‫Just to make this section more complete, I want to rewrite our dynamic's equations in a slightly different
‫form.
‫I have seen both forms in the literature and I want to make sure that you recognize them both and realize
‫that they are actually the same thing.
‫So you can rewrite this entire thing.
‫Like this and pay attention that this m super be now it's this matrix here.
‫All right.
‫So it's not a moment.
‫It's a mass and mass moment of inertia matrix this six by six matrix.
‫This is what it is.
‫Then this vector here, it's this then this entire second term here.
‫Which appears because your body frame is rotating with respect to your initial frame, it's sometimes
‫in the literature rewritten like this and then this lumbar superscript B, it's this one here where
‫you have your net force and net moment vectors.
‫So the only big difference here is that they have taken this second term that appears because your body
‫frame is rotating and they have rewritten it in a different way.
‫Where this sea superscript B is a six by six matrix and then this small V, B is a six by one vector
‫that contains information about the translational motion velocities in the body frame you the and W
‫and the angular velocities in the body frame as well, P, Q and R.
‫And so this c.B matrix ends up being this one here, the six by six matrix and then this V is this vector
‫here.
‫And to get this form, it's actually not that hard.
‫All you have to do, you just have to apply cross product like we learned before.
‫So you first take this one here.
‫You will have a three by one vector cross and then the other term it will also be a three by one vector
‫and then you apply the cross product to it.
‫If you remember, you had your three by three matrix.
‫That looks something like this, and then you took the determinant of it, and then you would find your
‫cross product and then you would do the same thing for this one, and then you would just manipulate
‫your variables in such a way that in all cases you extract your P, Q, R elements.
‫So this is what I have seen in the literature, and I just wanted to point it out so that when you see
‫it in the literature, so that you would know that it's actually the same thing, it's just you take
‫this second term and then you apply across product of it.
‫You do some algebraic manipulations and then you put this form into this form.
‫But they're the same thing.
‫There's nothing different here is just you apply mathematics to reformulate this term and put it like
‫this.
‫In fact, why don't you try it as an exercise and I'm going to give you the solutions in the next video,
‫try to put this form into this form.
‫See you in the next video.