Augmented proportional navigation — gravity compensation

80/20 core: Pure PN steers on the line-of-sight (LOS) rotation only. But gravity constantly bends the missile's flight, adding a spurious LOS rate that PN "chases" — wasting acceleration and causing miss. Augmented PN (APN) with gravity compensation simply adds a term that cancels the known gravitational acceleration so the guidance law only reacts to the real target maneuver.

WHY this note exists

• WHAT: Add a command term equal and opposite to the component of gravity acting perpendicular to the LOS (the part that curves the trajectory).
• WHY: So the effective closed-loop system behaves like a gravity-free PN engagement → zero-effort-miss driven only by target maneuver.
• HOW: Take standard PN, add the target-maneuver term (this is "augmentation"), then add a gravity-cancelling term.

Building the law from first principles

Step 1 — Recall pure Proportional Navigation

Why this form? If $\dot\lambda = 0$, the LOS direction is fixed in inertial space → this is the constant-bearing, decreasing-range collision condition. PN drives $\dot\lambda \to 0$, guaranteeing intercept.

Step 2 — Where does gravity break it?

Write the missile's acceleration as commanded + environmental: $\vec a_{\text{missile}} = \vec a_c + \vec g$

The LOS-rate dynamics (linearized, planar) are governed by the acceleration perpendicular to the LOS. Let $n$ denote the direction normal to the LOS. Then: $\dot\lambda \text{ evolves under } (a_{c,n} + g_n - a_{T,n})$ where $a_{T,n}$ is target acceleration $\perp$ LOS and $g_n$ is gravity's projection $\perp$ LOS.

Step 3 — Augment for target maneuver

If the target accelerates with known/estimated $a_T$, add the term that keeps zero-effort-miss at zero: $a_c = N' V_c \dot\lambda + \frac{N'}{2} a_T$

Why the $\tfrac{N'}{2}$? Solve the linear ZEM (zero-effort-miss) equation. For a constant target acceleration over time-to-go $t_{go}$, the miss integrates as $\tfrac{1}{2}a_T t_{go}^2$. Feeding this forward requires the factor $N'/2$ to exactly cancel it. (This is standard APN.)

Step 4 — Add gravity compensation

Now treat gravity as another known acceleration we must not let corrupt $\dot\lambda$. We command an extra lateral term that cancels the gravity component perpendicular to LOS:

Why the minus sign and why $g_n$ (not full $g$)?

• The full $\vec g$ has a component along the LOS (changes closing speed, harmless to $\dot\lambda$) and a component perpendicular ($g_n$, the guilty party). Only $g_n$ bends the LOS, so only $g_n$ needs cancelling.
• We subtract it because we want $a_{c,n} + g_n = (\text{PN term})$, i.e. the commanded normal accel must supply an extra $-g_n$ so that after gravity adds back $+g_n$, the net perpendicular acceleration equals exactly what pure PN wants.

HOW to compute $g_n$ in practice: If the LOS elevation angle is $\lambda$ (from horizontal) and gravity is $\vec g = -g\,\hat z$, then $g_n = g\cos\lambda \quad(\text{component }\perp\text{ LOS in the vertical plane}).$ When the LOS is horizontal ($\lambda = 0$), $g_n = g$ (worst case — all of gravity bends the LOS). When shooting straight up/down ($\lambda = 90°$), $g_n = 0$ (gravity is along LOS, harmless).

Worked examples

Flashcards

Connections

• Proportional Navigation (PN) — the base law this augments.
• Zero-Effort-Miss (ZEM) guidance — derivation source of the $N'/2$ factor.
• Line-of-Sight rate estimation — how $\dot\lambda$ is measured (seeker gimbal / Kalman filter).
• Missile Autopilot & Acceleration limits — why saving g-capacity by not chasing gravity matters.
• Coordinate frames & projections — how $g_n = \vec g\cdot\hat n$ is computed onboard.
• Ballistic trajectory & gravity turn — same droop physics, uncontrolled case.

Concept Map

Hinglish (regional understanding)