1 00:00:00,750 --> 00:00:01,620 ‫Welcome back. 2 00:00:02,160 --> 00:00:12,120 ‫So now that we have covered the three conventions and derived the rotation matrix for them now, I think 3 00:00:12,120 --> 00:00:21,390 ‫it's a good idea to finalize our discussion about these conventions by going back to this exercise here. 4 00:00:21,960 --> 00:00:28,050 ‫And I think after this video, your intuition about these conventions will be much better. 5 00:00:28,530 --> 00:00:39,540 ‫So in the last videos, we derived rotation matrices for X, Y, Z, Z, Y, X and X, Y, X. 6 00:00:40,200 --> 00:00:48,210 ‫In each case, the final rotation matrix is different because you multiply different matrices together, 7 00:00:48,870 --> 00:00:55,950 ‫or if you use the same matrices, then you simply multiply them together in a different order. 8 00:00:56,580 --> 00:01:05,160 ‫For example, here it's our Z times are Y times R, X, and then here are X, times are Y, times are 9 00:01:05,160 --> 00:01:05,440 ‫Z. 10 00:01:05,880 --> 00:01:08,730 ‫So obviously these matrices are different. 11 00:01:09,290 --> 00:01:19,500 ‫OK, so some videos ago we chose a convention are X, Y, Z, and then we said that all the angles, 12 00:01:20,010 --> 00:01:23,970 ‫Fifita and Passi, they were all power to radians. 13 00:01:24,450 --> 00:01:30,360 ‫And because of this convention, our body frame ended up in this configuration. 14 00:01:30,900 --> 00:01:42,300 ‫And then we said that our drone had this angular velocity, its p q r components were all one radians 15 00:01:42,300 --> 00:01:46,140 ‫per second in each body frame dimension. 16 00:01:46,710 --> 00:01:55,260 ‫And then we found out that if we take this matrix, our Z times are Y times R X, which is essentially 17 00:01:55,530 --> 00:02:00,080 ‫this matrix here because you see the same convention, X, Y, Z. 18 00:02:00,720 --> 00:02:11,100 ‫And if we substitute all the angles with PI over to Radians, then this big matrix here will boil down 19 00:02:11,490 --> 00:02:13,150 ‫to this matrix here. 20 00:02:13,710 --> 00:02:20,910 ‫And so you took the components of your purple vector in the body frame that where one, one, one. 21 00:02:21,570 --> 00:02:30,660 ‫And once you multiplied this three by three matrix by it, you found the coordinates of the drones angular 22 00:02:30,660 --> 00:02:33,430 ‫velocity vector in the inertia frame. 23 00:02:33,930 --> 00:02:41,250 ‫So in the inertial frame, which is this white one, which is fixed in this frame, this purple vector 24 00:02:41,850 --> 00:02:52,260 ‫has a value of one in the X direction, a value of one in the Y direction and a value of minus one in 25 00:02:52,260 --> 00:02:53,310 ‫the Z direction. 26 00:02:53,790 --> 00:03:01,560 ‫And by inspection, you could verify that it's true because if all your angles are pi over two radians, 27 00:03:02,190 --> 00:03:05,450 ‫then it's very easy to visually inspect all this. 28 00:03:06,060 --> 00:03:14,160 ‫But what would happen if instead of this convention, I used this convention here, Zwick's. 29 00:03:14,700 --> 00:03:25,980 ‫What if I had taken this matrix here, this rotation matrix, and used this one to multiply by this 30 00:03:25,980 --> 00:03:27,740 ‫green one one one vector? 31 00:03:28,320 --> 00:03:34,720 ‫Well, your intuition should tell you that you would get a wrong answer, but we can verify it. 32 00:03:35,310 --> 00:03:45,510 ‫Let's take this rotation matrix here and substitute all the angles with PI over to so our Phi Theta 33 00:03:45,510 --> 00:03:47,850 ‫and PSI, they're all pi over to radians. 34 00:03:48,180 --> 00:03:53,940 ‫Or if we choose to name those angles, gamma, beta and alpha still pi over to radians. 35 00:03:54,390 --> 00:04:01,560 ‫The good thing about power to Radians is that whenever you see a cosine then you know that that term 36 00:04:01,560 --> 00:04:02,040 ‫is zero. 37 00:04:02,090 --> 00:04:05,430 ‫So for example here you know it will be zero. 38 00:04:06,000 --> 00:04:09,290 ‫Then here you have a cosine here and you have a cosine here. 39 00:04:09,840 --> 00:04:17,190 ‫So this element will also be zero and here you will have two sides and then here you have cosines. 40 00:04:17,710 --> 00:04:20,930 ‫So this part will be one and then this part will be zero. 41 00:04:21,480 --> 00:04:24,720 ‫So you will have zero zero and one. 42 00:04:25,410 --> 00:04:26,880 ‫Then here you have a cosine. 43 00:04:26,880 --> 00:04:30,650 ‫So zero then here you have to cosine zero. 44 00:04:30,990 --> 00:04:37,890 ‫But then here you have sine so you will have one with a minus sign and then here you will have zero 45 00:04:38,220 --> 00:04:42,470 ‫and then here you will have zero as well because you have cosigns here. 46 00:04:42,960 --> 00:04:46,580 ‫So you will have a zero minus one and zero. 47 00:04:47,160 --> 00:04:51,770 ‫And finally you have a sign here and here and here you have cosines. 48 00:04:52,140 --> 00:04:56,270 ‫So these two elements will be Xeros, but here you will have one. 49 00:04:56,790 --> 00:04:59,730 ‫So you will have one zero and zero. 50 00:05:00,310 --> 00:05:06,910 ‫So now if you take this matrix and multiply it by this peak, you are matrix, then you will get one 51 00:05:06,910 --> 00:05:08,500 ‫minus one and one. 52 00:05:09,100 --> 00:05:14,420 ‫So you should have positive X, negative Y and then positive Z. 53 00:05:15,040 --> 00:05:21,120 ‫Obviously, it's not true because you have positive X and positive Y and negative Z. 54 00:05:21,700 --> 00:05:30,820 ‫And it makes sense because this body frame, it's in this orientation because we followed the X, Y, 55 00:05:30,820 --> 00:05:39,070 ‫Z convention, we first rotated power to radians about the inertial X axis, then pi over to radians 56 00:05:39,280 --> 00:05:45,000 ‫about the inertial y axis and then power to Radians about the inertial Z axis. 57 00:05:45,460 --> 00:05:49,830 ‫And that's why your body frame ended up in this configuration. 58 00:05:50,380 --> 00:05:58,060 ‫And so when you have your drone's angular velocity, then you measure this omega B in the body frame, 59 00:05:58,600 --> 00:06:00,650 ‫which is in this configuration. 60 00:06:01,240 --> 00:06:08,170 ‫Now what you need to understand is that your drone's angular velocity vector does not depend on which 61 00:06:08,170 --> 00:06:09,440 ‫convention you choose. 62 00:06:09,880 --> 00:06:16,660 ‫In this sense that if this vector points in this direction, then it will point in this direction, 63 00:06:16,840 --> 00:06:20,500 ‫regardless of which convention you choose. 64 00:06:20,980 --> 00:06:29,230 ‫So, for example, if this is my x axis and this is my Y axis and let's say that I'm walking in this 65 00:06:29,230 --> 00:06:36,070 ‫direction, then that means that I'm walking in the positive X direction and then the Y dimension is 66 00:06:36,070 --> 00:06:37,570 ‫not affected at all. 67 00:06:38,200 --> 00:06:46,960 ‫But now, if I all of a sudden decide to change my convention, then I'm still walking in this direction. 68 00:06:47,540 --> 00:06:50,400 ‫If this is my home, then I want to go home. 69 00:06:50,950 --> 00:06:53,880 ‫And that means that I have to walk in this direction. 70 00:06:54,430 --> 00:06:59,690 ‫So if I change my convention, I will still walk towards my home. 71 00:07:00,250 --> 00:07:09,940 ‫I'm not going to change the direction of my motion just because all of a sudden the convention changed. 