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‫Welcome back.

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

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‫because this is what we're going to use for our course.

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‫So, again, the law states the sum of the moments, but now in the body frame equals the time derivative

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‫of the angular momentum.

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‫But again, now in the body frame.

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‫And you can rewrite it like this, where this is your mass moment of inertia matrix or your inertia

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‫Tancer.

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‫But now in the body frame, so the mass moments of inertia and products of inertia, they're are all

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‫measured with respect to the body frame.

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‫And this is your angular velocity vector.

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‫So instead of this vector here, since we are in the body frame, we are now using this vector p, q,

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‫r radians per second.

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‫OK, so let's rewrite our angular momentum vector in a different way.

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‫I can rewrite it like this where I used the lowercase letters now to indicate that I'm now working in

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‫the body frame in assuming that our products of inertia are all zeros.

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‫I can rewrite all this like this, where all these mass moments of inertia are with respect to the body

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‫frame.

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‫And now I have the body frame, angular velocity components P, Q and R, and then these are the unit

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‫vectors I, J and K, and that comes from this vector matrix notation here.

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‫If you write it out then this is what you will get.

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‫OK, but now let's take the time derivative of our angular momentum.

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‫So it's going to be h dot vector in the body frame from the product rule in calculus.

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‫You know that first of all, you have to take the time derivative of all the first variables, the mass

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‫moments of inertia.

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‫I dot I y y dot and dot.

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‫Plus, after you have taken the time derivatives of the mass moments of inertia, then you take the

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‫time derivatives of your P, Q and R, and finally you take the time.

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‫Derivatives of your I, J and K.

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‫And remember, everything now is in the body frame now in the inertia frame, we made our last roll

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‫zero because our IDOT Jadot and K that they were all zeros, but then the first and the second row,

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‫they were not zeros.

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‫However, now since we are in the body frame then now the time derivatives of our mass moments of inertia,

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‫now they are zero.

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‫So the first rule actually becomes zero like this because this is zero, this is zero and this is zero.

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‫The second and the third rule, however, they won't be zero because now you're either Jadot and K that

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‫they are not zeros.

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‫They change because the body frame rotates with respect to the inertia frame.

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‫And you already know that IDOT equals the angular velocity vector cross I j that equals the angular

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‫velocity vector cross J and K that equals the angular velocity vector crus k like this.

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‫And so you can rewrite the instantaneous time, change of your angular momentum vector like this, where

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‫these three terms they are.

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‫The Mars moment of inertia matrix in the body frame times the time derivative of the angular velocity

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‫vector in the body frame as well.

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‫So this is equivalent.

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‫And then the fourth, fifth and the sixth term would be this one here where this is I thought this would

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‫be Jadot and this would be K Dot.

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‫So it's this row now.

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‫The green row was this one and the red row is this one.

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‫Again, you can treat P, Q and R like constants and then you can move them onto the other side of the

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‫cross product.

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‫And the same thing you can do with this I X, X, Y, Y and Z scalars.

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‫Right.

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‫Because now these specific elements, they are constants.

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‫And so you can also move them on the other side of the cross product because they are just scalars,

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‫they are numbers within the mass moment of inertia matrix.

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‫And so you can rewrite this term here like this, you see now I've taken these two constants and I've

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‫put them here, I do the same thing here and here you have your eyes and, ah, you can factor out the

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‫angular velocity vector like this.

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‫We have w superscript B cross I sex P I plus.

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‫I wonder why.

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‫Q Jay plus ises times are times like.

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‫Like this and of course, these mass moments of inertia are all in the body frame, and since you're

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‫products of inertia are zeros, you can rewrite this role like this WOUB Cross.

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‫And then I'm just going to put a mass moment of inertia matrix here times the angular velocity vector

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‫in the body frame, because essentially what you have here in the brackets is the same thing like this

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‫here in yellow.

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‫And so you can rewrite this entire thing like this.

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‫I superscript B and then this is your angle velocity vector and you have it here like this.

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‫And now this red part takes into account the fact that your body frame is rotating with respect to the

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‫inertia frame.

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‫And so a compact way to rewrite the time derivative of the angular momentum vector is like this, where

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‫this is your first term in green and this is your second term in red.

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‫Again, just to reiterate, the green term assumes that your body frame is not rotating with respect

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‫to the inertia frame.

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‫However, the Retter, this is the correction factor to correct for the fact that your body frame is

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‫rotating with respect to the inertia frame.

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‫And so the most compact way to rewrite this equation is like this.

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‫So this is the entire time derivative of the angular momentum vector.

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‫However, if our body frame is not rotating, then our time derivative of the angular momentum vector

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‫is this.

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‫But we're going to put here X, Y, Z in order not to confuse these two and then the return.

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‫Can simply be written like this, where I superscreen be times the angle of velocity vector in the body

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‫frame is the angular momentum vector in the body frame.

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‫And so since our law said that the time derivative of H is the sum of the moments, then all that equals

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‫this, the sum of the moments in the body frame or the net moment vector in the body frame, or let's

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‫just call it like this.

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‫All like this, if you want.