Foundations — Augmented proportional navigation — gravity compensation

This page assumes you know nothing. We will build every letter, arrow, and symbol the parent note uses, one at a time, each earning its place before the next arrives.

0 — The scene: two moving dots and a string between them

Before any symbol, picture the physical situation.

There are two objects: the missile (our chaser) and the target (what we chase). Stretch an imaginary elastic string between them. That string is the Line of Sight (LOS) — literally the direction the missile "looks" to see the target.

Everything else is built on this one picture: the string, how long it is, and — crucially — which way it points.

1 — Angle $\lambda$: which way the string points

A direction in a flat plane is captured by a single number: an angle. We measure the LOS direction as an angle up from the horizontal ground line.

Why an angle and not, say, coordinates? Because the whole game is about the string rotating. A rotation is a change in an angle. To talk about "the string turning," we first need a number for "which way it points" — that number is $\lambda$.

• If the target is straight ahead on the horizon, $\lambda =0$.
• If the missile shoots straight up, $\lambda =90°$.
• $\lambda$ can be any value; it lives on the picture as the wedge between the horizontal dashed line and the red LOS.

2 — The dot on top: $\dot{\lambda }$, how fast the string spins

Now the star of the show. Put a small dot above a symbol and it means "the rate at which that thing changes each second." This dot notation is called a time derivative — it answers the question "how fast?"

Why do we need a rate and not just the angle? Here is the central insight of the entire chapter, shown in one picture:

Look at the two cases:

• Left (string rotating, $\dot{\lambda }\ne 0$): the target is drifting sideways across the missile's view. They will miss.
• Right (string keeps the same direction, $\dot{\lambda }=0$): the target sits still in the missile's view even as the gap shrinks. They collide.

So $\dot{\lambda }$ is the ONE number a guided missile obsesses over. Make it zero → intercept.

3 — Speeds: closing velocity $V_c$

Two objects approaching each other close the gap at some speed. We name it.

Why do we care about closing speed specifically (not the missile's ground speed)? Because guidance is about time left until impact. A faster close means less time to fix errors, so the urgency of any correction scales with $V_c$. That is exactly why it multiplies the LOS rate in the PN law you will meet next.

4 — Acceleration: the letter $a$ and the arrow on top

Speed tells you how fast you move; acceleration tells you how fast your velocity changes — i.e. how hard you are being pushed sideways or forwards.

A missile steers by pushing itself sideways — turning its velocity arrow without necessarily changing its speed. That sideways push is a lateral (side-to-side) acceleration.

We will meet three flavours of acceleration, each earning its own subscript:

5 — Arrows over letters: vectors and $\vec{g}$

Some quantities need only a size (a "how much" number) — those are scalars, like $V_c$. Others need a size and a direction — those are vectors, and we draw a little arrow over them.

Why bother separating direction from size? Because gravity's trouble depends entirely on its direction relative to the string. The same downward pull hurts a lot in one geometry and not at all in another — and only vectors let us say that precisely.

6 — Splitting a vector: "along the string" vs "across the string"

Here is the mathematical tool the whole gravity-compensation idea rests on: projection. Any arrow can be split into two arrows at right angles — one along a chosen direction, one across it. Adding those two back gives the original.

Now split gravity $\vec{g}$ against the LOS:

• Along the LOS — this part only makes the string longer or shorter (changes $V_c$). It does not rotate the string. Harmless.
• Across the LOS ($\perp$) — this part swings the string sideways, so it creates a $\dot{\lambda }$. This is the guilty component.

Why "$\cdot$" (the dot product)? The dot product is precisely the tool that answers "how much of this arrow lies along that direction?" Feeding it $\vec{g}$ and the perpendicular direction $\hat{n}$ extracts exactly the across-the-string slice of gravity — nothing more.

In the vertical plane, geometry gives the clean result the parent uses: $g_n=g\cos \lambda$

Read the two extremes off the picture:

• Horizontal LOS ($\lambda =0$): $g_n=g\cos 0=g$. All of gravity is across the string — worst case.
• Vertical LOS ($\lambda =90°$): $g_n=g\cos 90°=0$. Gravity points along the string — completely harmless.

7 — The navigation constant $N^{\prime }$

One more symbol. When the computer sees the string rotating, how aggressively should it push back? That gain is a chosen number.

Why a specific 3–5 window? Too small and the missile reacts sluggishly and misses; too large and it over-reacts, saturating its steering and amplifying seeker noise. The 3–5 range is the sweet spot that both control theory and decades of practice settle on.

Putting the alphabet together

You now own every symbol in the boxed law of the parent note: $a_c=N^{\prime }{V}_c\dot{\lambda }+\frac{N^{\prime }}{2}{a}_T-g_n$

Read left to right in plain words: "the sideways push I command = (my aggressiveness) × (approach speed) × (how fast the string spins), plus a share of the target's own manoeuvre, minus the part of gravity that would spin the string." No symbol here is unfamiliar anymore.

The map reads bottom-to-top into the parent topic: the string gives an angle, the angle gives a rate, the rate gives the collision rule; speed, acceleration, and the projected gravity term all feed the final command.

Equipment checklist

Test yourself — cover the right side and answer each before revealing.

Connections

• Augmented proportional navigation — gravity compensation — the parent topic these foundations unlock.
• Proportional Navigation (PN) — the base law built directly on $\dot{\lambda }$, $V_c$, $N^{\prime }$.
• Line-of-Sight rate estimation — how $\dot{\lambda }$ is actually measured.
• Coordinate frames & projections — the projection/dot-product machinery for $g_n$.
• Ballistic trajectory & gravity turn — why $\vec{g}$ droops the flight path in the first place.
• Zero-Effort-Miss (ZEM) guidance — where the $N^{\prime }/2$ target term comes from.