Gyroscope — angular velocity measurement, bias, noise: Question bank

3.5.14 · D5 · Physics › Guidance, Navigation & Control (GNC) › Gyroscope — angular velocity measurement, bias, noise

Ek cheez yaad rakho jo yahan sab kuch anchor karti hai: ek rate gyro angular velocity $ω$ (rad/s) measure karta hai, aur orientation $θ$ tabhi milta hai jab tum $ω$ ko time ke saath integrate karo — aur yahi woh jagah hai jahan bias aur noise leak hoti hai.

Definition Is page par use ki gayi Notation (pehle padho)

$ω_{true}$ — woh actual rotation rate jis par duniya ghoom rahi hai.
$\tilde{ω}$ — tilde ka matlab hai "measured": yeh woh raw number hai jo gyro chip report karta hai, jo truth plus imperfections ke barabar hota hai. Tilde "sensor ne kya kaha" ke liye standard symbol hai, na ki "kya sach hai".
$b (t)$ — bias, ek slow offset jo reading mein hota hai tab bhi jab kuch rotate nahi ho raha.
$n (t)$ — white noise, reading par fast zero-mean jitter.
$n_{b} (t)$ — ek doosra, alag tiny white noise jo bias ko wander karne par drive karta hai. Yeh reading-noise $n$ nahi hai; yeh woh chota random push hai jo $b$ ko khud time ke saath change karta hai, $\dot{b} = n_{b}$ likha jata hai (bias rate = ek random nudge).
$S$ — scale factor: woh gain jo true rate ko reported rate mein convert karta hai, ideally exactly $1$ .

Scale factor ke saath fuller measurement model yeh hai: $\tilde{ω} = S ω_{true} + b + n, \dot{b} = n_{b} .$

Neeche do figures teen error families ko concrete banati hain — reveals se pehle inhe study karo.

Bias ramp vs. random walk. Dekho kaise ek constant offset (coral) ek seedhi line climb karta hai jabki integrated noise (lavender) $t$ ki tarah wander karti hai:

Figure — Gyroscope — angular velocity measurement, bias, noise

Allan-deviation "bathtub." Girta hua mint branch average kiya ja sakne wala white noise hai; flat butter floor bias-instability limit hai jise tum beat nahi kar sakte; utha hua coral branch drift hai jo jeet raha hai:

True or false — justify

Ek gyroscope directly tumhara orientation angle output karta hai.

False. Yeh angular rate

ω

output karta hai; angle hai

θ_{0} + \int ω d t

, aur woh integration hi woh jagah hai jahan tiny errors drift mein accumulate hoti hain.

Bias error aur noise error time ke saath same speed se badhte hain.

False. Constant bias ek ramp mein integrate hota hai (

\propto t

); zero-mean white noise ek random walk mein integrate hoti hai (

\propto t

), jo bahut slow badhti hai.

Zyada time tak gyro output ko average karna hamesha error reduce karta hai.

False. Yeh sirf Allan curve ke white-noise (

- 1/2

slope) region mein help karta hai; minimum ke baad, bias instability dominate karti hai aur zyada averaging cheezein worse bana deti hai.

Bias term

b

ek fixed constant hai jo tum ek baar measure karke hamesha ke liye subtract kar sakte ho.

False. Bias temperature aur time ke saath slowly wander karta hai (

\dot{b} = n_{b}

, tiny random driving noise), isliye ise ek baar subtract karne ki bajay continuously re-estimate karna padta hai.

Gyro reading of zero guarantee karta hai ki device perfectly still hai.

False. Rest par output

b (t) + n (t)

hota hai, zero nahi; ek momentary zero bas noise ho sakti hai jo bias offset se through cross kar rahi ho.

Scale-factor error aur bias same kind ki error hain.

False. Bias

b

zero rate par bhi constant offset add karta hai; scale-factor error ka matlab hai

S \neq = 1

, isliye yeh rate ke saath badh*ti hai — yeh tabhi dikhti hai jab tum actually rotate kar rahe ho aur

ω_{true}

ke proportional hoti hai.

Angle Random Walk (ARW) angle estimate ka wander describe karta hai, chahe yeh rate noise se aaye.

