Augmented proportional navigation — gravity compensation

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

80/20 core: Pure PN sirf line-of-sight (LOS) rotation pe steer karta hai. Lekin gravity missile ki flight ko continuously bend karti hai, ek spurious LOS rate add karti hai jise PN "chase" karta hai — acceleration waste hoti hai aur miss ho jaata hai. Augmented PN (APN) with gravity compensation bas ek aisa term add karta hai jo jaani-hui gravitational acceleration ko cancel kare, taaki guidance law sirf asli target maneuver pe react kare.

YE NOTE KYUN HAI

KYA: Ek command term add karo jo LOS ke perpendicular act karne wali gravity component ke equal aur opposite ho (woh part jo trajectory ko curve karta hai).
KYUN: Taaki effective closed-loop system ek gravity-free PN engagement ki tarah behave kare → zero-effort-miss sirf target maneuver se driven ho.
KAISE: Standard PN lo, target-maneuver term add karo (yahi "augmentation" hai), phir ek gravity-cancelling term add karo.

Law ko first principles se banana

Ye form kyun? Agar $\dot{λ} = 0$ hai, toh LOS direction inertial space mein fixed hai → yahi constant-bearing, decreasing-range collision condition hai. PN $\dot{λ} \to 0$ drive karta hai, intercept guarantee karta hai.

Step 2 — Gravity ise kahan todti hai?

Missile ki acceleration ko commanded + environmental likho: $a_{missile} = a_{c} + g$

LOS-rate dynamics (linearized, planar) LOS ke perpendicular acceleration se govern hoti hain. Maano $n$ LOS ke normal direction ko denote karta hai. Tab: $\dot{λ} evolves under (a_{c, n} + g_{n} - a_{T, n})$ jahaan $a_{T, n}$ target acceleration $⊥$ LOS hai aur $g_{n}$ gravity ka projection $⊥$ LOS hai.

Step 3 — Target maneuver ke liye Augment karo

Agar target known/estimated $a_{T}$ se accelerate karta hai, toh woh term add karo jo zero-effort-miss ko zero pe rakhti hai: $a_{c} = N^{'} V_{c} \dot{λ} + \frac{N ^{'}}{2} a_{T}$

$\frac{N ^{'}}{2}$ kyun? Linear ZEM (zero-effort-miss) equation solve karo. Time-to-go $t_{g o}$ pe ek constant target acceleration ke liye, miss $\frac{1}{2} a_{T} t_{g o}^{2}$ integrate hoti hai. Ise forward feed karne ke liye exactly cancel karne ke liye $N^{'} /2$ factor chahiye. (Yahi standard APN hai.)

Step 4 — Gravity compensation add karo

Ab gravity ko ek aur jaani-hui acceleration maano jise hum $\dot{λ}$ corrupt nahi karne de sakte. Hum ek extra lateral term command karte hain jo LOS ke perpendicular gravity component ko cancel kare:

Minus sign kyun hai aur $g_{n}$ kyun (poori $g$ nahi)?

Poori $g$ ka ek component LOS ke saath hai (closing speed change karta hai, $\dot{λ}$ ke liye harmless) aur ek component perpendicular hai ( $g_{n}$ , asli culprit). Sirf $g_{n}$ LOS ko bend karta hai, isliye sirf $g_{n}$ ko cancel karna hai.
Hum ise subtract karte hain kyunki hum chahte hain $a_{c, n} + g_{n} = (PN term)$ , yaani commanded normal accel ko extra $- g_{n}$ supply karna hai taaki gravity $+ g_{n}$ wapas add karne ke baad, net perpendicular acceleration exactly wahi ho jo pure PN chahta hai.

Practice mein $g_{n}$ kaise compute karo: Agar LOS elevation angle $λ$ hai (horizontal se) aur gravity $g = - g \overset{z}{^}$ hai, tab $g_{n} = g cos λ (vertical plane mein LOS ke ⊥ component) .$ Jab LOS horizontal hai ( $λ = 0$ ), $g_{n} = g$ (worst case — poori gravity LOS ko bend karti hai). Jab seedha upar/neeche shoot kar rahe ho ( $λ = 90°$ ), $g_{n} = 0$ (gravity LOS ke saath hai, harmless).

Figure — Augmented proportional navigation — gravity compensation

Worked examples

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek chhalta hua toy car pe dart phenk rahe ho. Dart udne ke dauran thoda girta hai. Agar tum seedha car ke abhi wale position par aim karte raho, toh girta hua dart hamesha neeche land karta hai. Ek smart dart-thrower girne ko cancel karne ke liye thoda upar aim karta hai, taaki sirf car ki movement ki chinta rehe. Gravity compensation iska matlab hai missile apne aap "thoda upar" aim karna — use exactly pata hota hai gravity use kitna neeche khichegi aur opposite push add karta hai, taaki uska brain 100% asli target par focus kar sake.

Flashcards

Pure PN kaun si LOS quantity ko zero drive karne ki koshish karta hai?

Line-of-sight angular rate

\dot{λ}

(constant-bearing collision condition).

Uncompensated gravity PN ko kyun hurt karti hai?

Iska perpendicular component

g_{n}

continuously trajectory ko bend karta hai, ek persistent nonzero

\dot{λ}

create karta hai jise PN wastefully chase karta hai → steady miss aur wasted g-capacity.

Full APN-with-gravity-compensation command likho.

a_{c} = N^{'} V_{c} \dot{λ} + \frac{N ^{'}}{2} a_{T} - g_{n}

Sirf

g_{n}

kyun, poori

g

kyun nahi?

Sirf LOS-perpendicular component LOS ko bend karta hai; LOS-parallel component sirf closing speed change karta hai aur

\dot{λ}

affect nahi karta.

Vertical plane mein LOS elevation

λ

ke liye

g_{n}

kya hai?

g_{n} = g cos λ

(max =

g

jab horizontal,

0

jab vertical).

Target-maneuver term par

N^{'} /2

factor kyun hai?

Kyunki ek constant target accel apni miss

\frac{1}{2} a_{T} t_{g o}^{2}

integrate karti hai;

N^{'} /2

feed-forward exactly is zero-effort-miss ko cancel karta hai.

Hum

g_{n}

SUBTRACT kyun karte hain?

Taaki commanded + actual gravity (

a_{c, n} + g_{n}

) pure-PN demand ke equal ho; hum droop ko baad mein correct karne ki bajaye pre-empt karte hain.

Seedha-upar shot mein kitna gravity compensation chahiye?

Zero — gravity LOS ke saath hai (

g_{n} = g cos 90° = 0

Connections

Proportional Navigation (PN) — woh base law jise yeh augment karta hai.
Zero-Effort-Miss (ZEM) guidance — $N^{'} /2$ factor ka derivation source.
Line-of-Sight rate estimation — $\dot{λ}$ kaise measure hota hai (seeker gimbal / Kalman filter).
Missile Autopilot & Acceleration limits — gravity chase na karke g-capacity bachana kyun matter karta hai.
Coordinate frames & projections — onboard $g_{n} = g \cdot \overset{n}{^}$ kaise compute hota hai.
Ballistic trajectory & gravity turn — same droop physics, uncontrolled case.