‫Welcome back.
‫Let's continue with our rotational motion equations of motion derivation.
‫There is a law that states that the sum of the moments or talks the same thing equals the time derivative
‫of the angular momentum.
‫And right now, we are in the inertial frame.
‫And of course, you can also write it down like this, because this part here is the angular momentum,
‫the mass movement of inertia, matrix times, the angular velocity vector.
‫And so the sum of the moments is also the net moment that we can write down like this.
‫Now, let's rewrite our angular momentum differently.
‫Let's rewrite it like this.
‫Each vector in the inertia frame equals H X times the unit vector plus h y times J unit vector plus
‫H Z times the K unit vector.
‫Assuming that the products of inertia are all a zero kilogram times meter squared.
‫Like this.
‫Then we can rewrite our angular momentum vector in the following way, I excel in the inertia frame
‫times by DOT and then the AI unit vector plus AI y y in the inertia frame times setar dot times J unit
‫vector plus I z z in the initial frame times side dot and then the K unit vector.
‫It comes from this here, so if you write it out, then you will get this thing.
‫And now let's take the time derivative of the angular momentum vector, according to the product rule,
‫you would first take the derivatives of IEX sex I y y and then IDM and then you would leave the other
‫variables as they are.
‫So you can see that I taking the derivative of X, X, I, y, y, and then I now I've taken the derivative
‫of five dot fita dot and that, and now you have the double derivatives here.
‫And finally I've taken the derivatives of I, J and K with respect of time, since we are in the inertia
‫frame that is fixed then our I dot Jadot and K Dot will be zero and so this final term will be entirely
‫zero as well.
‫Here's the problem, though, right now we are in the inertia frame and unlike in the linear momentum
‫case where your M dot was zero kilograms per second.
‫Now, when you treat the time derivatives of your mass moments of inertia, then they are not zero.
‫And why is that?
‫Well, let's take a look at our drone.
‫Let's put it center of mass, where the origin of the inertia frame is, and this is the inertia from
‫here, X, Y and Z.
‫And the origin of the inertia frame is where the center of mass of the drone is.
‫So this configuration is that time equals zero seconds.
‫And now we will rotate the drone about the Z axis.
‫So this is a time equals, let's say, one second as an example, and you can see that the more one
‫is now here and then more to the words here is now here and the entire region has rotated about the
‫Z axis.
‫And now if you start measuring the mass moments of inertia about the inertia X, Y and Z axis.
‫For the drone, for this object, then now you can see that the mass moments of inertia are different.
‫You can see that eye x X is different, I y Y is different because now the mass of this object with
‫respect to the axis is distributed differently.
‫The mass of the drone is the same.
‫But this mass is now positioned with respect to these axes in a different way.
‫And in fact, now you have a case where your object about the axis might not be mass symmetric anymore.
‫So that means that also you have to worry about products of inertia and all that happens because you're
‫measuring your mass moment of inertia with respect to fixed axis X, Y and Z.
‫They are fixed.
‫They are not moving.
‫They are not rotating.
‫They are just fixed to the ground.
‫But as the drone rotates at different times, it is positioned with respect to those acts differently,
‫every time sample you take the eye XXIII.
‫Why are different when the drone is not fixed?
‫And so now, if your time derivative of your mass moment of inertia, I doubt if it's not equal to zero
‫kg time's meter squared per second, if that's the case, then that's a big problem because mass moments
‫of inertia computations are not that easy.
‫And you saw it from the formulas, but they have to be done.
‫However, if you have to do them continuously every time you rotate the drone, then that would be extremely
‫impractical.
‫Imagine that you have to do it all the time continuously.
‫You move the drone a little bit and you have to compute the mass moments of inertia again.
‫So that's really not a good thing.
‫And even if you don't rotate your drone, but you simply fly away from the origin that even then your
‫mass moments of inertia will change.
‫So imagine that first, your drone is like this, where the center of mass of your drone is, where
‫the origin of the inertia frame is, and then the orientation of your drone stays the same.
‫But it will simply fly away from the inertia frame.
‫So the position of the drones center of mass will change with respect to the inertia frame, but the
‫orientation stays the same.
‫Even then, your mass moments of inertia will change, right?
‫Why?
‫Because you're measuring with respect to the axis of the inertia frame and the mass movement of inertia
‫depends on two things.
‫It depends on the mass and how far that mass is from the axis that you're measuring it about.
‫So now all the point masses on this drone are further away from these axis and therefore your total
‫mass movement of inertia will change.
‫And that is one of the big reasons why are we switch from the inertia frame to the body frame.
‫The other reason, of course, is that the control force and control moments of the drone are usually
‫given in the body frame as well, which further simplifies our lives.
‫If you work in the body frame, then.
‫IDOT equals zero kg's times meter squared over second.
‫Why?
‫Because the body frame is glued to the drone.
‫If you look at it, then first of all, your drone is like this, this is your body frame now X, Y
‫and Z.
‫And now we're going to rotated about the body from Z axis.
‫And now your body frame is like this, where this is your body from x axis, Y axis and Z axis.
‫You can see that the body frame moves with the drone, that means that when you move it, then the distances
‫of point masses with respect to the body frame axis do not change.
‫So let's say that this little D.M. here, with respect to the body frame will not change.
‫And that's why now your eye dot equals zero.
‫So let's now derive the equations of motion for the rotational motion in the body frame.
‫Just to stress one thing, the origin of the body frame is placed where the center of mass of the drone
‫is right here.
‫Also, we assume that the drone is mass symmetric about all of its body frame access.
‫Therefore, the products of inertia are all zero and the matrix.
‫Is diagonal.
‫All these assumptions result in the fact that the principal inertia axis exacts I, Y, Y and Z.
‫Beautifully aligned with the body frame, X, Y and Z axis, making our muscles much simpler.
‫So in the next video, we will start with the derivations, so thank you very much and see you there.