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‫Welcome back.

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‫Let's see now how we can connect R and Cidade radians per second.

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‫So if you look at how we went from side that to R, then you can see that BPCI that rotates about the

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‫inertia Z axis, which is equal to small Z one axis.

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‫Then from Z one we went to Z two and then from Z two we went to Z B.

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‫Now we already know how to connect the green and then the red reference frame and now we need to figure

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‫out how we connect the purple and the green reference frame here.

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‫Again, you can pretend that this purple reference frame is the inertial reference frame, even though

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‫it's not in the global picture, but if we take this small picture out of the context, then what you

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‫see is that you see that the purple reference frame is addressed and then the green reference frame

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‫is rotating.

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‫With respect to the purple reference frame, if that's the case, then we can apply a rotation matrix

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‫here.

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‫Now you can see that we are rotating at an angle, Seeta and if we pretend that the purple reference

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‫frame is the inertial reference frame, then that means that this Y one, we are pretending that it's

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‫an inertial y axis and then we're like rotating about the inertial y axis.

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‫It means that we are taking this are sub y rotation matrix and we are multiplying it by this green reference

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‫frame which is here.

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‫And then whatever we are measuring in the green reference frame, we're getting this quantity in the

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‫purple reference frame, as you can see here, and you can write it down like that as well.

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‫Of course, this is the R sub Y matrix and we already know how to connect the green reference frame

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‫with the red reference frame.

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‫Right.

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‫It's like this, just like we saw it in the last video.

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‫But that also means that I can connect the purple reference frame and the red reference frame like this.

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‫And this part here, it's this one is the green reference frame.

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‫But we could write our green reference frame in terms of our red reference frame, which is actually

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‫connected to the drone like this.

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‫And so we can just substitute it.

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‫And now we have an equation that connects our purple reference frame with our red reference frame.

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‫So our rotation matrices now connect this red reference frame with this purple reference frame.

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‫And that's what we need because the rotation happens about the red reference frame Z axis inside.

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‫That happens about the purple reference frame, the Z axis.

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‫That means that if you're on rotation happens in the red reference frame, which again is actually attached

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‫to the drone, then you can very simply connect them with these two rotation matrices like this.

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‫Or if you want to go back and you have your information on BPCI DOT will, then you very simply take

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‫their inverse and that's it.

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‫And now what we want to do, we want to write the vector P, Q, R, which are the rotational velocities

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‫about the red reference frame axis, which is the real body frame that is attached to the drone.

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‫We're going to write it out like this.

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‫And I can do that.

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‫Right.

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‫Linear algebra allows me to add vectors like that.

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‫However, if you recall, then this vector could be written like this and that's good because now we

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‫have an oilor angle time derivative here.

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‫So let's do that.

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‫Are sub X inverse times zero feet and that in zero.

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‫Now if you also recall then this vector here was equal to this vector here, which is again very good

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‫because we have another time derivative of one of our Euler angles.

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‫So let's write it here.

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‫Five dot zero and zero.

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‫And remember, the reason why they were equal was because here X2 and X B, they were in the same direction.

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‫And finally, this vector, if you recall, could be written like this, which is also very good because

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‫we have another oilor angle time derivative here.

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‫There you go.

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‫The multiplication of these two rotation matrices.

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‫And if you take an inverse of their product and multiply it by this purple vector, then you can get

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‫this vector.

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‫And so we can just add them together and we will have P, Q, R, which is the same like this one.

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‫And there you go.

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‫We have written our P Q R rotational velocities, in other words, the rotational velocities, but the

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‫body frame, X, Y, Z axis, we have written them in terms of our oilor angle changes with respect

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‫to time.

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‫So now we know how the oilor angle changes and the angular velocities about the drones axis, how they

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‫are related to each other.

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‫But this still is not our main goal, our main goal was to find this transfer matrix and the inverse

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‫of it so that we could conveniently switch between the body frame angular velocities and the oil angle

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‫changes with respect to time.

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‫We want the transfer matrix so that we could have these kind of relationships like we have here.

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‫And to get that, we still need to work on this equation.

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‫And essentially what we need to do, we keep the peak.

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‫You are vector the same.

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‫However, our oilor angle time derivatives have to be in this form, just like we saw before, to make

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‫the connection here.

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‫We need the inverse of the transfer matrix.

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‫And so this is the form that we want because this form allows us to find our transfer matrix inverse.

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‫And once we have our transfer matrix inverse, we just take the inverse of that and we get our transfer

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‫matrix.

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‫But in order to do that, you have to take this entire thing here and you have to put it into this form

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‫so that will test your mathematics skills.

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‫So I will leave it to you as an exercise and then you will see the solution in the next video.

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‫But try doing it yourself first and just challenge yourself, see if you can do it.

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‫If not, not a problem, but at least give it a try.

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‫So thank you very much and see you in the next video.