orientation

Orientation is which way a body is turned, regardless of where it is. Position tells you where a thing sits; orientation tells you how it is rotated once it is there. Hold a book at a fixed spot in the air and slowly twist it — the position never changes, but the orientation does. A coffee cup right-side-up and the very same cup tipped on its side share a position but have completely different orientations, and to a robot that wants to pour without spilling, the orientation is the part that matters.

Here is the subtle part: describing orientation needs three independent numbers, just like position, but they behave far more awkwardly. You can keep adding a metre of position forever, but spin something a full turn and it is right back where it started, so the numbers wrap around. Different turns also do not stack up the way distances do — twisting then flipping a box lands it differently from flipping then twisting. Because rotation is this slippery, engineers use several different bookkeeping schemes to write it down, each with its own strengths and headaches.

Those schemes are the next several ideas in this chapter: a rotation matrix (a tidy grid of nine numbers), Euler angles (three intuitive tilt-and-spin angles, which can jam in a trap called gimbal lock), axis–angle (one direction to spin around plus how far), and the quaternion (four numbers that turn smoothly and never jam). They are all just different languages for saying the same thing — the orientation — and a robot constantly translates among them. Pin down a body's position and its orientation together and you have its full pose: everything you need to say exactly how it is placed in space.