Terminal descent — velocity vector alignment, touchdown constraints: Foundations

3.5.54 · D1 · Physics › Guidance, Navigation & Control (GNC) › Terminal descent — velocity vector alignment, touchdown cons

Parent note padhne se pehle, usme diye har symbol ko obvious feel hona chahiye. Yeh page unhe ek ek karke, bilkul scratch se build karta hai. Agar tumne pehle kabhi $v$ jaisa koi bold arrow-letter nahi dekha, toh yahan se shuru karo aur kuch bhi skip mat karo.

0. Stage: position, aur "upar" ka matlab kya hai

Sab kuch ordinary 3-D space mein hota hai, jaise ek room. Yeh batane ke liye ki lander kahan hai, hum ground pe teen number-lines (axes) fix karte hain: $x$ (east), $y$ (north), $z$ (up). Koi bhi point teen numbers hai: kitna east, kitna north, kitna upar.

Figure — Terminal descent — velocity vector alignment, touchdown constraints

Figure 1 — Teen axes jisme $x$ horizontal hai aur $z$ upar ki taraf point kar raha hai. Lander (amber square) pad ke upar hai; uska motion arrow neeche ki taraf point karta hai, isliye uska $z$ -component negative hai. Gravity $g$ (white arrow) bhi exactly isi wajah se neeche point karta hai. Is picture ko use karo taaki "down = negative" iss topic mein fix ho jaye.

Yeh abhi kyun fix karein? Parent page pe tumne $v_{z} \to - v_{t d}$ aur $g = (0, 0, - g)$ dekha — dono negative hain ek hi wajah se: woh neeche point karte hain, aur neeche negative $z$ direction hai. Convention galat samjho toh har answer ka sign palat jaayega.

1. Ek vector: "arrow" objects jo bold mein likhe jaate hain

Parent note mein $r$ , $v$ , $a$ , $T$ , $g$ jaise bold letters hain. Bold ka matlab hai woh cheez ek vector hai: ek number nahi, balki ek arrow jo direction aur length dono carry karta hai.

Figure 2 — Ek velocity arrow $v$ (amber) apne horizontal part $v_{x}$ aur vertical part $v_{z}$ (cyan) mein split hua. Dono components right angle pe milte hain, aur arrow diagonal hai — isliye neeche length formula sirf Pythagorean theorem hai. Jab bhi "components" abstract lage tab yeh figure padhna: woh literally woh do sides hain jis box ko arrow span karta hai.

$r$ → position vector: pad se lander tak ka arrow. Uski length distance-to-go hai.
$v$ → velocity vector: arrow jo dikhata hai kaunsi taraf, aur kitni fast, lander abhi move kar raha hai.
$a$ → acceleration vector: velocity arrow abhi kaise change ho raha hai.
$T$ → thrust vector: engine jo push karta hai, vehicle ke bottom se bahar point karta hua.
$g$ → gravity vector: constant downward pull, $g = (0, 0, - g)$ .

2. Dot aur double-dot: rate of change

Parent mein $\dot{r}$ aur $\ddot{r}$ likhte hain. Upar ek dot shorthand hai "rate of change per second" ke liye — yeh calculus se aata hai, lekin ise purely ek picture ki tarah padhaa ja sakta hai.

$\ddot{r} = a_{c m d} + g$

plain words mein: lander ka acceleration equals engine ka commanded push, plus gravity ka constant downward pull.

3. Subscripts: kaunsa moment, kaunsa target

Symbols ke neeche chhote labels hote hain. Woh object ki type kabhi nahi badlte — sirf kaunsa object hai yeh batate hain.

Definition Jo subscripts milenge

$r$ , $v$ (koi subscript nahi) → abhi ki value.
$r_{f}$ , $v_{f}$ → final / target value jo hum touchdown pe chahte hain ( $f$ = final).
$r_{t a r g e t}$ → pad ki location. Yeh $r_{f}$ ke same point hai: pad wahi hai jahan hum final time pe hona chahte hain, isliye $r_{f} = r_{t a r g e t}$ . Dono naam sirf "us ground pe spot jise hum aim kar rahe hain" kehne ke do tarike hain.
$v_{t d}$ → touchdown vertical speed (ek small positive number jaise 1–2 m/s; arrow neeche point karta hai isliye yeh $- v_{t d}$ ke roop mein appear hota hai).
$v_{h}^{m a x}$ , $v_{t d}^{m a x}$ , $θ_{m a x}$ , $ω_{m a x}$ → sabse zyada allowed values (" $max$ " ceilings jiske andar touchdown envelope rehni chahiye). Note karo $v_{h}^{m a x}$ ek ceiling hai; actual horizontal speed jo yeh limit karta hai woh $v_{h}$ hai, section 5 mein define ki gayi hai.

