Proportional navigation guidance — N·V_c·λ̇, derivation: Question bank

3.5.50 · D5 · Physics › Guidance, Navigation & Control (GNC) › Proportional navigation guidance — N·V_c·λ̇, derivation

Shuru karne se pehle core law yaad karo:

Sahi ya galat — justify karo

Agar PN, LOS rate ko zero kar deta hai, toh missile seedha target ki taraf point kar raha hai

Galat — ek constant LOS angle (zero

\dot{λ}

) kisi bhi bearing pe hold ho sakta hai; PN sirf sightline ki direction ko freeze karta hai, yeh missile ki velocity heading ko us direction pe force nahi karta. Dekho Pure Pursuit vs Proportional Navigation.

Ghatti hui range ke saath constant bearing collision guarantee karta hai

Sahi — zero

\dot{λ}

ka matlab hai zero transverse relative velocity (

V_{⊥} = R \dot{λ} = 0

), isliye closing motion seedha sightline ke neeche chalti hai aur

R \to 0

target ko hit karta hai.

PN law ko

a_{c}

compute karne ke liye target ki absolute position chahiye

Galat — isko sirf

V_{c}

aur

\dot{λ}

chahiye, dono relative measurables hain seeker se; PN ko kabhi world frame mein target ki location jaanna zaroori nahi.

N = 2

ke saath LOS rate phir bhi intercept par धीरे धीरे zero ho jaati hai

Galat — exponent

N - 2 = 0

deta hai

\dot{λ} \propto R^{0} =

constant, isliye

\dot{λ}

kabhi decay nahi karta;

N = 2

bilkul stability boundary pe baitha hai. Dekho Navigation Constant Selection and Actuator Limits.

Closing velocity ko double karna same

\dot{λ}

ke liye commanded acceleration ko double kar deta hai

Sahi —

a_{c} = N V_{c} \dot{λ}

V_{c}

mein linear hai, isliye ek faster closing engagement mein same LOS drift ko null karne ke liye proportionally zyada lateral acceleration chahiye.

Agar shuru mein

\dot{λ} = 0

ho, toh PN zero acceleration command karta hai aur missile coast karta hai

Sahi (agar target non-maneuvering ho) — koi LOS drift nahi hai toh correct karne ko kuch hai hi nahi, aur ODE

\dot{λ} = 0

rakhta hai; missile already collision course pe hai.

N

badhana hamesha missile ko bina kisi downside ke zyada accurately hit karata hai

Galat — zyada

N

intercept ke paas

\dot{λ}

ko tezi se null karta hai lekin pehle badi accelerations demand karta hai aur seeker noise ko amplify karta hai, isliye yeh actuators ko saturate kar sakta hai. Dekho Navigation Constant Selection and Actuator Limits.

Coriolis term

2 \dot{R} \dot{λ}

ko chod sakte hain kyunki yeh small hai

Galat — yeh negligible nahi hai; factor of 2 drop karne se stability condition sahi

N > 2

se galat

N > 1

ho jaati hai, jo change kar deta hai ki kaun se gains actually kaam karte hain.

Error dhundo

"Transverse relative acceleration hai

\dot{V}_{⊥} = \dot{R} \dot{λ} + R \ddot{λ}

Error —

\dot{V}_{⊥}

yeh ignore karta hai ki transverse unit vector

\hat{θ}

rotate karta hai; sahi value hai

R \ddot{λ} + 2 \dot{R} \dot{λ}

, rotating basis se ek aur

\dot{R} \dot{λ}

ke saath. Dekho Coriolis Term in Polar Coordinate Acceleration.

"Kyunki

V_{c} = - \dot{R}

aur hum close kar rahe hain,

\dot{R} > 0

Error — closing ka matlab hai range shrink ho rahi hai, isliye

\dot{R} < 0

aur issiliye

V_{c} = - \dot{R} > 0

; sign flip hi

V_{c}

define karne ka poora point hai. Dekho Closing Velocity and Range Rate.

"PN,

a_{c}

ko line of sight ke along set karta hai missile ko target ki taraf push karne ke liye."

Error —

a_{c}

LOS ke perpendicular (lateral) hai; ek purely radial push

\dot{λ}

nahi badal sakta, jo ki wo quantity hai jise control karne ke liye PN exist karta hai.

\ddot{λ} / \dot{λ} = (N - 2) \dot{R} / R

se integrate karke

\dot{λ} \propto R^{N}

milta hai."

Error —

ln \dot{λ} = (N - 2) ln R + C

integrate karne par

\dot{λ} \propto R^{N - 2}

milta hai,

R^{N}

nahi; exponent

N - 2

hai, jo exactly

N > 2

condition force karta hai.

"Non-maneuvering target ke liye transverse relative acceleration

+ a_{c}

ke barabar hai."

Error — missile ka lateral command transverse motion ko oppose karta hai, isliye

a_{⊥} = - a_{c}

; yahi sign hai jo feedback ko negative aur stabilizing banata hai.

"PN open-loop hai: yeh ek preset acceleration profile fire karta hai."

