1 00:00:00,330 --> 00:00:09,210 ‫And now to get the transfer matrix itself, you just take the inverse of the inverse of the transfer 2 00:00:09,210 --> 00:00:17,580 ‫matrix and now you have to be careful because this is not an all the normal matrix. 3 00:00:18,200 --> 00:00:24,930 ‫It means that you cannot just take the transpose of it to get the inverse like you did with rotation 4 00:00:24,930 --> 00:00:25,610 ‫matrices. 5 00:00:26,070 --> 00:00:34,740 ‫You have to use traditional linear algebra methods for that or use a symbolic math tool such as Mathematica 6 00:00:35,130 --> 00:00:38,760 ‫Matlab or the sci fi library in Python. 7 00:00:39,240 --> 00:00:46,560 ‫And after you take the inverse of the inverse of the transfer matrix, you will find that your transferring 8 00:00:46,560 --> 00:00:54,580 ‫matrix looks like this, and that's the transfer matrix you've been looking for during all this time. 9 00:00:55,290 --> 00:01:05,400 ‫You now have a matrix that connects the changes of the Euler angles with respect to time to the angular 10 00:01:05,400 --> 00:01:07,670 ‫velocities in the body frame. 11 00:01:08,010 --> 00:01:11,100 ‫And that's the matrix that makes the connection. 12 00:01:11,370 --> 00:01:15,260 ‫And if you want to go back, then you have the inverse of it. 13 00:01:15,750 --> 00:01:21,640 ‫This matrix allows you to switch between these variables very conveniently. 14 00:01:22,320 --> 00:01:32,910 ‫Now, keep in mind that this transfer matrix is only exclusively valid for our convention, which is 15 00:01:33,180 --> 00:01:44,700 ‫R, sub Z, Y, X, using the oilor angle approach, meaning that we are rotating about the moving body 16 00:01:44,700 --> 00:01:46,020 ‫frame axis. 17 00:01:46,440 --> 00:01:55,350 ‫If you remember, then the rotation matrix for our convention was the same like for the convention are 18 00:01:55,680 --> 00:02:00,920 ‫sub X, Y, Z using the fixed angle approach. 19 00:02:01,440 --> 00:02:09,540 ‫But now thanks to the transfer matrix, you can see that it matters which one of those two you choose. 20 00:02:10,050 --> 00:02:17,480 ‫Their product of the rotation matrices might be the same, which was this one here. 21 00:02:18,270 --> 00:02:22,370 ‫This product was valid for both conventions. 22 00:02:22,860 --> 00:02:31,410 ‫However, the transfer matrix would be different for this fixed angle approach y because when you use 23 00:02:31,410 --> 00:02:37,190 ‫the fixed angle approach, then the path from inertial to body frame is different. 24 00:02:37,800 --> 00:02:48,690 ‫I have tried to draw a rotation sequence here for the convention are sub X, Y, Z for the fixed angle 25 00:02:48,690 --> 00:02:49,470 ‫approach. 26 00:02:49,950 --> 00:02:57,630 ‫So we first rotate about the inertial X axis, then about the inertial y axis and then about the inertial 27 00:02:57,960 --> 00:02:58,530 ‫axis. 28 00:02:59,130 --> 00:03:09,960 ‫And you can clearly see that in order to connect p q r angular velocity with Fido's standard and Poseidon's, 29 00:03:10,530 --> 00:03:13,980 ‫the transfer matrix would be different. 30 00:03:14,430 --> 00:03:23,490 ‫For example, in our case, our angular velocity p, which was the rotation about the body frame x axis 31 00:03:23,970 --> 00:03:30,040 ‫P radians per second was equal to fly dot radians per second. 32 00:03:30,390 --> 00:03:35,850 ‫However, in this approach here, fi, that would happen here. 33 00:03:36,270 --> 00:03:36,780 ‫Right. 34 00:03:37,230 --> 00:03:46,740 ‫But you can see that the direction here is different from the direction here because P is rotation about 35 00:03:46,740 --> 00:03:48,300 ‫the body frame x axis. 36 00:03:48,600 --> 00:03:55,140 ‫But here 5.0 is angular velocity about the inertial x axis. 37 00:03:55,650 --> 00:03:59,840 ‫Therefore here P cannot be equal to fly dot. 38 00:04:00,330 --> 00:04:07,520 ‫So this is one example Y you can already see why the transfer matrix for this convention will be different. 39 00:04:08,330 --> 00:04:08,810 ‫All right. 40 00:04:09,180 --> 00:04:11,700 ‫And this is all for the transfer matrix. 41 00:04:12,000 --> 00:04:20,280 ‫And in the next lecture, we will wrap up the kinematics part where we apply both our rotation and transfer 42 00:04:20,280 --> 00:04:23,670 ‫matrices to our specific situation. 43 00:04:24,240 --> 00:04:25,260 ‫Thank you very much. 44 00:04:25,650 --> 00:04:27,480 ‫And I'll see you in the next video.