Foundations — Quaternion kinematics — q̇ = ½ Ξ(q) ω

This page builds every squiggle the topic uses, in order, from nothing. Each one gets: what it says, what it looks like, and why the topic needs it. We will meet the target equation $\dot{q}=\frac{1}{2}\Xi (q)\omega$ only at the very end, once each of its four symbols — $\dot{q}$, $\frac{1}{2}$, $\Xi (q)$, $\omega$ — has been earned. For now, treat it as a locked door; we are cutting the keys one at a time.

0 · What does "attitude" even mean?

Picture a satellite floating in space. It isn't moving anywhere, but it can turn. The complete description of "how turned is it right now" is its attitude.

Figure 1 shows why we need two sets of axes. The body's own three axes (its body frame, orange) are rotated away from the fixed lab axes (the inertial frame, gray). The insight the picture gives: attitude is not a thing the body has, it is the rotation that carries gray onto orange. Everything in this topic tracks that one rotation as it changes over time.

1 · A rotation = an axis + an angle

• $\theta$ (the Greek letter "theta") = how far you turn, measured in radians.
• $\hat{n}$ (an "n" with a hat) = the axis you turn around. The hat $\hat{}$ always means "this vector has length 1" — it only carries direction, not size.

This axis–angle idea is the bridge to quaternions and it links directly to Rodrigues rotation formula, which turns an axis and angle into an actual rotation of a vector.

2 · The length of a vector: $\Vert \cdot \Vert$

Why the topic needs it: a quaternion must always have length exactly $1$. The whole final section of the parent ("why the norm stays 1") is checking $\Vert q\Vert$ never drifts. If you can't read $\Vert q\Vert$, that entire safety argument is invisible.

3 · The dot product $\cdot$ and cross product $\times$

These two symbols appear all over the Hamilton product. They are the only two ways to "multiply" 3D vectors, and they answer different questions.

Figure 2 makes the split visible: the blue and orange arrows are the inputs $\mathbf{a},\mathbf{b}$; the green marker (cross product) points straight out of the plane they span; the shaded parallelogram's area equals $\Vert \mathbf{a}\times \mathbf{b}\Vert$; and the boxed number is the dot product $\mathbf{a}\cdot \mathbf{b}$. The one geometric lesson: dot = a number living in the plane, cross = a vector leaving the plane.

4 · The quaternion $q$ and its two parts

Figure 3 puts the two halves side by side: on the left, the four numbered slots ($q_0$ blue, $q_1{q}_2{q}_3$ orange); on the right, what those slots mean geometrically — a blue angle $\theta$ and an orange axis arrow. The takeaway the picture teaches: one column of four numbers = one angle glued to one axis.

5 · The half-angle $\frac{\theta }{2}$ — where the ½ is born

Notice the angle is always halved inside the quaternion: $\cos \frac{\theta }{2}$, $\sin \frac{\theta }{2}$.

Why the topic needs this NOW: when you later differentiate $\cos \frac{\theta }{2}$ with respect to time, the chain rule pulls out a $\frac{1}{2}$. That single $\frac{1}{2}$ is the entire mystery of the target equation. It was hiding in the half-angle all along.

$\frac{d}{dt}\cos \frac{\theta }{2}=-\frac{1}{2}\sin \frac{\theta }{2}\cdot \dot{\theta }⟸\text{the }\frac{1}{2}\text{ comes from the }\frac{\theta }{2}.$

6 · The overdot $\dot{q}$ — rate of change in time

Picture $q$ as a point sliding along a curve; $\dot{q}$ is the velocity arrow of that point — direction it's heading and how fast.

7 · Angular velocity $\omega$

A gyroscope on the spacecraft measures exactly this. Because gyros are bolted to the body, they report $\omega$ in the body frame — a distinction that decides multiplication order later. Deepen this in Angular velocity and the body frame.

8 · Body frame vs inertial frame (and why order matters)

The same physical arrow gets different numbers depending on which frame you read it in. This is the whole subject of Euler angles and gimbal lock and Rotation matrices and SO(3).

9 · Multiplication $\otimes$ and the "identity" quaternion

Why the topic needs it: the derivation writes the tiny rotation $\delta q$ as "identity plus a small correction." That correction, factored cleanly, is the right-hand side of the target equation.

10 · A matrix acting on a vector: $\Xi (q)\omega$

Assembling the key: reading the target equation

Now every symbol is defined, so the locked door opens: $\dot{q}=\frac{1}{2}\Xi (q)\omega .$ Read left to right: $\dot{q}$ (how the attitude moves, §6) equals $\frac{1}{2}$ (born from the half-angle, §5) times $\Xi (q)$ (the Hamilton product $q\otimes (0,\cdot )$ as a grid, §9–10) acting on $\omega$ (the body-frame spin from the gyros, §7–8).

How it all feeds the topic

Equipment checklist

Cover the right side and see if you can state each from memory.

Ready? Head back to the parent topic and every symbol will now read like plain English.