with the help of the post changing frame, we are now good to use the Euler angle.

recall

for the rotating frame axis is rotating positive with the static frame.

the Euler angle is performed on 3 steps

- rotate on , the z-axis with , which is . the x-axis and the y-axis is now different, we notate this frame with a 1 .
- rotate on , the y-axis in the 1- frame by angle , which is . the new axis is notated by 2.
- rotate on , the z-axis in the 2-frame by angle , which is . the new axis is notated by R.

The rotating frame is related with the static frame by:

or

for each rotation is on a new frame, the computation will be ugly, since, after each rotation, we have to use the rotation matrix in new coordinate.

There is another representation, notice that:

which mean, the rotating on y1 -axis by is equal to rotate it back to on zS -axis and rotated it by on yS – axis, then rotate back the to on zS – axis.

i use a and b for the axis between the transform.

and we have it for the z2-axis.

by using these 2 equation and notice that the z1-axis is equal to zS-axis.

which act only on the the same frame.