‫Welcome back.
‫Let's do this exercise together now, so first of all, in the next case, we have this convention here,
‫so we first rotate about the x axis by 90 degrees.
‫You get this configuration, then you rotate about the inertial y axis.
‫You get this thing here and then you rotate about the inertial Z axis and that's your final configuration.
‫And now let's do the Oilor case.
‫But now we are going to use this convention here.
‫First we rotate about the body frame Z axis by 90 degrees.
‫You get this configuration here, then you rotate about the body frame y axis.
‫And according to the right hand rule, the positive rotation about the body frame y axis will be like
‫this.
‫When you grab this axis with the right hand, then your thumb shift point towards the positive body
‫frame y axis.
‫Then you have this configuration here and finally you rotate about the positive body frame x axis,
‫which is like this.
‫And if you do that, you will have this configuration.
‫By the way, I made the rotations a bit clearer now so that you wouldn't confuse in which way I'm rotating
‫the body frame.
‫So what do you see here?
‫Will you see that both here and here?
‫The body frames are in the same configuration.
‫You see that if you rotate about the X, Y, Z axis using the fixed angle approach and then you rotate
‫the same amount of radians using the oilor angle approach.
‫But the order of rotation axis is the opposite.
‫Now the sequence is a Z, Y, X, then your body frame ends up in the same orientation, like in the
‫case of the fixed angles with the rotation sequence of X, Y, Z.
‫And in fact, this is not an exception.
‫It is actually true for all the conventions say that you rotate your body frame using the fixed angle
‫approach like this are sub Y, Z, X, that is your convention.
‫And since these are the capital letters, then you can see that these are the inertial frame axis.
‫So you are using the fixed angle approach and there are many notations for these kind of things, but
‫I find it convenient to use this one.
‫And so first you rotate about the Y axis and let's say you rotate twenty two degrees, then you rotate
‫about the Z axis and you rotate forty five degrees and then you rotate about the X axis and you rotate
‫50 degrees in vector matrix form.
‫This entire thing would look like this.
‫You first rotate about the fixed y axis by 20 degrees, then about the fixed Z axis and then but the
‫fixed x axis.
‫And so your body frame ends up in a certain orientation.
‫However, it will end up in the same orientation.
‫If you use the oilor angle approach and you first rotate 50 degrees about the body frame x axis, then
‫you rotate forty five degrees about the body frame the axis, and then you rotate twenty two degrees
‫about the body frame y axis.
‫So here the sequence was Y, then Z, then X.
‫But if you have the sequence in reverse, X, Y, Z and Y, but you're rotating about the body frame
‫axis, then you will end up in the same configuration.
‫So that's just an interesting phenomena of rotation matrices.
‫Now if that's true and when you use the body frame approach and your convention is are sub X, Z and
‫Y and you end up in the same orientation, like when you use the convention are sub Y, Z, X, but
‫using the fixed angle approach, if that's true, then that means that the vector matrix form for this
‫case here is also this one.
‫In other words, the total product of rotation matrices that you have here, this total product can
‫be used for this fixed angle approach and the same product can be used for the oilor angle approach.
‫All right.
‫Provided that when you use the oilor angle approach, then the sequence is opposite to the fixed angle
‫approach.
‫So I just want to make one thing very, very clear.
‫When you use an oilor angle approach, when you rotate about the moving body frame and not about the
‫fixed inertial frame in your rotation sequence, is this that you first rotate about the body frame
‫x axis, then about the body from Z axis and then about the body frame Y axis, then it would be a mistake
‫to have something like this.
‫This for this convention here, using the oilor angle approach would be wrong, even though that could
‫be your first intuition.
‫You first rotate about the X axis, then about the Z axis and then about the Y axis.
‫So why wouldn't you write it like this?
‫Well, you can't because you are rotating about the moving body frame axis and not about the fixed inertia
‫frame axis and what we are doing here.
‫This thing applies to when we rotate about the inertial frame axis.
‫That's what this operation applies to.
‫However, since rotation matrices have an interesting phenomena that when I rotate about the moving
‫body from Axis and my sequence is opposite to the one in the fixed angle case, then my body frame ends
‫up in the same orientation.
‫And that means that this triple matrix product here can also be used for this convention, provided
‫that the sequence here, when I rotate about the body frame axis is opposite to the sequence here when
‫I rotate about the inertia frame axis.
‫So when you use the oilor angle approach using this sequence, then you can use the product of those
‫three matrices here that follow a fixed angle approach using this sequence.
‫So I hope it's clear now.