1 00:00:00,740 --> 00:00:01,570 ‫Welcome back. 2 00:00:02,060 --> 00:00:13,040 ‫And so in this video, we will derive the product of rotation matrices for our X, Y, X, so that's 3 00:00:13,040 --> 00:00:13,890 ‫our convention. 4 00:00:13,910 --> 00:00:16,450 ‫Now it's X, Y, X. 5 00:00:17,030 --> 00:00:26,300 ‫However, even though we've changed the convention in the fixed angle approach, we will not change 6 00:00:26,300 --> 00:00:28,340 ‫the order of the angles. 7 00:00:28,610 --> 00:00:38,000 ‫So it will still be Gummo, Betar and Alpha, just like in the previous cases and just like in the previous 8 00:00:38,000 --> 00:00:43,900 ‫cases, Gamma Phi, Beta Theta and then Alpha SPSSI. 9 00:00:44,360 --> 00:00:50,380 ‫So I could also write here Fi and then Theta and BPCI. 10 00:00:50,840 --> 00:01:00,020 ‫So what we are saying here now is that first rotate about the inertial x frame axis by Gamma Radians 11 00:01:00,020 --> 00:01:09,290 ‫or Phi Radians, then rotate about the inertia frame y axis by better radians or theta radians, and 12 00:01:09,290 --> 00:01:16,420 ‫then rotate about the inertia from x axis by alpha radians or BPCI radians. 13 00:01:16,850 --> 00:01:20,510 ‫And so this is our product of our rotation matrices. 14 00:01:21,020 --> 00:01:27,470 ‫You first rotate about the inertial x axis, then about the inertial y axis and then about the inertial 15 00:01:27,470 --> 00:01:28,610 ‫x axis again. 16 00:01:29,180 --> 00:01:41,780 ‫But now the angles, Gamma or PHI, they will enter into this matrix here, then the angles, beta or 17 00:01:41,900 --> 00:01:51,530 ‫theta, they enter into this matrix here and then the angles alpha or PSI. 18 00:01:52,100 --> 00:01:54,710 ‫They enter into this matrix here. 19 00:01:55,280 --> 00:02:06,800 ‫So you will have one R X matrix with Gamma or Phi Angles and then another R X Matrix with Alpha or PSI 20 00:02:06,830 --> 00:02:07,640 ‫Radians. 21 00:02:08,000 --> 00:02:12,500 ‫So if you write these matrices out, then you will have it like this. 22 00:02:13,130 --> 00:02:17,790 ‫You see two are X matrices, but you have different angles in them. 23 00:02:18,350 --> 00:02:27,550 ‫So again, we first multiply these matrices and then to that product we will add another matrix. 24 00:02:28,160 --> 00:02:37,760 ‫And so if I multiply our Y times our X, then I will get this matrix here and then I add this are X, 25 00:02:38,360 --> 00:02:40,100 ‫which will be this one here. 26 00:02:40,730 --> 00:02:48,700 ‫And then if you multiply them all together, then this is your final product of your rotation matrices. 27 00:02:49,100 --> 00:02:56,360 ‫And now again, this product is valid for this convention here. 28 00:02:56,990 --> 00:03:08,720 ‫When you rotate about fixed axis and it is also valid for our small X, small Y, small X, which is 29 00:03:08,720 --> 00:03:16,010 ‫when you rotate about the moving frame axis, but then the order of angles, of course, will change. 30 00:03:16,520 --> 00:03:20,980 ‫Now it will be Alpha, Beta and gamma. 31 00:03:21,470 --> 00:03:25,820 ‫So this is when you rotate about moving axis and then. 32 00:03:25,820 --> 00:03:34,730 ‫Well in our course it would have been BPCI here, Theta here and then Phi here. 33 00:03:35,270 --> 00:03:44,510 ‫So this is the procedure that is followed in order to derive all the 12 products of rotation matrices 34 00:03:44,690 --> 00:03:47,120 ‫for all the twenty four conventions. 35 00:03:47,600 --> 00:03:55,790 ‫And in the book that I had suggested you in Appendix B, you will have those products for all the conventions, 36 00:03:56,240 --> 00:03:57,920 ‫but now you know the procedure. 37 00:03:58,470 --> 00:04:05,720 ‫However, in our course we will only use this convention and of course once you choose the convention, 38 00:04:05,720 --> 00:04:07,600 ‫you should stick with it. 39 00:04:08,210 --> 00:04:12,620 ‫You should not change the conventions in one project. 40 00:04:13,040 --> 00:04:15,140 ‫So this will be our convention. 41 00:04:15,770 --> 00:04:25,910 ‫We first rotate about the moving from Z axis by BPCI Radians, then about the moving frame Y axis by 42 00:04:25,910 --> 00:04:35,370 ‫Theta Radians and then about the moving from x axis by five radians or alpha beta gamma in this nomenclature. 43 00:04:35,930 --> 00:04:42,650 ‫And so this will be the product of rotation matrices that we will use in this course. 44 00:04:43,070 --> 00:04:44,030 ‫Thank you very much.