True. Rate par white noise, ek baar integrate hone par, angle mein ek Wiener process ban jaata hai jiska standard deviation

ARW \cdot t

ki tarah badhta hai.

Ek zyada expensive gyro jiska ARW lower hai, automatically uski bias instability bhi lower hogi.

False. ARW (high-frequency noise) aur bias instability (low-frequency flicker floor) alag processes hain; ek gyro quiet ho sakta hai phir bhi drift kare, ya noisy ho phir bhi stable rahe.

Woh Coriolis force jise MEMS gyro exploit karta hai ek "real" push hai jo tum inertial frame mein feel kar sakte ho.

False. Yeh ek frame-dependent effect hai: rotating body frame mein vibrating mass sideways shoved appear hota hai, aur woh apparent deflection hi

ω

encode karti hai. Dekho Coriolis Force.

Ek optical gyro (RLG/FOG) aur ek MEMS gyro fundamentally alag quantities measure karte hain.

False. Dono same angular rate

ω

measure karte hain; sirf physics alag hai — MEMS Coriolis deflection use karta hai, optical Sagnac phase use karta hai.

Pure-integration time ko double karne se ARW angle uncertainty double ho jaati hai.

False. Uncertainty

t

ki tarah scale karti hai, isliye time double karne par woh

2 \approx 1.41

se multiply hoti hai, 2 se nahi — random increments partially cancel ho jaate hain.

Spot the error

"Gyro quiet hai, isliye main poori flight ke liye akele ise integrate kar lunga."

Error hai long-term integration par trust karna: even ek quiet gyro ka small bias bina bound ke ramp up karta hai (

b t

), isliye tumhe GPS/mag/accel ke saath filter ke zariye reset karna hi padega.

"White noise ka ek mean hota hai, isliye ise integrate karne se bias ki tarah steady drift milti hai."

White noise zero-mean hoti hai; iska integral ek random walk hai jo dono taraf wander karta hai, steady one-directional ramp nahi. Sirf nonzero bias hi directed drift produce karta hai.

"Maine rest par reading zero ki, toh gyro fully calibrated hai."

Rest par zero karna sirf us instant ka bias remove karta hai; scale-factor error (

S \neq = 1

) ke liye kuch nahi karta, jo sirf actual rotation ke dauran reveal hoti hai aur still rehte hue chupi rehti hai.

"Allan deviation ka rising (

+ 1/2

) branch matlab hai sensor time ke saath noisier ho raha hai."

Rise low-frequency flicker/bias instability / random walk ke takeover ko reflect karta hai, high-frequency noise ko nahi; bathtub minimum se aage averaging zyada der tak kaam karna band kar deti hai.

0.02 °/ s

ka bias negligible hai — yeh ek tiny number hai."

Integrate karne par, woh "tiny" rate

b t

ban jaata hai; ek ghante mein yeh

72°

ka heading error hai, isliye small rates navigation ka #1 dushman hain.

"Drift fix karne ke liye mujhe gyro ki sample rate faster karni chahiye."

Faster sampling zyada high-frequency detail capture karta hai lekin bias ramp ke baare mein kuch nahi karta; ilaaj hai drifting bias ko estimate aur subtract karna, jaise Kalman Filter ke saath.

"Allan bathtub ka bottom woh noise hai jo main average karke zero kar sakta hun."

Bathtub floor bias instability hai jo flicker noise se set hoti hai — exactly woh part jise tum average away NAHI kar sakte; yeh device ki practical stability limit hai.

"3-axis gyro akela mujhe full position aur heading de sakta hai."

Gyro sirf orientation rate deta hai; position ke liye ek accelerometer se acceleration integration chahiye, aur dono milke ek Inertial Measurement Unit (IMU) banate hain jo dead reckoning ko feed karta hai.

Why questions

Bias time ke saath linearly kyun badhta hai lekin noise sirf

t

ki tarah?

Ek constant ko integrate karne par ramp milta hai (

\int b d t = b t

), jabki independent random noise increments jab sum hote hain toh partially cancel ho jaate hain, isliye unka spread

t

ki tarah badhta hai (ek Wiener process — dekho Random Walk & Wiener Process).

