Foundations — Optimal guidance — ZEM - ZEV formulation

Why this page exists

The parent note freely writes things like $\mathbf{r}(t)$, $t_{go}$, $\int (t_f-\tau )\mathbf{a}d\tau$, and "Lagrange multipliers." If any of those look like magic runes to you, you cannot follow the derivation. So here we earn each symbol one at a time: plain words → a picture → why the topic needs it. Read top to bottom; each idea leans on the one above it.

1. Position, velocity, acceleration — the three moving arrows

Imagine a spacecraft as a single glowing dot in space. To describe where it is and how it moves we need three quantities. Because motion happens in more than one direction (up/down, left/right), each is an arrow — a vector — not just a number.

• $\mathbf{r}(t)$ — the position arrow: points from a chosen origin to where the vehicle is at time $t$. Picture: the arrow tip sits on the dot.
• $\mathbf{v}(t)$ — the velocity arrow: how fast and in which direction the position is changing. Picture: an arrow riding on the dot, pointing the way it's heading; longer arrow = faster.
• $\mathbf{a}(t)$ — the acceleration arrow: how the velocity itself changes. This is our steering control — the thrust the engine commands. Picture: a push applied to the velocity arrow.

Why the topic needs these: guidance is the art of choosing $\mathbf{a}(t)$ so that $\mathbf{r}$ and $\mathbf{v}$ arrive where we want. You literally cannot state the problem without these three arrows.

2. Gravity $\mathbf{g}$ — the free push we didn't ask for

$\mathbf{g}$ is the gravitational acceleration arrow: a constant pull the vehicle feels whether or not the engine fires. On the Moon its length is $1.62{\text{m/s}}^2$ pointing down; on Earth $9.81{\text{m/s}}^2$.

3. Time symbols: $t$, $t_f$, and $t_{go}$

Three moments matter:

• $t$ — now, the current instant.
• $t_f$ — the final time, the deadline when we must have arrived (landing touchdown, intercept moment).
• $t_{go}$ — the time-to-go, how many seconds are left: $t_{go}=t_f-t.$

Why the topic needs it: every coefficient in the guidance law ($6/t_{go}^2$, $2/t_{go}$) is written in $t_{go}$, and the dramatic "commands blow up as $t_{go}\to 0$" behaviour is entirely about this shrinking clock.

4. "Coasting" and the double integrator — free-fall prediction

If we coast, the standard schoolbook motion formulas apply. These come from integrating $\dot{\mathbf{v}}=\mathbf{g}$ twice — a system called the double integrator (velocity is one integral of accel, position is two). See Double Integrator Dynamics.

${\mathbf{v}}_{coast}(t_f)=\mathbf{v}+\mathbf{g}{t}_{go},$ ${\mathbf{r}}_{coast}(t_f)=\mathbf{r}+\mathbf{v}{t}_{go}+\frac{1}{2}\mathbf{g}{t}_{go}^2.$

Why the topic needs it: ZEM and ZEV are defined entirely from the coast prediction. No coast → no ZEM/ZEV.

5. ZEM and ZEV — the two "if I do nothing" errors

Now the star symbols. We want to end at target position ${\mathbf{r}}_f$ and (for soft landing) target velocity ${\mathbf{v}}_f$. Compare wanted with predicted-if-coasting:

Substituting the coast formulas gives the parent's boxed expressions: $\mathbf{ZEM}={\mathbf{r}}_f-\mathbf{r}-\mathbf{v}{t}_{go}-\frac{1}{2}\mathbf{g}{t}_{go}^2,$ $\mathbf{ZEV}={\mathbf{v}}_f-\mathbf{v}-\mathbf{g}{t}_{go}.$

Why the topic needs them: the entire guidance law is ${\mathbf{a}}_c=\frac{6}{t_{go}^2}\mathbf{ZEM}-\frac{2}{t_{go}}\mathbf{ZEV}$. These two arrows are the inputs.

6. The integral sign $\int$ and the cost $J$

The guidance is optimal because it minimizes control effort: $J=\frac{1}{2}{\int }_t^{t_f}\Vert \mathbf{a}(\tau ){\Vert }^{2}d\tau .$

Why the topic needs it: without a cost to minimize there are infinitely many ways to hit the target. $J$ is what makes one answer the answer.

7. The lever-arm weight $(t_f-\tau )$

In the parent's Step 1, position gets a weighted integral $\int (t_f-\tau )\mathbf{a}d\tau$. Why the weight?

Why the topic needs it: this weight is why the position constraint is $\int (t_f-\tau )\mathbf{a}d\tau =\mathbf{ZEM}$ while velocity is the plain $\int \mathbf{a}d\tau =\mathbf{ZEV}$.

8. Lagrange multipliers ${\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2$

The heavier machinery behind this — that a quadratic cost forces the optimal ${\mathbf{a}}^{\ast }(\tau )$ to be a straight line in time — comes from Calculus of Variations & Pontryagin's Minimum Principle. You do not need to master it to use ZEM/ZEV; you only need to trust that it hands you the profile ${\mathbf{a}}^{\ast }(\tau )={\mathbf{\lambda }}_1(t_f-\tau )+{\mathbf{\lambda }}_2$.

Why the topic needs it: the multipliers are the bridge from "minimize $J$" to the clean $2\times 2$ system whose solution gives the coefficients $6$ and $2$.

9. The dot product $\cdot$

The Lagrangian in the parent uses $\mathbf{\lambda }\cdot \mathbf{a}$.

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