Foundations — Augmented proportional navigation — gravity compensation

3.5.51 · D1 · Physics › Guidance, Navigation & Control (GNC) › Augmented proportional navigation — gravity compensation

Yeh page yeh maanta hai ki aapko kuch nahi pata. Hum parent note ke har letter, arrow, aur symbol ko ek ek karke build karenge, har ek apni jagah banayega phir agla aayega.

0 — Scene: do chalte hue dots aur unke beech ek string

Kisi bhi symbol se pehle, physical situation ko picture mein socho.

Do objects hain: missile (hamara chaser) aur target (jise hum chase karte hain). Unke beech ek khayal ki elastic string stretch karo. Woh string hi Line of Sight (LOS) hai — literally woh direction jisme missile target ko "dekhta" hai.

Baaki sab kuch isi ek picture par bana hai: string, woh kitni lambi hai, aur — sabse zaroori — woh kis taraf point karti hai.

1 — Angle $\lambda$: string kis taraf point karti hai

Ek flat plane mein ek direction ko ek single number se capture kiya ja sakta hai: ek angle se. Hum LOS direction ko horizontal ground line se upar ke angle ke roop mein measure karte hain.

Angle kyun, aur, say, coordinates kyun nahi? Kyunki poora khel string ke ghoomne ke baare mein hai. Ek rotation angle mein badlaav hota hai. "String ghoom rahi hai" kehne ke liye, pehle humein "woh kis taraf point karti hai" ka ek number chahiye — woh number hai $\lambda$.

• Agar target seedha horizon par aage hai, toh $\lambda =0$.
• Agar missile seedha upar shoot karta hai, toh $\lambda =90°$.
• $\lambda$ koi bhi value ho sakti hai; woh picture mein horizontal dashed line aur red LOS ke beech ke wedge ke roop mein dikhti hai.

2 — Upar dot: $\dot{\lambda }$, string kitni tezi se ghoomti hai

Ab show ka star. Kisi symbol ke upar ek chhota dot rakho aur iska matlab hai "woh cheez har second kitni badlti hai." Yeh dot notation time derivative kehlati hai — yeh sawaal ka jawaab deti hai "kitni tezi se?"

Hume sirf angle ki nahi, ek rate ki zaroorat kyun hai? Yahan poore chapter ki central insight hai, ek picture mein dikhaya gaya hai:

Do cases dekho:

• Left (string ghoom rahi hai, $\dot{\lambda }\ne 0$): target missile ki nazar mein side mein drift kar raha hai. Woh miss kar denge.
• Right (string same direction mein rehti hai, $\dot{\lambda }=0$): target missile ki nazar mein still rehta hai chahe gap kitna bhi kam ho. Woh takraenge.

Toh $\dot{\lambda }$ woh EK number hai jis par ek guided missile obsess karta hai. Isse zero karo → intercept.

3 — Speeds: closing velocity $V_c$

Do objects ek doosre ki taraf aate hue apne beech ka gap kuch speed se kam karte hain. Hum use naam dete hain.

Hum specifically closing speed ki kyun parwah karte hain (missile ki ground speed ki nahi)? Kyunki guidance impact tak baccha hua time ke baare mein hai. Zyada tezi se close hona matlab errors theek karne ka waqt kam hota hai, isliye kisi bhi correction ki urgency $V_c$ ke saath scale hoti hai. Yahi wajah hai ki yeh PN law mein LOS rate ke saath multiply hota hai jo aap aage miloge.

4 — Acceleration: letter $a$ aur upar arrow

Speed batati hai tum kitni tezi se chalte ho; acceleration batati hai tumhari velocity kitni tezi se badlti hai — yaani tumhe kitna sideway ya aage dhakka lag raha hai.

Ek missile khud ko sideways dhakka dekar steer karta hai — apni velocity arrow ko bina necessarily speed badlaye ghuma leta hai. Woh sideway dhakka ek lateral (side-to-side) acceleration hai.

