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‫Welcome back.

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‫Let's now derive that transfer matrix in the last video, we saw that when we had our inertial frame

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‫in white in our body frame, which was attached to the drone in red, then I showed you how you go from

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‫the inertial frame configuration or from the inertial frame orientation to the body frame orientation.

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‫What you do, you first take the body frame and you completely align it with the inertia frame.

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‫And then according to our convention, we first rotate about the body frame Z axis by an angle psi.

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‫Then we rotate about the body frame Y axis by an angle theta, and then we rotate about the body frame

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‫x axis by an angle phi.

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‫And that's how we reach our body frame, which is the same thing that we have here, which is attached

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‫to the drone.

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‫And now our job is to find this transfer matrix that connects the angular velocities about the body

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‫frame, access to the time derivative of our oilor angles.

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‫Now earlier we defined that the rotation about the body frame x axis has an angular velocity of P radians

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‫per second.

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‫However, in our last rotation, when we rotated about the body frame x2 axis at an angle phi, then

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‫that had an angle of velocity of phi that radians per second.

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‫And that is the time change of one of our oilor angles.

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‫But now look, x2 equals x b and this means that the angular velocity P equals phi dot radians per second.

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‫In other words, the angular velocity about the body frame x b axis is the same that the time derivative

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‫of the phi oilor angle.

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‫And that's true only in this convention because we rotated about the moving frame x axis last.

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‫Now you can write it down like this or like this and you will see later why I wrote it down like this.

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‫Now things are different when we look at angular velocity Q radians per second, which is the rotation

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‫of the drone above the body frame y b axis.

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‫And if we also look at the time derivative of the oilor angle, theta or theta, that radians per second,

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‫whose rotation happens about the body frame y two axis.

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‫So it's here theta that radians per second.

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‫Now, the Y two axis is not equal to Y b axis, and therefore Q cannot be equal to see date.

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‫So how do we connect them?

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‫Well, in order to get from Y to to Y B, what did you do now?

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‫If you look at it, then you could take this picture out of context and you could pretend that this

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‫frame in green X to Y to Z two, you can treat it as an inertial frame, even though it's not.

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‫But just for this specific case, you could pretend that this is the inertia frame and in that case,

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‫the red frame X B will be in Zebb.

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‫This is the rotated frame that you rotated by applying the rotation matrix R, sub X, and you could

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‫do that, right?

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‫If you take this thing out of context and you just pretend that X to Y two and Z two is the inertia

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‫frame and you multiply this are sub X matrix by the red body frame, then you will get your inertial

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‫frame.

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‫In other words, if you measure something in the body frame and then you multiply this R sub X matrix

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‫by it, then whatever you're measuring you will have it in this green x2, y2 and Z to frame.

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‫And if you forget about everything else, then you could pretend that this is your inertia frame.

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‫But the point is that the rotation matrix R sub X is the connection between these two frames and you

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‫can also write it down like this.

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‫And now we can apply this logic to our situation because our goal was to connect CU with Theta Dot and

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‫we can do it like this.

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‫This is our rotation matrix, our sub X, and you know that it's our sub X because it has the angle

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‫phi in it which is here, and the x axis of both frames are equal, which is why you have one here.

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‫So that's why you know that it's our sub X and so we can multiply this matrix by CU, which is here

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‫in this red reference frame so we can write it down like this zero Q zero.

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‫And if we multiply this matrix by this red vector then we will get zero theta dot inside that.

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‫So now you see why I wrote these variables in this form.

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‫And of course we can write it down like this as well, where we just use this abbreviation instead of

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‫this entire matrix.

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‫And suppose that you want to go back and you have information about theta that radians per second,

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‫then you very simply take the R X matrix and you take the inverse of it and then you multiply it by

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‫this green vector and that's how you get this vector here.

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‫And now finally, how about R, which is the angular velocity of the drone about the Zebb axis?

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‫How would we connect our to the time derivative of the oil angle.

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‫BPCI how do we connect are with BPCI Dots and remember sai that is what happens here according to our

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‫convention and I will leave it to you as an exercise.

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‫Try doing it yourself and then you will see the solutions in the next video.

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‫Thank you very much and see you there.