1 00:00:00,490 --> 00:00:01,360 ‫Welcome back. 2 00:00:01,660 --> 00:00:08,920 ‫So we have now seen how to derive and apply the rotation matrices and how to obtain the transfer matrix. 3 00:00:09,310 --> 00:00:15,340 ‫Let's now see how we can apply the transfer matrix along with the rotation matrix. 4 00:00:15,580 --> 00:00:21,790 ‫In our specific situation here, you can see the inertial frame and then the body frame and you can 5 00:00:21,790 --> 00:00:31,120 ‫see the linear velocities of the drone in the direction of the body frame axis you the N.W. in meters 6 00:00:31,120 --> 00:00:31,990 ‫per second. 7 00:00:32,470 --> 00:00:40,270 ‫And you can also see the angular velocity of the drone, or in other words, about the body frame axes 8 00:00:40,270 --> 00:00:44,110 ‫that are attached to the drone PKU. 9 00:00:44,110 --> 00:00:47,020 ‫You are in radians per second. 10 00:00:47,350 --> 00:00:51,220 ‫Earlier we defined a vector for the drones position. 11 00:00:51,610 --> 00:00:57,700 ‫That vector was the big gamma and you had here X, Y, Z. 12 00:00:58,240 --> 00:01:06,400 ‫This is the position vector for the inertial frame and we could also define a vector for the oilor angles, 13 00:01:06,430 --> 00:01:09,670 ‫which was theta superscript e arrow. 14 00:01:10,120 --> 00:01:14,740 ‫And it looked like this Phi Theta and PSI. 15 00:01:15,130 --> 00:01:17,950 ‫We can combine them in one vector like this. 16 00:01:18,460 --> 00:01:22,360 ‫You have big gamma here and you have big theta here. 17 00:01:22,780 --> 00:01:26,510 ‫We transpose it and let's call it Epsilon. 18 00:01:26,980 --> 00:01:36,430 ‫We can also write it out like this big X, big Y, big Z and then Phi Beta and PSI. 19 00:01:36,820 --> 00:01:41,680 ‫And then we transpose it because actually it's a column vector, not a row vector. 20 00:01:42,070 --> 00:01:49,840 ‫But I just want to say some space, if we take a time derivative of this epsilon vector and let's put 21 00:01:49,840 --> 00:01:55,150 ‫arrows here as well to indicate that it's a vector, then it looks like this. 22 00:01:55,690 --> 00:02:00,760 ‫All the elements here will become the time derivatives of these variables. 23 00:02:01,440 --> 00:02:08,440 ‫These are the linear or the translational velocities in the inertial frame, which are in meters per 24 00:02:08,440 --> 00:02:09,040 ‫second. 25 00:02:09,460 --> 00:02:15,640 ‫And these are angular velocities in the inertia frame or the time derivatives of the oil or angles which 26 00:02:15,640 --> 00:02:18,340 ‫are radians per second. 27 00:02:19,590 --> 00:02:26,710 ‫Earlier, we also defined the vector for the linear velocities in the body frame, which was this huge 28 00:02:26,730 --> 00:02:34,950 ‫VW and that was the vector, and we also had the angular velocities in the body frame that looked like 29 00:02:34,950 --> 00:02:35,370 ‫this. 30 00:02:35,940 --> 00:02:44,550 ‫And so we can also combine them in one vector like this or like this, where, again, this part is 31 00:02:44,550 --> 00:02:51,820 ‫in meters per second and this part is in radians per second. 32 00:02:52,260 --> 00:03:00,060 ‫The only difference between this vector and this vector is that they're measured in different reference 33 00:03:00,060 --> 00:03:00,610 ‫frames. 34 00:03:01,500 --> 00:03:09,330 ‫We also saw how to connect the linear velocities in the inertial frame to the linear velocities in the 35 00:03:09,330 --> 00:03:17,850 ‫body frame using this rotation matrix product or let's just call it R and we followed the convention 36 00:03:18,060 --> 00:03:27,450 ‫are some small Z, Y, X, meaning that we first rotated about the Z axis, then Y axis and then X axis 37 00:03:27,450 --> 00:03:32,540 ‫using the moving body frame approach or the oilor angle approach. 38 00:03:33,030 --> 00:03:40,860 ‫And in the same way we can connect the oilor time derivatives and the body frame angular velocities 39 00:03:41,160 --> 00:03:44,540 ‫like this using the transfer matrix. 40 00:03:45,120 --> 00:03:49,920 ‫And in fact you can write all that even in a more compact way. 41 00:03:50,430 --> 00:03:55,590 ‫You can define a matrix that we call J that looks like this. 42 00:03:56,040 --> 00:03:58,020 ‫You have rotation matrix here. 43 00:03:58,530 --> 00:04:00,400 ‫You have transfer matrix here. 44 00:04:01,050 --> 00:04:04,680 ‫Now the rotation matrix is a three by three matrix. 45 00:04:04,680 --> 00:04:05,070 ‫Right. 46 00:04:05,430 --> 00:04:07,310 ‫And so is the transfer matrix. 47 00:04:07,920 --> 00:04:15,510 ‫So this entire matrix will be a six by six matrix, which means that you need a three by three zero 48 00:04:15,510 --> 00:04:18,300 ‫matrix here that looks like this. 49 00:04:18,870 --> 00:04:24,770 ‫And also here, that's how this three by three zero matrix looks like. 50 00:04:25,290 --> 00:04:33,660 ‫And then using this J matrix, you can have this kind of relationship epsilon, dot, vector equals 51 00:04:34,470 --> 00:04:38,250 ‫J matrix and then the vector here. 52 00:04:38,760 --> 00:04:40,680 ‫You can also write it out like this. 53 00:04:41,100 --> 00:04:47,160 ‫And now you see why this J matrix looks like this, because if you write out this entire equation, 54 00:04:47,160 --> 00:04:56,880 ‫then you see that this rotation matrix only affects you, VW and the transfer matrix only affects P, 55 00:04:56,880 --> 00:04:59,430 ‫Q, R and then R times you. 56 00:04:59,430 --> 00:05:11,760 ‫VW will give you these values here, these ones and T times P Q are will give you these values here, 57 00:05:12,210 --> 00:05:13,860 ‫which is exactly what you want.