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‫Welcome back.

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‫In order to write less, I'm going to make some abbreviations instead of cosine.

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‫I'm just going to write C and instead of sine I'm just going to write S.

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‫So let's say cosine theta would simply be C theta and sine theta would simply be as theta and the same

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‫thing with tangent.

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‫I'm just going to say that it's t so tangent theta would be t theta.

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‫So let's start driving our product of rotation matrices for this convention and the nomenclature for

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‫this convention is like this are X, Y, Z and then parentheses Gamma Better and alpha.

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‫Now what it means is that first of all, rotate gamma radiance about the inertial x axis.

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‫Then after that, rotate beta radiance about the inertial y axis and then finally rotate alpha radiance

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‫about the inertial Z axis.

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‫And in our course, of course, it would have been like this.

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‫This would be Phi, this would be Theta, and then this would be BPCI, because we said that Gamma equals

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‫Phi, Beta equals Theta and then Alpha equals PSI.

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‫This is our our Z Matrix.

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‫This is our r y matrix.

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‫And this is our, our X matrix.

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‫So our X, Y, Z, Gamma, better and Alpha, it equals this product.

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‫And notice if we choose this convention then the angle gamma goes into this r x matrix, the angle better

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‫goes into this r y matrix and then the angle alpha goes into this r z matrix y because we first rotate

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‫about the X axis, then Y axis and then Z axis, and they are of course inertial axis and the order

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‫of the angles in this parenthesis, better and Alpha must follow the order of this axis here.

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‫So X, Y, Z, Gamma, better in Alpha and that's what this convention means.

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‫You first rotate about the inertial x axis by Gamma Radiance, then about the inertial y axis by better

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‫radians and then about the inertia axis by alpha gradients.

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‫And so if you write these matrices out, then this is what you need to multiply.

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‫And so we will first multiply this thing and the same thing here as well, and then we will multiply

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‫this thing and the same thing here like this and so are Y times are X equals this matrix here.

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‫And so the red ones, they are the components from this matrix and then the white ones are the components

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‫from this matrix here.

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‫And then we just put our R Z here.

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‫And so now we take this blue matrix and then we're going to multiply by this one here.

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‫And so if you multiply these matrices together, then you will get this matrix here and this is your

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‫final product matrix.

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‫And this product matrix is valid for this convention.

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‫When we rotate about the inertia from Axis.

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‫And it is also valid for this convention if we rotate about the moving frame axis.

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‫But of course, the rotation order is in the opposite direction.

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‫And now this is also very important here in the parentheses.

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‫I will write Alpha, Beta and Gamma.

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‫So notice what just happened.

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‫When we go from fixed angles to oilor angles, we change the order of axis and we also change the order

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‫of the angles.

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‫So now what I'm saying is that first rotate about the moving frame, the axis by Alpha Radians, then

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‫about the moving frame y axis by better radians, and then about the moving from X axis.

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‫I got my radiance, so that's what you need to keep in mind if you want to apply this product of mattresses

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‫to both conventions, this and this.

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‫In other words, if you write something like this, ah, and then lowercase letters, the Y X.

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‫So we're talking about the moving frame axis.

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‫But if you don't change the order of the angles and you still write Gummo Better and Alpha, that would

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‫be wrong because you would be saying rotate about the body from axis by Gamma Radiance and then here

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‫you would say rotate about the inertia from x axis by Gamma Radiance and then here rotate about the

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‫moving from y axis by better radians and then here rotate about the inertia frame y axis bebetter radians

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‫and then again here rotate about the moving frame x axis by alpha radians and then here rotate about

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‫the inertia frame Z axis by Alpha Radians.

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‫This would be wrong.

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‫You could not apply this product of matrices to this convention.

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‫It would be wrong.

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‫So this product is valid for these two conventions if you change the order of axis and the angles.

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‫So this convention is for fixed axis, this for moving axis, and for you to know in this course for

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‫our drone we will be using this convention here.

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‫This will be our convention, which means that this will be our rotation matrix product.

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‫And by the way, some videos ago we did this exercise, we used this convention, and then we rotated

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‫about the inertial X, Y and then Z axis and all rotations were over to radians.

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‫So your Fifita and BPCI, they were all over to radians.

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‫In this case, your gamma, beta and Alpha would be pi over to radians as well.

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‫So if you substitute all these angles in this product of rotation matrices with PI over to radians,

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‫if you make all the angles here power to radians, then you will see that this matrix, it will become

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‫this one here because they are the same thing is just here.

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‫You are representing this rotation matrix in terms of angle variables and here you have taken a specific

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‫value for all the angles, which was power to radians.

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‫And so you got a specific matrix.

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‫So just try it out, substitute all the angles here with PI over to Radiance, and then you will see

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‫that this matrix will become this RCR Y and our X matrix.

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‫You will have minus one here, one here and then one here.

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‫And then the rest of the elements will be zeros.

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‫And remember, in this course, gamma is our fire irradiance, then beta is our theta radiance and then

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‫Alpha is our psi radians.

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‫So if you want to know this product of rotation matrices in terms of Phi Theta and PSI, then you just

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‫replace these angles instead of sine gamma you would write sine PHI and that's it.

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‫And now in the next video we will derive this product, but for another convention for R, A, Z, Y,

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‫X, where we first rotate about the inertial frames axis, then inertial frame Y axis and then inertial

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‫frame x axis.

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‫See you in the next video.