axis–angle representation

The axis–angle representation describes any orientation as one single twist: pick a line to spin around, then say how far to spin. The line is given as a unit axis, an arrow of length one pointing in some direction, and the amount is just an angle in degrees or radians. This rests on a tidy fact, known as Euler's rotation theorem, that no matter how complicated a reorientation looks, it can always be reproduced by a single rotation about a single fixed axis. So 'turn 90 degrees about the vertical axis' is a complete answer, not a simplification.

This view is wonderfully intuitive because it matches how we naturally talk about turning things: one direction, one amount. It is also compact, needing only an axis and a number, and it makes 'how much did the robot rotate' easy to read off directly. Often the axis and angle are folded into a single rotation vector, an arrow whose direction is the axis and whose length is the angle, which is handy for math. Its main weakness is that combining two rotations is awkward in this form, and the axis becomes meaningless when the angle is zero (there is no 'direction' to a non-turn). For that reason robots frequently convert axis–angle into a quaternion or rotation matrix for actual computation, while still using axis–angle to express things like a commanded turn or the error between two orientations.