Proportional navigation guidance — N·V_c·λ̇, derivation: Foundations

3.5.50 · D1 · Physics › Guidance, Navigation & Control (GNC) › Proportional navigation guidance — N·V_c·λ̇, derivation

Is page par kuch bhi assume nahi kiya gaya. PN derivation padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumpar fire karta hai. Hum unhe ek-ek karke build karte hain, har ek ek picture se anchor kiya gaya hai, har ek agli ke liye deserving hai.

0 — Do players aur unke beech ki line

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

Figure dekho: missile corner par baitha hai, target plane mein bahar baitha hai, aur unhe join karne wali burnt-orange line LOS hai. Jab target move karta hai, woh line do cheezein ek saath karti hai — woh chhoti/badi ho sakti hai aur ghadi ki sui ki tarah swing kar sakti hai. Poora topic in do motions ko alag karne aur control karne ke baare mein hai.

1 — $R$ : target kitna door hai (range)

Figure mein orange line dekho: $R$ simply batata hai woh kitni lambi hai. Kuch aur nahi.

2 — Dot: rate of change (derivatives par pehli nazar)

Agle symbols se pehle, hume woh chhoti si dot earn karni hogi jo unpar baithi hai.

$\dot{R} > 0$ → range badh raha hai → target bhaag raha hai.
$\dot{R} < 0$ → range shrink ho raha hai → hum close ho rahe hain.
$\dot{R} = 0$ → range is instant ke liye frozen hai.

Do dots, jaise $\ddot{λ}$ , matlab rate-of-change of the rate-of-change (ek acceleration-jaisi quantity). Hume baad mein woh chahiye hogi; abhi ke liye bas jaano ki second dot = "pehli-dot wali quantity kitni tez khud change ho rahi hai."

(Agar "derivative" word naya hai, yeh dot hi time ke saath derivative hai — poori machinery ke liye Polar Coordinate Kinematics dekho.)

3 — $V_{c}$ : closing velocity (kyun hum $\dot{R}$ ka sign palatete hain)

Situation	$\dot{R}$	$V_{c} = - \dot{R}$
Tezi se close ho raha hai	$- 900$	$+ 900$ (closing)
Door ja raha hai (miss)	$+ 400$	$- 400$ (closing nahi)
Range frozen	$0$	$0$

Intercept timing ke liye $V_{c}$ (raw speed nahi) kyun matter karta hai, Closing Velocity and Range Rate dekho.

4 — $λ$ : line-of-sight angle (aur iska sign convention)

Figure mein, dashed grey line fixed reference hai (space mein pin ki gayi ek direction, missile se attached nahi). Us dashed line se orange LOS tak ka angle $λ$ hai. Radians mein measure kiya (ek poora chakkar = $2 π$ radians).

5 — $\dot{λ}$ : LOS rotation rate (show ka star)

Figure do nearby instants par LOS dikhata hai (solid orange, phir faded orange). Unke beech ka chhota wedge, counter-clockwise sweep kiya, elapsed time se divide kiya, ek positive $\dot{λ}$ hai. Ise "apni window ke paas target ka drift" samjho.

Yahi "glass par target pinned vs drifting" picture hai jo parent note kholti hai — glass par drift hi $\dot{λ} \neq = 0$ hai.

6 — Motion ko "along" aur "across" mein split karna (unit vectors aur signs)

Missile aur target ke beech koi bhi relative motion do perpendicular pieces mein tod ja sakta hai, aur yeh split poori derivation ka engine hai. Signs track karne ke liye (sirf sizes nahi) hume un pieces ki directions name karni hogi.

Figure mein target ki velocity $V$ (plum arrow) teal $\overset{r}{^}$ axis par decompose ki gayi hai ( $\dot{R} \overset{r}{^}$ deta hai) aur teal $\hat{θ}$ axis par ( $R \dot{λ} \hat{θ}$ deta hai). Ab signs unambiguous hain: $+ \hat{θ}$ ke saath ek component positive $R \dot{λ}$ hai, hence positive $\dot{λ}$ .

7 — Coriolis ka surprise (kyun factor of 2 aata hai)

Parent derivation ek non-obvious fact use karta hai jise tumhe abhi milna chahiye taaki baad mein shock na lage.

