Proportional navigation guidance — N·V_c·λ̇, derivation

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

KYA cheez guide ho rahi hai?

YEH shape kyun? Hum derive karenge ki missile ka lateral acceleration $V_{c} \dot{λ}$ ke proportional hona chahiye agar hum $\dot{λ} \to 0$ chahte hain. Neeche sab kuch geometry se build hota hai — kuch bhi assume nahi kiya gaya.

Collision-course idea (pehle intuition banao)

Figure — Proportional navigation guidance — N·V_c·λ̇, derivation

KAISE — first principles se derivation

Hum plane mein kaam karte hain. Missile ko origin par rakho aur target ko polar coordinates $(R, λ)$ se describe karo, missile ke relative.

Step 1 — LOS frame mein relative velocity components

Relative position vector (missile→target) ki length $R$ hai aur angle $λ$ hai. Iska time derivative do components deta hai:

LOS ke along \dot{R}, LOS ke across R \dot{λ}

Yeh step kyun? Polar coordinates mein koi bhi velocity ek radial part $\dot{R} \overset{r}{^}$ aur ek transverse part $R \dot{λ} \hat{θ}$ mein split hoti hai. Transverse relative speed exactly $V_{⊥} = R \dot{λ}$ hoti hai. Woh transverse component hi "miss-building" motion hai — wahi sightline ko rotate karati hai.

Step 2 — "Collision course par hona" ka matlab

Agar missile aur target perfect collision course par hain, toh LOS angle constant hai: $\dot{λ} = 0 ⟹ V_{⊥} = R \dot{λ} = 0.$

Yeh step kyun? Zero transverse relative velocity ke saath, target seedha sightline ke down close hota hai — ek guaranteed intercept (jab $R \to 0$ ). Toh guidance ka goal $\dot{λ} \to 0$ force karna hai.

Step 3 — LOS dynamics (Coriolis ke saath poori polar acceleration)

Hume dekhna hai ki $\dot{λ}$ khud kaise evolve karta hai. Polar coordinates mein relative acceleration ka transverse component sirf $R \ddot{λ}$ nahi hota — usme Coriolis term $2 \dot{R} \dot{λ}$ bhi hoti hai:

a_{⊥} = R \ddot{λ} + 2 \dot{R} \dot{λ} .

Yeh step kyun? Transverse velocity $V_{⊥} = R \dot{λ}$ ko ek baar differentiate karne par $\dot{V}_{⊥} = \dot{R} \dot{λ} + R \ddot{λ}$ milta hai, lekin $\dot{V}_{⊥}$ physical transverse acceleration nahi hai — transverse unit vector $\hat{θ}$ khud rotate ho raha hai, aur uski rotation ek doosra $\dot{R} \dot{λ}$ contribute karta hai. Dono jodne par standard polar result $a_{⊥} = R \ddot{λ} + 2 \dot{R} \dot{λ}$ milta hai. Factor of 2 miss karna classic error hai.

Ab $a_{⊥}$ ko net transverse acceleration (target minus missile) ke barabar set karo. Non-maneuvering target ke liye ( $a_{⊥, target} = 0$ ) aur missile lateral command $a_{c}$ jo LOS ke perpendicular hai, $a_{⊥} = - a_{c}$ :

R \ddot{λ} + 2 \dot{R} \dot{λ} = - a_{c} ⟹ \ddot{λ} = \frac{- a _{c} - 2 R ˙ λ ˙}{R} .

Yeh step kyun? Yeh correct LOS dynamics equation hai — yeh batata hai ki hamara command $a_{c}$ us quantity $\dot{λ}$ par kaise feedback karta hai jise hum khatam karna chahte hain.

Step 4 — $a_{c}$ choose karo taaki $\dot{λ}$ decay kare

PN law $a_{c} = N V_{c} \dot{λ} = - N \dot{R} \dot{λ}$ substitute karo (kyunki $V_{c} = - \dot{R}$ ):

\ddot{λ} = \frac{N R ˙ λ ˙ - 2 R ˙ λ ˙}{R} = \frac{R ˙ λ ˙ ( N - 2 )}{R} .

$\dot{λ}$ mein ek first-order ODE ki tarah rearrange karo:

\frac{λ ¨}{λ ˙} = (N - 2) \frac{R ˙}{R} .

Yeh step kyun? Dono sides integrate karne par:

ln \dot{λ} = (N - 2) ln R + const ⟹ \dot{λ} \propto R^{N - 2} .

