Gyroscope — angular velocity measurement, bias, noise

3.5.14 · Physics › Guidance, Navigation & Control (GNC)

Gyroscope ASAL mein kya measure karta hai?

Navigation ke liye angular velocity kyun zaroori hai? Kyunki orientation (attitude) angular velocity ka time-integral hai. Agar aap jaante hain aapne kahan se start kiya aur $ω (t)$ bilkul theek jaante hain, toh aap apni orientation hamesha ke liye jaante hain: $θ (t) = θ_{0} + \int_{0}^{t} ω (τ) d τ$

Physical gyros $ω$ kaise sense karte hain? Do common principles:

MEMS Coriolis gyro (phones, drones): ek chota proof mass vibrate karta hai. Jab device rotate karta hai, Coriolis force $F_{c} = - 2 m ω \times v$ mass ko sideways push karta hai. Woh sideways deflection capacitively measure hoti hai aur $\propto ω$ hoti hai.
Optical (RLG / FOG): Sagnac effect use karta hai — ek rotating loop mein counter-propagating light beams ek phase difference accumulate karte hain jo $\propto ω$ hoti hai.

Gyro measurement model SCRATCH se derive karna

Hum kabhi bhi raw sensor par trust nahi karte. Hum ek model banate hain ki sensor asal mein kya output karta hai, phir use invert karte hain.

Step 1 — Ideal se shuru karo. Ek perfect gyro true rate output karta hai: $\tilde{ω} = ω_{true}$ Yeh step kyun? Imperfections add karne se pehle target establish karo.

Step 2 — Ek dheere-dheere varying offset (bias) add karo. Real electronics mein zero rotation par bhi ek offset hota hai. Use $b (t)$ kahte hain: $\tilde{ω} = ω_{true} + b (t)$ Yeh step kyun? Rest par measured output zero nahi hoti — woh constant-ish part bias hai.

Step 3 — Random measurement noise add karo. Thermal/electronic fluctuations ek jitter $n (t)$ add karte hain, jise white noise ki tarah model kiya jaata hai: $\tilde{ω} (t) = ω_{true} (t) + b (t) + n (t)$ Yeh step kyun? Koi bhi real reading smooth nahi hoti; upar high-frequency randomness hoti hai.

Step 4 — Bias ko khud model karo. Bias bilkul perfectly constant nahi hoti; yeh bhatakti rehti hai. Standard model ek random walk hai jo apne white noise $n_{b}$ se drive hota hai: $\dot{b} (t) = n_{b} (t)$ Yeh step kyun? Minutes/hours ke dauran offset dheere-dheere drift karta hai — yeh ek chota white noise integrate karke capture hota hai.

Toh poora gyro error model do coupled equations hain: ek fast noise $n$ aur ek slow drifting bias $b$ .

Errors drift mein kaise badalte hain (woh derivation jo matter karti hai)

Maan lo true rate zero hai ( $ω_{true} = 0$ ) aur hum reading ko integrate karke angle estimate $\hat{θ}$ paate hain: $\hat{θ} (t) = \int_{0}^{t} \tilde{ω} d τ = \int_{0}^{t} b d τ + \int_{0}^{t} n d τ$

Bias ka contribution. Agar $b$ kabhi kabhi constant hai: $θ_{bias} (t) = b t \Rightarrow error time mein LINEARLY badhti hai$ Kyun? Ek constant offset integrate hone par ek ramp deta hai. $0.01°/ s$ bias → ek ghante baad $36°$ error. Isliye bias #1 dushman hai.

Noise ka contribution (Angle Random Walk). White noise integrate hone par ek random walk banta hai. Uski standard deviation $t$ ki tarah badhti hai: $σ_{θ} (t) = ARW \cdot t$

Do growth laws ka summary:

Error source	Badhta hai	Jaisa lagta hai
Bias $b$	$\propto t$ (linear ramp)	steady slow tilt
White noise (ARW)	$\propto t$ (random walk)	jittery wander

Figure — Gyroscope — angular velocity measurement, bias, noise

Allan Deviation — gyro ka spec sheet padhna

Slope $- 1/2$ region → white noise / ARW dominate karta hai. Zyaada der average karna help karta hai (error girta hai).
Curve ka minimum → best achievable stability.
Slope $+ 1/2$ region → bias instability / random walk dominate karta hai. Zyaada der average karna NUKSAN karta hai (drift jeet jaata hai).

