Foundations — Reaction control system — thruster selection, plume impingement limits

This is a foundations page. The parent note the RCS topic throws around arrows, cross products, matrices, and cosines as if you already own them. Here we build every one of those from nothing, in an order where each brick rests on the brick before it. If you have never seen an arrow with a hat on it, start at line one — you lose nothing.

0. The picture the whole topic lives in

Look at the figure. A boxy spacecraft floats in black. A dot in the middle marks its balance point. Small nozzles stick out of the corners, and one is firing — a colored spray leaves the nozzle, and the ship gets nudged the opposite way. That single image contains almost every symbol we are about to name. Keep glancing back at it.

1. A point in space — position, and the arrow that points to it

The picture: in the figure above, draw an arrow from the center dot to a nozzle. That arrow is ${\mathbf{r}}_i$ — where thruster number $i$ is bolted, measured from the balance point.

Why the topic needs it: a force applied far from the balance point twists the ship; a force applied right at it only shoves it. So the topic can never talk about turning without knowing where each nozzle sits. That "where" is ${\mathbf{r}}_i$.

2. Direction only — the unit vector with a hat $\hat{\mathbf{u}}$

The picture: shrink or stretch any arrow until it's exactly one grid-square long, keeping its heading. That shortened arrow is the hat version. In the figure, the short colored arrow at each nozzle showing which way the exhaust leaves is ${\hat{\mathbf{u}}}_i$.

Why the topic needs it: each thruster fires in a fixed direction. We want to store that direction once (${\hat{\mathbf{u}}}_i$) and separately dial the strength ($F_i$) up and down. Splitting "direction" from "amount" is the whole trick that later lets us write force as $F_i{\hat{\mathbf{u}}}_i$.

3. Force $\mathbf{F}$ and the number $F$ — how hard, which way

The picture: the recoil arrow on the ship in the figure. Long arrow = hard push; short arrow = gentle nudge.

We can now write the clean split promised above: ${\mathbf{F}}_i=F_i{\hat{\mathbf{u}}}_i.$ Read it out loud: "the force of thruster $i$ equals its strength $F_i$ times its direction ${\hat{\mathbf{u}}}_i$." Strength scales the unit arrow.

Why $F\ge 0$ (one-sidedness): a nozzle can only blow gas out — it can push, never pull. So $F$ can shrink to $0$ (off) but never go negative. This single fact ($F\ge 0$) forces the ship to carry nozzles pointing opposite ways, because you cannot "reverse a jet"; you fire its neighbour instead.

4. Torque $\mathbf{\tau }$ — turning, and why we need the cross product

Now the heart of it: a force off to the side makes the ship spin. The measure of "spinning effort" is torque, written $\mathbf{\tau }$ (Greek letter tau, bold because it too is an arrow).

Look at the figure. From the center, arrow $\mathbf{r}$ reaches the nozzle. At the nozzle, force arrow $\mathbf{F}$ pushes. The turning depends on the sideways part of $\mathbf{F}$ relative to $\mathbf{r}$. When $\mathbf{F}$ lines up with $\mathbf{r}$ (points straight out from center), the twist is zero — pure shove. When $\mathbf{F}$ is perpendicular to $\mathbf{r}$, the twist is biggest.

WHY a cross product and not ordinary multiplication? Ordinary multiplication of two lengths can't know about angle between them, and can't produce a new direction (the spin axis). We need a machine that eats two arrows and outputs:

• a size that is largest when the arrows are perpendicular and zero when parallel, and
• a direction = the axis the ship spins about.

That exact machine is the cross product, written $\mathbf{r}\times \mathbf{F}$.

Reading a cross product's number: its length is $|\mathbf{r}\times \mathbf{F}|=|\mathbf{r}||\mathbf{F}|\sin \theta ,$ where $\theta$ is the angle between the two arrows. The $\sin \theta$ is exactly "how sideways" — it is $0$ when parallel ($\theta =0$) and $1$ when perpendicular ($\theta =90^{\circ }$), matching the wrench intuition. (This is why the topic later cares about sines and angles.)