Error — PN ek closed-loop feedback hai jo continuously error signal

\dot{λ}

ko null karta hai;

a_{c}

har instant live LOS rate se recompute hota hai. Dekho Feedback Control — Nulling an Error Signal.

Why wale sawaal

PN,

\dot{λ}

(na ki

λ

khud) ko kyun null karta hai as error signal?

Kyunki ek constant LOS angle already collision course ka matlab hai; rotation

\dot{λ}

hi miss distance build karta hai, isliye rate ko — angle ko nahi — zero drive karna intercept guarantee karta hai.

Transverse velocity

R \dot{λ}

"miss-building" motion kaise measure karta hai?

Yeh relative velocity ka component hai sightline ke across; koi bhi nonzero crossing speed LOS ko swing karti hai aur, unchecked,

R

ke zero reach karne se pehle target ko collision line se le jaati hai.

PN, command ko

V_{c}

se kyun scale karta hai, given

\dot{λ}

ke liye fixed turn command karne ki jagah?

Kyunki same

\dot{λ}

higher closing speed pe ek faster-developing miss correspond karta hai aur correct karne ka time kam hota hai, isliye required acceleration genuinely

V_{c}

ke saath badhti hai — yeh law uss urgency ko encode karta hai.

Stability condition

N > 2

kyun aata hai,

N > 0

nahi?

Coriolis "2" LOS dynamics mein appear karta hai, isliye effective decay exponent

N - 2

hai; tabhi jab yeh zero se zyada ho,

\dot{λ} \propto R^{N - 2} \to 0

R \to 0

PN target ka path predict kiye bina automatically "lead" kyun karta hai?

\dot{λ}

null karke yeh steadily sightline jahan point kare udhar steer karta hai, jo hai hi future collision point; geometry prediction karta hai, koi explicit forecast zaroori nahi. Dekho Zero-Effort-Miss and Augmented Proportional Navigation.

Same

\dot{λ}

ek engagement mein bahut alag accelerations kyun demand kar sakta hai?

Kyunki

a_{c} = N V_{c} \dot{λ}

V_{c}

pe depend karta hai, jo geometry aur speed ke saath change karta hai; high

V_{c}

ke saath flight mein late mein ek modest LOS rate badi lateral

g

command kar sakta hai.

Edge cases

PN kya command karta hai jab

\dot{λ} > 0

ho aur missile close kar raha ho (

V_{c} > 0

a_{c} = N V_{c} \dot{λ} > 0

— increasing

λ

ki direction mein ek lateral acceleration, missile ko counterclockwise-drifting sightline pakadne ke liye turn karta hai aur drift cancel karta hai.

Agar range rate reverse ho jaaye toh

a_{c}

ka kya hota hai, matlab target open ho raha ho (

V_{c} < 0

a_{c}

ka sign flip ho jaata hai, isliye command direction reverse ho jaata hai; PN sirf closing geometry ke liye design kiya gaya hai, aur opening engagement ka matlab hai intercept fail ho raha hai.

Intercept ke paas

R \to 0

hone par term

2 \dot{R} \dot{λ} / R

kya karta hai, aur

a_{c}

infinite kyun nahi hota?

1/ R

LOS dynamics ko stiff banata hai, lekin kyunki

N > 2

\dot{λ} \to 0

faster hota hai than

R

shrinks, product

V_{c} \dot{λ}

bounded rehta hai aur ideal PN mein

a_{c}

blow up nahi karta.

Agar exactly

N = 3

ho aur range one-tenth ho jaaye toh LOS rate profile kya hai?

\dot{λ} \propto R^{N - 2} = R^{1}

, isliye

\dot{λ}

apne starting value ka one-tenth ho jaata hai — range mein linear decay, practical

N \in [3, 5]

band ka slowest.

\dot{λ} = 0

lekin

V_{c} = 0

(matched speeds, frozen bearing) guidance ke liye kya imply karta hai?

a_{c} = 0

, lekin koi closing nahi toh koi intercept bhi nahi — ek co-moving standoff; PN na correct karta hai na collide karta hai kyunki range kabhi shrink nahi hoti.

Agar target maneuver kare (nonzero transverse target acceleration), kya plain PN phir bhi

\dot{λ}

null karta hai?

Cleanly nahi — plain PN assume karta hai

a_{⊥, target} = 0

, isliye ek maneuver ek persistent LOS rate inject karta hai jiske liye augmented PN chahiye (target-acceleration term add karke). Dekho Zero-Effort-Miss and Augmented Proportional Navigation.

Recall Jaane se pehle self-check

PN jo error signal null karta hai ::: LOS rotation rate $\dot{λ}$ , LOS angle $λ$ nahi. $\dot{λ} \propto R^{?}$ mein exponent ::: $N - 2$ , isliye convergence ke liye $N > 2$ chahiye. Commanded acceleration ki direction ::: line of sight ke perpendicular (lateral). Jo factor stability ko $N > 2$ banata hai $N > 1$ nahi ::: Coriolis $2 \dot{R} \dot{λ}$ term.

Related: Line-of-Sight Geometry and Kinematics, Polar Coordinate Kinematics.