Augmented proportional navigation — gravity compensation: Question bank

3.5.51 · D5 · Physics › Guidance, Navigation & Control (GNC) › Augmented proportional navigation — gravity compensation

Stage set karna — pehle har symbol earn karo

Kisi bhi trap se pehle, yahan woh picture hai jisme poora page jeeta hai. Do cheezein ek vertical plane mein move karti hain: missile aur target. Missile se target tak ka seedha arrow line of sight (LOS) hai — socho jaise unhe ek tight dhaga jod raha ho.

Figure — Augmented proportional navigation — gravity compensation

Ab woh second-order dynamics build karo jo "cancel $a_{n}$ " ko kuch matlab dete hain.

Formula Kyun acceleration

$⊥$ LOS, $\dot{λ}$ ko control karta hai — differential equation Polar (LOS) coordinates mein sightline ke perpendicular acceleration classic transverse term hai $a_{n} = r \ddot{λ} + 2 \overset{r}{˙} \dot{λ},$ jahan $r$ range hai aur $a_{n}$ net perpendicular acceleration hai (missile minus target). Sightline ke angular acceleration ke liye rearrange karo: $\ddot{λ} = \frac{a _{n} - 2 r ˙ λ ˙}{r} = \frac{a _{n}}{r} + 2 \frac{V _{c}}{r} \dot{λ},$ $\overset{r}{˙} = - V_{c}$ use karke. Ise $\dot{λ}$ ke liye ek driven equation ki tarah padho: term $2 \frac{V _{c}}{r} \dot{λ}$ (closing ke dauran positive) destabilising hai, isliye ek nonzero $\dot{λ}$ badhta rehta hai jab tak $a_{n}$ push back na kare. Net perpendicular accel $a_{n} = a_{c, n} + g_{n} - a_{T, n}$ substitute karo aur PN command $a_{c, n} = N^{'} V_{c} \dot{λ}$ choose karo: loop phir $\dot{λ} \to 0$ force karta hai (yeh standard result hai jo Proportional Navigation (PN) mein quoted hai). Toh $a_{n}$ ko pure-PN value tak drive karna exactly wahi hai jo $\dot{λ}$ ko collapse karta hai — aur $a_{n}$ ke andar koi bhi uncancelled $g_{n}$ ya $a_{T, n}$ $\dot{λ}$ ko kabhi settle nahi hone deta.

Formula Kyun half-factor

$\frac{N ^{'}}{2}$ hai — quoted nahi, shown Zero-effort-miss (ZEM) woh jagah hai jahan target miss hoga agar dono sides abhi steering band kar dein. Constant target accel $a_{T}$ ke liye jo baaki $t_{g o}$ time tak act kare, kinematics (distance $= \frac{1}{2} a t^{2}$ ) woh lateral miss deta hai jo build hoti hai: $ZEM_{target} = \frac{1}{2} a_{T} t_{g o}^{2} .$ PN ka apna feedback kisi bhi ZEM ka ek fraction $N^{'}$ se set rate par remove karta hai; kisi maneuver-driven ZEM ko fully null karne ke liye feed-forward command yeh honi chahiye: $a_{c, aug} = \frac{N ^{'}}{r _{g o}} \cdot ZEM = \frac{N ^{'}}{V _{c}^{2} t _{g o}^{2}} \cdot \frac{1}{2} a_{T} t_{g o}^{2} \cdot V_{c}^{2} = \frac{N ^{'}}{2} a_{T},$ jahan miss se $t_{g o}^{2}$ command gain mein chhupe $t_{g o}^{2}$ ko cancel karta hai, clean $\frac{N ^{'}}{2}$ chodkar. Toh $\frac{1}{2}$ freshman kinematics ka $\frac{1}{2} a t^{2}$ hai jo cancellation ke baad survive karta hai (poori derivation Zero-Effort-Miss (ZEM) guidance mein).