Reading its direction — the right-hand rule: point your right fingers along the first arrow $\mathbf{r}$, curl them toward the second arrow $\mathbf{F}$; your thumb points along $\mathbf{\tau }$, the spin axis. This is why opposite-side thrusters can twist the ship the same way — both give a thumb pointing the same direction. (See the parent's worked "pure roll couple.")

For the deeper machinery of turning bodies, see Cross Product & Rigid-Body Torque and Attitude Dynamics — Euler's Equations.

5. Stacking force and torque — the wrench $\mathbf{w}$ and the vector notation

We now have two arrows describing one thruster: the shove $\mathbf{F}$ (3 numbers) and the twist $\mathbf{\tau }$ (3 numbers). The topic stacks them into a single tall list of 6 numbers:

The picture: imagine a control panel with 6 sliders — 3 for "how much to shove (in each of 3 directions)" and 3 for "how much to twist (about each of 3 axes)." Any command the pilot could want is one setting of those 6 sliders. That setting is ${\mathbf{w}}_{\mathrm{cmd}}$ (the commanded wrench).

Why bundle them: because every control command — nudge forward, roll, hold still — is fully described by these 6 numbers. Bundling lets us write the whole selection problem in one clean line instead of six scattered ones.

6. The matrix $B$ — a table that turns "which nozzles" into "what wrench"

Each thruster, fired at strength $1$, contributes its own little wrench: $[{\hat{\mathbf{u}}}_i;{\mathbf{r}}_i\times {\hat{\mathbf{u}}}_i]$ — a column of 6 numbers. Line up all $N$ thrusters' columns side by side and you get a table of numbers, the matrix $B$:

$B=\left[\begin{array}{ccc}{\hat{\mathbf{u}}}_1& \cdots & {\hat{\mathbf{u}}}_N\\ {\mathbf{r}}_1\times {\hat{\mathbf{u}}}_1& \cdots & {\mathbf{r}}_N\times {\hat{\mathbf{u}}}_N\end{array}\right](6\times N).$

The picture: think of $B$'s columns as $N$ ingredient arrows and $\mathbf{f}$ as how much of each ingredient to pour in. $B\mathbf{f}$ is the final mixed wrench. This is exactly the "sum over all thrusters" from the parent, written compactly.

This is the machinery of Control Allocation & Pseudo-inverse: usually there are more nozzles than the 6 things to control ($N>6$), so many mixes work and we pick the cheapest — see Rocket Equation & Specific Impulse for why "cheapest" means "least propellant."

7. Angles and the plume — $\theta$, $\cos$, and the $1/d^2$ falloff

The exhaust doesn't disappear at the nozzle — it fans out into a spray cone. Two new symbols describe where that spray goes.

• $d$ = the distance from the nozzle mouth to whatever the spray hits (a panel, a target ship). In the figure, the length along the cone's center.
• $\theta$ = the angle off the centerline — how far to the side of "straight ahead" a surface sits. $\theta =0$ means dead-center in the spray; larger $\theta$ means out toward the cone's rim.

Putting the two together gives the parent's pressure law: $P_{\mathrm{imp}}(d,\theta )\approx P_0{\left(\frac{d_0}{d}\right)}^2{\cos }^k\theta .$

Why the topic needs all this: this single formula is the safety gate. If $P_{\mathrm{imp}}$ on a fragile surface exceeds its limit $P_{\max }$, that nozzle is forbidden — its strength is locked to $0$ before selection even runs. That's the "keep-out cone." Deeper modeling lives in Rarefied Gas Dynamics / Plume Modeling, and it matters most during close approach — see Rendezvous and Docking.

8. How the foundations feed the topic

Every arrow in that map is a "before you can understand this, understand that." The two streams — the allocation stream (left) and the plume-safety stream (right) — meet at the final selection.

Equipment checklist

Cover the right side and answer aloud before revealing.