Integration kyun hai, sensor khud nahi, "drift" ka source?

Raw rate error bounded rehti hai; yeh accumulation

\int (b + n) d t

hi hai jo even small errors ko bina limit ke growing orientation error mein pile up karne deta hai.

Model mein sirf

ω

ki jagah tilde

\tilde{ω}

kyun appear karta hai?

Tilde measured value ko flag karta hai — woh jo chip report karta hai — taaki hum ise truth plus imperfections (

S ω_{true} + b + n

) ke roop mein likh sakein aur phir truth recover karne ke liye model invert kar sakein.

Bias ke liye

n

ko reuse karne ki bajay alag alag noise

n_{b}

kyun introduce karte hain?

n

reading par fast jitter hai, jabki

n_{b}

slow random push hai jo bias ko wander karne par drive karta hai (

\dot{b} = n_{b}

); yeh alag timescales par kaam karte hain aur alag math chahiye, isliye inhe mix karne se distinction erase ho jaati.

Allan deviation

τ

averaging time ke against log-log plot kyun use karta hai?

Alag noise processes characteristic slope ki seedhi lines ke roop mein appear hote hain (

- 1/2

ARW ke liye,

+ 1/2

random walk ke liye), isliye log-log unhe cleanly alag karta aur identify karta hai — dekho Allan Variance Analysis.

Averaging bias instability ko white noise ki tarah remove kyun nahi kar sakta?

Bias instability slow frequencies par concentrated flicker (

1/ f

) noise se aati hai; averaging fast fluctuations suppress karta hai lekin slow ones ko untouched chhod deta hai, isliye flicker floor survive karta hai.

Short-duration application mein noise se pehle bias calibrate kyun karein?

Kuch seconds mein hi bias ramp (

\propto t

) noise wander (

\propto t

) se zyada ho jaata hai, isliye bias pehla dominant villain hai — 80/20 fix.

Kalman filter bias ko ek baar ki bajay continuously kyun re-estimate karta hai?

Kyunki

\dot{b} = n_{b}

ka matlab hai bias khud temperature aur time ke saath drift karta hai; ek baar ka correction stale ho jaata hai, isliye filter ise GPS/mag/accel se update karta rehta hai.

Edge cases

Agar true rate exactly zero hai, toh kya integrated angle zero rehta hai?

Nahi —

ω_{true} = 0

ke saath estimate

\int (b + n) d t

ban jaata hai, jo phir bhi bias se ramp karta hai aur noise se wander karta hai, isliye perfect rest par bhi angle drift karta hai.

ω_{true} = 0

ke saath, kya scale-factor error kuch contribute karta hai?

Nahi — scale-factor error

S ω_{true}

hai, jo zero ho jaata hai jab true rate zero ho, isliye yeh rest par invisible hai aur sirf high-rate maneuvers ke dauran bite karta hai.

t = 0

par ARW angle uncertainty ka kya hota hai?

Yeh exactly zero hoti hai, kyunki

σ_{θ} = ARW \cdot t \to 0

; uncertainty sirf tab accumulate hoti hai jab tum elapsed time ke saath integrate karte ho.

Agar bias truly perfectly constant aur known hota, toh kya drift bilkul nahi hoti?

Subtract hone par bias ramp vanish ho jaata, lekin noise-driven random walk (

\propto t

) baki rehti, isliye kuch slow angle wander persist karta.

Exact Allan-deviation minimum par, kya averaging help kar rahi hai ya hurt?

Na kuch — yeh woh turnover point hai jahan falling white-noise branch aur rising flicker/bias-instability branch balance karti hain, best achievable stability deti hain.

Extremely long averaging time ki limit mein, kaunsa process error dominate karta hai?

Random-walk / bias-instability branch, jiska Allan deviation

τ

ke saath rise karta hai, isliye bahut long averaging hamesha estimate ko improve karne ki bajay worsen karta hai.

Agar do gyros ka ARW equal hai lekin ek ka bias double hai, toh kya woh 2-hour flight ke liye equally good hain?

Nahi — ghanton mein linear bias ramp dominate karta hai, isliye higher-bias unit identical noise specs ke bawajood bahut zyada drift karta hai.