Visual walkthrough — Gyroscope — angular velocity measurement, bias, noise

3.5.14 · D2 · Physics › Guidance, Navigation & Control (GNC) › Gyroscope — angular velocity measurement, bias, noise

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Step 1 — "Angular velocity" ka matlab kya hai

KYA HAI. Ek gyroscope har axis ke liye ek number report karta hai: angular velocity $\omega$. Yeh turning ki speed hai, radians per second (rad/s) mein measure ki jaati hai. Ek radian bas ek angle-size hai: ek radian ghoomne se utna hi arc sweep hota hai jitna radius hai. Ek poora circle $2\pi \approx 6.28$ radians hota hai.

YAHAN SE KYUN SHURU KAREN. "Angle mein error" ki baat tab tak nahi ho sakti jab tak aap agree na kar lo ki gyro aapko actually kya deta hai. Woh aapko angle $\theta$ nahi deta. Woh deta hai yeh ki $\theta$ kitni tezi se badal raha hai. Yahi distinction har drift problem ki jad hai.

PICTURE. Neeche, red arrow ek spinning object hai. $\omega$ yeh hai ki arrow har second kitne radians sweep karta hai. Angle $\theta$ (black wedge) woh jagah hai jahan arrow hai; $\omega$ yeh hai ki woh kitna tezi se move karta hai.

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Step 2 — Angle recover karna: integrate kyun karna padta hai

KYA HAI. Kyunki $\omega =d\theta /dt$, time $t$ par angle shuruat se ab tak $\omega$ ka running total ("curve ke neeche ka area") hai: $\theta (t)=\underset{\text{jahan se shuru kiya}}{\underbrace{{\theta }_0}}+\underset{\text{sabhi chote turns ka sum}}{\underbrace{{\int }_0^t\omega (\tau )d\tau }}$

INTEGRATE KI KYON, AUR KYA NAHI? Integration woh tool hai jo derivative ko reverse karta hai — yeh sawaal ka jawab deta hai "agar main har pal ki rate jaanta hoon, toh accumulated total kya hai?" Derivative poochta hai "kitni tezi se?"; integral poochta hai "kul kitna?" Hum total turn chahte hain, isliye integrate karte hain. Symbol ${\int }_0^t\dots d\tau$ ka matlab literally hai "time $0\to t$ ko $d\tau$ width ke slices mein kaato, har slice ki rate $\omega$ ko uski width se multiply karo ek tiny turn paane ke liye, aur sab jod do." ($\tau$ sirf running time ka ek dummy naam hai sum ke andar, taaki end-time $t$ se confuse na ho.)

PICTURE. Angle woh shaded area hai $\omega$-vs-time curve ke neeche. Har patli red column ek sliver $\omega (\tau )d\tau$ hai.

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Step 3 — Real reading $\omega$ nahi hai: bias aur noise enter karte hain

KYA HAI. Ek perfect gyro $\tilde{\omega }={\omega }_{\text{true}}$ report karta (tilde $\stackrel{~}{}$ ka matlab sirf "measured version" hai). Ek real gyro teen cheezein stack karke report karta hai: $\underset{\text{jo hum padhte hain}}{\underbrace{\tilde{\omega }(t)}}=\underset{\text{woh sach jo chahiye}}{\underbrace{{\omega }_{\text{true}}(t)}}+\underset{\text{bias: ek slow offset}}{\underbrace{b(t)}}+\underset{\text{noise: fast jitter}}{\underbrace{n(t)}}$

YEH DO EXTRAS KYUN? Har electronic sensor mein hota hai (1) standing offset — woh kuch read karta hai jab kuch bhi move nahi ho raha, jaise bathroom scale jo empty mein $1\text{kg}$ dikhaye — ise bias $b$ kehte hain; aur (2) thermal jitter — heat-agitated electrons se random flicker — ise noise $n$ kehte hain. Yeh physically alag cheezein hain, aur hum inhe deliberately alag rakhte hain.

PICTURE. Same true rate, teen signals: clean truth (black), truth jo steady bias se upar shift ho gayi (red offset), aur truth jo fuzzy noise mein dab gayi (grey fuzz).

