Foundations — Attitude control modes — spin stabilization, 3-axis active

This page builds every symbol the parent note uses, starting from a reader who has never seen a letter with a little overbar-arrow on it. We go in order: each idea uses only ideas already built.

0. What the little marks mean

Before any physics, three pieces of notation appear everywhere. Let's earn them.

0b. The right-hand rule — read it once, use it everywhere

Several arrows below ($\vec{\omega }$, $\vec{\tau }$, and the cross product) point along an axis, and we need an unambiguous rule for which way along that axis. That rule is always the same:

1. Angle $\theta$ — the thing we actually control

Figure s01 — the pointing-error angle.

Look at figure s01: the pale-yellow arrow is the direction we want; the pink arrow is where we actually point; the wedge between them is $\theta$. When $\theta =0$ the two arrows lie on top of each other — mission accomplished.

$\dot{\theta }$ then means "how fast that wedge is opening or closing," and $\ddot{\theta }$ means "how fast that rate itself is changing" — you will meet both in the PID law.

2. Angular velocity $\vec{\omega }$ — spin as an arrow

Turning has a speed (how fast) and an axis (about which line). One arrow captures both.

Figure s02 — the spin arrow from the right-hand rule.

In figure s02 the disc turns counter-clockwise, so by the right-hand rule $\vec{\omega }$ (chalk-blue) sticks up out of the disc. Spin the other way and the arrow flips to point down. (The special case where the whole craft spins fast about one axis gets its own name, ${\omega }_s$, once we reach §7.)

3. Torque $\vec{\tau }$ — the "twist" you apply

To change spin you need a twisting push. That is torque.

4. Angular momentum $\vec{L}$ — stored spin, the star of the show

This single sentence is the engine of both control modes:

• Spin stabilization: make $\vec{L}$ huge, so a small disturbance torque ${\vec{\tau }}_{\text{ext}}$ barely moves it. Stiff.
• 3-axis active: deliberately create torques with actuators to change $\vec{L}$ on demand.

5. The moment-of-inertia tensor $\mathbf{I}$ — the machine linking spin to stored spin

Here is why we need a machine, not just a number.

Figure s03 — same spin, different axis, different stored spin.

Figure s03 shows the same disc spun two ways: about the fat flat axis (big $I$, long stored-spin arrow) versus a thin edge axis (small $I$, short arrow) — same $|\vec{\omega }|$, different $|\vec{L}|$. That difference is the whole plot.

6. The cross product $\times$ — how a spinning frame twists arrows, and the Euler equation

The rotation law contains the term $\vec{\omega }\times (\mathbf{I}\vec{\omega })$. What does the "$\times$" mean?

Now we can finally write the equation those symbols live in. Starting from ${\vec{\tau }}_{\text{ext}}=\dot{\vec{L}}$ (§4) and rewriting the rate of change in the body's own spinning frame gives Euler's rotational equation of motion. Because every torque here is external, we drop the "ext" label (§3) and write plain $\vec{\tau }$:

$\vec{\tau }=\mathbf{I}\dot{\vec{\omega }}+\vec{\omega }\times (\mathbf{I}\vec{\omega }).$

Read it in plain words: the applied twist $\vec{\tau }$ does two jobs — the first term $\mathbf{I}\dot{\vec{\omega }}$ speeds the spin up or slows it down, and the second term $\vec{\omega }\times (\mathbf{I}\vec{\omega })$ is the free sideways swing the spinning frame adds all by itself. Full derivation lives in Euler's rotational equations of motion; here we just want every symbol in it to now be readable.

7. Rate of change over time: the derivative $\frac{d}{dt}$

8. Kinetic energy of rotation $T$ — the tiebreaker for stability

First, name the "fattest" axis explicitly, because the stability rule leans on it.

How these feed the topic

Equipment checklist

Cover the right side and answer out loud; reveal to check.