Visual walkthrough — Proportional navigation guidance — N·V_c·λ̇, derivation

3.5.50 · D2 · Physics › Guidance, Navigation & Control (GNC) › Proportional navigation guidance — N·V_c·λ̇, derivation

Hum ek pursuer (missile) aur ek target (plane) draw karenge. Poori kahaani ek line ke baare mein hai jo unhe join karti hai — line of sight — aur yeh hai ki kya woh line rotate karti hai ya nahi.

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Step 1 — Do dots aur unke beech ki line

KYA: humne "ek missile aur ek plane" ko sirf do numbers, $R$ aur $\lambda$, se replace kar diya.

KYUN: target ko hit karne ke baare mein har sawaal actually yeh sawaal hai ki yeh do numbers time ke saath kaise change hoti hain. Agar hum $R$ aur $\lambda$ track karein, toh hum sab kuch track karte hain.

PICTURE: figure mein, blue dot missile hai, orange dot target hai, gray line LOS hai. Angle $\lambda$ dashed reference line se LOS tak measure hota hai. Kisi symbol ke upar chhota dot ka matlab hai "change per second ka rate" — toh $\dot{\lambda }$ hai gray line kitni tezi se swing karti hai, aur $\dot{R}$ hai do dots kitni tezi se paas ya door ho rahe hain.

Full frame setup ke liye dekho Line-of-Sight Geometry and Kinematics.

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Step 2 — Target ki motion ko do arrows mein todna

KYA: humne ek velocity arrow ko ek red "along" arrow aur ek green "across" arrow mein toda.

KYUN: hum specifically inhi do directions ki parwah karte hain kyunki har ek hamare do numbers mein se ek ko control karta hai. Along part $R$ change karta hai; across part $\lambda$ change karta hai. Koi bhi velocity is tarah likhi ja sakti hai — yeh sirf perpendicular axes choose karna hai jo LOS ke saath ride karte hain, fixed East/North axes ki jagah. Yeh exactly Polar Coordinate Kinematics hai.

Term by term, across-LOS piece mein:

• $\dot{\lambda }$ = radians/second mein line kitni tezi se ghoomti hai.
• $R$ = line ki length. Ek rod ke end par ek point jo $\dot{\lambda }$ rad/s par ghoom raha hai woh sideways (length)×(turn rate) = $R\dot{\lambda }$ metres/second ki speed se move karta hai. Isliye across-speed $R\dot{\lambda }$ hai, na ki sirf $\dot{\lambda }$.

Hum across-speed ko $V_{\perp }=R\dot{\lambda }$ naam dete hain.

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Step 3 — "Collision course par hona" kaisa dikhta hai

KYA: humne pucha "dots kab collide karte hain?" aur jawaab mila $V_{\perp }=0$.

KYUN: collision matlab gap $R$ zero ho jaata hai bina line ke apni bearing se slide kiye. Sliding bearing ($\dot{\lambda }\ne 0$) matlab target tumhare aage ya peeche cross kar raha hai — ek miss.

PICTURE: left panel — green across-arrow nonzero hai, toh LOS har frame mein ek naye angle tak sweep karta hai (target "glass par drift karta hai"): miss. Right panel — across-arrow gaya, sari motion red/along hai, sightlines parallel aur stacked rehti hain: collision.

Yeh error signal hai jo hum null karna chahte hain: hamara error $\dot{\lambda }$ khud hai.

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Step 4 — $\dot{\lambda }$ khud kitni tezi se change hota hai? (chhupa hua factor of 2)

$\dot{\lambda }$ ko zero drive karne ke liye hum jaanna chahte hain ki $\dot{\lambda }$ ek push ke response mein kaise react karta hai. Iska matlab hai across-velocity $V_{\perp }=R\dot{\lambda }$ ko differentiate karna — lekin carefully.

KYA: picture dikhata hai kyun ek term missing hai. Green "across" direction $\hat{\theta }$ khud rotate ho raha hai jab LOS swing karta hai. Ek instant se doosre tak, across-arrow thoda alag direction mein point karta hai.

KYUN extra term: radial motion $\dot{R}$ velocity carry karta hai, aur kyunki across-direction $\dot{\lambda }dt$ se rotate hua hai, us radial velocity ka ek hissa across-direction mein leak ho jaata hai. Woh leak ek doosra $\dot{R}\dot{\lambda }$ add karta hai. Yeh leaked term Coriolis term hai — dekho Coriolis Term in Polar Coordinate Acceleration.

PICTURE: figure mein do curved arrows mein se har ek $\dot{R}\dot{\lambda }$ ki ek copy represent karta hai; saath mein woh $2$ banaate hain.

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Step 5 — Missile ka command wire karna

Missile apne thrusters LOS ke sideways fire karta hai. Woh commanded acceleration $a_c$ hai. Ek target ke liye jo khud across-LOS maneuver nahi kar raha, sirf across-acceleration missile ki apni hai, drift ko oppose karne ki taraf:

KYA: humne physical across-acceleration $a_{\perp }$ ko $-a_c$ ke barabar set kiya (minus kyunki command growing $\lambda$ ke against push karta hai).

KYUN: yeh woh moment hai jab control input geometry mein enter karta hai. Is line se pehle, $a_{\perp }$ pure kinematics tha; ab hum keh rahe hain "missile ise supply karta hai."

PICTURE: orange arrow $a_c$ hai jo missile ko uski current heading se push kar raha hai, LOS ko wapas fixed bearing ki taraf bend kar raha hai.