"Bathtub" ka bottom bias instability deta hai, woh floor jo aap average out nahi kar sakte.

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum ek office chair par aankhein band karke ghoom rahe ho aur andaaza lagaane ke liye "ek-Mississippi" count kar rahe ho ki kitna muda. Ek gyroscope ek chota gadget hai jo feel karta hai tum kitni tezi se ghoom rahe ho. Yeh jaanne ke liye ki tum finally kis direction face kar rahe ho, yeh saare "kitni tezi" ko time ke saath add karta rehta hai. Mushkil: yeh gadget hamesha sochta hai ki yeh thoda ghoom raha hai jab bhi band hota hai (woh bias hai — jaise ek scale jo kuch nahi hone par bhi 1 kg dikhata hai), aur yeh thoda kaampata bhi hai (woh noise hai). Woh chota bias add hota rehta hai, isliye kuch der baad aapka "main kis direction face kar raha hoon" ka andaaza drift ho jaata hai — isliye phones aur drones bhi GPS ya compass se dekh lete hain drift theek karne ke liye.

Flashcards

Gyroscope directly kya physical quantity measure karta hai?

Angular velocity

ω

(rad/s), angle nahi.

Standard gyro measurement model likho.

\tilde{ω} = ω_{true} + b (t) + n (t)

, bias

b

aur white noise

n

ke saath.

Bias ko dynamically khud kaise model kiya jaata hai?

Ek random walk ki tarah:

\dot{b} = n_{b}

(chote white noise ka integral).

Rate ko angle mein integrate karne par bias error time ke saath kaise badhta hai?

Linearly,

θ_{bias} = b t

White-noise (ARW) angle error time ke saath kaise badhti hai?

t

ki tarah:

σ_{θ} = ARW \cdot t

(ek random walk).

Integrated white noise

t

ki tarah kyun scale hoti hai?

Var [θ] = \int_{0}^{t} \int_{0}^{t} Q δ (τ - τ^{'}) d τ d τ^{'} = Qt

, toh

σ \propto t

MEMS gyro rotation sense karne ke liye kaun sa physical effect use karta hai?

Vibrating proof mass par Coriolis force

F_{c} = - 2 m ω \times v

Optical (RLG/FOG) gyros kaun sa effect use karte hain?

Sagnac effect — counter-propagating beams ke beech phase difference

\propto ω

Allan deviation plot mein slope

- 1/2

kya indicate karta hai?

Angle Random Walk (white noise) region; zyaada der average karna help karta hai.

Allan deviation curve ka minimum kya deta hai?

Bias instability — woh noise floor jo aap average out nahi kar sakte.

ARW ke liye typically kaunse units use hote hain?

°/ h

(degrees per root-hour).

0.02°/ s

bias ko 120 s integrate karne par kaun sa angle error milta hai?

0.02 \times 120 = 2.4°

Bias ko ek baar measure karke nahi, balki continuously re-estimate kyun karna padta hai?

Yeh temperature/time ke saath drift karta hai (

\dot{b} = n_{b}

), toh ek static subtraction purani pad jaati hai.

Connections

Coriolis Force — woh force jo MEMS gyros exploit karte hain.
Sagnac Effect — optical gyro ka operating principle.
Accelerometer — specific force, gravity — complementary IMU sensor.
Inertial Measurement Unit (IMU) — gyro + accel package.
Kalman Filter — drifting bias estimate aur remove karta hai sensor fusion se.
Attitude Estimation / Dead Reckoning — jahan gyro drift feel hoti hai.
Allan Variance Analysis — sensor noise characterize karna.
Random Walk & Wiener Process — $t$ growth ke peeche ka math.