Now vs final

koi subscript nahi matlab present state; subscript

f

matlab desired state at time

t_{f}

Kya r_f aur r_target ek hi hain?

haan — pad location aur final position target ek hi point hai.

4. Time symbols: $t_{f}$ aur $t_{g o}$

5. Tilt / drift angle ke peeche right triangle

Parent $arctan (v_{h} / v_{t d})$ aur $θ$ use karta hai. Inhe padhne ke liye ek chhota triangle chahiye — aur pehle, symbol $v_{h}$ .

Figure 3 — Touchdown velocity ek right triangle ke roop mein: downward leg $v_{t d}$ aur sideways leg $v_{h}$ (cyan), velocity arrow hypotenuse ke roop mein (amber). Amber arc vertical se angle mark karta hai, jiska tangent $v_{h} / v_{t d}$ hai. Jab $v_{h} = 0$ arrow downward leg ke saath flat parta hai aur angle $0^{\circ}$ hai — perfectly vertical touchdown.

$θ$ → tilt angle: vehicle ki thrust axis straight-up se kitni dur lean karti hai.
$θ_{t i p}$ → woh tilt jisme center of mass legs ke bahar girti hai aur lander tip ho jaata hai.
$ω$ → angular rate: body kitni fast rotate (spin) kar raha hai, degrees ya radians per second mein.

6. Boxed guidance formula ko loud padhna

Ab tumhare paas

$a_{c m d} = \frac{6}{t _{g o}^{2}} (r_{f} - r) - \frac{2}{t _{g o}} (2 v + v_{f}) - g$

ke har symbol hain.

Plain reading: "Jo push main command karta hoon = (meri position kitni off hai) times kitna kam time bacha hai, minus (meri current aur target velocity ka blend) times time bacha, minus gravity taaki woh already cancel ho jaaye." Har piece ek arrow hai; arrows ko tip-to-tail add karke final commanded push milti hai.

Prerequisite map

Yeh aage deeper machinery mein feed hote hain: command $a_{c m d}$ ek thrust order ban jaati hai jo Attitude Control & Inner Loop handle karta hai, two-point boundary idea Powered Descent Guidance (PDG) aur Convex Optimization Landing (lossless convexification) mein generalize hota hai, aur "target ko chase karo" wali instinct Proportional Navigation se connect hoti hai.

Equipment checklist

Right side cover karo aur khud test karo. Agar koi bhi jawab surprising lage, toh parent note se pehle woh section dobara padho.

v

jaisa ek bold letter matlab

ek vector — direction aur length dono wala arrow, koi single number nahi.

v_{z}

component ka negative hona matlab

lander neeche move kar raha hai, kyunki upar positive

z

choose kiya gaya tha.

∣ v ∣ = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

compute karta hai

magnitude, yaani velocity arrow ki length (uski overall speed).

v_{h}

ka matlab

horizontal speed,

v_{x}^{2} + v_{y}^{2}

— velocity arrow ke sideways part ki length.

\dot{r}

aur

\ddot{r}

ka matlab

position ka rate of change (velocity) aur velocity ka rate of change (acceleration).

v_{f}

mein subscript

f

mark karta hai

woh final / target value jo hum touchdown time

t_{f}

pe chahte hain.

r_{f}

aur

r_{t a r g e t}

refer karte hain

ek hi point ko — pad ki location, jo exactly final position target hai.

t_{g o}

hai

time-to-go,

t_{f} - t

, touchdown tak bache seconds, zero ki taraf count karte hue.

t_{g o}

se divide karne pe command badi ho jaati hai jab

kam time bacha ho, taaki engine errors fix karne ke liye zyada push kare.

tan (angle) = v_{h} / v_{t d}

describe karta hai

velocity arrow kitna slanted hai — sideways part over downward part.

arctan

ek ratio leta hai aur return karta hai

woh angle jo us slant wala hai (yeh

tan

undo karta hai).

θ

aur

arctan (v_{h} / v_{t d})

alag hain kyunki

θ

body ka tilt hai; arctan term velocity arrow ka tilt hai — dono chhote rehne chahiye.

Do vectors geometrically add karne ka matlab

unhe tip-to-tail rakh do; component-wise, matching numbers add karo.

g = (0, 0, - g)

z

mein negative hai kyunki

gravity neeche pull karti hai aur neeche negative

z

direction hai.