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

3.5.53 · Physics › Guidance, Navigation & Control (GNC)

80/20 CORE: G-FOLD us problem ko — "minimum fuel use karte hue rocket ko pinpoint land kaise karein, bina thrust limits, glide-slope, aur physics violate kiye" — ek convex optimization mein convert kar deta hai — jise computers reliably aur fast solve kar sakte hain global optimum tak. Iska trick (Nobel-worthy move) ek change of variables + lossless convexification hai jo non-convex thrust lower bound ko hata deta hai.

The Problem (WHAT are we solving?)

WHY convex? Non-convex problems mein kai local minima ho sakte hain; ek solver ek aisi "landing" return kar sakta hai jo best nahi hai, ya descent ke dauran milne wale ~1 second mein converge karne mein fail kar sakta hai. Ek convex problem mein ek hi minimum hota hai, aur interior-point methods use guaranteed polynomial time mein dhundte hain. 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 $m(t)$.

Gravity $\mathbf{g}$ (constant, planet ke paas) aur thrust $\mathbf{T}$ ke saath Newton ka 2nd law:

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

Fuel thrust magnitude ke proportional burn hota hai (rocket equation, differential form). Exhaust velocity $v_e=I_{sp}{g}_0$ ke saath, $\alpha =1/v_e$ define karo:

$\dot{m}=-\alpha \Vert \mathbf{T}\Vert$

Villain: constraint $\Vert \mathbf{T}\Vert \ge T_{\min }$ (engines ek floor se neeche throttle nahi kar sakte) ek sphere ke bahar ka region define karta hai — ek non-convex set. Saath hi $m$ ka $\dot{\mathbf{v}}$ ke saath multiply hona dynamics ko nonlinear banata hai.

Step 1 — Change of Variables (nonlinearity khatam karo)

Acceleration equation ko $m$ se divide karo. Define karo:

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

Tab $\dot{\mathbf{v}}=\mathbf{a}+\mathbf{g}$ — nayi control $\mathbf{a}$ mein linear hai.

Mass: $\dot{z}=\frac{\dot{m}}{m}=-\alpha \frac{\Vert \mathbf{T}\Vert }{m}=-\alpha \Gamma$ — yeh bhi linear hai.

Thrust bounds, $m$ se divide karne par, ban jaate hain $\frac{T_{\min }}{m}\le \Vert \mathbf{a}\Vert \le \frac{T_{\max }}{m}$, aur hum slack $\Gamma$ introduce karte hain:

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

Step 2 — Lossless Convexification (genius move)

Bacha hua nonlinearity yeh hai: $\Gamma \ge \frac{T_{\min }}{m}=T_{\min }{e}^{-z}$. Hum exponential ko ek second-order Taylor form mein linearize karte hain jo phir bhi convex hai:

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

jahan $z_0(t)$ ek reference (no-thrust) trajectory par mass hai. Saare constraints ab convex hain (second-order cone / linear).

Step 3 — The Convex Program

Kaise solve karein: time ko $N$ nodes mein discretize karo → ek finite-dimensional SOCP → interior-point method se solve karo (ECOS, SDPT3). $t_f$ (flight time) par line search karo kyunki problem fixed $t_f$ ke liye convex hai; fuel-vs-$t_f$ curve unimodal hoti hai.

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

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

Q: Kya yeh ground se pehle ruk sakta hai? Available max deceleration $=12-3.71=8.29$ m/s² (thrust upar, gravity neeche). 20 m/s se rukne ki distance: $d=\frac{v^2}{2a}=\frac{400}{2(8.29)}=24.1$ m $<100$ m. ✔ Yeh step kyun? Hum "full thrust par shortest stopping distance" ko available altitude se compare karte hain — agar fit hota hai, toh fuel-optimal solution exist karta hai.

Fuel-optimal shape: minimum-fuel ⇒ $\Gamma$ mein bang-bang: coast (ya min-thrust) phir last moment par full-thrust brake. Kyun? Pehle burn karna propellant barbad karta hai kyunki gravity zyada der tak lad ti hai; optimizer burn delay karta hai — G-FOLD classic "suicide burn" reproduce karta hai.

Worked Example B — Glide-slope kyun matter karta hai

Target origin par; lander side se 30° above horizontal approach kar raha hai, lekin ek ridge 40° se neeche ke angles mein hai. ${\theta }_{\text{gs}}=45°$ set karo taaki $\Vert [r_x,r_y]\Vert \le \tan 45°(r_z)=r_z$.

$r_z=200$ m par lander 200 m horizontal radius ke andar hona chahiye. Yeh step kyun? Cone trajectory ko ek safe descent funnel ke upar rehne par majboor karta hai — ek convex constraint (cone convex hota hai), isliye yeh SOCP mein tractability ko bina koi nuksan pahunchaye seedha fit ho jaata hai.

Common Mistakes (Steel-man + Fix)

Flashcards

Connections

• Pontryagin Minimum Principle — bang-bang aur lossless result prove karta hai.
• Convex Optimization aur Second-Order Cone Programming — solver machinery.
• Tsiolkovsky Rocket Equation — $\dot{m}=-\alpha \Vert \mathbf{T}\Vert$ ka source.
• Interior-Point Methods (ECOS/SDPT3) — polynomial-time global solve.
• Apollo Lunar Descent Guidance — polynomial-thrust predecessor jise G-FOLD improve karta hai.
• Model Predictive Control — G-FOLD har cycle mein re-solve karna = landing ke liye MPC.
• SpaceX Falcon 9 Landing aur Mars EDL — real deployments.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps