Exercises — Augmented proportional navigation — gravity compensation

• $N^{\prime }{V}_c\dot{\lambda }$ — the pure Proportional Navigation (PN) term. It reacts to LOS rotation and drives $\dot{\lambda }\to 0$ (constant-bearing collision).
• $\frac{N^{\prime }}{2}{a}_T$ — the augmentation term. It feeds forward the target's own maneuver so the missile does not have to wait for that maneuver to show up as LOS rotation. Its origin is the Zero-Effort-Miss (ZEM) guidance integral.
• $-g_n$ — the gravity-compensation term. It is subtracted so that after the real gravity adds $+g_n$ back on, the net perpendicular acceleration equals exactly what pure PN wanted.

$g_n=g\cos \lambda$.

• $\cos \lambda$ is largest at $\lambda =0^{\circ }$ (horizontal LOS): $g_n=g\cos 0^{\circ }=9.81{\text{m/s}}^2$ — worst case, all of gravity bends the LOS.
• $\cos \lambda =0$ at $\lambda =90^{\circ }$ (straight up/down): $g_n=9.81\cos 90^{\circ }=0$ — best case, gravity runs along the LOS and cannot bend it.

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

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

$g_n=9.81\cos 60^{\circ }=4.905{\text{m/s}}^2$. Rearrange the law: $\dot{\lambda }=\frac{a_c+g_n-\frac{N^{\prime }}{2}{a}_T}{N^{\prime }{V}_c}=\frac{50+4.905-0}{4(900)}=\frac{54.905}{3600}=0.01525\text{rad/s}$ Note we add $g_n$ when back-solving, because the law had $-g_n$: moving it across flips the sign.

Perpendicular direction bookkeeping: gravity adds $+g_n=+9.81$ (it pulls the missile off the LOS in the droop direction).

• (a) Uncompensated: commands $24$, gravity adds $9.81$ ⇒ net perpendicular $=24+9.81=33.81{\text{m/s}}^2$ — more than PN asked for, so the missile over-turns and $\dot{\lambda }$ never truly nulls.
• (b) Compensated: commands $24-9.81=14.19$; gravity adds $9.81$ ⇒ net $=14.19+9.81=24{\text{m/s}}^2$.
• (c) The compensated case gives exactly the intended $24{\text{m/s}}^2$. That is the whole point: $a_{c,n}+g_n=$ pure-PN demand.

(a) Effective PN demand $=N^{\prime }{V}_c\dot{\lambda }=4(800)(0.011)=35.2{\text{m/s}}^2$. This is below the $40$ limit — achievable. (b) Without compensation the autopilot must itself fight the droop: it would need to command the PN value plus enough to overcome gravity's continuous bending, effectively demanding $35.2+9.81=45.01{\text{m/s}}^2$, which exceeds the $40{\text{m/s}}^2$ limit ⇒ saturation, degraded miss. With compensation the commanded value is $35.2-9.81=25.39{\text{m/s}}^2$ and gravity adds back to give the required $35.2$ — comfortably inside the limit. Compensation recovered $\approx 9.8{\text{m/s}}^2$ of usable capacity.

Substitute the constant-maneuver ZEM into the PN-on-ZEM form: $a_c=\frac{N^{\prime }}{t_{go}^2}\left(\frac{1}{2}{a}_T{t}_{go}^2\right)=\frac{N^{\prime }}{t_{go}^2}\cdot \frac{a_T{t}_{go}^2}{2}=\frac{N^{\prime }}{2}{a}_T.$ The $t_{go}^2$ cancels exactly, leaving a constant feed-forward $\frac{N^{\prime }}{2}{a}_T$ independent of time-to-go — which is why it can be added straight into the command with no timer. This is the origin cited in Zero-Effort-Miss (ZEM) guidance.

Project gravity: $g_n=9.81\cos 40^{\circ }=9.81(0.76604)=7.515{\text{m/s}}^2$. $a_c=5(1100)(-0.003)+\frac{5}{2}(25)-7.515$ $=-16.5+62.5-7.515=38.485{\text{m/s}}^2\approx 38.49{\text{m/s}}^2.$ The negative $\dot{\lambda }$ pulls the PN term negative (steer the other way), but the strong maneuver term dominates, giving a net positive command. Gravity compensation trims $7.515$ off.

Convert $\omega t$ to degrees or keep radians consistently. Use $\lambda (t)$ in radians:

• $\lambda (0)=20^{\circ }=0.34907\text{rad}$. $g_n(0)=9.81\cos (0.34907)=9.81(0.93969)=9.219{\text{m/s}}^2$ ⇒ 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 $=-8.369{\text{m/s}}^2$. Why recompute: $g_n$ depends on the current LOS angle, which moves as the geometry evolves. A fixed compensation set at launch would over- or under-cancel gravity as $\lambda$ drifts, re-introducing exactly the persistent $\dot{\lambda }$ we set out to kill. The autopilot therefore reads $\lambda$ (from Line-of-Sight rate estimation / seeker geometry, resolved in the Coordinate frames & projections frame) each guidance cycle and re-projects gravity.

The driver of $\dot{\lambda }$'s change is the net perpendicular acceleration $D=a_{c,n}+g_n-a_{T,n}$.

• Uncompensated: $a_{c,n}=0$, $a_{T,n}=0$ ⇒ $D=0+g_n-0=g_n\ne 0$. A nonzero $D$ forces $\dot{\lambda }$ away from zero next instant — the droop appears, and PN must then chase it after the fact.
• Compensated: $a_{c,n}=-g_n$ ⇒ $D=-g_n+g_n-0=0$. The net perpendicular acceleration is zero, so $\dot{\lambda }$ stays put at $0$. The missile holds a true collision course. This proves compensation is predictive: it nulls the disturbance before it becomes an LOS-rate error, rather than reacting to a droop that has already opened a miss. Contrast with the Ballistic trajectory & gravity turn where the droop is simply accepted.

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 $-9.219,-8.369$. If yours differ, recheck the $\cos \lambda$ projection and the $-g_n$ sign.