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

3.5.53 · D1 · Physics › Guidance, Navigation & Control (GNC) › Powered descent guidance — G-FOLD algorithm (convex optimiza

Parent note padhne se pehle, tumhe uska har letter apna banana hoga. Yeh page har ek ko ek picture se build karta hai. Upar se neeche padho — har symbol apne upar wale pe lean karta hai.

0. Stage: position, velocity, acceleration

Sab kuch ek lander ke saath hota hai jo ek target ke upar ud raha hai. Iske baare mein baat karne ke liye hume teen ideas chahiye jo stack hote hain.

Figure dekho. Arrow $\mathbf{r}$ ka ground pe ek shadow hai (lander kitna sideways hai) aur ek height hai (kitna upar hai). Dono ko ek saath likhne ke liye hum arrow ko teen fixed directions — axes — ke saath pieces mein tod dete hain.

Ab, cheezein chalti hain. Velocity yeh hai ki position kitni tezi se change hoti hai.

Dot ko phir stack karo: $\dot{\mathbf{v}}=\ddot{\mathbf{r}}$ acceleration hai — velocity kitni tezi se change ho rahi hai. Engine fire karne se velocity change hoti hai, toh acceleration wahi jagah hai jahan engine ki push dikhti hai.

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

Figure dekho: thrust arrow $\mathbf{T}$ ek right triangle ka diagonal hai jiske legs components $T_x$ aur $T_z$ hain. Dashed legs right angle pe milte hain, aur arrow ki length hypotenuse $\sqrt{T_x^2+T_z^2}$ hai — Pythagoras in action. Dhyan do ki arrow kisi bhi direction mein swing kar sakta hai jabki usi length ko rakhta hai; woh "same length, any direction" exactly wahi hai jo double-bar capture karta hai.

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

3. Fuel loss ki rate: $\dot{m}$, $v_e$, $I_{sp}$, $\alpha$

Fuel exhaust ke roop mein nikalta hai. Exhaust jitni tezi se aur jitna zyaada ho, push utni hi zyaada hogi.

4. Logarithm $\ln$ aur exponential $e^x$

Parent ka Step 1 mass ke log ko track karega mass ki jagah. Ise follow karne ke liye pehle tumhe jaanna hoga ki $\ln$ kya hai aur kya undo karta hai.

Figure dekho: $e^x$ (lavender) aur $\ln x$ (mint) dashed diagonal line ke across mirror images hain. Ek number ko ek curve mein daalo, phir doosre mein, aur tum wapas wahan pahunch jaate ho jahan shuru kiya tha — yahi "exact inverse" dikhta hai. Dhyan do ki mint $\ln$ curve sirf zero ke daayein rehti hai: $x\le 0$ ke liye iska koi output nahi hai, domain restriction $x>0$ ki picture hai.

5. Naye controls: $\mathbf{a}$ aur slack $\Gamma$

Yahaan algebra hai, step by step, taaki tum dekh sako kyun problem door ho jaati hai. Newton's law se shuru karo aur har term ko mass $m$ se divide karo:

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

Left side pe dono $m$ cancel ho jaate hain, sirf $\dot{\mathbf{v}}$ bachta hai. Right side pe, $\mathbf{T}/m$ hamara naya naam $\mathbf{a}$ hai, aur $m\mathbf{g}/m$ cancel hokar $\mathbf{g}$ ban jaata hai:

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

Ab throttle limits ko us divide se guzarte huye dekho. Hardware band se shuru karo aur teeno parts ko positive number $m$ se divide karo (positive number se divide karne par $\le$ signs same direction mein rehte hain):

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

Lekin $\Vert \mathbf{T}\Vert /m=\Vert \mathbf{T}/m\Vert =\Vert \mathbf{a}\Vert$ (ek arrow ko positive number se divide karne par uski length usi factor se scale hoti hai), toh:

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

Aakhir mein middle ko slack ke hawaale karo. Hum set karte hain $\Vert \mathbf{a}\Vert \le \Gamma$ aur hardware band ko $\Gamma$ pe daal dete hain:

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

Yahi transformed throttle box hai jo parent use karta hai. (Agar hum floor hataa dein, toh yeh simply $0\le \Gamma \le T_{\max }/m$ padha jaata.)

6. Convex vs. non-convex — woh shape jo sab kuch decide karti hai

Yahi heart hai kyun parent apni saari gymnastics karta hai.

7. Glide-slope cone aur $\tan {\theta }_{\text{gs}}$

Parent ka aakhri constraint lander ko ek safe funnel ke upar rakhta hai tangent use karke. (Section 1 se yaad karo ki $[r_x,r_y]$ do ground coordinates bundle karta hai, toh $\Vert [r_x,r_y]\Vert$ sideways distance hai — shadow ki length.)

Figure dekho: mint funnel allowed positions ka set hai. Uski walls ground se ${\theta }_{\text{gs}}$ angle banati hain. Kisi bhi height $h=r_z-r_{z,\text{land}}$ pe funnel ka radius $\tan {\theta }_{\text{gs}}\cdot h$ hai — toh allowed sideways room (coral segment) seedha zero tak simat jaata hai jaise lander pad ke paas aata hai, ek clean vertical touchdown force karta hai. Red dot pe lander funnel ke bahar constraint violate kar raha hoga (woh ridge mein fly kar sakta hai); green dot andar safe hai.

8. Bhaari machinery (woh names jo tumhe milenge, ek picture mein har ek)

Parent teen technical names drop karta hai. Tumhe unhe yahaan master karne ki zaroorat nahi — bas ek one-line mental picture rakho taaki baad mein woh magic na lagein.

Hum teeno ko is foundations page pe trusted black boxes maante hain; parent aur baad ke deep-dives unhe kholte hain.

Prerequisite map

Equipment checklist

Khud ko test karo — tum parent note ke liye ready ho sirf tab jab har reveal obvious lage.