72 00:07:10,510 --> 00:07:18,160 ‫No, I will still keep walking towards my home no matter what is just if I use this convention, then 73 00:07:18,160 --> 00:07:26,230 ‫I'm walking in the positive X direction and then if I change my convention, then all that means is 74 00:07:26,230 --> 00:07:33,940 ‫that now I'm walking in the negative X direction and the same thing with the angular velocity of the 75 00:07:33,940 --> 00:07:34,360 ‫drone. 76 00:07:35,020 --> 00:07:43,870 ‫If that's the vector of the angle of velocity of the drone, if that's how fast the drone rotates and 77 00:07:43,870 --> 00:07:51,790 ‫if that's the direction of the drones rotation, then that's how this vector will be, regardless of 78 00:07:51,970 --> 00:07:53,590 ‫which convention I choose. 79 00:07:54,100 --> 00:08:00,580 ‫So if I all of a sudden want to change my convention and let's say that I want to use this convention, 80 00:08:00,580 --> 00:08:08,980 ‫Tsewang X will, then first of all, I need to understand how this body frame will be oriented with 81 00:08:08,980 --> 00:08:10,630 ‫respect to the initial frame. 82 00:08:11,290 --> 00:08:17,750 ‫If I use that convention and I rotate power to radiance about all my inertial axis. 83 00:08:18,250 --> 00:08:28,570 ‫So this here, the lower one, this is my Zwi X convention, so I first rotate power to Radians about 84 00:08:28,570 --> 00:08:37,570 ‫the inertial Z axis, then I rotate power to radiance about the inertial Y axis and then power to radians 85 00:08:37,570 --> 00:08:39,100 ‫about the inertial x axis. 86 00:08:39,640 --> 00:08:48,790 ‫So if I use the R, Z, Y, X convention, then my body frame will be in this orientation. 87 00:08:49,450 --> 00:08:56,830 ‫If I rotate about all the inertial axis by pi over to radians, if the angles are different, then of 88 00:08:56,830 --> 00:09:00,870 ‫course this body frame will be oriented differently. 89 00:09:01,450 --> 00:09:10,330 ‫But in case of power to radians using the R, Z, Y, X convention, my body frame will be oriented 90 00:09:10,930 --> 00:09:19,370 ‫in this way, not in this way that we had for R, X, Y, Z, but in this way our Zywiec. 91 00:09:20,140 --> 00:09:29,200 ‫So this is my inertial frame and now our body frame is in this configuration because we used this convention 92 00:09:29,830 --> 00:09:33,190 ‫and all our angles were pyra to radians. 93 00:09:33,790 --> 00:09:45,090 ‫You see, it's like here, here you have X, B then Y be and then A Zebb XP, Y, B and Zebb. 94 00:09:45,670 --> 00:09:55,210 ‫But like I said, the drone itself physically will not start rotating differently just because you went 95 00:09:55,210 --> 00:09:59,350 ‫from R, X, Y, Z to R, Z, Y, X. 96 00:10:00,180 --> 00:10:10,170 ‫The angular velocity vector of this drone is still like this, just like it's here in this picture. 97 00:10:10,770 --> 00:10:13,550 ‫Its components are still in the same direction. 98 00:10:14,070 --> 00:10:22,440 ‫And this is your Omega B, but because of the fact that you changed your convention, then you have 99 00:10:22,440 --> 00:10:25,220 ‫to make some modifications to your components. 100 00:10:25,590 --> 00:10:28,880 ‫So, OK, this is your body frame x axis. 101 00:10:28,920 --> 00:10:36,170 ‫So that means that this component is still your P, then this is your body frame Y axis. 102 00:10:36,690 --> 00:10:40,590 ‫So this is your Q and this is your body from Z axis. 103 00:10:41,130 --> 00:10:42,370 ‫So this is your arm. 104 00:10:42,900 --> 00:10:50,760 ‫So if you compare these two situations, then you can see that you have P here and then you also call 105 00:10:50,770 --> 00:10:57,680 ‫this P, then you have Q here and you also call this Q and then you have your R here. 106 00:10:58,290 --> 00:11:01,640 ‫And in this case your R is also here. 107 00:11:02,100 --> 00:11:06,340 ‫So you did not have to rename your components. 