‫Welcome back.
‫In order to write less, I'm going to make some abbreviations instead of cosine.
‫I'm just going to write C and instead of sine I'm just going to write S.
‫So let's say cosine theta would simply be C theta and sine theta would simply be as theta and the same
‫thing with tangent.
‫I'm just going to say that it's t so tangent theta would be t theta.
‫So let's start driving our product of rotation matrices for this convention and the nomenclature for
‫this convention is like this are X, Y, Z and then parentheses Gamma Better and alpha.
‫Now what it means is that first of all, rotate gamma radiance about the inertial x axis.
‫Then after that, rotate beta radiance about the inertial y axis and then finally rotate alpha radiance
‫about the inertial Z axis.
‫And in our course, of course, it would have been like this.
‫This would be Phi, this would be Theta, and then this would be BPCI, because we said that Gamma equals
‫Phi, Beta equals Theta and then Alpha equals PSI.
‫This is our our Z Matrix.
‫This is our r y matrix.
‫And this is our, our X matrix.
‫So our X, Y, Z, Gamma, better and Alpha, it equals this product.
‫And notice if we choose this convention then the angle gamma goes into this r x matrix, the angle better
‫goes into this r y matrix and then the angle alpha goes into this r z matrix y because we first rotate
‫about the X axis, then Y axis and then Z axis, and they are of course inertial axis and the order
‫of the angles in this parenthesis, better and Alpha must follow the order of this axis here.
‫So X, Y, Z, Gamma, better in Alpha and that's what this convention means.
‫You first rotate about the inertial x axis by Gamma Radiance, then about the inertial y axis by better
‫radians and then about the inertia axis by alpha gradients.
‫And so if you write these matrices out, then this is what you need to multiply.
‫And so we will first multiply this thing and the same thing here as well, and then we will multiply
‫this thing and the same thing here like this and so are Y times are X equals this matrix here.
‫And so the red ones, they are the components from this matrix and then the white ones are the components
‫from this matrix here.
‫And then we just put our R Z here.
‫And so now we take this blue matrix and then we're going to multiply by this one here.
‫And so if you multiply these matrices together, then you will get this matrix here and this is your
‫final product matrix.
‫And this product matrix is valid for this convention.
‫When we rotate about the inertia from Axis.
‫And it is also valid for this convention if we rotate about the moving frame axis.
‫But of course, the rotation order is in the opposite direction.
‫And now this is also very important here in the parentheses.
‫I will write Alpha, Beta and Gamma.
‫So notice what just happened.
‫When we go from fixed angles to oilor angles, we change the order of axis and we also change the order
‫of the angles.
‫So now what I'm saying is that first rotate about the moving frame, the axis by Alpha Radians, then
‫about the moving frame y axis by better radians, and then about the moving from X axis.
‫I got my radiance, so that's what you need to keep in mind if you want to apply this product of mattresses
‫to both conventions, this and this.
‫In other words, if you write something like this, ah, and then lowercase letters, the Y X.
‫So we're talking about the moving frame axis.
‫But if you don't change the order of the angles and you still write Gummo Better and Alpha, that would
‫be wrong because you would be saying rotate about the body from axis by Gamma Radiance and then here
‫you would say rotate about the inertia from x axis by Gamma Radiance and then here rotate about the
‫moving from y axis by better radians and then here rotate about the inertia frame y axis bebetter radians
‫and then again here rotate about the moving frame x axis by alpha radians and then here rotate about
‫the inertia frame Z axis by Alpha Radians.
‫This would be wrong.
‫You could not apply this product of matrices to this convention.
‫It would be wrong.
‫So this product is valid for these two conventions if you change the order of axis and the angles.
‫So this convention is for fixed axis, this for moving axis, and for you to know in this course for
‫our drone we will be using this convention here.
‫This will be our convention, which means that this will be our rotation matrix product.
‫And by the way, some videos ago we did this exercise, we used this convention, and then we rotated
‫about the inertial X, Y and then Z axis and all rotations were over to radians.
‫So your Fifita and BPCI, they were all over to radians.
‫In this case, your gamma, beta and Alpha would be pi over to radians as well.
‫So if you substitute all these angles in this product of rotation matrices with PI over to radians,
‫if you make all the angles here power to radians, then you will see that this matrix, it will become
‫this one here because they are the same thing is just here.
‫You are representing this rotation matrix in terms of angle variables and here you have taken a specific
‫value for all the angles, which was power to radians.
‫And so you got a specific matrix.
‫So just try it out, substitute all the angles here with PI over to Radiance, and then you will see
‫that this matrix will become this RCR Y and our X matrix.
‫You will have minus one here, one here and then one here.
‫And then the rest of the elements will be zeros.
‫And remember, in this course, gamma is our fire irradiance, then beta is our theta radiance and then
‫Alpha is our psi radians.
‫So if you want to know this product of rotation matrices in terms of Phi Theta and PSI, then you just
‫replace these angles instead of sine gamma you would write sine PHI and that's it.
‫And now in the next video we will derive this product, but for another convention for R, A, Z, Y,
‫X, where we first rotate about the inertial frames axis, then inertial frame Y axis and then inertial
‫frame x axis.
‫See you in the next video.