1 00:00:00,770 --> 00:00:01,640 ‫Welcome back. 2 00:00:02,150 --> 00:00:14,210 ‫So now we are going to derive our product of rotation mattresses for the convention, Z, Y, X, these 3 00:00:14,210 --> 00:00:15,470 ‫are capital letters. 4 00:00:15,710 --> 00:00:19,840 ‫So we are rotating about the inertia from Axis. 5 00:00:20,330 --> 00:00:21,380 ‫But here's the thing. 6 00:00:21,980 --> 00:00:32,540 ‫When we change our rotation sequence, then the angles, gamma, beta and then alpha, they still stay 7 00:00:32,540 --> 00:00:40,220 ‫in the same order, just like in the previous case when we had this convention, X, Y, Z, the order 8 00:00:40,220 --> 00:00:43,550 ‫of our angles was Gamma, Beta and alpha. 9 00:00:44,090 --> 00:00:49,010 ‫But now we change the order of our axis. 10 00:00:49,460 --> 00:00:54,060 ‫But we did not change the order of the angles. 11 00:00:54,560 --> 00:01:01,970 ‫Now, in the previous case here you see that we did change the order of our angles, but that's because 12 00:01:01,970 --> 00:01:07,600 ‫we went from the fixed axis approach to the moving axis approach. 13 00:01:08,210 --> 00:01:14,440 ‫When you go from fixed angles to oilor angles, then you change the order of these angles. 14 00:01:14,840 --> 00:01:22,490 ‫But when you go from a fixed angle approach with one convention to another fixed angle approach with 15 00:01:22,490 --> 00:01:26,880 ‫another convention, then you do not change the order of your angles. 16 00:01:27,410 --> 00:01:37,490 ‫So in the previous case, we said first rotate gamma radians about the inertial x axis, then beta radians 17 00:01:38,060 --> 00:01:44,490 ‫about the inertial y axis, and then alpha radians about the inertia axis. 18 00:01:45,020 --> 00:01:53,510 ‫Now we are saying rotate gamma radians about the inertial Z axis, then better radians about the inertial 19 00:01:53,510 --> 00:01:58,730 ‫Y axis and then alpha radians about the inertial x axis. 20 00:01:59,240 --> 00:02:10,280 ‫So if we want to use the angles used in this course, Phi Theta and PSI, then if we use this convention, 21 00:02:10,580 --> 00:02:19,400 ‫then we are saying first rotate five radians above the inertia Z axis, then theta radians about the 22 00:02:19,400 --> 00:02:24,860 ‫inertial Y axis and then Pi radians about the inertial x axis. 23 00:02:25,340 --> 00:02:30,020 ‫So in this convention, the order of matrices is like this. 24 00:02:30,620 --> 00:02:39,860 ‫In the previous case it was our Z times are Y, times are X, because you first rotate about the inertial 25 00:02:39,860 --> 00:02:43,760 ‫X axis, then the inertial Y axis and then the inertia axis. 26 00:02:44,330 --> 00:02:52,460 ‫But now the product is our X times, our Y times, our Z, because you are first rotating about the 27 00:02:52,460 --> 00:03:00,920 ‫inertial Z axis, then the inertial Y axis, then the inertial X axis, and now the gumba or the fire 28 00:03:00,920 --> 00:03:12,770 ‫angle enters into this matrix or Z, then beta or theta angle enters into this matrix R, Y, and then 29 00:03:12,770 --> 00:03:17,690 ‫Alpha or Kayangel enters into this matrix. 30 00:03:17,690 --> 00:03:19,520 ‫Here are acts. 31 00:03:20,090 --> 00:03:24,560 ‫It doesn't really matter in this course because we are not using this convention. 32 00:03:25,190 --> 00:03:32,460 ‫But just for you to know, when you change a convention, then these angles enter into different matrices. 33 00:03:32,930 --> 00:03:37,460 ‫So now we have to multiply together these three matrices. 34 00:03:37,940 --> 00:03:40,850 ‫As you can see, it's a little bit different here. 35 00:03:40,850 --> 00:03:44,780 ‫The order and then the matrices were like this. 36 00:03:45,350 --> 00:03:48,110 ‫And now for this convention, they are like this. 37 00:03:48,710 --> 00:03:57,920 ‫And again, we first multiply these two matrices and then we would multiply our X by the product of 38 00:03:58,100 --> 00:04:04,080 ‫R, Y and Z, and that would be the product of our Y times, our Z. 39 00:04:04,670 --> 00:04:09,400 ‫And now here we put our X and here as well. 40 00:04:09,980 --> 00:04:17,050 ‫And so if you multiply these matrices together, then this will be your final product matrix. 41 00:04:17,630 --> 00:04:22,460 ‫And so this product matrix is valid for this convention. 42 00:04:22,850 --> 00:04:31,520 ‫This is the fixed axis approach and then for this convention, which is the moving axis approach. 43 00:04:32,120 --> 00:04:39,120 ‫But now, since we go from fixed to moving, the order of these angles will change. 44 00:04:39,140 --> 00:04:43,730 ‫So I will put here Alpha, Beta and Gamma. 45 00:04:44,270 --> 00:04:56,180 ‫And in our case, it would be Phi Beta and then PSI and then here you would have PSI, Theta and Phi. 46 00:04:56,660 --> 00:05:00,140 ‫So, for example, in the previous case, the. 47 00:05:00,210 --> 00:05:09,680 ‫Our X matrix contained the gamma angles or in our case, the fire angles, but in this case the R X 48 00:05:09,690 --> 00:05:16,180 ‫matrix contains the alpha angles or the sci angles in our case. 49 00:05:16,590 --> 00:05:26,550 ‫And so in conclusion, in this case, we say rotate about the inertial Z axis by Gamma or Phi Radians, 50 00:05:27,010 --> 00:05:37,140 ‫then about the inertial frame Y axis by better or theta radians and then by the inertia frame x axis 51 00:05:37,590 --> 00:05:39,660 ‫by Alpha PSI radians. 52 00:05:40,140 --> 00:05:44,160 ‫Then you use this product and you can also use this product. 53 00:05:44,160 --> 00:05:53,610 ‫If you first rotate about the moving frame x axis by Alpha or PSI radians, then you rotate about the 54 00:05:53,610 --> 00:06:05,490 ‫moving frame y axis by beta or theta radians and then you rotate about the moving frame Z axis by Gamma 55 00:06:06,090 --> 00:06:07,500 ‫or Phi Radians. 56 00:06:07,950 --> 00:06:18,330 ‫And in the next video we will have one more convention are sub X, Y, X, so we will not even rotate 57 00:06:18,960 --> 00:06:21,390 ‫about the inertia frame. 58 00:06:21,390 --> 00:06:29,250 ‫Z axis will first rotate about the inertia from X axis, then about the inertia from Y axis and then 59 00:06:29,250 --> 00:06:31,980 ‫about the inertial frame x axis again. 60 00:06:32,580 --> 00:06:34,350 ‫And we'll see that in the next video. 61 00:06:34,920 --> 00:06:35,880 ‫Thank you very much.