108 00:11:06,750 --> 00:11:15,210 ‫However, there is something different in this case, X, Y, Z, which was our original case, this 109 00:11:15,210 --> 00:11:21,110 ‫purple vector, it's coordinates in the body frame where one one one. 110 00:11:21,690 --> 00:11:31,590 ‫But since we changed our convention, now the coordinates of our angular velocity vector in the body 111 00:11:31,590 --> 00:11:35,010 ‫frame is not one one one anymore. 112 00:11:35,580 --> 00:11:44,640 ‫Now you see that your P is minus one because it's in the opposite direction to the positive body frame 113 00:11:44,640 --> 00:11:45,480 ‫x axis. 114 00:11:46,020 --> 00:11:53,010 ‫Then the same thing with Q it's also minus one and then your R is positive one. 115 00:11:53,580 --> 00:12:02,580 ‫Since you changed your convention, the drone still continues to rotate as it had rotated when you used 116 00:12:02,580 --> 00:12:03,590 ‫this convention. 117 00:12:04,200 --> 00:12:07,920 ‫So the physical emotional of the drone hasn't changed. 118 00:12:08,430 --> 00:12:16,530 ‫But since you changed your convention, you now describe this physical motion in your body frame differently 119 00:12:17,130 --> 00:12:18,280 ‫in this convention. 120 00:12:18,990 --> 00:12:22,290 ‫This motion is this minus one, minus one and one. 121 00:12:22,890 --> 00:12:30,600 ‫And now if we take this matrix that corresponds to this convention, why X? 122 00:12:31,140 --> 00:12:40,350 ‫And then we multiply it by the coordinates of the angular velocity vector in the body frame that corresponds 123 00:12:40,530 --> 00:12:49,980 ‫to this convention, then you will get one here, one here and minus one here. 124 00:12:50,520 --> 00:12:56,070 ‫And these are your inertial frame coordinates and you see now you get the right stuff. 125 00:12:56,580 --> 00:13:05,820 ‫Now this purple vector indeed has a positive X component, positive Y component and the negative Z component 126 00:13:05,820 --> 00:13:07,050 ‫in the inertia frame. 127 00:13:07,590 --> 00:13:12,150 ‫And we got the same result when we use this X, Y, Z convention. 128 00:13:12,810 --> 00:13:20,160 ‫It's just in the case of X, Y, Z, the body frame coordinates of this purple vector where one, one, 129 00:13:20,160 --> 00:13:29,010 ‫one, and then the corresponding rotation matrix for this convention provided that all the angles are 130 00:13:29,010 --> 00:13:31,530 ‫pi over to radians was this. 131 00:13:32,130 --> 00:13:37,740 ‫And that's how you found out the inertia from coordinates for this purpose vector. 132 00:13:38,340 --> 00:13:46,320 ‫But now since we changed our convention, then our matrix changed and our body frame coordinates changed. 133 00:13:46,800 --> 00:13:53,130 ‫But if we stick to the convention that we choose, we will still get the right inertia frame coordinates. 134 00:13:53,460 --> 00:14:00,240 ‫So you see, when you move from one convention to another, your inertia friend coordinates do not change. 135 00:14:00,900 --> 00:14:07,470 ‫What do change are your rotation matrix and your body frame coordinates. 136 00:14:07,890 --> 00:14:13,920 ‫And now to really solidify this understanding, let's go through the same thing. 137 00:14:13,920 --> 00:14:17,820 ‫But for this convention are X, Y, X. 138 00:14:18,390 --> 00:14:20,170 ‫Let's do that in the next video. 139 00:14:20,760 --> 00:14:21,630 ‫Thank you very much.