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‫Welcome back.

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‫Let's continue with our rotational motion equations of motion derivation.

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‫There is a law that states that the sum of the moments or talks the same thing equals the time derivative

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‫of the angular momentum.

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‫And right now, we are in the inertial frame.

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‫And of course, you can also write it down like this, because this part here is the angular momentum,

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‫the mass movement of inertia, matrix times, the angular velocity vector.

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‫And so the sum of the moments is also the net moment that we can write down like this.

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‫Now, let's rewrite our angular momentum differently.

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‫Let's rewrite it like this.

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‫Each vector in the inertia frame equals H X times the unit vector plus h y times J unit vector plus

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‫H Z times the K unit vector.

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‫Assuming that the products of inertia are all a zero kilogram times meter squared.

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‫Like this.

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‫Then we can rewrite our angular momentum vector in the following way, I excel in the inertia frame

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‫times by DOT and then the AI unit vector plus AI y y in the inertia frame times setar dot times J unit

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‫vector plus I z z in the initial frame times side dot and then the K unit vector.

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‫It comes from this here, so if you write it out, then you will get this thing.

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‫And now let's take the time derivative of the angular momentum vector, according to the product rule,

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‫you would first take the derivatives of IEX sex I y y and then IDM and then you would leave the other

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‫variables as they are.

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‫So you can see that I taking the derivative of X, X, I, y, y, and then I now I've taken the derivative

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‫of five dot fita dot and that, and now you have the double derivatives here.

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‫And finally I've taken the derivatives of I, J and K with respect of time, since we are in the inertia

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‫frame that is fixed then our I dot Jadot and K Dot will be zero and so this final term will be entirely

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‫zero as well.

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‫Here's the problem, though, right now we are in the inertia frame and unlike in the linear momentum

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‫case where your M dot was zero kilograms per second.

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‫Now, when you treat the time derivatives of your mass moments of inertia, then they are not zero.

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‫And why is that?

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‫Well, let's take a look at our drone.

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‫Let's put it center of mass, where the origin of the inertia frame is, and this is the inertia from

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‫here, X, Y and Z.

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‫And the origin of the inertia frame is where the center of mass of the drone is.

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‫So this configuration is that time equals zero seconds.

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‫And now we will rotate the drone about the Z axis.

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‫So this is a time equals, let's say, one second as an example, and you can see that the more one

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‫is now here and then more to the words here is now here and the entire region has rotated about the

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‫Z axis.

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‫And now if you start measuring the mass moments of inertia about the inertia X, Y and Z axis.

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‫For the drone, for this object, then now you can see that the mass moments of inertia are different.

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‫You can see that eye x X is different, I y Y is different because now the mass of this object with

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‫respect to the axis is distributed differently.

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‫The mass of the drone is the same.

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‫But this mass is now positioned with respect to these axes in a different way.

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‫And in fact, now you have a case where your object about the axis might not be mass symmetric anymore.

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‫So that means that also you have to worry about products of inertia and all that happens because you're

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‫measuring your mass moment of inertia with respect to fixed axis X, Y and Z.

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‫They are fixed.

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‫They are not moving.

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‫They are not rotating.

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‫They are just fixed to the ground.

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‫But as the drone rotates at different times, it is positioned with respect to those acts differently,

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‫every time sample you take the eye XXIII.

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‫Why are different when the drone is not fixed?

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‫And so now, if your time derivative of your mass moment of inertia, I doubt if it's not equal to zero

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‫kg time's meter squared per second, if that's the case, then that's a big problem because mass moments

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‫of inertia computations are not that easy.

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‫And you saw it from the formulas, but they have to be done.

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‫However, if you have to do them continuously every time you rotate the drone, then that would be extremely

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‫impractical.

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‫Imagine that you have to do it all the time continuously.

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‫You move the drone a little bit and you have to compute the mass moments of inertia again.

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‫So that's really not a good thing.

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‫And even if you don't rotate your drone, but you simply fly away from the origin that even then your

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‫mass moments of inertia will change.

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‫So imagine that first, your drone is like this, where the center of mass of your drone is, where

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‫the origin of the inertia frame is, and then the orientation of your drone stays the same.

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‫But it will simply fly away from the inertia frame.

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‫So the position of the drones center of mass will change with respect to the inertia frame, but the

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‫orientation stays the same.

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‫Even then, your mass moments of inertia will change, right?

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‫Why?

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‫Because you're measuring with respect to the axis of the inertia frame and the mass movement of inertia

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‫depends on two things.

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‫It depends on the mass and how far that mass is from the axis that you're measuring it about.

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‫So now all the point masses on this drone are further away from these axis and therefore your total

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‫mass movement of inertia will change.

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‫And that is one of the big reasons why are we switch from the inertia frame to the body frame.

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‫The other reason, of course, is that the control force and control moments of the drone are usually

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‫given in the body frame as well, which further simplifies our lives.

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‫If you work in the body frame, then.

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‫IDOT equals zero kg's times meter squared over second.

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‫Why?

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‫Because the body frame is glued to the drone.

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‫If you look at it, then first of all, your drone is like this, this is your body frame now X, Y

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‫and Z.

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‫And now we're going to rotated about the body from Z axis.

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‫And now your body frame is like this, where this is your body from x axis, Y axis and Z axis.

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‫You can see that the body frame moves with the drone, that means that when you move it, then the distances

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‫of point masses with respect to the body frame axis do not change.

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‫So let's say that this little D.M. here, with respect to the body frame will not change.

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‫And that's why now your eye dot equals zero.

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‫So let's now derive the equations of motion for the rotational motion in the body frame.

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‫Just to stress one thing, the origin of the body frame is placed where the center of mass of the drone

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‫is right here.

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‫Also, we assume that the drone is mass symmetric about all of its body frame access.

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‫Therefore, the products of inertia are all zero and the matrix.

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‫Is diagonal.

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‫All these assumptions result in the fact that the principal inertia axis exacts I, Y, Y and Z.

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‫Beautifully aligned with the body frame, X, Y and Z axis, making our muscles much simpler.

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‫So in the next video, we will start with the derivations, so thank you very much and see you there.