Foundations — Gyroscope — angular velocity measurement, bias, noise

3.5.14 · D1 · Physics › Guidance, Navigation & Control (GNC) › Gyroscope — angular velocity measurement, bias, noise

Parent note Gyroscope — angular velocity, bias, noise bahut saari notation ek saath throw karta hai: $\omega$, $\int$, $\dot{b}$, $\delta$, PSD $Q$, $\sqrt{t}$. Neeche hum har ek ko earn karte hain, us order mein jisme ek doosre pe depend karte hain. Koi bhi cheez use karne se pehle draw ki jaayegi.

1. Angle $\theta$ — "kitna ghuma"

Picture. Ek arrow draw karo (drone ki "naak"). Ab use rotate karo. Jahan se shuru hua aur jahan ab point kar raha hai — beech ka wedge hi $\theta$ hai.

Topic ko yeh kyun chahiye. Navigation ultimately is sawaal ka jawaab chahta hai ki "main kis taraf face kar raha hoon?" — woh jawaab ek angle hai. Gyro jo kuch bhi karta hai woh $\theta$ paane ke liye karta hai.

2. Time $t$ aur rate ka idea

Ek rate ka jawaab hota hai "har second mein kitna?". Speed, position ki rate hai (metres per second). Hum abhi angle ki rate milne waale hain.

3. Angular velocity $\omega$ — show ka star

Picture. Agar $\theta$ hai naak kahan point karti hai, toh $\omega$ hai naak kitni jaldi swing ho rahi hai. Tez spin = bada $\omega$; still baithna = $\omega =0$.

Derivative kyun aur kuch simple kyun nahi? Kyunki turning speed har moment change ho sakti hai. $\Delta \theta /\Delta t$ jaisa ratio sirf time ke ek chunk ka average deta hai; derivative $\frac{d\theta }{dt}$ instantaneous rate deta hai — exactly wahi jo gyro har tick pe report karta hai.

4. Integration $\int$ — saare chote turns jodna

Kyunki gyro sirf $\omega$ deta hai (kitni tezi), hum $\theta$ (kitna door) recover karte hain $\omega$ ko time ke saath accumulate karke. Woh accumulation hi integration hai.

Picture. $\omega$ ko time ke against draw karo. Us curve ke neeche time $t$ tak ka area hi $\theta (t)$ hai. Chauda-aur-ooncha region → zyada angle accumulate.

Integration mein trouble kyun hai. Agar har $\omega$ reading ek fixed amount se thodi zyada ho, toh sum us error ko slice by slice pile karta jaata hai — error accumulate hoti hai. Yahi drift ka seed hai, parent note ka central villain.

5. Bias $b$ — woh offset jo hona hi nahi chahiye

Topic ko yeh kyun chahiye. Jab tum non-zero $b$ ko integrate karte ho, ek fake angle accumulate hoti hai jo steadily badhti hai — parent note ka linear drift $bt$.

6. Dot $\dot{b}$ — "rate of change of"

7. Noise $n$ — kaanpti jitter

Picture. Bias poori reading ko ek fixed step se upar shift karta hai; noise us shifted value ke around use fuzzy banata hai.

8. Averages, variance, aur wiggle-size $\sigma$

Random cheezein discuss karne ke liye humein ek tarika chahiye "wiggle kitni badi hai" measure karne ka.

Pehle square kyun phir square root? Squaring over-shoots aur under-shoots dono ko positive spread count karaata hai; final square root number ko sensible units mein wapas laata hai ($°$ instead of ${°}^2$).

9. Delta spike $\delta$ aur power spectral density $Q$

Parent ke ARW derivation mein do aur symbols hain. Yahan hain, seedha saadha.

10. Square root ​ aur log-log plots (Allan chart ke liye)

Foundations topic ko kaise feed karti hain

Har arrow kehta hai "right box samajhne se pehle left box samajhna zaroori hai." Poori chain drift pe khatam hoti hai, jisse parent note apni poori energy se ladta hai.

Equipment checklist

Jawaab cover karo aur khud test karo. Agar koi line fuzzy lage, parent note kholne se pehle uska section upar dobara padho.

Ready ho? Ab parent note ka model $\tilde{\omega }={\omega }_{\text{true}}+b+n$ aur uske drift laws seedha English jaisa padhenge. Related tools jo in foundations pe build karte hain: Coriolis Force, Sagnac Effect, Random Walk & Wiener Process, Allan Variance Analysis, Kalman Filter, Accelerometer — specific force, gravity, Inertial Measurement Unit (IMU), Attitude Estimation / Dead Reckoning.