Foundations — Powered descent guidance — G-FOLD algorithm (convex optimization)

Before you can read the parent note, you must own every letter it writes. This page builds each one from a picture. Read top to bottom — each symbol leans on the one above it.

0. The stage: position, velocity, acceleration

Everything happens to a lander flying above a target. To talk about it we need three ideas that stack.

Look at the figure. The arrow $\mathbf{r}$ has a shadow on the ground (how far sideways the lander is) and a height (how far up). To write both at once we split the arrow into pieces along three fixed directions — the axes.

Now, things move. Velocity is how fast position changes.

Stack the dot again: $\dot{\mathbf{v}}=\ddot{\mathbf{r}}$ is acceleration — how fast the velocity is changing. Firing the engine changes velocity, so acceleration is where the engine's push shows up.

1. Magnitude: the double-bar $\Vert \cdot \Vert$

Look at the figure: the thrust arrow $\mathbf{T}$ is the diagonal of a right triangle whose legs are the components $T_x$ and $T_z$. The dashed legs meet at a right angle, and the arrow's length is the hypotenuse $\sqrt{T_x^2+T_z^2}$ — Pythagoras in action. Notice the arrow could swing to any direction while keeping the same length; that "same length, any direction" is precisely what the double-bar captures.

2. Forces: thrust $\mathbf{T}$, gravity $\mathbf{g}$, mass $m$

3. The rate of fuel loss: $\dot{m}$, $v_e$, $I_{sp}$, $\alpha$

Fuel leaves as exhaust. The faster the exhaust and the more of it, the harder the push.

4. The logarithm $\ln$ and the exponential $e^x$

The parent's Step 1 will track the log of the mass instead of the mass itself. To follow it you first need what $\ln$ is and undoes.

Look at the figure: $e^x$ (lavender) and $\ln x$ (mint) are mirror images across the dashed diagonal line. Feed a number into one curve, then into the other, and you land back where you started — that is what "exact inverse" looks like. Notice the mint $\ln$ curve lives only to the right of zero: it has no output for $x\le 0$, the picture of the domain restriction $x>0$.

5. The new controls: $\mathbf{a}$ and the slack $\Gamma$

Here is the algebra, step by step, so you see why the trouble disappears. Start from Newton's law and divide every term by the mass $m$:

$m\dot{\mathbf{v}}=\mathbf{T}+m\mathbf{g}\stackrel{÷m}{\to }\frac{m\dot{\mathbf{v}}}{m}=\frac{\mathbf{T}}{m}+\frac{m\mathbf{g}}{m}.$

On the left the two $m$'s cancel, leaving just $\dot{\mathbf{v}}$. On the right, $\mathbf{T}/m$ is our new name $\mathbf{a}$, and $m\mathbf{g}/m$ cancels to $\mathbf{g}$:

$\dot{\mathbf{v}}=\mathbf{a}+\mathbf{g}.$

Now watch the throttle limits ride through the same division. Start from the hardware band and divide all three parts by the positive number $m$ (dividing by a positive number keeps the $\le$ signs pointing the same way):

$T_{\min }\le \Vert \mathbf{T}\Vert \le T_{\max }\stackrel{÷m}{\to }\frac{T_{\min }}{m}\le \frac{\Vert \mathbf{T}\Vert }{m}\le \frac{T_{\max }}{m}.$

But $\Vert \mathbf{T}\Vert /m=\Vert \mathbf{T}/m\Vert =\Vert \mathbf{a}\Vert$ (dividing an arrow by a positive number scales its length by the same factor), so:

$\frac{T_{\min }}{m}\le \Vert \mathbf{a}\Vert \le \frac{T_{\max }}{m}.$

Finally hand the middle over to the slack. We set $\Vert \mathbf{a}\Vert \le \Gamma$ and put the hardware band on $\Gamma$ instead:

$\frac{T_{\min }}{m}\le \Gamma \le \frac{T_{\max }}{m},\Vert \mathbf{a}\Vert \le \Gamma .$

That is the transformed throttle box the parent uses. (If we dropped the floor, it would read simply $0\le \Gamma \le T_{\max }/m$.)

6. Convex vs. non-convex — the shape that decides everything

This is the heart of why the parent does all its gymnastics.

7. The glide-slope cone and $\tan {\theta }_{\text{gs}}$

The parent's last constraint keeps the lander above a safe funnel using a tangent. (Recall from Section 1 that $[r_x,r_y]$ bundles the two ground coordinates, so $\Vert [r_x,r_y]\Vert$ is the sideways distance — the length of the shadow.)

Look at the figure: the mint funnel is the set of allowed positions. Its walls make angle ${\theta }_{\text{gs}}$ with the ground. At any height $h=r_z-r_{z,\text{land}}$ the funnel's radius is $\tan {\theta }_{\text{gs}}\cdot h$ — so the allowed sideways room (coral segment) shrinks straight down to zero as the lander nears the pad, forcing a clean vertical touchdown. A lander at the red dot outside the funnel would be violating the constraint (it could fly into the ridge); the green dot inside is safe.

8. The heavier machinery (names you'll meet, in one picture each)

The parent drops three technical names. You don't need to master them here — just hold a one-line mental picture so they don't feel like magic later.

We treat all three as trusted black boxes on this foundations page; the parent and later deep-dives open them up.

Prerequisite map

Equipment checklist

Test yourself — you are ready for the parent note only if each reveal feels obvious.