# Lesson 7: Chaining turns: the object's axes or the world's axes

Source: https://quaternion.steppeschool.com/combine-rotations-body-world-frame/ (part of "Quaternions, Visually" by Steppeschool, free)

When we add one more turn to the object, there are two different questions: turn about the object's own axis, or about the fixed world axis? The answer decides which side we multiply on.

## 7.1 Two kinds of "turn about x"

Imagine the object already turned 90° about z. Now "turn about x". The object’s own x has moved with it, but the world’s x is still where it was. These are two different turns, and they give different results.

> **Interactive visual: The object's x or the world's X.** **Try it:** the object has already turned 90° about z. Press the first button, go back to the start, then press the second. Same angle, same letter x, two different results.

Here q is the orientation so far, and Δq (delta q) is the quaternion of the new turn. The arrow ← means: replace q by what stands on the right.

$$\text{about the object’s axes:}\ \ q \leftarrow q \otimes \Delta q \qquad\qquad \text{about the world axes:}\ \ q \leftarrow \Delta q \otimes q$$

**With numbers.** Start with q = 90° about z = [0.707, 0, 0, 0.707], and add Δq = 90° about x = [0.707, 0.707, 0, 0]:

$$q \otimes \Delta q = [0.5,\ 0.5,\ 0.5,\ 0.5] \qquad\qquad \Delta q \otimes q = [0.5,\ 0.5,\ -0.5,\ 0.5]$$

Two different results, so two different orientations.

**Why this way round?** From lesson 5, the quaternion on the right acts first. In $\Delta q \otimes q$, the old q acts first and the new turn comes after it, about the world axes. In $q \otimes \Delta q$, the new turn acts first, while the object is still in its start pose, where its own axes and the world axes are the same. Then q carries the whole thing along. So the new turn ends up about the object’s own axes.

> **Interactive visual: Chain as many turns as you like.** **Try it:** press **+90° about z**, then **+90° about x**, with "the object's axes". Remember the pose. Press **Reset**, switch to "the world axes" and do the same two steps. The object ends up in a different place. Watch the faint X, Y, Z world axes: they never move.

## 7.2 Euler angles are a chain of three turns

Now we can write the ZYX Euler angles as one product:

$$q = q_{yaw} \otimes q_{pitch} \otimes q_{roll}$$

Read it left to right: yaw, then pitch about the new y, then roll about the newest x. Those are turns about the object’s own axes, multiplied on the right. Read it right to left: roll, pitch, yaw about the fixed world axes. Both readings give the same result. That’s why both descriptions of Euler angles you see in books are correct.

> **Interactive visual: Euler angles are three turns in a chain.** **Try it:** press play. First yaw about z, then pitch about the y axis the object has now, then roll about the x axis it has now. The dashed line is the axis of the current step.

## 7.3 Which one do I need?

Ask where the turn is measured. A gyroscope is fixed to the object, and a pilot or a player thinks "pitch up" from the cockpit. Those are turns about the object’s own axes, so we multiply on the right: $q \leftarrow q \otimes \Delta q$. A user dragging a 3D model on the screen turns it about the fixed screen axes, so we multiply on the left. We use the first one in lesson 9.

**One more thing:** Every multiplication adds a tiny rounding error, and after thousands of steps the length of q drifts away from 1. So after chaining, we divide q by its length. One square root and four divisions, and q is a clean rotation again.

**Remember:**
- Object axes (many books say body axes): multiply the new turn on the right, q ⊗ Δq.

- World axes: multiply on the left, Δq ⊗ q.

- Turn rates measured on the object itself are object-axis turns.

- Normalise q after many multiplications.

## ✓ Check yourself

**Q: A player pushes the stick to pitch the plane. Do we multiply the new turn on the left or on the right?**

On the right, q ⊗ Δq. The player thinks from inside the plane, so it is a turn about the object’s own axes.

**Q: Is "yaw 90° then roll 90° about the object's axes" the same as "roll 90° then yaw 90° about the world axes"?**

Yes. Both are q_yaw ⊗ q_roll. Try it in the lab.

## Sources and further reading

- J. Solà, "Quaternion kinematics for the error-state Kalman filter", 2017 <https://arxiv.org/abs/1711.02508>: chapter 2 and 4: composing rotations, local and global