Foundations — Terminal descent — velocity vector alignment, touchdown constraints

Before you can read the parent note, you need every symbol it throws at you to feel obvious. This page builds them one at a time, from nothing. If you have never seen a bold arrow-letter like $\mathbf{v}$ before, start here and do not skip.

0. The stage: position, and what "up" means

Everything happens in ordinary 3-D space, like a room. To talk about where the lander is, we pin three number-lines (axes) to the ground: $x$ (east), $y$ (north), $z$ (up). Any point is then three numbers: how far east, how far north, how far up.

Why fix this now? On the parent page you saw $v_z\to -v_{td}$ and $\mathbf{g}=(0,0,-g)$ — both are negative for the same reason: they point down, and down is the negative $z$ direction. Get the convention wrong and every answer flips sign.

1. A vector: the "arrow" objects written in bold

The parent note writes things like $\mathbf{r}$, $\mathbf{v}$, $\mathbf{a}$, $\mathbf{T}$, $\mathbf{g}$ in bold. Bold means the thing is a vector: not one number, but an arrow carrying both a direction and a length.

• $\mathbf{r}$ → position vector: the arrow from the pad to the lander. Its length is distance-to-go.
• $\mathbf{v}$ → velocity vector: the arrow showing which way, and how fast, the lander is moving right now.
• $\mathbf{a}$ → acceleration vector: how the velocity arrow is changing right now.
• $\mathbf{T}$ → thrust vector: the push the engine makes, pointing out the bottom of the vehicle.
• $\mathbf{g}$ → gravity vector: the constant downward pull, $\mathbf{g}=(0,0,-g)$.

2. The dot and double-dot: rate of change

The parent writes $\dot{\mathbf{r}}$ and $\ddot{\mathbf{r}}$. A dot on top is shorthand for "rate of change per second" — it comes from calculus, but you can read it purely as a picture.

$\ddot{\mathbf{r}}={\mathbf{a}}_{cmd}+\mathbf{g}$

reads in plain words: the lander's acceleration equals the push the engine commands, plus the constant downward pull of gravity.

3. Subscripts: which moment, which target

Symbols carry little labels below them. They never change the kind of object — only which one.

4. Time symbols: $t_f$ and $t_{go}$

5. The right triangle behind the tilt / drift angle

The parent uses $\arctan (v_h/v_{td})$ and $\theta$. To read those, you need one small triangle — and first, the symbol $v_h$.

• $\theta$ → tilt angle: how far the vehicle's thrust axis leans away from straight-up.
• ${\theta }_{tip}$ → the tilt at which the center of mass falls outside the legs and the lander tips.
• $\omega$ → angular rate: how fast the body is rotating (spinning), in degrees or radians per second.

6. Reading the boxed guidance formula out loud

You now own every symbol in

${\mathbf{a}}_{cmd}=\frac{6}{t_{go}^2}({\mathbf{r}}_f-\mathbf{r})-\frac{2}{t_{go}}(2\mathbf{v}+{\mathbf{v}}_f)-\mathbf{g}.$

Plain reading: "The push I command = (how far off my position is) scaled by how little time is left, minus (a blend of my current and target velocity) scaled by time left, minus gravity so it's already cancelled." Each piece is an arrow; you add arrows tip-to-tail to get the final commanded push.

Prerequisite map

These feed forward into the deeper machinery: the command ${\mathbf{a}}_{cmd}$ becomes a thrust order handled by Attitude Control & Inner Loop, the two-point boundary idea generalizes into Powered Descent Guidance (PDG) and Convex Optimization Landing (lossless convexification), and the "chase a target" instinct connects to Proportional Navigation.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, re-read that section before the parent note.