Exercises — Augmented proportional navigation — gravity compensation

• $N^{\prime }{V}_c\dot{\lambda }$ — yeh pure Proportional Navigation (PN) term hai. Yeh LOS rotation par react karta hai aur $\dot{\lambda }\to 0$ drive karta hai (constant-bearing collision).
• $\frac{N^{\prime }}{2}{a}_T$ — yeh augmentation term hai. Yeh target ki apni maneuver ko feed-forward karta hai taaki missile ko wait na karna pade ki woh maneuver LOS rotation ke roop mein dikhaye. Iska origin Zero-Effort-Miss (ZEM) guidance integral hai.
• $-g_n$ — yeh gravity-compensation term hai. Ise subtract kiya jaata hai taaki jab real gravity $+g_n$ wapas add kare, tab net perpendicular acceleration bilkul wahi ho jo pure PN chahta tha.

$g_n=g\cos \lambda$.

• $\cos \lambda$ sabse bada hota hai $\lambda =0^{\circ }$ par (horizontal LOS): $g_n=g\cos 0^{\circ }=9.81{\text{m/s}}^2$ — worst case, gravity ka poora hissa LOS ko bend karta hai.
• $\cos \lambda =0$ hota hai $\lambda =90^{\circ }$ par (seedha upar/neeche): $g_n=9.81\cos 90^{\circ }=0$ — best case, gravity LOS ke saath saath chalti hai aur use bend nahi kar sakti.

Horizontal ⇒ $g_n=g\cos 0^{\circ }=9.81{\text{m/s}}^2$. $a_c=4(750)(0.008)+0-9.81=24-9.81=14.19{\text{m/s}}^2$

$g_n=9.81\cos 30^{\circ }=9.81(0.86603)=8.496{\text{m/s}}^2$. $a_c=3(1200)(0.005)+\frac{3}{2}(40)-8.496=18+60-8.496=69.50{\text{m/s}}^2$

$g_n=9.81\cos 60^{\circ }=4.905{\text{m/s}}^2$. Law rearrange karo: $\dot{\lambda }=\frac{a_c+g_n-\frac{N^{\prime }}{2}{a}_T}{N^{\prime }{V}_c}=\frac{50+4.905-0}{4(900)}=\frac{54.905}{3600}=0.01525\text{rad/s}$ Note karo hum back-solve karte waqt $g_n$ add karte hain, kyunki law mein $-g_n$ tha: use doosri taraf move karne se sign flip ho jaata hai.

Perpendicular direction bookkeeping: gravity $+g_n=+9.81$ add karta hai (woh missile ko LOS se droop direction mein pull karta hai).

• (a) Uncompensated: $24$ command karta hai, gravity $9.81$ add karti hai ⇒ net perpendicular $=24+9.81=33.81{\text{m/s}}^2$ — PN ne jo maanga us se zyada, toh missile over-turn karta hai aur $\dot{\lambda }$ kabhi truly null nahi hota.
• (b) Compensated: $24-9.81=14.19$ command karta hai; gravity $9.81$ add karti hai ⇒ net $=14.19+9.81=24{\text{m/s}}^2$.
• (c) Compensated case exactly intended $24{\text{m/s}}^2$ deta hai. Yahi toh poora point hai: $a_{c,n}+g_n=$ pure-PN demand.

(a) Effective PN demand $=N^{\prime }{V}_c\dot{\lambda }=4(800)(0.011)=35.2{\text{m/s}}^2$. Yeh $40$ limit se kam hai — achievable. (b) Compensation ke bina autopilot ko khud droop se ladna padta: use PN value plus itna command karna padta jo gravity ki continuous bending ko overcome kare, effectively $35.2+9.81=45.01{\text{m/s}}^2$ demand karna padta, jo $40{\text{m/s}}^2$ limit exceed karta hai ⇒ saturation, degraded miss. Compensation ke saath commanded value hai $35.2-9.81=25.39{\text{m/s}}^2$ aur gravity wapas add hokar required $35.2$ deti hai — limit ke andar aaram se. Compensation ne $\approx 9.8{\text{m/s}}^2$ usable capacity recover ki.