Neeche use hone wale har symbol ka quick refresher:

$\dot{λ}$ = LOS angular rate; $\ddot{λ}$ = uski rate of change; $λ$ = LOS elevation angle (horizontal se upar).
$\overset{n}{^}$ = LOS-normal direction; $g_{n} = ∣ g \cdot \overset{n}{^} ∣ = g cos λ$ = perpendicular gravity ka magnitude.
$a_{c, n} = a_{c} \cdot \overset{n}{^}$ = commanded accel $\overset{n}{^}$ par project ki gayi; $a_{T, n} = a_{T} \cdot \overset{n}{^}$ = target accel $\overset{n}{^}$ par project ki gayi.
$N^{'}$ = navigation constant; $V_{c} = - \overset{r}{˙}$ = closing speed; $a_{T}$ = target accel $⊥$ LOS; $t_{g o}$ = time-to-go.
Net perpendicular dynamics: $a_{n} = r \ddot{λ} + 2 \overset{r}{˙} \dot{λ}$ , with $a_{n} = a_{c, n} + g_{n} - a_{T, n}$ .
Full law: $a_{c} = N^{'} V_{c} \dot{λ} + \frac{N ^{'}}{2} a_{T} - g_{n}$ .

True or false — justify karo

Commanded acceleration

a_{c}

mein gravity ka effect already include hai, isliye humein

- g_{n}

term ki zaroorat nahi.

False —

a_{c}

sirf wahi hai jo autopilot command karta hai; gravity uske upar se act karti hai. Physical normal accel

a_{c, n} + g_{n}

hai, isliye us sum ko pure-PN demand

N^{'} V_{c} \dot{λ}

ke barabar banane ke liye

g_{n}

pre-subtract karna zaroori hai.

Gravity compensation missile ko straight line mein fly karata hai.

False — yeh effective guidance loop ko aise behave karata hai jaise gravity-free ho; missile physically phir bhi droops karta hai, lekin guidance us droop ko ab target motion ki tarah nahi padhti.

Agar target kabhi maneuver na kare, augmentation term

\frac{N ^{'}}{2} a_{T}

vanish ho jaata hai lekin

- g_{n}

term generally nahi hota.

True —

a_{T} = 0

augmentation ko khatam karta hai, phir bhi gravity LOS ko bend karti rehti hai jab tak shot vertical na ho, isliye

g_{n}

compensation abhi bhi required hai.

Ek poore ballistic arc mein gravity average out ho jaati hai, isliye homing ke dauran ise ignore karne ka koi cost nahi.

False — terminal homing ke dauran missile actively close ho raha hai; downward

g_{n}

ka continuous effect ek persistent

\dot{λ}

produce karta hai jo ek steady miss mein accumulate hoti hai, yeh average out nahi hoti.

Bada navigation constant

N^{'}

target-maneuver term ko increase karta hai lekin gravity term ko untouched chodta hai.

True — gravity term

- g_{n}

hai,

N^{'}

se independent; sirf PN aur augmentation terms

N^{'}

ke saath scale karti hain.

Compensation reactive hai: miss aane ke baad droop fix karta hai.

False — yeh predictive hai; advance mein

- g_{n}

supply karke woh

\dot{λ} \equiv 0

hold karta hai taaki droop-induced error kabhi appear hi na ho.

Purely vertical (seedha-upar) engagement ke liye gravity-compensation term zero hai.

True — gravity phir poori tarah LOS ke saath (

∥ \overset{r}{^}

) lie karti hai, isliye

g_{n} = g cos 9 0^{\circ} = 0

aur kuch bhi perpendicular cancel karne ki zaroorat nahi.

Gravity ka LOS ke saath wala component guidance ke liye harmless hai.

True (

\dot{λ}

ke liye) —

g \cdot \overset{r}{^}

sirf closing speed

V_{c} = - \overset{r}{˙}

ko change karta hai; yeh sightline ko rotate nahi karta, isliye woh LOS-rate corrupt nahi karta jis par PN steer karta hai.

Transverse equation

a_{n} = r \ddot{λ} + 2 \overset{r}{˙} \dot{λ}

mein,

g_{n}

a_{n}

ke andar chodne par bhi

\dot{λ}

zero par settle ho sakta hai.

False — ek uncancelled

g_{n}

ek constant forcing term ki tarah kaam karta hai, isliye

\ddot{λ}

(hence

\dot{λ}

) kabhi steady zero nahi pahunchta; yahi woh persistent rate hai jise PN wastefully chase karta hai.