Hum acceleration ke teen flavours milenge, har ek apna subscript leke:

5 — Letters par Arrows: vectors aur $\vec{g}$

Kuch quantities ko sirf ek size chahiye (ek "kitna" number) — woh scalars hain, jaise $V_c$. Doosron ko size aur direction chahiye — woh vectors hain, aur hum unke upar ek chhota arrow draw karte hain.

Direction ko size se alag karne ki zahmat kyun? Kyunki gravity ki takleef poori tarah uski direction string ke relative par depend karti hai. Wahi neeche ka pull ek geometry mein bahut zyada hurt karta hai aur doosri mein bilkul nahi — aur sirf vectors hi humein yeh precisely kehne dete hain.

6 — Vector ko todna: "string ke along" vs "string ke across"

Yahan woh mathematical tool hai jis par poori gravity-compensation idea tiki hai: projection. Koi bhi arrow do arrows mein tod a ja sakta hai jo right angles par hoon — ek chosen direction ke saath, ek uske aakad. Un dono ko wapis jodne se original milta hai.

Ab gravity $\vec{g}$ ko LOS ke against split karo:

• LOS ke along — yeh part sirf string ko lambi ya chhoti karata hai ($V_c$ badlata hai). Yeh string ko rotate nahi karta. Harmless.
• LOS ke across ($\perp$) — yeh part string ko sideways swing karta hai, toh yeh ek $\dot{\lambda }$ create karta hai. Yeh guilty component hai.

"$\cdot$" (dot product) kyun? Dot product precisely woh tool hai jo jawaab deta hai "is arrow ka kitna hissa us direction mein hai?" Ise $\vec{g}$ aur perpendicular direction $\hat{n}$ dene par gravity ka exactly across-the-string slice milta hai — kuch aur nahi.

Vertical plane mein, geometry woh clean result deti hai jo parent use karta hai: $g_n=g\cos \lambda$

Picture se do extremes padho:

• Horizontal LOS ($\lambda =0$): $g_n=g\cos 0=g$. Gravity ka poora hissa string ke across hai — worst case.
• Vertical LOS ($\lambda =90°$): $g_n=g\cos 90°=0$. Gravity string ke along point karti hai — bilkul harmless.

7 — Navigation constant $N^{\prime }$

Ek aur symbol. Jab computer string ko ghoomte dekhta hai, use kitni aggressively wapas dhakka dena chahiye? Woh gain ek chosen number hai.

Specifically 3–5 window kyun? Bahut chhota ho toh missile sluggishly react karta hai aur miss karta hai; bahut bada ho toh yeh over-react karta hai, apni steering saturate karta hai aur seeker noise amplify karta hai. 3–5 range woh sweet spot hai jahan control theory aur decades ki practice dono settle hoti hain.

Alphabet ko ek saath rakhna

Ab parent note ke boxed law ke har symbol ke malik tum ho: $a_c=N^{\prime }{V}_c\dot{\lambda }+\frac{N^{\prime }}{2}{a}_T-g_n$

Left se right plain words mein padho: "woh sideways push jo main command karta hun = (meri aggressiveness) × (approach speed) × (string kitni tezi se ghoomti hai), plus target ki apni manoeuvre ka ek hissa, minus gravity ka woh part jo string ko spin karta." Yahan koi bhi symbol ab anjaan nahi hai.

Yeh map parent topic mein bottom-to-top padhta hai: string ek angle deti hai, angle ek rate deta hai, rate collision rule deta hai; speed, acceleration, aur projected gravity term sab milke final command feed karte hain.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle har ek ka jawaab do.

Connections

• Augmented proportional navigation — gravity compensation — woh parent topic jise yeh foundations unlock karte hain.
• Proportional Navigation (PN) — woh base law jo directly $\dot{\lambda }$, $V_c$, $N^{\prime }$ par bana hai.
• Line-of-Sight rate estimation — $\dot{\lambda }$ actually kaise measure kiya jaata hai.
• Coordinate frames & projections — $g_n$ ke liye projection/dot-product machinery.
• Ballistic trajectory & gravity turn — $\vec{g}$ pehli jagah flight path ko kyun droop karta hai.
• Zero-Effort-Miss (ZEM) guidance — $N^{\prime }/2$ target term kahan se aata hai.