Foundations — Reference frames — body frame, inertial frame; rotation between them

3.5.1 · D1 · Physics › Guidance, Navigation & Control (GNC) › Reference frames — body frame, inertial frame; rotation betw

Is page par assume kiya gaya hai ki aapne kuch bhi nahi dekha. Hum har symbol build karte hain jaise parent note aap par throw karta hai, aisi order mein jahan har ek sirf usse pehle waale par depend karta hai. Upar se neeche padho.

1. Arrow (vector) kya hota hai?

Topic ko yeh kyun chahiye: ek spacecraft ka thrust, ek star ki direction, ek angular velocity — yeh sab arrows hain. Agar hum arrows ke baare mein baat nahi kar sakte, toh pointing ke baare mein bilkul baat nahi kar sakte.

2. Axes, ek frame, aur components

Arrow ko numbers mein badle ke liye, pehle teen reference directions neeche rakhte hain measure karne ke liye.

Topic ko yeh kyun chahiye: ek tumbling spacecraft par bolted gyro ek tilted corner se measure karta hai; star map ek seedhe corner mein likhi hoti hai. Same star, do triples. Hume unke beech convert karna hoga.

3. Orthonormal — well-behaved frame ka type

Topic ko yeh kyun chahiye: sirf orthonormal frames ke liye hi sasta "invert karne ke liye bas transpose karo" trick (Section 8) kaam karti hai. Real spacecraft frames hamesha is tarah purpose se banaye jaate hain.

4. Dot product — ek machine jo "kitna along" measure karti hai

Yeh page par sabse important tool hai, isliye hum ise dhire-dhire build karte hain aur batate hain ki kyun yeh tool aur koi nahi.

Har piece ko unpack karte hain:

• $\Vert \vec{a}\Vert$ — arrow $\vec{a}$ ki length (magnitude), ek plain positive number, jaise ek stick ki length.
• $\theta$ — do arrows ke beech ka angle, jahan unki tails milti hain wahan measure kiya jaata hai.
• $\cos \theta$ — cosine: ek number jo jawaab deta hai "yeh do directions kitni aligned hain?" Yeh $+1$ se chalta hai (bilkul parallel, $\theta =0$), $0$ se guzarta hai (perpendicular, $\theta =90^{\circ }$), $-1$ tak (opposite, $\theta =180^{\circ }$).

Topic ko yeh kyun chahiye: parent har component ko $v_x^{\mathcal{B}}=\vec{v}\cdot {\hat{x}}_{\mathcal{B}}$ ke roop mein derive karta hai. Woh yahi special case hai, kuch aur nahi.

5. Perpendicular axes zero dete hain — cleanup rule

6. Kronecker delta ${\delta }_{ij}$ — "same ya different" ka shorthand

Topic ko yeh kyun chahiye: parent ka proof ki $R{R}^{\mathsf{T}}=I$ kehta hai "entry $(i,j)$ hai ${\hat{e}}_i\cdot {\hat{e}}_j={\delta }_{ij}$" — woh exactly yahi symbol hai jo 1s-aur-0s pattern ko package karta hai.

7. Matrix — numbers ka ek rack jo ek arrow ko transform karta hai

Topic ko yeh kyun chahiye: DCM $R_{\mathcal{B}\mathcal{I}}$ aisi hi ek matrix hai; frames ke beech arrow ke numbers convert karna exactly ek matrix–vector multiply hai. Aur "$R^{\mathsf{T}}R=I$" woh property hai jo tumhe ek rotation free mein undo karne deti hai.

8. Subscripts ko kaam par lagana: $R_{\mathcal{B}\mathcal{I}}$

9. Prerequisite map

Sab kuch usi ek destination mein funnel ho jaata hai: ek arrow ke numbers ko body ki attitude aur fixed stars ke beech convert karna. Yahan se tum quaternions, gyro sensing, aur orbit frames ke liye ready ho.

Equipment checklist

Right side ko cover karo aur zor se jawab do. Agar koi ek bhi ruk jaaye, toh parent note se pehle us section ko dobara padho.