‫Welcome back.
‫So let's now derive the equations of motion for the rotational motion, but now in the body frame,
‫because this is what we're going to use for our course.
‫So, again, the law states the sum of the moments, but now in the body frame equals the time derivative
‫of the angular momentum.
‫But again, now in the body frame.
‫And you can rewrite it like this, where this is your mass moment of inertia matrix or your inertia
‫Tancer.
‫But now in the body frame, so the mass moments of inertia and products of inertia, they're are all
‫measured with respect to the body frame.
‫And this is your angular velocity vector.
‫So instead of this vector here, since we are in the body frame, we are now using this vector p, q,
‫r radians per second.
‫OK, so let's rewrite our angular momentum vector in a different way.
‫I can rewrite it like this where I used the lowercase letters now to indicate that I'm now working in
‫the body frame in assuming that our products of inertia are all zeros.
‫I can rewrite all this like this, where all these mass moments of inertia are with respect to the body
‫frame.
‫And now I have the body frame, angular velocity components P, Q and R, and then these are the unit
‫vectors I, J and K, and that comes from this vector matrix notation here.
‫If you write it out then this is what you will get.
‫OK, but now let's take the time derivative of our angular momentum.
‫So it's going to be h dot vector in the body frame from the product rule in calculus.
‫You know that first of all, you have to take the time derivative of all the first variables, the mass
‫moments of inertia.
‫I dot I y y dot and dot.
‫Plus, after you have taken the time derivatives of the mass moments of inertia, then you take the
‫time derivatives of your P, Q and R, and finally you take the time.
‫Derivatives of your I, J and K.
‫And remember, everything now is in the body frame now in the inertia frame, we made our last roll
‫zero because our IDOT Jadot and K that they were all zeros, but then the first and the second row,
‫they were not zeros.
‫However, now since we are in the body frame then now the time derivatives of our mass moments of inertia,
‫now they are zero.
‫So the first rule actually becomes zero like this because this is zero, this is zero and this is zero.
‫The second and the third rule, however, they won't be zero because now you're either Jadot and K that
‫they are not zeros.
‫They change because the body frame rotates with respect to the inertia frame.
‫And you already know that IDOT equals the angular velocity vector cross I j that equals the angular
‫velocity vector cross J and K that equals the angular velocity vector crus k like this.
‫And so you can rewrite the instantaneous time, change of your angular momentum vector like this, where
‫these three terms they are.
‫The Mars moment of inertia matrix in the body frame times the time derivative of the angular velocity
‫vector in the body frame as well.
‫So this is equivalent.
‫And then the fourth, fifth and the sixth term would be this one here where this is I thought this would
‫be Jadot and this would be K Dot.
‫So it's this row now.
‫The green row was this one and the red row is this one.
‫Again, you can treat P, Q and R like constants and then you can move them onto the other side of the
‫cross product.
‫And the same thing you can do with this I X, X, Y, Y and Z scalars.
‫Right.
‫Because now these specific elements, they are constants.
‫And so you can also move them on the other side of the cross product because they are just scalars,
‫they are numbers within the mass moment of inertia matrix.
‫And so you can rewrite this term here like this, you see now I've taken these two constants and I've
‫put them here, I do the same thing here and here you have your eyes and, ah, you can factor out the
‫angular velocity vector like this.
‫We have w superscript B cross I sex P I plus.
‫I wonder why.
‫Q Jay plus ises times are times like.
‫Like this and of course, these mass moments of inertia are all in the body frame, and since you're
‫products of inertia are zeros, you can rewrite this role like this WOUB Cross.
‫And then I'm just going to put a mass moment of inertia matrix here times the angular velocity vector
‫in the body frame, because essentially what you have here in the brackets is the same thing like this
‫here in yellow.
‫And so you can rewrite this entire thing like this.
‫I superscript B and then this is your angle velocity vector and you have it here like this.
‫And now this red part takes into account the fact that your body frame is rotating with respect to the
‫inertia frame.
‫And so a compact way to rewrite the time derivative of the angular momentum vector is like this, where
‫this is your first term in green and this is your second term in red.
‫Again, just to reiterate, the green term assumes that your body frame is not rotating with respect
‫to the inertia frame.
‫However, the Retter, this is the correction factor to correct for the fact that your body frame is
‫rotating with respect to the inertia frame.
‫And so the most compact way to rewrite this equation is like this.
‫So this is the entire time derivative of the angular momentum vector.
‫However, if our body frame is not rotating, then our time derivative of the angular momentum vector
‫is this.
‫But we're going to put here X, Y, Z in order not to confuse these two and then the return.
‫Can simply be written like this, where I superscreen be times the angle of velocity vector in the body
‫frame is the angular momentum vector in the body frame.
‫And so since our law said that the time derivative of H is the sum of the moments, then all that equals
‫this, the sum of the moments in the body frame or the net moment vector in the body frame, or let's
‫just call it like this.
‫All like this, if you want.