Error dhundo

"Set karo

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

Wrong sign — gravity subtract honi chahiye.

g_{n}

add karna droop ko cancel karne ki jagah double kar deta, kyunki real gravity already physical normal accel mein

+ g_{n}

contribute karti hai.

"Poore

g

se compensate karo: likho

a_{c} = N^{'} V_{c} \dot{λ} - g

Over-correction — sirf LOS-normal part

g_{n} = g cos λ

hi sightline ko bend karta hai. Poora

g

subtract karna ek phantom perpendicular pull se ladna hoga jo wahan hai hi nahi aur control authority waste karega.

"Kyunki

g \cdot \overset{n}{^} = - g cos λ

negative hai, compensation term plain dot product se

+ g cos λ

honi chahiye."

Half-right, half-trap — dot product is negative hai kyunki

g

\overset{n}{^}

ke tilt ke opposite point karta hai, lekin

g_{n}

projection ke magnitude

g cos λ

ke roop mein define hai, aur command gravity ki downward direction explicit minus sign

- g_{n}

se carry karta hai; dono conventions ko mix karna sign ko double-count karta hai.

"LOS elevation

λ

par,

g_{n} = g sin λ

use karo."

Wrong trig function — is convention mein (

λ

horizontal se upar measure kiya gaya)

g_{n} = g cos λ

: horizontal LOS (

λ = 0

) maximum

g_{n} = g

deta hai kyunki

cos 0 = 1

, jabki

sin 0 = 0

galti se kehta ki koi compensation chahiye hi nahi.

"Kyunki abhi

\dot{λ} = 0

hai, bilkul koi command nahi chahiye, gravity term bhi nahi."

Error — inertial target aur momentarily fixed LOS ke saath, pure PN

0

deta hai, lekin gravity turant ek droop create kar deti; compensation phir bhi

- g_{n}

command karta hai

\dot{λ}

ko zero par rakhne ke liye.

"APN ka

\frac{N ^{'}}{2} a_{T}

gravity ko bhi handle karta hai, kyunki gravity bhi sirf ek aur acceleration hai."

Error — woh term target ke known maneuver ko zero-effort-miss ke zariye feed-forward karta hai; gravity ek self acceleration hai jo missile par act karti hai aur apne alag

- g_{n}

term ki zaroorat hai.

\frac{N ^{'}}{2} a_{T}

mein

\frac{1}{2}

upar-neeche gravity ko average karne se aaya hai."

Error — yeh target maneuver ke

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

kinematic ZEM se aaya hai;

t_{g o}^{2}

command gain ke against cancel hota hai aur

\frac{1}{2}

chodta hai, gravity se koi lena-dena nahi.

t_{g o}

ko poore engagement mein ek fixed constant ki tarah treat kiya ja sakta hai."

Error —

t_{g o} \approx r / V_{c}

impact par zero tak shrink hota hai; ise fixed maan lena kisi bhi ZEM-based feed-forward ko galat scale karega jab range collapse ho.

Why questions

Sirf gravity ka perpendicular component LOS rate ko kyun affect karta hai?

Kyunki

a_{n} = r \ddot{λ} + 2 \overset{r}{˙} \dot{λ}

mein sirf transverse accel

a_{n}

\ddot{λ}

drive karta hai;

\overset{r}{^}

ke along koi force radial equation mein jaati hai aur sirf range ko lengthens/shortens karti hai, sightline ko kabhi rotate nahi karti.

Gravity compensation ke liye horizontal LOS worst case kyun hai?

λ = 0

ke saath poora gravity vector LOS ke perpendicular hai, isliye

g_{n} = g cos 0 = g

— sightline ka maximum possible bending.

Uncompensated gravity missile ki g-capacity kyun waste karti hai?

PN continuously lateral accel command karta hai gravity-induced

\dot{λ}

ko null karne ke liye, isliye precious acceleration budget real target ki jagah ek ghost se ladne mein kharch hota hai.

g_{n}

ko engagement badhne ke saath recompute kyun karna chahiye, ek baar fix karne ki jagah?