Key setup, jise misleading shortcuts hamesha skip karte hain: hum ek vector differentiate kar rahe hain jiska basis rotate karta hai. §6 se yaad karo ki $(\overset{r}{^}, \hat{θ})$ ek rotating frame hai jo $\dot{λ}$ par spin karta hai. Jab ek unit vector ghoomta hai, uski apni time-derivative zero nahi hoti — aur woh do derivatives hi hain jahan se cross-terms aate hain. Hamare CCW frame ke liye standard rules hain:

\dot{\overset{r}{^}} = \dot{λ} \hat{θ}, \dot{\hat{θ}} = - \dot{λ} \overset{r}{^} .

Inhe physically padho: jab LOS ek tiny angle CCW rotate karta hai, $\overset{r}{^}$ ki tip $+ \hat{θ}$ direction mein sideways move karti hai (hence $\dot{\overset{r}{^}} = \dot{λ} \hat{θ}$ ); aur $\hat{θ}$ , $9 0^{\circ}$ aage hone ki wajah se, $- \overset{r}{^}$ ki taraf tip karta hai (hence $\dot{\hat{θ}} = - \dot{λ} \overset{r}{^}$ ). Yeh factor of 2 ka engine hain.

Figure exactly yeh do unit-vector derivatives dikhata hai: teal $\overset{r}{^}$ arrow orange $+ \hat{θ}$ direction mein nudge karta hai (woh nudge niche Contribution A hai), aur teal $\hat{θ}$ arrow plum $- \overset{r}{^}$ direction mein tip karta hai — $\dot{R}$ se fed hone par uska $+ \hat{θ}$ -pointing consequence Contribution B hai.

Intuition 2 ki sahi polar-coordinate derivation

§6 se rotating frame mein likhi relative velocity se shuru karo: $V = \dot{R} \overset{r}{^} + R \dot{λ} \hat{θ} .$ Term by term product rule se differentiate karo, aur kabhi mat bhulo ki $\overset{r}{^}$ aur $\hat{θ}$ khud bhi change ho rahe hain:

a = \frac{d V}{d t} = (\ddot{R} \overset{r}{^} + \dot{R} \dot{\overset{r}{^}}) + (\dot{R} \dot{λ} \hat{θ} + R \ddot{λ} \hat{θ} + R \dot{λ} \dot{\hat{θ}}) .

Ab $\dot{\overset{r}{^}} = \dot{λ} \hat{θ}$ aur $\dot{\hat{θ}} = - \dot{λ} \overset{r}{^}$ substitute karo:

a = (\ddot{R} \overset{r}{^} + \dot{R} \dot{λ} \hat{θ}) + (\dot{R} \dot{λ} \hat{θ} + R \ddot{λ} \hat{θ} - R \dot{λ}^{2} \overset{r}{^}) .

$\hat{θ}$ ki taraf point karne wale pieces (sideways wale) collect karo:

a_{⊥} = A: \dot{\overset{r}{^}} se \dot{R} \dot{λ} + B: V ke \dot{R} term se \dot{R} \dot{λ} + angle speeds up R \ddot{λ} = R \ddot{λ} + 2 \dot{R} \dot{λ} .

Contribution A radial unit vector $\overset{r}{^}$ ke ghoomne ka derivative hai ( $\dot{R} \dot{\overset{r}{^}} = \dot{R} \dot{λ} \hat{θ}$ ). Contribution B transverse velocity term ke andar radial coefficient ka product-rule derivative hai ( $\dot{R} \dot{λ} \hat{θ}$ ). Do alag, honestly-different $\dot{R} \dot{λ}$ 's — yahi factor of 2 hai. (Bacha hua $- R \dot{λ}^{2} \overset{r}{^}$ centripetal term hai; yeh $\overset{r}{^}$ ke saath point karta hai, toh $a_{⊥}$ mein enter nahi karta.)

8 — $a_{c}$ aur $N$ : command, uski direction, aur gain

Kyunki $a_{c}$ , $\dot{λ}$ ke saath $\hat{θ}$ axis share karta hai, PN law ke signs ab cleanly padte hain: agar $V_{c} > 0$ (closing) aur $\dot{λ} > 0$ (LOS CCW rotate ho rahi hai), toh $a_{c} = N V_{c} \dot{λ} > 0$ , yaani missile CCW accelerate karta hai sightline ki rotation ko pakadne aur null karne ke liye. $\dot{λ}$ ko negative karo aur $a_{c}$ bhi flip ho jaata hai — hamesha us direction mein jo drift hatata hai.