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Step 4 — Error ko isolate karne ke liye truth ko freeze karo

KYA HAI. Sirf errors ki damage study karne ke liye, true rate zero set karo: ${\omega }_{\text{true}}=0$. Gyro bilkul still baitha hai. Ab reading pure error hai: $\tilde{\omega }(t)=b(t)+n(t)$ Ise integrate karo woh angle paane ke liye jo gyro sochta hai usne turn kiya hai (use zero sochna chahiye): $\hat{\theta }(t)={\int }_0^t\tilde{\omega }d\tau =\underset{\text{bias drift}}{\underbrace{{\int }_0^{t}bd\tau }}+\underset{\text{noise drift}}{\underbrace{{\int }_0^{t}nd\tau }}$

TRUTH ZERO KYUN SET KAREN? Yeh ek controlled experiment hai. Agar object sach mein turn nahi kar raha, toh gyro jo bhi angle report kare woh 100% error hai — aur kisi cheez ko blame karne ka mauka nahi. Yeh drift ko do independent pieces mein saaf split karta hai jo hum ek ek karke study kar sakte hain. ($\hat{}$ hat ka matlab hai "hamaara estimate.")

PICTURE. Ek stationary object (bilkul turn nahi kar raha) jis par gyro laga hua hai; dial dheere dheere zero se creep karta jaata hai jabki kuch bhi move nahi hua.

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Step 5 — Bias integrate hoke → ek seedhi ramp (badhta hai $t$ ki tarah)

KYA HAI. Sirf bias term lo aur assume karo yeh constant hai, $b(t)=b$. Integral mein se ek constant bahar aata hai: ${\theta }_{\text{bias}}(t)={\int }_0^{t}bd\tau =b\underset{=t}{\underbrace{{\int }_0^{t}d\tau }}=\underset{\text{slope}}{\underbrace{b}}\cdot \underset{\text{time}}{\underbrace{t}}$

CONSTANT SEEDHI LINE KYUN BANTA HAI? Ek flat, unchanging value ko integrate karna sirf use kitne time se add kar rahe ho se multiply karta hai. Har second $b$ add karo $t$ seconds ke liye aur paoge $b\cdot t$. Graph par yeh origin se guzarti seedhi line hai slope $b$ ke saath — ek unstoppable ramp. Koi cancellation nahi, kyunki offset hamesha same direction mein push karta hai.

PICTURE. Constant bias (flat red line, left) integrate hokar seedhi rising ramp ban jaati hai (right). Ek flat line ke neeche ka area ek aisa rectangle hai jo bas chauda hota jaata hai.

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Step 6 — Noise integrate hoke → ek random walk (badhta hai $\sqrt{t}$ ki tarah)

KYA HAI. Ab sirf noise term. White noise $n$ random aur zero-mean hai, toh yeh ek hi direction mein pile up nahi ho sakti — har moment yeh kisi random direction mein push karta hai. Iska integral ek random walk hai (ek "drunkard's walk"): random steps se bana ek path. Jahan aap end up karte hain uski spread $t$ ki tarah nahi, balki $\sqrt{t}$ ki tarah badhti hai.

$\sqrt{t}$ KYUN, $t$ NAHI? Kyunki random steps partially cancel karte hain. Agar aap coin flip karo aur left/right step lo, $N$ steps baad aap typically ghar se $\sqrt{N}$ steps door hote ho — $N$ nahi, kyunki roughly aadhe lefts rights ko undo karte hain. Zyada time = zyada steps, par cancellation wandering ko slow rakhti hai. Exact statement noise power spectral density $Q$ use karta hai (gyro kitni jitter-power per Hz output karta hai, units $(\text{rad/s}{)}^2/\text{Hz}$): $\mathrm{Var}[{\theta }_{\text{noise}}(t)]=Qt\Rightarrow {\sigma }_{\theta }(t)=\sqrt{Q}\sqrt{t}$ Yahan $\mathrm{Var}$ spread squared hai aur ${\sigma }_{\theta }$ standard deviation hai (typical error size). Square-root of time ek random walk ki pehchaan hai.

DELTA FUNCTION KA MATLAB KYA HAI. Neeche ki derivation mein symbol $\delta (\tau -{\tau }^{\prime })$ use hota hai, Dirac delta. Ise ek infinitely tall, infinitely thin spike samjho jo $\tau ={\tau }^{\prime }$ par baitha hai, jiski area exactly $1$ hai. Yeh mathematician ka tarika hai yeh kehne ka ki "sirf ek instant par nonzero." White noise ke liye yeh ek physical fact encode karta hai: do alag times par jitter bilkul unrelated hai — ab ka noise jaankar ek millisecond baad ka noise nahi pata chalta. Toh $\mathbb{E}[n(\tau )n({\tau }^{\prime })]$ (do times par noise ke product ka average) zero hai jab tak $\tau ={\tau }^{\prime }$ na ho, aur spike ki area strength $Q$ set karti hai.

