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

80/20 CORE: G-FOLD turns the "how do I land a rocket on a pinpoint using minimum fuel while never violating thrust limits, glide-slope, and physics" problem into a convex optimization — which computers can solve reliably and fast to the global optimum. The trick (its Nobel-worthy move) is a change of variables + lossless convexification that removes the non-convex thrust lower bound.

The Problem (WHAT are we solving?)

WHY convex? Non-convex problems can have many local minima; a solver might return a "landing" that's not the best, or fail to converge in the ~1 second you have during descent. A convex problem has one minimum, and interior-point methods find it in guaranteed polynomial time. G-FOLD = "Guidance for Fuel Optimal Large Diverts."

Dynamics From First Principles (HOW)

State: position $\mathbf{r}(t)\in\mathbb{R}^3$, velocity $\mathbf{v}=\dot{\mathbf r}$, mass $m(t)$.

Newton's 2nd law with gravity $\mathbf g$ (constant, near-planet) and thrust $\mathbf T$:

$m\,\dot{\mathbf v} = \mathbf T + m\,\mathbf g$

Fuel is burned proportional to thrust magnitude (rocket equation, differential form). With exhaust velocity $v_e=I_{sp}\,g_0$, define $\alpha = 1/v_e$:

$\dot m = -\alpha\,\|\mathbf T\|$

The villain: the constraint $\|\mathbf T\| \ge T_{\min}$ (engines can't throttle below a floor) defines the region outside a sphere — a non-convex set. Also $m$ multiplying $\dot{\mathbf v}$ makes the dynamics nonlinear.

Step 1 — Change of Variables (kill the nonlinearity)

Divide the acceleration equation by $m$. Define:

$\mathbf a = \frac{\mathbf T}{m}\quad(\text{acceleration command}), \qquad \Gamma = \frac{\|\mathbf T\|}{m}\ (\text{slack}), \qquad z = \ln m$

Then $\dot{\mathbf v} = \mathbf a + \mathbf g$ — linear in the new control $\mathbf a$.

Mass: $\dot z = \dfrac{\dot m}{m} = -\alpha\,\dfrac{\|\mathbf T\|}{m} = -\alpha\,\Gamma$ — also linear.

The thrust bounds, divided by $m$, become $\frac{T_{\min}}{m}\le \|\mathbf a\| \le \frac{T_{\max}}{m}$, and we introduce slack $\Gamma$ with:

$\|\mathbf a\| \le \Gamma, \qquad \frac{T_{\min}}{m}\le \Gamma \le \frac{T_{\max}}{m}$

Step 2 — Lossless Convexification (the genius move)

The remaining nonlinearity is $\Gamma \ge \frac{T_{\min}}{m}=T_{\min}e^{-z}$. We linearize the exponential to a second-order Taylor form that is still convex:

$T_{\min}e^{-z_0}\Big[1-(z-z_0)+\tfrac12(z-z_0)^2\Big] \le \Gamma \le T_{\max}e^{-z_0}\big[1-(z-z_0)\big]$

where $z_0(t)$ is the mass along a reference (no-thrust) trajectory. All constraints are now convex (second-order cone / linear).

Step 3 — The Convex Program

How to solve: discretize time into $N$ nodes → a finite-dimensional SOCP → solve with an interior-point method (ECOS, SDPT3). Do a line search over $t_f$ (flight time) since the problem is convex for fixed $t_f$; the fuel-vs-$t_f$ curve is unimodal.

Worked Example A — Vertical 1-D landing (feel the physics)

Lander at $r_z=100$ m, $v_z=-20$ m/s, $g=-3.71$ m/s² (Mars). Ignore mass change; $T_{\max}/m = 12$ m/s², $T_{\min}/m=3$ m/s².

Q: Can it stop before ground? Max deceleration available $= 12-3.71=8.29$ m/s² (thrust up, gravity down). Distance to stop from 20 m/s: $d=\frac{v^2}{2a}=\frac{400}{2(8.29)}=24.1$ m $<100$ m. ✔ Why this step? We compare "shortest stopping distance at full thrust" against available altitude — if it fits, a fuel-optimal solution exists.

Fuel-optimal shape: minimum-fuel ⇒ bang-bang in $\Gamma$: coast (or min-thrust) then full-thrust brake at the last moment. Why? Burning early wastes propellant fighting gravity longer; the optimizer delays the burn — G-FOLD reproduces the classic "suicide burn."

Worked Example B — Why glide-slope matters

Target at origin; lander approaching from the side at 30° above horizontal, but a ridge occupies angles below 40°. Set $\theta_{\text{gs}}=45°$ so $\|[r_x,r_y]\| \le \tan45°\,(r_z)= r_z$.

At $r_z = 200$ m the lander must be within 200 m horizontal radius. Why this step? The cone forces the trajectory to stay above a safe descent funnel — a convex constraint (a cone is convex), so it slots straight into the SOCP with no penalty to tractability.

Common Mistakes (Steel-man + Fix)

Flashcards

Connections

• Pontryagin Minimum Principle — proves the bang-bang & lossless result.
• Convex Optimization and Second-Order Cone Programming — the solver machinery.
• Tsiolkovsky Rocket Equation — source of $\dot m=-\alpha\|\mathbf T\|$.
• Interior-Point Methods (ECOS/SDPT3) — polynomial-time global solve.
• Apollo Lunar Descent Guidance — the polynomial-thrust predecessor G-FOLD improves on.
• Model Predictive Control — G-FOLD re-solved each cycle = MPC for landing.
• SpaceX Falcon 9 Landing and Mars EDL — real deployments.

Concept Map

Hinglish (regional understanding)