and if you rewrite your input vector in the initial 107 00:11:14,930 --> 00:11:18,850 ‫frame, then the entire thing looks like this. 108 00:11:19,370 --> 00:11:23,970 ‫And now what you can do, you can simply cancel out the masses. 109 00:11:23,980 --> 00:11:32,690 ‫So if I cancel out the mass here, then that means that I will have a mass here you want divided by 110 00:11:32,690 --> 00:11:33,170 ‫mass. 111 00:11:33,530 --> 00:11:40,490 ‫And also here you one divided by mass and also here you one divided by mass. 112 00:11:41,210 --> 00:11:44,720 ‫And I will have it here in the denominator as well. 113 00:11:44,720 --> 00:11:47,780 ‫But then I can cancel them out. 114 00:11:48,800 --> 00:11:51,440 ‫And that's your final answer. 115 00:11:52,250 --> 00:11:53,080 ‫And there you go. 116 00:11:53,690 --> 00:12:00,100 ‫That's how you get the equations of motion for position control in the inertial frame. 117 00:12:00,770 --> 00:12:04,550 ‫Not that they are not states based equations. 118 00:12:05,090 --> 00:12:10,130 ‫You simply relate the accelerations x double date. 119 00:12:10,130 --> 00:12:12,600 ‫Why double date and Z double that. 120 00:12:12,620 --> 00:12:19,360 ‫And now these accelerations are in the inertial X, Y and Z direction. 121 00:12:19,820 --> 00:12:29,960 ‫You simply relate them with Phi Theta and Passi angles and you want control input. 122 00:12:30,710 --> 00:12:35,110 ‫And in this section you don't need to put them in the state's space form. 123 00:12:35,630 --> 00:12:43,400 ‫We're going to use these equations directly for position control and we will start with that in the 124 00:12:43,400 --> 00:12:44,030 ‫next video. 125 00:12:44,330 --> 00:12:44,840 ‫Thank you.