Foundations — Optimal guidance — ZEM - ZEV formulation

3.5.52 · D1 · Physics › Guidance, Navigation & Control (GNC) › Optimal guidance — ZEM - ZEV formulation

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Parent note mein freely likha hai jaise $\mathbf{r}(t)$, $t_{go}$, $\int (t_f-\tau )\mathbf{a}d\tau$, aur "Lagrange multipliers." Agar yeh sab aapko magic runes lagte hain, toh aap derivation follow nahi kar sakte. Toh yahan hum har symbol ek ek karke kamate hain: plain words → ek picture → topic ko yeh kyun chahiye. Upar se neeche padho; har idea upar wale par lean karta hai.

1. Position, velocity, acceleration — teen moving arrows

Ek spacecraft ko space mein ek glowing dot samjho. Yeh describe karne ke liye ki woh kahan hai aur kaise move kar raha hai, humein teen quantities chahiye. Kyunki motion ek se zyada direction mein hoti hai (upar/neeche, left/right), har ek ek arrow hai — ek vector — sirf ek number nahi.

• $\mathbf{r}(t)$ — position arrow: ek chosen origin se vehicle ki location ki taraf point karta hai time $t$ par. Picture: arrow ki tip dot par baithti hai.
• $\mathbf{v}(t)$ — velocity arrow: position kitni fast aur kis direction mein change ho rahi hai. Picture: dot par sawaar ek arrow, jis taraf ja raha hai usi taraf point karta hai; lamba arrow = zyada fast.
• $\mathbf{a}(t)$ — acceleration arrow: velocity khud kaise change hoti hai. Yeh hamara steering control hai — thrust jo engine command karta hai. Picture: velocity arrow par apply kiya ek push.

Topic ko yeh kyun chahiye: guidance ek art hai $\mathbf{a}(t)$ choose karne ki taaki $\mathbf{r}$ aur $\mathbf{v}$ wahan pahunchen jahan hum chahte hain. Bina in teen arrows ke aap problem state hi nahi kar sakte.

2. Gravity $\mathbf{g}$ — woh free push jo humne maangi nahi

$\mathbf{g}$ gravitational acceleration arrow hai: ek constant pull jo vehicle feel karta hai chahe engine fire kare ya na kare. Moon par iska length $1.62{\text{m/s}}^2$ neeche ki taraf hai; Earth par $9.81{\text{m/s}}^2$.

3. Time symbols: $t$, $t_f$, aur $t_{go}$

Teen moments matter karte hain:

• $t$ — abhi, current instant.
• $t_f$ — final time, deadline jab tak hum pahunch jaane chahiye (landing touchdown, intercept moment).
• $t_{go}$ — time-to-go, kitne seconds baaki hain: $t_{go}=t_f-t.$

Topic ko yeh kyun chahiye: guidance law ke har coefficient ($6/t_{go}^2$, $2/t_{go}$) mein $t_{go}$ likha hai, aur "commands blow up as $t_{go}\to 0$" wala dramatic behaviour poori tarah is shrinking clock ke baare mein hai.

4. "Coasting" aur double integrator — free-fall prediction

Agar hum coast karein, toh standard schoolbook motion formulas apply hote hain. Yeh $\dot{\mathbf{v}}=\mathbf{g}$ ko do baar integrate karne se aate hain — ek system jise double integrator kehte hain (velocity accel ka ek integral hai, position do). Dekhein Double Integrator Dynamics.

${\mathbf{v}}_{coast}(t_f)=\mathbf{v}+\mathbf{g}{t}_{go},$ ${\mathbf{r}}_{coast}(t_f)=\mathbf{r}+\mathbf{v}{t}_{go}+\frac{1}{2}\mathbf{g}{t}_{go}^2.$

Topic ko yeh kyun chahiye: ZEM aur ZEV poori tarah coast prediction se define hote hain. Coast nahi → ZEM/ZEV nahi.

5. ZEM aur ZEV — do "agar main kuch na karun" errors

Ab star symbols. Hum chahte hain ki target position ${\mathbf{r}}_f$ aur (soft landing ke liye) target velocity ${\mathbf{v}}_f$ par pahunchen. Chaahi gayi cheez ko predicted-if-coasting se compare karo:

Coast formulas substitute karne par parent ke boxed expressions milte hain: $\mathbf{ZEM}={\mathbf{r}}_f-\mathbf{r}-\mathbf{v}{t}_{go}-\frac{1}{2}\mathbf{g}{t}_{go}^2,$ $\mathbf{ZEV}={\mathbf{v}}_f-\mathbf{v}-\mathbf{g}{t}_{go}.$

Topic ko yeh kyun chahiye: poora guidance law hai ${\mathbf{a}}_c=\frac{6}{t_{go}^2}\mathbf{ZEM}-\frac{2}{t_{go}}\mathbf{ZEV}$. Yeh do arrows hi inputs hain.

6. Integral sign $\int$ aur cost $J$

Guidance optimal isliye hai kyunki yeh control effort minimize karta hai: $J=\frac{1}{2}{\int }_t^{t_f}\Vert \mathbf{a}(\tau ){\Vert }^{2}d\tau .$

Topic ko yeh kyun chahiye: minimize karne ke liye koi cost na ho toh target hit karne ke infinitely many tarike hain. $J$ hi woh cheez hai jo ek answer ko the answer banata hai.

7. Lever-arm weight $(t_f-\tau )$

Parent ke Step 1 mein, position ko ek weighted integral milta hai $\int (t_f-\tau )\mathbf{a}d\tau$. Yeh weight kyun?

Topic ko yeh kyun chahiye: yahi weight hai jiske karan position constraint hai $\int (t_f-\tau )\mathbf{a}d\tau =\mathbf{ZEM}$ jabki velocity wala plain hai $\int \mathbf{a}d\tau =\mathbf{ZEV}$.

8. Lagrange multipliers ${\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2$

Is ke peeche ki heavy machinery — ki ek quadratic cost optimal ${\mathbf{a}}^{\ast }(\tau )$ ko time mein ek straight line hone par majboor karta hai — Calculus of Variations & Pontryagin's Minimum Principle se aati hai. ZEM/ZEV use karne ke liye aapko ise master karne ki zaroorat nahi; aapko bas yeh trust karna hai ki yeh profile ${\mathbf{a}}^{\ast }(\tau )={\mathbf{\lambda }}_1(t_f-\tau )+{\mathbf{\lambda }}_2$ deta hai.

Topic ko yeh kyun chahiye: multipliers hi "minimize $J$" se us clean $2\times 2$ system tak ka bridge hain jiska solution coefficients $6$ aur $2$ deta hai.

9. Dot product $\cdot$

Parent ke Lagrangian mein $\mathbf{\lambda }\cdot \mathbf{a}$ use hota hai.

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