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‫Welcome back.

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‫Let's do this exercise together now, so first of all, in the next case, we have this convention here,

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‫so we first rotate about the x axis by 90 degrees.

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‫You get this configuration, then you rotate about the inertial y axis.

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‫You get this thing here and then you rotate about the inertial Z axis and that's your final configuration.

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‫And now let's do the Oilor case.

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‫But now we are going to use this convention here.

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‫First we rotate about the body frame Z axis by 90 degrees.

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‫You get this configuration here, then you rotate about the body frame y axis.

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‫And according to the right hand rule, the positive rotation about the body frame y axis will be like

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‫this.

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‫When you grab this axis with the right hand, then your thumb shift point towards the positive body

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‫frame y axis.

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‫Then you have this configuration here and finally you rotate about the positive body frame x axis,

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‫which is like this.

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‫And if you do that, you will have this configuration.

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‫By the way, I made the rotations a bit clearer now so that you wouldn't confuse in which way I'm rotating

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‫the body frame.

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‫So what do you see here?

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‫Will you see that both here and here?

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‫The body frames are in the same configuration.

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‫You see that if you rotate about the X, Y, Z axis using the fixed angle approach and then you rotate

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‫the same amount of radians using the oilor angle approach.

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‫But the order of rotation axis is the opposite.

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‫Now the sequence is a Z, Y, X, then your body frame ends up in the same orientation, like in the

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‫case of the fixed angles with the rotation sequence of X, Y, Z.

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‫And in fact, this is not an exception.

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‫It is actually true for all the conventions say that you rotate your body frame using the fixed angle

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‫approach like this are sub Y, Z, X, that is your convention.

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‫And since these are the capital letters, then you can see that these are the inertial frame axis.

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‫So you are using the fixed angle approach and there are many notations for these kind of things, but

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‫I find it convenient to use this one.

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‫And so first you rotate about the Y axis and let's say you rotate twenty two degrees, then you rotate

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‫about the Z axis and you rotate forty five degrees and then you rotate about the X axis and you rotate

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‫50 degrees in vector matrix form.

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‫This entire thing would look like this.

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‫You first rotate about the fixed y axis by 20 degrees, then about the fixed Z axis and then but the

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‫fixed x axis.

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‫And so your body frame ends up in a certain orientation.

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‫However, it will end up in the same orientation.

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‫If you use the oilor angle approach and you first rotate 50 degrees about the body frame x axis, then

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‫you rotate forty five degrees about the body frame the axis, and then you rotate twenty two degrees

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‫about the body frame y axis.

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‫So here the sequence was Y, then Z, then X.

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‫But if you have the sequence in reverse, X, Y, Z and Y, but you're rotating about the body frame

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‫axis, then you will end up in the same configuration.

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‫So that's just an interesting phenomena of rotation matrices.

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‫Now if that's true and when you use the body frame approach and your convention is are sub X, Z and

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‫Y and you end up in the same orientation, like when you use the convention are sub Y, Z, X, but

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‫using the fixed angle approach, if that's true, then that means that the vector matrix form for this

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‫case here is also this one.

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‫In other words, the total product of rotation matrices that you have here, this total product can

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‫be used for this fixed angle approach and the same product can be used for the oilor angle approach.

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‫All right.

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‫Provided that when you use the oilor angle approach, then the sequence is opposite to the fixed angle

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‫approach.

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‫So I just want to make one thing very, very clear.

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‫When you use an oilor angle approach, when you rotate about the moving body frame and not about the

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‫fixed inertial frame in your rotation sequence, is this that you first rotate about the body frame

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‫x axis, then about the body from Z axis and then about the body frame Y axis, then it would be a mistake

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‫to have something like this.

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‫This for this convention here, using the oilor angle approach would be wrong, even though that could

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‫be your first intuition.

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‫You first rotate about the X axis, then about the Z axis and then about the Y axis.

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‫So why wouldn't you write it like this?

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‫Well, you can't because you are rotating about the moving body frame axis and not about the fixed inertia

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‫frame axis and what we are doing here.

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‫This thing applies to when we rotate about the inertial frame axis.

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‫That's what this operation applies to.

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‫However, since rotation matrices have an interesting phenomena that when I rotate about the moving

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‫body from Axis and my sequence is opposite to the one in the fixed angle case, then my body frame ends

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‫up in the same orientation.

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‫And that means that this triple matrix product here can also be used for this convention, provided

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‫that the sequence here, when I rotate about the body frame axis is opposite to the sequence here when

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‫I rotate about the inertia frame axis.

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‫So when you use the oilor angle approach using this sequence, then you can use the product of those

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‫three matrices here that follow a fixed angle approach using this sequence.

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‫So I hope it's clear now.