Rendezvous and proximity operations — Clohessy-Wiltshire equations

3.2.40 · Physics › Orbital Mechanics & Astrodynamics

1. Frame setup karna (KYO rotating frame?)

KYO rotating? Kyunki target khud accelerate ho raha hai (Earth ke around curve kar raha hai). Agar hum non-rotating frame use karte toh target ud jaata. Co-rotate karne se target apni jagah rehta hai aur sirf relative motion bachti hai — bilkul wahi jo ek docking pilot dekhta hai.

Circular target orbit ka radius $R_{0}$ aur mean motion hai $n = \frac{μ}{R _{0}^{3}},$ jo sirf Kepler's law hai: gravity centripetal force provide karti hai, $\frac{μ}{R _{0}^{2}} = n^{2} R_{0}$ .

2. Scratch se derivation (KAISE)

Ek frame mein jo angular velocity $ω = n \overset{z}{^}$ se rotate kar raha hai, position $r$ pe ek point ka acceleration (Earth ke center se measure kiya hua) Newton ko follow karta hai, lekin rotating frame mein likhe jaane par use fictitious terms milti hain:

\ddot{r}_{rot} = g (r) - 2 ω \times \dot{r}_{rot} - ω \times (ω \times r) .

Teeno added pieces (left se) hain gravity, Coriolis, aur centrifugal.

Chaser ko $r = (R_{0} + x) \overset{x}{^} + y \overset{y}{^} + z \overset{z}{^}$ pe maano (target $R_{0} \overset{x}{^}$ pe hai).

Step A — Gravity, linearized. KYO? Gravity $- μ r / r^{3}$ hai; hum sirf $(x, y, z)$ mein first order chahte hain.

g = - \frac{μ}{r ^{3}} r, r = (R_{0} + x)^{2} + y^{2} + z^{2} .

$r^{- 3} \approx R_{0}^{- 3} (1 - \frac{3 x}{R _{0}})$ ko first order tak expand karo. Phir

g_{x} \approx - \frac{μ}{R _{0}^{2}} + \frac{2 μ}{R _{0}^{3}} x, g_{y} \approx - \frac{μ}{R _{0}^{3}} y, g_{z} \approx - \frac{μ}{R _{0}^{3}} z .

KYO yeh step? $- μ / R_{0}^{2}$ constant target pe centrifugal se exactly cancel hota hai, aur $μ / R_{0}^{3} = n^{2}$ hai.

Step B — Coriolis $- 2 ω \times \dot{r}$ , $ω = n \overset{z}{^}$ ke saath:

- 2 n \overset{z}{^} \times (\overset{x}{˙}, \overset{y}{˙}, \overset{z}{˙}) = (+ 2 n \overset{y}{˙}, - 2 n \overset{x}{˙}, 0) .

Step C — Centrifugal $- ω \times (ω \times r) = n^{2} (x, y, 0)$ (orbital plane mein baahir ki taraf point karta hai).

Step D — Inhe add karo. Component by component, $μ / R_{0}^{3} = n^{2}$ use karke:

$x$ : $\overset{x}{¨} = (- n^{2} R_{0} + 2 n^{2} x) + 2 n \overset{y}{˙} + n^{2} x + n^{2} R_{0}$ $\Rightarrow \overset{x}{¨} - 2 n \overset{y}{˙} - 3 n^{2} x = 0$ ( $- n^{2} R_{0}$ gravity se aur $+ n^{2} R_{0}$ centrifugal se cancel hote hain — isliye target hover karta hai!)
$y$ : $\overset{y}{¨} = - n^{2} y - 2 n \overset{x}{˙} + n^{2} y \Rightarrow \overset{y}{¨} + 2 n \overset{x}{˙} = 0$
$z$ : $\overset{z}{¨} = - n^{2} z \Rightarrow \overset{z}{¨} + n^{2} z = 0$

3. Solve karna (KAISE — closed form)

$z$ equation SHM hai: $z (t) = z_{0} cos n t + \frac{z ˙ _{0}}{n} sin n t$ .