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Step 6 — PN law plug in karo aur $\dot{\lambda }$ ko collapse hote dekho

Ab PN law $a_c=N{V}_c\dot{\lambda }$ substitute karo. Kyunki $V_c=-\dot{R}$ (dekho Closing Velocity and Range Rate), woh hai $a_c=-N\dot{R}\dot{\lambda }$.

$$\ddot{\lambda }=\frac{-(-N\dot{R}\dot{\lambda })-2\dot{R}\dot{\lambda }}{R}=\frac{\dot{R}\dot{\lambda }(N-2)}{R}$$

KYA: command term aur Coriolis term ek single clean factor $(N-2)$ mein merge ho gaye.

KYUN: $\dot{\lambda }$ se divide karne par yeh ratios ki kahaani ban jaati hai:

$$\frac{\ddot{\lambda }}{\dot{\lambda }}=(N-2)\frac{\dot{R}}{R}.$$

Har side "change ki rate ÷ current value" hai — exactly woh shape jiska integral logarithm hai. Isliye agle step mein logarithm aata hai: yeh "growth proportional to current size" ka natural tool hai.

PICTURE: kai $N$ ke liye $\dot{\lambda }$ versus $R$ ke curves, sab right par same start se pinned, right-to-left padhe jaate hain jab engagement run hoti hai ($R$ 0 ki taraf shrink ho raha hai).

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Step 7 — Stability condition graph se padho (sab cases)

Jab missile pahunchta hai, $R\to 0$. $\dot{\lambda }\propto R^{N-2}$ ka kya hota hai yeh poori tarah exponent $N-2$ ke sign par depend karta hai:

Har dikhaya gaya case:

• $N>2$ (exponent positive): $R^{N-2}\to 0$ jab $R\to 0$. Swing-rate zero ho jaata hai — collision course achieve! ✓ (green curve zero tak dive karti hai)
• $N=2$ (exponent zero): $R^0=1$, toh $\dot{\lambda }$ constant rehta hai — kabhi decay nahi karta. Boundary par, koi fayda nahi. (gray flat line)
• $N<2$ (exponent negative): $R^{N-2}\to \infty$ jab $R\to 0$. Swing-rate blow up ho jaata hai — miss target ke paas badhta hai. ✗ (red curve shoot up karti hai)

KYUN yeh sab kuch close karta hai: Step 3 mein humne kaha tha "kaam hai $\dot{\lambda }\to 0$ force karna." Steps 4–7 dikhate hain ki specific form $N{V}_c\dot{\lambda }$ — $N>2$ ke saath — exactly wahi automatically karta hai, geometry khud se drive hokar. Yeh law ek guess nahi hai; yeh woh choice hai jo graph ko dive karaati hai.

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Ek-picture summary

Yeh single panel poori chain stack karta hai: do dots → motion split karo → collision matlab green across-arrow khatam ho jaata hai → command $a_c$ se $\dot{\lambda }$ tak loop → power law → $N>2$ ke liye zero tak dive.

Recall Feynman retelling — plain words mein bolo

Socho ek plane tumhari side window ke bahar hai. Us invisible line ko draw karo jo tumse uss tak jaati hai. Us line ke saath do cheezein ho sakti hain: woh chhoti ho sakti hai (tum close in kar rahe ho) ya woh tumhari window ke across swing kar sakti hai (plane aage ya peeche drift kar raha hai). Agar line sirf chhoti hoti hai aur kabhi swing nahi karti, toh tum dono ek hi spot ki taraf ja rahe ho — ek crash. Toh kisi cheez ko hit karne ki poori trick yeh hai: swing ko khatam karo.

Swing ko khatam karne ke liye tumhe jaanna hoga ki sideways nudge ise kaise change karta hai. Jab tum woh calculate karte ho toh tumhe ek sneaky extra term milta hai — Coriolis wala, do ka factor — kyunki "sideways" direction khud ghoom raha hai jab tum measure kar rahe ho. Woh miss karo aur tumhara poora jawab galat hoga.

Phir tum apne missile ko swing-rate ke proportion mein sideways push karaate ho, is scale par ki tum kitni tezi se close kar rahe ho, times ek number $N$. Algebra karo aur swing-rate remaining distance ke saath ek power, $N-2$, se tied ho jaata hai. Jab tum last few metres close karte ho, agar woh power positive hai (matlab $N$ 2 se bada hai), swing bilkul zero ho jaata hai jab tum pohuchte ho — ek clean hit. Agar $N$ 2 hota toh swing coast karta rehta; 2 se kam aur woh tumhare muh par explode ho jaata. Isliye har real missile $N$ 3 aur 5 ke beech use karta hai.

Recall Quick self-check

$R\dot{\lambda }$ physically kya hai? ::: Across-LOS relative speed — kitni tezi se sightline sideways sweep karti hai; yeh miss-building motion hai. Across-acceleration mein 2 ka factor kyun hai? ::: Ek $\dot{R}\dot{\lambda }$ product rule se, ek aur isliye kyunki transverse basis vector khud rotate karta hai (Coriolis). $N>2$ kyun zaroori hai? ::: Kyunki $\dot{\lambda }\propto R^{N-2}$; sirf positive exponent $N-2>0$ hi $\dot{\lambda }\to 0$ bhejta hai jab $R\to 0$. PN kaunsa error signal null karta hai? ::: LOS rotation rate $\dot{\lambda }$.