Assembled, poora vocabulary woh law produce karta hai jo tum abhi derive karne wale ho: $a_{c} = N V_{c} \dot{λ} (gain) \times (closing speed) \times (LOS drift) .$

Prerequisite map

Top-to-bottom padho: humble overdot aur range/angle do rates feed karte hain; CCW-positive angle hume wrapping ke baare mein bhi warn karta hai; rates closing velocity aur transverse split feed karte hain; woh plus Coriolis correction aur gain $N$ PN law assemble karte hain, jiska purpose $\dot{λ}$ ko zero karna hai.

Equipment checklist

Right side cover karo aur zor se jawab do. Agar koi stumps kare, parent note kholne se pehle us section ko dobara padho.

Ek symbol par overdot ka matlab kya hota hai?

"Rate of change per second" — time derivative (jaise

\dot{R}

= range har second kitna change hota hai).

Line of sight (LOS) kya hai?

Missile se target tak seedhi khayal ki line.

R

kya measure karta hai, aur

R = 0

ka matlab kya hai?

LOS ki length (range);

R = 0

matlab hit.

V_{c} = - \dot{R}

kyun hai (minus kyun)?

Kyunki closing ka matlab hai

R

decrease ho raha hai (

\dot{R} < 0

); minus closing velocity ko positive number banata hai.

λ

kahan se measure hota hai, aur kaunsi direction positive hai?

ek fixed (inertial) reference direction se, counter-clockwise = positive measure kiya jaata hai.

\dot{λ} > 0

physically kya matlab hai?

Sightline counter-clockwise rotate ho rahi hai (hamare CCW-positive convention ke saath).

Angle wrap par kya galat hota hai, aur fix?

Jab

λ

reference cross karta hai toh woh jump karta hai (jaise

2 π \to 0

), ek huge false

\dot{λ}

deta hai; fix: differentiate karne se pehle angle unwrap karo.

§6 mein

V

exactly kya hai?

Target ki velocity missile ke relative, inertial (non-rotating) frame mein padhi gayi.

Do LOS unit vectors aur unki directions batao.

\overset{r}{^}

missile→target point karta hai;

\hat{θ}

hai

\overset{r}{^}

9 0^{\circ}

CCW rotate kiya (increasing

λ

ki direction).

(\overset{r}{^}, \hat{θ})

frame inertial hai ya rotating?

Rotating — yeh LOS ke saath

\dot{λ}

rate par spin karta hai, yahi wajah hai ki unit-vector derivatives appear hote hain.

Relative velocity split directions ke saath likho.

V = \dot{R} \overset{r}{^} + R \dot{λ} \hat{θ}

Rotating unit vectors ke derivatives kya hain?

\dot{\overset{r}{^}} = \dot{λ} \hat{θ}

aur

\dot{\hat{θ}} = - \dot{λ} \overset{r}{^}

Transverse speed

R \dot{λ}

kyun hai, sirf

\dot{λ}

kyun nahi?

\dot{λ}

ek angular rate hai; radius

R

se multiply karna spin ko sideways speed mein convert karta hai.

Poori transverse acceleration kya hai, aur do

\dot{R} \dot{λ}

kahan se aate hain?

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

; ek

\dot{R} \dot{\overset{r}{^}}

se (radial unit vector ka ghoomna), ek transverse velocity term mein radial coefficient ke product-rule derivative se.

Positive

a_{c}

kis taraf point karta hai?

+ \hat{θ}

ki taraf (CCW, increasing

λ

ki direction).

N

kya hai aur iska typical range kya hai?

Dimensionless navigation gain, usually

3 \leq N \leq 5

PN law words mein batao.

Commanded acceleration = gain

\times

closing velocity

\times

LOS rotation rate,

a_{c} = N V_{c} \dot{λ}

Recall Self-test: woh single quantity kaunsi hai jise PN zero tak drive karne ki koshish karta hai, aur kyun?

$\dot{λ}$ , LOS rotation rate — kyunki frozen sightline plus shrinking range exactly ek collision course hai.