In-plane pair ke liye, $y$ -equation ko ek baar integrate karo: $\overset{y}{˙} + 2 n x = \overset{y}{˙}_{0} + 2 n x_{0} (const) .$ $x$ -equation mein substitute karo taaki $\overset{x}{¨} + n^{2} x = const$ mile, jo ek aur driven SHM hai. Solve karke:

Figure — Rendezvous and proximity operations — Clohessy-Wiltshire equations

4. Worked examples

Worked example Example 3 — Radial offset use karke along-track gap close karne ka time

Tum $y_{0} = - 1$ km peeche ho, aage aana chahte ho. Ek phasing trick: thoda neeche orbit mein drop karo by giving a radial offset $x_{0} < 0$ taaki secular term $6 (sin n t - n t) x_{0}$ tumhe aage le jaaye. Net drift per full orbit. Ek period $T = 2 π / n$ ke baad, oscillatory $sin n t$ zero pe wapas aa jaata hai, isliye net along-track shift sirf $- n t$ piece se aati hai: $Δ y = 6 (= 0 sin n T - n T) x_{0} = - 6 n T x_{0} = - 12 π x_{0} .$ KYO yeh step? $n T = 2 π$ hai, isliye $- 6 n T x_{0} = - 12 π x_{0}$ per orbit — ek chhote se radial offset se bada along-track walk. (Note: $Δ y > 0$ jab $x_{0} < 0$ : ek lower orbit tumhe aage move karti hai.) $1000$ m ko ek orbit mein close karne ke liye hume chahiye $∣ - 12 π x_{0} ∣ = 1000 \Rightarrow ∣ x_{0} ∣ = \frac{1000}{12 π} \approx 26.5$ m. KYO yeh step? Bas $Δ y = - 12 π x_{0}$ ko invert karo; $n$ aur $T$ pure number $12 π$ mein cancel ho gaye, isliye required offset orbit altitude se independent hai.

5. Steel-manned mistakes

6. Active recall

CW equations kis frame mein likhi jaati hain?

LVLH / Hill frame mein — target pe centered, mean motion

n

pe rotate karta hua (x radial, y along-track, z cross-track).

Teeno CW equations batao.

\overset{x}{¨} - 2 n \overset{y}{˙} - 3 n^{2} x = 0

;

\overset{y}{¨} + 2 n \overset{x}{˙} = 0

;

\overset{z}{¨} + n^{2} z = 0

Target origin pe hover kyun karta hai?

Constant gravity term

- n^{2} R_{0}

radial equation mein centrifugal term

+ n^{2} R_{0}

se exactly cancel ho jaata hai.

n

μ, R_{0}

ke terms mein kya hai?

n = μ / R_{0}^{3}

(

μ / R_{0}^{2} = n^{2} R_{0}

se).

Kaun si motion ek decoupled SHM hai?

Cross-track

z

\overset{z}{¨} + n^{2} z = 0

, frequency

n

pe oscillate karta hua.

x aur y ke beech coupling ki wajah kya hai?

Coriolis force (

\pm 2 n \dot{\cdot}

) rotating frame mein kaam karne ki wajah se.

Bounded (drift-free) in-plane motion ke liye condition?

\overset{y}{˙}_{0} = - 2 n x_{0}

Bounded relative motion ki shape kya hoti hai?

Ek 2:1 ellipse ("football/racetrack") — along-track amplitude radial amplitude se double.

Secular along-track drift kahaan se aati hai?

- 6 n t x_{0}

aur

- 3 n t \overset{y}{˙}_{0}

terms se; ek radial offset orbital period badalta hai → endless along-track walk.

Radial offset

x_{0}

se ek orbit mein net along-track drift kitna?

Δ y = - 6 n T x_{0} = - 12 π x_{0}

(kyunki

n T = 2 π

CW ke do key assumptions kya hain?

Circular target orbit (constant

n

) aur chhota relative separation (gravity ka linearization).

Recall Feynman: 12-saal ke bachche ko explain karo

Do race cars ek circular track ke around jaate hain. Ek (target) apni lane mein rehta hai; tum (chaser) bilkul uske paas ho. Poora track dekhne ki bajay, target car pe ek camera lagao aur sirf dekho ki tum uske paas kaise slide karte ho. Space mein, agar tum "upar" nudge karo (Earth se door), tumhari lane slower ho jaati hai, isliye tum dheere dheere peeche khisak jaate ho. Agar tum "aage" nudge karo, tum upar climb karte ho aur phir peeche reh jaate ho! CW equations is weird sliding ka simple rulebook hai jab tum orbit mein ek dost ke paas hote ho. Upar-neeche uchhalna ek simple bounce hai; upar-aage wali motions ek doosre se ulajhi hui hain.

7. Connections

Two-Body Problem — exact nonlinear system jise humne linearize kiya.
Kepler's Laws — $n = μ / R_{0}^{3}$ ka source aur kyun uunchi orbits slower hoti hain.
Rotating Reference Frames — Coriolis and Centrifugal — coupling terms ka origin.
Tschauner–Hempel Equations — eccentric-orbit generalization.
Orbital Maneuvers — Hohmann Transfer — local phasing ka global cousin.
Linearization and Taylor Expansion — derivation ka mathematical engine.
State Transition Matrix — CW solution ko guidance ke liye package karna.