‫So let's define a vector now, big Gummo, superscript E equals big X, big Y and big Z.
‫This is our position vector in the inertial frame.
‫And that's why you have this letter E here, because you can also call it the earth frame.
‫If we take the time derivative of this vector, this output, that means time derivative, the change
‫of this vector with respect to time.
‫If we do that, then we obtain the velocity vector in the initial frame.
‫So as you can see, all the elements here have dots now, X, Y and Z dot.
‫And now let's define a velocity vector in the body frame, which will be small X dot, small Y dot and
‫small Z dot.
‫However, like we have already mentioned in this course, we will use this notation small U, small
‫V in small W, so that will be our velocity vector in the body frame.
‫In other words, you ve and W on the body frame velocities in the body frame X, Y and Z direction respectively.
‫And now we use the rotation matrix to make the connection between the body frame and the inertial frame
‫velocities.
‫If you have this information and you want to go here and obtain this information, then you need this
‫rotation matrix, our sub body frame, Z, Y, X.
‫So we are using the oilor angle approach where rotating about the moving body frame first about the
‫Z axis, then Y and then X.
‫However, if we have this information here and we want to obtain this information will then very simply
‫we take this rotation matrix and we take the inverse of it.
‫And remember, since rotation matrices are also normal matrices, then that means that you can just
‫take the transpose of this matrix.
‫And if you apply it to this vector, then you will obtain this vector here.
‫By the way, I will also add an arrow here and here.
‫So it just means that we're dealing with vectors.
‫Sometimes I forget these arrows, but if you really want to be very correct with notation, then you
‫should put these arrows on top of these letters in order to say that you're dealing with vectors.
‫And so this entire thing here can also be represented like this gamma dot, superscript e vector.
‫And this is what you are looking for.
‫And let's suppose that you have the velocity vector in the body frame.
‫And since we are using this convention here and we want to get this rotation matrix here, then you
‫can get this rotation matrix by first putting a rotation matrix about the initial frame X here, then
‫a rotation matrix about the initial frame Y here, and then a rotation matrix about the inertia frame,
‫the Z here.
‫Remember, this triple multiplication was for this convention originally.
‫When you rotate about the inertia frames, however, it is also suitable for our case because we are
‫rotating about the moving frame and our sequences opposite to the one that happens in the inertia frame.
‫So if you rotate about the inertia frame axis here in this sequence, X, Y, Z, then if you rotate
‫about the body frame Z, Y, X axis, then the orientation of the body frame will be the same.
‫And therefore you can use this multiplication for this case and for our case.
‫And then you multiplied by the velocity vector in the body frame and you get your velocities in the
‫inertia frame.
‫And if you want to go back, then you simply take this matrix multiplication and you take the inverse
‫of it or transpose.
‫And from linear algebra, you know that you can also write it down like this.
‫So if you have this triple multiplication here and you take the inverse of it, then if you want to
‫open.
‫These parentheses, then the order of these matrices will change, meaning that I can write it down
‫like this are sub X inverse times or sub Y inverse times are sub Z inverse times.
‫Our government dot in the inertia frame.
‫So that comes just from linear algebra.
‫And one final remark.
‫Sometimes I can write this B letter or this E letter over here because I've seen both annotations and
‫sometimes I use one notation, sometimes I use another notation.
‫There is no difference whatsoever whether I put this B here or here, it doesn't matter.
‫It's just to let you know, are we measuring the velocities in the body frame or are we measuring the
‫velocities in the inertia frame?
‫So don't get confused by that.
‫If sometimes I write it like this, the same thing that if sometimes I don't put an arrow here, but
‫from the context, you know, that it's a vector, then it's the same thing.
‫So now you can clearly see why you need these rotation matrices you needed in order to get the velocities
‫of the drone in the inertial frame, because initially you have them in the body frame, either from
‫calculations or from sensors, and then you use the rotation matrices to have them in the initial frame
‫and you need them in the inertia frame in order to compute your position in the inertia frame, something
‫like this that we did in the previous course in this series.
‫But more on that later in the next video, we will find out that we also need to connect the angular
‫velocities in the body frame p, q, r with how the oilor angles change with respect time, fi, dot,
‫theta dot inside dot.
‫And in order to make this connection, we need to derive something called a transfer matrix.
‫And in the next video we will derive that transfer matrix.
‫And then you will also see that there is a difference whether you choose this convention or this oilor
‫angle approach convention.
‫Even though the rotation matrices for both of these cases are the same, the transfer matrix will be
‫different.
‫And so we will derive the transfer matrix for our oilor approach case.
‫And once you understand the method of how to derive this transfer matrix, then you will see, and I
‫will explicitly show you that the transfer matrix for this convention would be way different than what
‫we will get for this convention here.
‫So thank you very much and see you in the next video.