Since the azimuth axis carries the altitude axis with it, but not the other way around, the standard Euler rotation is valid as long as the pan is done first. It seems that the transformation advantage of the gimbal is that only the final pan and tilt angles are needed since intermediate altitude changes don't affect the azimuth. That is, if a series of rotations are made: pan, tilt, tilt, pan, tilt, pan, pan, tilt, only the cumulative rotations are needed to find the new orientation, whereas if the axes were not gimbaled, a transformation would have to be made every time a rotation in either axis occurred.
Perhaps I'm in the wrong forum or sub-forum for this subject. Can someone suggest another web site?