‫Welcome back.
‫Let's see now how we can connect R and Cidade radians per second.
‫So if you look at how we went from side that to R, then you can see that BPCI that rotates about the
‫inertia Z axis, which is equal to small Z one axis.
‫Then from Z one we went to Z two and then from Z two we went to Z B.
‫Now we already know how to connect the green and then the red reference frame and now we need to figure
‫out how we connect the purple and the green reference frame here.
‫Again, you can pretend that this purple reference frame is the inertial reference frame, even though
‫it's not in the global picture, but if we take this small picture out of the context, then what you
‫see is that you see that the purple reference frame is addressed and then the green reference frame
‫is rotating.
‫With respect to the purple reference frame, if that's the case, then we can apply a rotation matrix
‫here.
‫Now you can see that we are rotating at an angle, Seeta and if we pretend that the purple reference
‫frame is the inertial reference frame, then that means that this Y one, we are pretending that it's
‫an inertial y axis and then we're like rotating about the inertial y axis.
‫It means that we are taking this are sub y rotation matrix and we are multiplying it by this green reference
‫frame which is here.
‫And then whatever we are measuring in the green reference frame, we're getting this quantity in the
‫purple reference frame, as you can see here, and you can write it down like that as well.
‫Of course, this is the R sub Y matrix and we already know how to connect the green reference frame
‫with the red reference frame.
‫Right.
‫It's like this, just like we saw it in the last video.
‫But that also means that I can connect the purple reference frame and the red reference frame like this.
‫And this part here, it's this one is the green reference frame.
‫But we could write our green reference frame in terms of our red reference frame, which is actually
‫connected to the drone like this.
‫And so we can just substitute it.
‫And now we have an equation that connects our purple reference frame with our red reference frame.
‫So our rotation matrices now connect this red reference frame with this purple reference frame.
‫And that's what we need because the rotation happens about the red reference frame Z axis inside.
‫That happens about the purple reference frame, the Z axis.
‫That means that if you're on rotation happens in the red reference frame, which again is actually attached
‫to the drone, then you can very simply connect them with these two rotation matrices like this.
‫Or if you want to go back and you have your information on BPCI DOT will, then you very simply take
‫their inverse and that's it.
‫And now what we want to do, we want to write the vector P, Q, R, which are the rotational velocities
‫about the red reference frame axis, which is the real body frame that is attached to the drone.
‫We're going to write it out like this.
‫And I can do that.
‫Right.
‫Linear algebra allows me to add vectors like that.
‫However, if you recall, then this vector could be written like this and that's good because now we
‫have an oilor angle time derivative here.
‫So let's do that.
‫Are sub X inverse times zero feet and that in zero.
‫Now if you also recall then this vector here was equal to this vector here, which is again very good
‫because we have another time derivative of one of our Euler angles.
‫So let's write it here.
‫Five dot zero and zero.
‫And remember, the reason why they were equal was because here X2 and X B, they were in the same direction.
‫And finally, this vector, if you recall, could be written like this, which is also very good because
‫we have another oilor angle time derivative here.
‫There you go.
‫The multiplication of these two rotation matrices.
‫And if you take an inverse of their product and multiply it by this purple vector, then you can get
‫this vector.
‫And so we can just add them together and we will have P, Q, R, which is the same like this one.
‫And there you go.
‫We have written our P Q R rotational velocities, in other words, the rotational velocities, but the
‫body frame, X, Y, Z axis, we have written them in terms of our oilor angle changes with respect
‫to time.
‫So now we know how the oilor angle changes and the angular velocities about the drones axis, how they
‫are related to each other.
‫But this still is not our main goal, our main goal was to find this transfer matrix and the inverse
‫of it so that we could conveniently switch between the body frame angular velocities and the oil angle
‫changes with respect to time.
‫We want the transfer matrix so that we could have these kind of relationships like we have here.
‫And to get that, we still need to work on this equation.
‫And essentially what we need to do, we keep the peak.
‫You are vector the same.
‫However, our oilor angle time derivatives have to be in this form, just like we saw before, to make
‫the connection here.
‫We need the inverse of the transfer matrix.
‫And so this is the form that we want because this form allows us to find our transfer matrix inverse.
‫And once we have our transfer matrix inverse, we just take the inverse of that and we get our transfer
‫matrix.
‫But in order to do that, you have to take this entire thing here and you have to put it into this form
‫so that will test your mathematics skills.
‫So I will leave it to you as an exercise and then you will see the solution in the next video.
‫But try doing it yourself first and just challenge yourself, see if you can do it.
‫If not, not a problem, but at least give it a try.
‫So thank you very much and see you in the next video.