Step 5 — Stability condition padhna

Jab intercept proceed hota hai, $R \to 0$ . Kyunki $\dot{λ} \propto R^{N - 2}$ :

N > 2 ⟹ \dot{λ} \to 0 as R \to 0. ✓

Yeh step kyun? $N > 2$ ke liye, exponent $N - 2 > 0$ hai, toh jab range collapse hoti hai LOS rate force hokar zero ho jaati hai — sightline exactly tab ghoomna band ho jaati hai jab hum target tak pahunchte hain. Yeh Step 2 ka collision course hai. Practice mein $N = 3 - 5$ (sab safely $> 2$ ) fast $\dot{λ}$ nulling ko actuator effort aur noise ke saath balance karta hai. Yeh loop close karta hai: $N V_{c} \dot{λ}$ ki form guess nahi hai — yeh woh choice hai jo $\dot{λ}$ ko collapse karati hai.

Worked examples

Common mistakes (steel-manned)

Recall Feynman: ise ek 12-saal ke bachche ko explain karo

Imagine karo tum ek bike par ho aur apne dost ko bump karna chahte ho jo bhi bike par hai. Apne dost ko ghoor ke seedha unki taraf steer mat karo — tum hamesha ek step peeche rahoge. Balki dekho ki tumhara dost background ke against kahan baith raha hai (ek ped, ek fence). Agar tumhara dost background ke against sideways slide karta rehta hai, toh tum miss karne wale ho — toh steer karo taaki woh slide karna band kar de. Jab tumhara dost background par same spot par glued reh jaata hai aur bas bada hota jaata hai... bonk! — tum milte ho. Ek guided missile exactly yahi karta hai: woh zyada tez turn karta hai jab uska target sky ke against faster slide karta hai, aur jitna woh turn karta hai woh $N$ times hota hai us sky-slide ki rate ( $\dot{λ}$ ) aur gap kitna tez close ho raha hai ( $V_{c}$ ) ka.

Active recall

Constant LOS angle with decreasing range kya imply karta hai?

Ek collision course — target seedha sightline ke down close hota hai, intercept guaranteed hai.

Proportional navigation guidance law batao.

a_{c} = N V_{c} \dot{λ}

(commanded lateral accel = nav constant × closing velocity × LOS rate).

R

aur

\dot{λ}

ke terms mein transverse relative velocity kya hai?

V_{⊥} = R \dot{λ}

Polar form mein relative acceleration ka full transverse component kya hai?

a_{⊥} = R \ddot{λ} + 2 \dot{R} \dot{λ}

(Coriolis term

2 \dot{R} \dot{λ}

hai).

Correct LOS dynamics se shuru karke PN

\dot{λ}

ke liye kya proportionality produce karta hai?

\dot{λ} \propto R^{N - 2}

N

par kya condition hai jo

\dot{λ} \to 0

banati hai jab

R \to 0

N > 2

Navigation constant

N

ka typical practical range?

3 \leq N \leq 5

(sab safely 2 se bade).

Threshold

N > 2

kyun hai,

N > 1

kyun nahi?

Kyunki transverse dynamics mein Coriolis term

2 \dot{R} \dot{λ}

included hai; factor of 2 exponent ko

N - 2

shift kar deta hai.

Huge

N

kyun use nahi karte?

Yeh

\dot{λ}

mein measurement noise amplify karta hai aur actuators ko early saturate karta hai, LOS-rate nulling tez hone ke bawajood.

Closing velocity

V_{c}

aur range rate

\dot{R}

ka relation?

V_{c} = - \dot{R}

(positive jab range decrease ho).

Near intercept par command

a_{c}

zero kyun fade ho jaata hai (for

N > 2

Kyunki

\dot{λ} \to 0

, aur

a_{c} \propto \dot{λ}

PN pure pursuit se alag kaise hai?

PN LOS rotation null karta hai (future collision point par aim karta hai / target ko lead karta hai); pure pursuit current target position par seedha point karta hai aur lag karta hai.

Connections

Line-of-Sight Geometry and Kinematics
Coriolis Term in Polar Coordinate Acceleration
Pure Pursuit vs Proportional Navigation
Closing Velocity and Range Rate
Navigation Constant Selection and Actuator Limits
Zero-Effort-Miss and Augmented Proportional Navigation
Feedback Control — Nulling an Error Signal
Polar Coordinate Kinematics

Proportional navigation guidance — N·V_c·λ̇, derivation

KYA cheez guide ho rahi hai?

Collision-course idea (pehle intuition banao)

KAISE — first principles se derivation

Step 1 — LOS frame mein relative velocity components

Step 2 — "Collision course par hona" ka matlab

Step 3 — LOS dynamics (Coriolis ke saath poori polar acceleration)

Step 4 — $a_{c}$ choose karo taaki $\dot{λ}$ decay kare

Step 5 — Stability condition padhna

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Deep Dive