Foundations — Attitude control modes — spin stabilization, 3-axis active

3.5.47 · D1 · Physics › Guidance, Navigation & Control (GNC) › Attitude control modes — spin stabilization, 3-axis active

Yeh page har woh symbol build karta hai jo parent note use karta hai, uss reader ke liye shuru karke jo kabhi overbar-arrow wala koi letter nahi dekha. Hum order mein jaate hain: har idea sirf pehle se bane ideas use karta hai.

0. Chhote marks ka matlab kya hai

Koi bhi physics se pehle, teen tarah ke notation har jagah dikhte hain. Aao inhe samjhte hain.

0b. Right-hand rule — ek baar padho, har jagah use karo

Neeche kuch arrows ($\vec{\omega }$, $\vec{\tau }$, aur cross product) ek axis ke saath point karte hain, aur hume ek clear rule chahiye us axis ke kaunse taraf. Woh rule hamesha same hoti hai:

1. Angle $\theta$ — woh cheez jo hum actually control karte hain

Figure s01 — pointing-error angle.

Figure s01 dekho: pale-yellow arrow woh direction hai jo hum chahte hain; pink arrow woh hai jahan hum actually point kar rahe hain; dono ke beech ki wedge $\theta$ hai. Jab $\theta =0$ toh dono arrows ek doosre ke upar ho jaate hain — mission accomplished.

$\dot{\theta }$ ka matlab hai "woh wedge kitni tezi se khul ya band ho rahi hai," aur $\ddot{\theta }$ ka matlab hai "woh rate khud kitni tezi se change ho rahi hai" — dono PID law mein milenge.

2. Angular velocity $\vec{\omega }$ — spin as an arrow

Turning ki speed hoti hai (kitna fast) aur ek axis hoti hai (kaunsi line ke baare mein). Ek arrow dono capture karta hai.

Figure s02 — right-hand rule se spin arrow.

Figure s02 mein disc counter-clockwise turn karti hai, toh right-hand rule se $\vec{\omega }$ (chalk-blue) disc se upar nikalta hai. Doosri taraf spin karo toh arrow neeche point karne ke liye flip ho jaata hai. (Woh special case jahan poora craft ek axis ke baare mein fast spin karta hai, uska apna naam hai ${\omega }_s$, jab hum §7 pahunchenge.)

3. Torque $\vec{\tau }$ — woh "twist" jo tum lagaate ho

Change karne ke liye spin, tumhe ek twisting push chahiye. Woh hai torque.

4. Angular momentum $\vec{L}$ — stored spin, show ka star

Yeh single sentence dono control modes ka engine hai:

• Spin stabilization: $\vec{L}$ ko bahut bada bana do, taki ek chhota disturbance torque ${\vec{\tau }}_{\text{ext}}$ ise barely move kare. Stiff.
• 3-axis active: demand par $\vec{L}$ change karne ke liye actuators se deliberately torques create karo.

5. Moment-of-inertia tensor $\mathbf{I}$ — woh machine jo spin ko stored spin se jodti hai

Yahan woh reason hai kyun hume ek machine chahiye, sirf ek number nahi.

Figure s03 — same spin, alag axis, alag stored spin.

Figure s03 mein same disc do tarik se spin hoti hai: fat flat axis ke baare mein (bada $I$, lamba stored-spin arrow) versus ek thin edge axis ke baare mein (chhota $I$, chhota arrow) — same $|\vec{\omega }|$, alag $|\vec{L}|$. Woh difference hi poora plot hai.

6. Cross product $\times$ — kaise ek spinning frame arrows ko twist karta hai, aur Euler equation

Rotation law mein term $\vec{\omega }\times (\mathbf{I}\vec{\omega })$ hai. "$\times$" ka kya matlab hai?

Ab hum finally woh equation likh sakte hain jisme woh symbols rehte hain. ${\vec{\tau }}_{\text{ext}}=\dot{\vec{L}}$ (§4) se shuru karke aur body ke apne spinning frame mein rate of change rewrite karne par Euler's rotational equation of motion milta hai. Kyunki har torque yahan external hai, hum "ext" label drop karte hain (§3) aur plain $\vec{\tau }$ likhte hain:

$\vec{\tau }=\mathbf{I}\dot{\vec{\omega }}+\vec{\omega }\times (\mathbf{I}\vec{\omega }).$

Plain words mein padho: applied twist $\vec{\tau }$ do kaam karta hai — pehla term $\mathbf{I}\dot{\vec{\omega }}$ spin ko speed up ya slow down karta hai, aur doosra term $\vec{\omega }\times (\mathbf{I}\vec{\omega })$ woh free sideways swing hai jo spinning frame akele hi add karta hai. Poori derivation Euler's rotational equations of motion mein hai; yahan hum sirf chahte hain ki uske har symbol ab readable ho.

7. Time ke saath rate of change: derivative $\frac{d}{dt}$

8. Rotational kinetic energy $T$ — stability ke liye tiebreaker

Pehle "fattest" axis explicitly naam do, kyunki stability rule isi par lean karti hai.

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