‫Welcome back.
‫So now we are going to derive our product of rotation mattresses for the convention, Z, Y, X, these
‫are capital letters.
‫So we are rotating about the inertia from Axis.
‫But here's the thing.
‫When we change our rotation sequence, then the angles, gamma, beta and then alpha, they still stay
‫in the same order, just like in the previous case when we had this convention, X, Y, Z, the order
‫of our angles was Gamma, Beta and alpha.
‫But now we change the order of our axis.
‫But we did not change the order of the angles.
‫Now, in the previous case here you see that we did change the order of our angles, but that's because
‫we went from the fixed axis approach to the moving axis approach.
‫When you go from fixed angles to oilor angles, then you change the order of these angles.
‫But when you go from a fixed angle approach with one convention to another fixed angle approach with
‫another convention, then you do not change the order of your angles.
‫So in the previous case, we said first rotate gamma radians about the inertial x axis, then beta radians
‫about the inertial y axis, and then alpha radians about the inertia axis.
‫Now we are saying rotate gamma radians about the inertial Z axis, then better radians about the inertial
‫Y axis and then alpha radians about the inertial x axis.
‫So if we want to use the angles used in this course, Phi Theta and PSI, then if we use this convention,
‫then we are saying first rotate five radians above the inertia Z axis, then theta radians about the
‫inertial Y axis and then Pi radians about the inertial x axis.
‫So in this convention, the order of matrices is like this.
‫In the previous case it was our Z times are Y, times are X, because you first rotate about the inertial
‫X axis, then the inertial Y axis and then the inertia axis.
‫But now the product is our X times, our Y times, our Z, because you are first rotating about the
‫inertial Z axis, then the inertial Y axis, then the inertial X axis, and now the gumba or the fire
‫angle enters into this matrix or Z, then beta or theta angle enters into this matrix R, Y, and then
‫Alpha or Kayangel enters into this matrix.
‫Here are acts.
‫It doesn't really matter in this course because we are not using this convention.
‫But just for you to know, when you change a convention, then these angles enter into different matrices.
‫So now we have to multiply together these three matrices.
‫As you can see, it's a little bit different here.
‫The order and then the matrices were like this.
‫And now for this convention, they are like this.
‫And again, we first multiply these two matrices and then we would multiply our X by the product of
‫R, Y and Z, and that would be the product of our Y times, our Z.
‫And now here we put our X and here as well.
‫And so if you multiply these matrices together, then this will be your final product matrix.
‫And so this product matrix is valid for this convention.
‫This is the fixed axis approach and then for this convention, which is the moving axis approach.
‫But now, since we go from fixed to moving, the order of these angles will change.
‫So I will put here Alpha, Beta and Gamma.
‫And in our case, it would be Phi Beta and then PSI and then here you would have PSI, Theta and Phi.
‫So, for example, in the previous case, the.
‫Our X matrix contained the gamma angles or in our case, the fire angles, but in this case the R X
‫matrix contains the alpha angles or the sci angles in our case.
‫And so in conclusion, in this case, we say rotate about the inertial Z axis by Gamma or Phi Radians,
‫then about the inertial frame Y axis by better or theta radians and then by the inertia frame x axis
‫by Alpha PSI radians.
‫Then you use this product and you can also use this product.
‫If you first rotate about the moving frame x axis by Alpha or PSI radians, then you rotate about the
‫moving frame y axis by beta or theta radians and then you rotate about the moving frame Z axis by Gamma
‫or Phi Radians.
‫And in the next video we will have one more convention are sub X, Y, X, so we will not even rotate
‫about the inertia frame.
‫Z axis will first rotate about the inertia from X axis, then about the inertia from Y axis and then
‫about the inertial frame x axis again.
‫And we'll see that in the next video.
‫Thank you very much.