Foundations — Autonomous GNC for reusable rockets — SpaceX approach overview

This page assumes nothing. If the parent note wrote a symbol, we build it here from the ground up, in an order where each idea leans only on the ones before it.

1. Position $\mathbf{r}$ — "where am I?"

The picture. Look at the blue arrow in the figure below. Its tail sits on the pad (the origin), its head sits on the rocket. Two numbers describe it here: how far sideways ($x$) and how far up ($y$). We write $\mathbf{r}=(x,y)$. In real flight there are three numbers ($x,y,z$) but two is enough to see everything.

Why the topic needs it. Guidance asks "where do I want to be?" and Navigation asks "where am I?" — both are questions about $\mathbf{r}$. You cannot aim at a barge without a way to name points in space.

2. Velocity $\mathbf{v}$ — "how fast, which way?"

The picture. In the figure, the orange arrow attached to the rocket is $\mathbf{v}$. If the rocket falls straight down, $\mathbf{v}$ points down. During descent $v_0$ (the parent's symbol) is just the length of this arrow at the start of the landing burn — the falling speed, e.g. $200$ m/s.

Why this tool (the derivative)? We want instantaneous speed, not "average over the whole flight." The derivative $\frac{d}{dt}$ answers exactly the question "how fast is this changing right now?" — the only speed that matters when you're 3 seconds from the ground.

3. Acceleration $\mathbf{a}$ and gravity $g$ — "how fast is the speed changing?"

The little $2$'s in $\frac{d^2\mathbf{r}}{d{t}^2}$ mean "take the rate-of-change twice": position → velocity → acceleration. This stacked symbol is exactly what the parent's $\ddot{\mathbf{r}}$ means — the two dots are shorthand for "differentiate twice with respect to time."

Why the topic needs $\mathbf{a}$. The engine's only job is to produce acceleration. The IMU (a sensor) measures acceleration directly. And the whole hoverslam is a story about choosing the right upward acceleration to cancel a downward velocity. Acceleration is where the rocket's control actually bites.

4. Angles $\theta$ and the tan/arctan tool — "which way am I tilted?"

The rocket doesn't just have a position — it has an orientation: is it nose-up, tilted, sideways? The simplest angle question is: given how far sideways and how far up something reaches, what is its tilt?

Why this tool and not another? We have two lengths (sideways and up) and we want an angle. Tangent is precisely the function that turns a length-ratio into a steepness, and $\arctan$ walks it back to the angle. Sine/cosine need a length too; tangent needs only the ratio, which is all a tilt is.

5. Integration $\int$ — "add up all the tiny pieces"

The parent recovers position from acceleration by writing $\int$. Time to earn that symbol.

The picture. In the figure, the blue curve is velocity over time. The distance travelled is the shaded area under the curve — a sum of thin rectangles. That summing-of-slices is what ${\int }_0^t\mathbf{v}dt$ means: add the contributions from time $0$ to time $t$.

Why the topic needs it. Acceleration is what the sensor feels; position is what guidance needs. Integration is the only bridge from one to the other. It also explains the drift problem: a tiny constant error $b$ in acceleration, integrated twice, grows like $\frac{1}{2}b{t}^2$ — small at first, enormous after a minute. That $t^2$ growth is why GPS must be fused in.

6. The feedback loop — "measure, compare, correct, repeat"

Why the topic needs it. Winds, exact mass, and engine performance are never known in advance. An open-loop plan (blindly replay a script) has no way to notice it's drifting off the barge. Only a closed loop — the $e$-measuring cycle above — can absorb surprises. Every letter in the parent's PID law ($K_{p}e+K_d\dot{e}+K_i\int edt$) is built from pieces on this page: $e$ (this section), $\dot{e}$ (a derivative, §2), $\int edt$ (an integral, §5).

How the foundations feed the topic

Each foundation links forward to a named prerequisite you'll meet in later pages: dead-reckoning feeds the Kalman Filter; the error loop feeds PID Control; angles feed Attitude Dynamics & Quaternions and Thrust Vectoring (Gimbal); the $v^2$ braking law is Kinematics — v²=v0²+2as; the fuel-optimal layer is Convex Optimization via Lossless Convexification; and the mass-vs-thrust story is the Rocket Equation (Tsiolkovsky). All roads lead back to the parent, Autonomous GNC for reusable rockets — SpaceX approach overview.

Equipment checklist

Cover the right side and check you can answer each before moving on.