‫And now, since we have our equations for our Idon, Jadot and K that we can write down our translational
‫equations of motion in the body frame.
‫So instead of writing the equation like this.
‫And we said that this entire thing here was the translational acceleration in the body frame.
‫We can right this entire thing like this, this part stays the same, but this part can be written like
‫this.
‫Where this is IDOT, this is Jadot, and this is K Dot, the derivatives of unit vectors with respect
‫to time, linear algebra allows you to rewrite this yellow part like this, where you, the N.W., they're
‫just like constants.
‫And so when you have this cross product, you can just take this constant and then you can put it here,
‫here and here.
‫And so now you have this form.
‫Remember, again, I'm going to emphasize it.
‫This W here, it's a scalar.
‫It's the translational velocity in the Z direction, in the body frame, in metres per second.
‫It is part of this Vehbi velocity vector.
‫It had three components you the N.W..
‫Like this, you see, you've N.W., it's this one.
‫But the W vector in the B frame, it's an angular velocity vector in the body frame in radians per second.
‫It's this one here.
‫It tells you the angular velocity is about the body frame, X, Y, Z axis, respectively.
‫And so we can take now this part here.
‫And we can just ride it like this, it's equivalent, right?
‫So you have your eye unit vector J unit vector and K unit vector.
‫And you can rewrite this portion in this form without the unit vectors and let's call it the vector
‫in the body frame X, Y, Z, in order not to confuse it with this one here, because this vector here,
‫you get this when you're either Jadot and Cadart are zero.
‫In other words, when your body frame is not rotating with respect to your inertial frame.
‫You get it when your frame is not rotating in which you're measuring your work.
‫And the other part which is here, we can write it like this.
‫We can just factor out our WB vector here and put it like this and then cross.
‫UI plus the J plus W look like this, or equivalently, we can also write this portion like this, you
‫the W transposed or even in a more compact way, WB Cross and then this thing, we just write it like
‫this, the vector in the body frame.
‫And so this term you get because your body frame is rotating.
‫With respect to the inertia frame, if your body frame didn't rotate with respect to the initial frame,
‫then this yellow term would be zero and you would only be left with this blue part.
‫And so the final form of our translational equation of motion is the following.
‫You have the net force vector equals mass, the acceleration of the velocity vector, assuming that
‫the body frame is not rotating with respect to the initial frame.
‫Plus, the correction factor that takes into account the fact that the body is rotating with respect
‫to the inertia frame, so this term corrects for that fact.
‫The first term assumes that the body frame is not rotating, but in reality it is.
‫So that's why the second term, it's called a correction factor and it corrects for the fact that the
‫body frame actually is rotating with respect to the inertia frame.
‫And so let me also just ride it out in a larger way so that it would be more intuitive for you.
‫This force vector looks like this.
‫And by the way, this is also in the body frame, so it has three elements here.
‫And you can see that it's the body frame, because I'm using small X, Y and Z letters here, and so
‫all that equals mass times.
‫This vector would look like this.
‫Plus, this is the angular velocity vector cross you, the N.W..
‫And there you go.
‫This is your dynamic's equation for translational motion in the body frame.
‫OK, so this equation is just to describe how the drone translates in space, however, if it only translated,
‫then it would only be a three degree of freedom system where your independent variables are X, Y and
‫Z.
‫But the drone doesn't only translate in space, it also changes its attitude.
‫You also have Phi Theta and BPCI Angles as your independent variables, which are oilor angles in our
‫convention and then makes the drone a six degree of freedom system three four position.
‫And three for orientation, and so we also need to derive fundamental dynamics, equations to describe
‫the drones attitude in space.
‫What we have done here was basically force equals mass times acceleration only in the body frame, which
‫rotated with respect to the inertia frame.
‫But now we also have to do moment equals mass moment of inertia times the angular acceleration, and
‫that can also be done in the inertial frame and in the body frame.
‫And we're going to look at both ways and then we're going to end up doing it in the body frame, which
‫also rotates with respect to the inertia frame.
‫And we will do that in the next video.
‫Thank you very much.
‫And see you there.