LOS elevation

λ

poori flight mein change hoti hai, isliye

g_{n} = g cos λ

bhi change hota hai; ek stale value geometry rotate hone par over- ya under-compensate karti.

Gravity ko sirf uski vertical value use karne ki jagah

\overset{n}{^}

par projection ke zariye kyun express karte hain?

Kyunki guidance LOS frame

(\overset{r}{^}, \overset{n}{^})

mein kaam karta hai; sirf LOS-normal direction

\overset{n}{^}

par projection hi

\dot{λ}

dynamics mein enter karti hai — dekho Coordinate frames & projections.

Half-factor

t_{g o}

se free ek clean number ke roop mein kyun survive karta hai?

Kyunki target miss

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

ki tarah grow karta hai jabki ZEM-to-command gain ek compensating

1/ t_{g o}^{2}

carry karta hai; dono

t_{g o}^{2}

cancel ho jaate hain, sirf

\frac{N ^{'}}{2} a_{T}

chodkar.

Loop ko "effectively gravity-free" banana non-maneuvering target par intercept guarantee kyun karta hai?

Kyunki gravity-free frame mein pure PN

\dot{λ} \to 0

drive karta hai, jo constant-bearing, decreasing-range collision condition hai — sightline rotate karna band kar deti hai isliye paths meet karte hain.

Edge cases

LOS elevation exactly

9 0^{\circ}

(target seedha upar, straight-up shot).

g_{n} = g cos 9 0^{\circ} = 0

; koi compensation term act nahi karta, lekin along-LOS gravity phir bhi closing speed

V_{c}

slow karti hai —

\dot{λ}

ke liye harmless lekin energy ke liye real.

Closing velocity

V_{c} \to 0

(missile ab close nahi ho raha,

\overset{r}{˙} \to 0

PN term

N^{'} V_{c} \dot{λ}

collapse ho jaata hai aur

t_{g o} \approx r / V_{c} \to \infty

, isliye command

\frac{N ^{'}}{2} a_{T} - g_{n}

se dominate hoti hai; geometrically yeh ek degenerate near-parallel engagement hai jahan PN apni intercept guarantee khota hai. Agar

V_{c} < 0

toh range badh rahi hai — overshoot/opening pass.

Target maneuver unknown / unestimated (

a_{T}

0

set kiya gaya jabki target turn kar raha hai).

Law PN plus gravity compensation tak reduce ho jaata hai; yeh true maneuver se lag karega aur ek residual miss chodega, kyunki feed-forward sirf wahi cancel kar sakta hai jo estimation provide kare.

\dot{λ} = 0

inertial target aur horizontal LOS ke saath.

Pure PN aur augmentation dono

0

dete hain, phir bhi

g_{n} = g

; command exactly

- g

hai, woh maximum predictive push jo true collision course ko drooping se rokta hai.

Descending LOS jahan target horizontal se neeche ho (

λ < 0

Kyunki

cos

even function hai,

g_{n} = g cos λ = g cos ∣ λ ∣

—

λ

ka sign compensation ka magnitude nahi badlata, sirf geometry badlati hai; ek shallow depression (

∣ λ ∣

chhota) abhi bhi nearly full

g

chahata hai, ek steep dive (

∣ λ ∣ \to 9 0^{\circ}

) almost kuch nahi chahata.

LOS elevation

9 0^{\circ}

se zyada (

λ \in (9 0^{\circ}, 18 0^{\circ})

, target above-and-behind).

cos λ

negative ho jaata hai, isliye projection

g cos λ

\overset{n}{^}

ke saath direction flip karta hai;

- g_{n}

convention ke zariye carry karne par compensation push automatically sign reverse karta hai — geometry cosine handle karta hai, haath se nahi.

Bahut kam time-to-go (

t_{g o} \to 0

impact par).

ZEM-based augmentation

\frac{N ^{'}}{2} a_{T}

finite rehta hai jabki

\dot{λ}

spike kar sakta hai; vanishing interval mein gravity ka contribution negligible ho jaata hai, isliye

- g_{n}

term bilkul end par sabse kam matter karta hai.