Foundations — Autonomous GNC for reusable rockets — SpaceX approach overview

3.5.55 · D1 · Physics › Guidance, Navigation & Control (GNC) › Autonomous GNC for reusable rockets — SpaceX approach overvi

Is page mein kuch bhi assumed nahi hai. Agar parent note ne koi symbol likha, toh hum use yahan se ground up build karte hain, uss order mein jahan har idea sirf pehle waale ideas pe rely karta hai.

1. Position $\mathbf{r}$ — "main kahan hoon?"

Picture. Neeche figure mein blue arrow dekho. Uski tail pad pe (origin pe) hai, uska head rocket pe. Yahan do numbers ise describe karte hain: kitna sideways ($x$) aur kitna upar ($y$). Hum likhte hain $\mathbf{r}=(x,y)$. Real flight mein teen numbers hote hain ($x,y,z$) lekin do se sab kuch samajh aata hai.

Topic ko yeh kyun chahiye. Guidance poochti hai "main kahan rehna chahta hoon?" aur Navigation poochti hai "main kahan hoon?" — dono questions $\mathbf{r}$ ke baare mein hain. Barge pe aim karna impossible hai bina space mein points ko naam diye.

2. Velocity $\mathbf{v}$ — "kitni tez, kis direction mein?"

Picture. Figure mein, rocket se attached orange arrow $\mathbf{v}$ hai. Agar rocket seedha neeche girta hai, toh $\mathbf{v}$ neeche point karta hai. Descent ke dauran $v_0$ (parent ka symbol) bas is arrow ki length hai landing burn ke start mein — girne ki speed, jaise $200$ m/s.

Yeh tool (derivative) kyun? Hum instantaneous speed chahte hain, na ki "poori flight ka average." Derivative $\frac{d}{dt}$ exactly woh question answer karta hai "yeh abhi is waqt kitni tez change ho raha hai?" — wahi speed matter karti hai jab tum ground se 3 second door ho.

3. Acceleration $\mathbf{a}$ aur gravity $g$ — "speed kitni tez change ho rahi hai?"

$\frac{d^2\mathbf{r}}{d{t}^2}$ mein chhote $2$ ka matlab hai "rate-of-change do baar lo": position → velocity → acceleration. Yeh stacked symbol exactly wahi hai jo parent ka $\ddot{\mathbf{r}}$ kehta hai — do dots shorthand hai "time ke saath do baar differentiate karo."

Topic ko $\mathbf{a}$ kyun chahiye. Engine ka ek hi kaam hai acceleration produce karna. IMU (ek sensor) seedha acceleration measure karta hai. Aur poora hoverslam ek story hai sahi upward acceleration choose karne ki jo downward velocity cancel kare. Acceleration wahan hai jahan rocket ka control actually kaata hai.

4. Angles $\theta$ aur tan/arctan tool — "main kitna tila hua hoon?"

Rocket ke paas sirf position nahi hoti — uski ek orientation bhi hoti hai: kya naak upar hai, tila hua hai, sideways hai? Sabse simple angle question hai: diya gaya kitna sideways aur kitna upar kuch pahunchta hai, toh uska tilt kya hai?

Yeh tool kyun aur koi nahi? Hamare paas do lengths hain (sideways aur upar) aur hum ek angle chahte hain. Tangent precisely woh function hai jo length-ratio ko steepness mein badalta hai, aur $\arctan$ use angle tak wapas le jaata hai. Sine/cosine ko bhi ek length chahiye; tangent ko sirf ratio chahiye, jo ki tilt hai hi.

5. Integration $\int$ — "saare tiny pieces add karo"

Parent acceleration se position recover karta hai $\int$ likhke. Ab yeh symbol earn karte hain.

Picture. Figure mein, blue curve time ke saath velocity hai. Tay ki gayi distance curve ke neeche shaded area hai — thin rectangles ka sum. Yeh slices-ka-summing wahi hai jo ${\int }_0^t\mathbf{v}dt$ kehta hai: time $0$ se time $t$ tak contributions add karo.

Topic ko yeh kyun chahiye. Acceleration woh hai jo sensor feel karta hai; position woh hai jo guidance ko chahiye. Integration ek se doosre tak ek hi bridge hai. Yeh drift problem bhi explain karta hai: acceleration mein ek tiny constant error $b$, do baar integrate hone ke baad $\frac{1}{2}b{t}^2$ ki tarah badata hai — pehle chhota, ek minute baad enormous. Yeh $t^2$ growth wahi reason hai ki GPS ko fuse karna padta hai.

6. Feedback loop — "measure karo, compare karo, correct karo, repeat"

Topic ko yeh kyun chahiye. Wind, exact mass, aur engine performance pehle se kabhi pata nahi hoti. Ek open-loop plan (blindly ek script replay karo) ke paas notice karne ka koi tarika nahi ki woh barge se drift ho raha hai. Sirf ek closed loop — upar wala $e$-measuring cycle — surprises absorb kar sakta hai. Parent ke PID law ka har letter ($K_{p}e+K_d\dot{e}+K_i\int edt$) is page ke pieces se bana hai: $e$ (yeh section), $\dot{e}$ (ek derivative, §2), $\int edt$ (ek integral, §5).

Foundations topic ko kaise feed karte hain

Har foundation aage ek named prerequisite se link karta hai jo tumhe baad ke pages mein milega: dead-reckoning Kalman Filter ko feed karta hai; error loop PID Control ko feed karta hai; angles Attitude Dynamics & Quaternions aur Thrust Vectoring (Gimbal) ko feed karte hain; $v^2$ braking law Kinematics — v²=v0²+2as hai; fuel-optimal layer Convex Optimization hai Lossless Convexification ke zariye; aur mass-vs-thrust story Rocket Equation (Tsiolkovsky) hai. Saari raahein parent tak jaati hain, Autonomous GNC for reusable rockets — SpaceX approach overview.

Equipment checklist

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