Constant-maneuver ZEM ko PN-on-ZEM form mein substitute karo: $a_c=\frac{N^{\prime }}{t_{go}^2}\left(\frac{1}{2}{a}_T{t}_{go}^2\right)=\frac{N^{\prime }}{t_{go}^2}\cdot \frac{a_T{t}_{go}^2}{2}=\frac{N^{\prime }}{2}{a}_T.$ $t_{go}^2$ exactly cancel ho jaata hai, ek constant feed-forward $\frac{N^{\prime }}{2}{a}_T$ chodta hai jo time-to-go se independent hai — isliye ise directly command mein kisi timer ke bina add kiya ja sakta hai. Yahi origin hai jo Zero-Effort-Miss (ZEM) guidance mein cited hai.

Gravity project karo: $g_n=9.81\cos 40^{\circ }=9.81(0.76604)=7.515{\text{m/s}}^2$. $a_c=5(1100)(-0.003)+\frac{5}{2}(25)-7.515$ $=-16.5+62.5-7.515=38.485{\text{m/s}}^2\approx 38.49{\text{m/s}}^2.$ Negative $\dot{\lambda }$ PN term ko negative pull karta hai (doosri taraf steer karo), lekin strong maneuver term dominate karta hai, ek net positive command deta hai. Gravity compensation $7.515$ trim off karta hai.

$\omega t$ ko degrees mein convert karo ya radians consistently rakho. $\lambda (t)$ radians mein use karo:

• $\lambda (0)=20^{\circ }=0.34907\text{rad}$. $g_n(0)=9.81\cos (0.34907)=9.81(0.93969)=9.219{\text{m/s}}^2$ ⇒ term $=-9.219{\text{m/s}}^2$.
• $\lambda (2)=0.34907+0.10(2)=0.54907\text{rad}(\approx 31.46^{\circ })$. $g_n(2)=9.81\cos (0.54907)=9.81(0.85310)=8.369{\text{m/s}}^2$ ⇒ term $=-8.369{\text{m/s}}^2$. Kyun recompute karein: $g_n$ current LOS angle par depend karta hai, jo geometry evolve hone ke saath move karta hai. Launch par set ki gayi fixed compensation gravity ko over- ya under-cancel karti jab $\lambda$ drift karta, exactly wahi persistent $\dot{\lambda }$ wapas laati jo hum khatam karne nikle the. Autopilot isliye $\lambda$ padhta hai (from Line-of-Sight rate estimation / seeker geometry, Coordinate frames & projections frame mein resolve hokar) har guidance cycle mein aur gravity ko re-project karta hai.

$\dot{\lambda }$ ke change ka driver net perpendicular acceleration $D=a_{c,n}+g_n-a_{T,n}$ hai.

• Uncompensated: $a_{c,n}=0$, $a_{T,n}=0$ ⇒ $D=0+g_n-0=g_n\ne 0$. Ek nonzero $D$ $\dot{\lambda }$ ko agle instant zero se dur force karta hai — droop appear hoti hai, aur PN ko phir use baad mein chase karna padta hai.
• Compensated: $a_{c,n}=-g_n$ ⇒ $D=-g_n+g_n-0=0$. Net perpendicular acceleration zero hai, toh $\dot{\lambda }$ $0$ par bani rehti hai. Missile ek true collision course hold karta hai. Yeh prove karta hai ki compensation predictive hai: woh disturbance ko pehle null karta hai isse pehle ki woh LOS-rate error bane, rather than ek droop par react karne ke jo already miss kho chuki ho. Contrast karo Ballistic trajectory & gravity turn se jahan droop simply accept ki jaati hai.

L2.1 $=14.19$ · L2.2 $=69.50$ · L2.3 $\dot{\lambda }=0.01525$ · L3.1 $33.81/24/24$ · L3.2 $35.2$ vs $45.01$ · L4.2 $=38.49$ · L5.1 $-9.219,-8.369$. Agar tumhara alag aaya, toh $\cos \lambda$ projection aur $-g_n$ sign recheck karo.