VAR LINEAR KYUN BADHTA HAI? Likho $\theta ={\int }_0^{t}nd\tau$, ise square karo (do dummy times $\tau ,{\tau }^{\prime }$ par double integral), aur delta rule use karo: $\mathrm{Var}[\theta ]={\int }_0^t{\int }_0^t\underset{\text{noise correlation}}{\underbrace{Q\delta (\tau -{\tau }^{\prime })}}d\tau d{\tau }^{\prime }=Q\underset{=t}{\underbrace{{\int }_0^{t}d\tau }}=Qt.$ Delta spike sirf diagonal $\tau ={\tau }^{\prime }$ par nonzero hai, isliye do integrals mein se ek collapse ho jaata hai (woh spike "kha" leta hai aur $1$ return karta hai), ek single ${\int }_0^{t}d\tau =t$ bachta hai.

PICTURE. Bahut se random-walk paths (grey) zero se fan out karte hain; red envelope $\pm \sqrt{Q}\sqrt{t}$ hai — ek sideways parabola, wide par seedhi ramp nahi.

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Step 7 — Dono ko side by side rakho: line vs curve

KYA HAI. Ek graph par, dono errors ko time ke against plot karo: bias ramp $bt$ (straight) aur ARW envelope $\text{ARW}\sqrt{t}$ (bending over). Chhote $t$ ke liye $\sqrt{t}$ term actually taller hoti hai (kyunki $\sqrt{t}>t$ jab $t<1$); par seedhi line eventually kisi bhi square-root curve ko overtake kar leti hai aur kabhi peeche nahi dekhti.

KYU MATTERS. Yeh crossover batata hai ki kis enemy se ladna hai. Seconds-to-minutes mein, bias ya noise dominate karta hai yeh actual numbers par depend karta hai — par kyunki ramp chadti rehti hai aur curve flatten ho jaata hai, bias hamesha end mein jeetta hai. Yahi engineering headline hai: pehle bias calibrate karo.

PICTURE. Red seedha ramp (bias) black bending curve (noise) ko cross karta hua. Crossover mark karo.

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Step 8 — Degenerate cases (koi bhi scenario uncovered mat chodo)

Teen edge cases picture complete karte hain:

Case A — zero bias, zero noise. Ek perfect gyro. Tab $\hat{\theta }(t)=0$ hamesha: flat lines, koi drift nahi. Yeh woh target hai jo Step 1 ne establish kiya tha.

Case B — sirf noise, $b=0$. Koi ramp nahi. Error ek pure random walk hai jo zero ke aas paas hover karta hai, spread $\sqrt{Q}\sqrt{t}$. Iska koi preferred direction nahi hota wander karne ka — bahut saare runs average karo toh zero milta hai. Yeh single regime hai jahan "average longer to improve" actually kaam karta hai.

Case C — bias bhi drift karta hai, $\dot{b}=n_b$. Reality mein bias khud bhi dheere dheere wander karta hai (temperature, ageing). Hum ise model karte hain yeh keh ke ki bias apne khud ke white noise $n_b$ se driven hai, jiski strength ek power spectral density $Q_b$ hai (units $({\text{rad/s}}^2{)}^2/\text{Hz}$, yaani slope khud kitna randomly kick hota hai, per Hz).

Layer 1 — bias ek random walk ban jaata hai. Yeh bilkul Step 6 hai dobara: white noise $n_b$ ek baar integrate karo aur ek random walk milta hai jiska spread $\sqrt{t}$ ki tarah badhta hai: $b(t)={\int }_0^t{n}_{b}d\tau \Rightarrow \mathrm{Var}[b(t)]=Q_{b}t\Rightarrow {\sigma }_b(t)=\sqrt{Q_b}\sqrt{t}.$

Layer 2 — us wandering bias ko angle mein feed karo. Angle error hai ${\theta }_{\text{bias}}(t)={\int }_0^{t}b(\tau )d\tau$, toh hum random walk ko ek baar aur integrate karte hain — white noise do baar integrate hua. Yahan WHY concrete hai: noise ke delta-correlated do nested integrals dete hain: $\mathrm{Var}[{\theta }_{\text{bias}}(t)]={\int }_0^t{\int }_0^t{Q}_b\min (\tau ,s)d\tau ds=\frac{Q_b{t}^3}{3}.$ Square root lo typical size paane ke liye: $\boxed{\underset{\text{drifting bias se typical angle error}}{\underbrace{{\sigma }_{\theta ,\text{bias}}(t)}}=\underset{\text{ek constant}}{\underbrace{\sqrt{\frac{Q_b}{3}}}}\underset{\sqrt{t}\text{se tezi badhta hai}}{\underbrace{t^{3/2}}}}$

$t^{3/2}$ INTUITIVELY KYUN? Har extra integration growth ko roughly ek power of $\sqrt{t}$ se multiply karta hai: white noise ek baar integrate hone par $t^{1/2}$ ki tarah wander karta hai (Step 6); doosri baar integrate hone par $t^{1/2}\cdot t=t^{3/2}$ ki tarah wander karta hai. Toh ek drifting bias angle ko fast noise ke $\sqrt{t}$ se bure tarike se corrupt karta hai, phir bhi fixed bias ke clean $t$ ramp se dheemal — yeh dono ke beech hai, same delta-integral machinery se bana, sirf do baar apply kiya. Aur kyunki $\dot{b}=n_b\ne 0$, aap ek baar bias subtract nahi kar sakte aur done ho jao: ise continuously re-estimate karna padta hai (ek Kalman filter jo GPS/accel/mag fuse kare).

PICTURE. Teen stacked panels: (A) flat perfect line, (B) zero-mean wander $\propto \sqrt{t}$, (C) ek ramp jiska slope khud dheere dheere drift karta hai, uska envelope $t^{3/2}$ ki tarah khulta hai.

Recall Teen growth laws ek jagah

Rate par white noise → angle error $\propto \sqrt{t}$ (white noise ek baar integrate karo). Constant bias → angle error $\propto t$ (ek constant integrate karo). Random-walk (drifting) bias → angle error $\propto t^{3/2}$ (white noise do baar integrate karo).

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Ek-picture summary

Is poore page ki cheezein ek hi frame mein: ek still gyro (true rate zero) integral se guzarta hai, seedhe red ramp (bias, $\propto t$) aur bending noise envelope ($\propto \sqrt{t}$) mein split hota hai, jo ek marked time par cross karte hain.

Recall Feynman style: poora walkthrough seedhe shabdon mein

Gyro ek "main kitni tezi se ghoom raha hoon" meter hai, na ki "main kis taraf face kar raha hoon" meter. Yeh jaanne ke liye ki aap kis taraf face kar rahe ho, aap sab "kitni tezi se" ko time ke upar add karte ho — yahi integrate karna hai. Ab yahan catch hai: meter hamesha thoda sa off read karta hai jab aap bilkul still baithe ho bhi. "Off" hone ke do type hain. Pehla hai ek steady lean, hamesha same direction mein — yahi bias hai, jaise ek scale jo empty mein 1 kg dikhaye. Ek steady lean baar baar add karo aur woh seedhe ramp ki tarah badhta hai: double time, double error. Doosra hai nervous shaking jo har instant randomly jump karta hai — yahi noise hai. Kyunki yeh dono taraf jump karta hai, aadha khud cancel ho jaata hai, isliye uska damage sirf time ki square-root ki tarah badhta hai (chaar guna zyada time = sirf do guna wander). Dono ko ek graph par rakho: ramp seedhi line hai, noise ek curve hai jo bend ho jaata hai aur flatten ho jaata hai. Seedhi line hamesha lamba race jeetti hai. Toh agar sirf ek cheez fix karni ho, bias fix karo — aur kyunki bias bhi temperature ke saath dheere dheere wander karta hai, ise ek baar fix karke bhool nahi sakte; GPS ya compass se hamesha re-check karte rehna hoga. (Yeh saara maths radians mein hai; hum degrees mein sirf tab flip karte hain jab number print karna ho.)

Recall Quick self-check

Q: Gyro angle measure karta hai ya angular velocity? A: Angular velocity — angle sirf integrate karne ke baad milta hai.

Q: Bias error kis tarah badhta hai? A: $t$ (seedha ramp).

Q: Noise (ARW) error kis tarah badhta hai? A: $\sqrt{t}$ (random walk).

Q: Ek drifting bias angle ko kis tarah corrupt karta hai? A: $t^{3/2}$ (white noise do baar integrate karo).

Q: Noise ke liye $t$ nahi $\sqrt{t}$ kyun? A: Random steps partially cancel karte hain, isliye spread count se dheemal badhta hai.

Q: Bias sirf ek baar subtract kyun nahi kar sakte? A: Kyunki $\dot{b}=n_b\ne 0$ — bias khud dheere dheere drift karta hai.

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Connected ideas: Coriolis Force · Sagnac Effect · Accelerometer — specific force, gravity · Inertial Measurement Unit (IMU) · Kalman Filter · Attitude Estimation / Dead Reckoning · Allan Variance Analysis · Random Walk & Wiener Process