Gyroscope — angular velocity measurement, bias, noise: Worked examples

3.5.14 · D3 · Physics › Guidance, Navigation & Control (GNC) › Gyroscope — angular velocity measurement, bias, noise

Kuch bhi shuru karne se pehle, ek quick unit reminder taaki koi symbol unclear na rahe.

Definition Hamare symbols, simple shabdon mein

$ω_{true}$ — body kitni tezi se actually rotate kar rahi hai, $°/ s$ mein (degrees per second).
$\tilde{ω}$ — raw gyro reading ("tilde" mark, $\sim$ , ka matlab sirf "measured, not true" hai). Parent ke model se, $\tilde{ω} = ω_{true} + b + n$ : true rate plus bias plus noise.
$b$ — bias: woh fake rotation jo gyro report karta hai jab bilkul still ho, yeh bhi $°/ s$ mein.
$n$ — har reading par fast white-noise jitter, $°/ s$ mein.
$ARW$ — Angle Random Walk, $°/ h$ mein quoted (degrees per square-root-hour). Yeh $t$ ke aage wala coefficient hai.
$Q$ — noise power spectral density, white noise ki "taakat". Parent ne dikhaya tha ki integrated noise ki variance $Q t$ hoti hai, aur $ARW = Q$ — toh $Q$ aur $ARW$ ek hi noise level ke do naam hain (ek squared, ek nahi).
$\hat{θ}$ — reading integrate karne ke baad hamara estimated angle.
$Δ θ$ — us estimate mein error.
$t$ — elapsed integration time.

Scenario matrix

Har gyro-error question inhi cells mein se ek hai. Neeche ke examples mein cell ka tag lagaya gaya hai, aur neeche wali map figure mein har cell ko time axis par letter se index kiya gaya hai.

#	Case class	Kya badalta hai	Covered by
A	Positive bias, integrate	$b > 0$ , ramp	Ex 1
B	Negative bias, integrate	$b < 0$ , sign of error	Ex 2
C	Zero bias, pure noise	$b = 0$ , only ARW	Ex 3
D	Small- $t$ limit	$t \to 0$ : kaun sa term jeetega?	Ex 4
E	Large- $t$ / limiting	$t \to \infty$ : kaun sa term jeetega?	Ex 4
F	Exact crossover	bias line = noise line	Ex 5
G	Nonzero true rate + bias	$ω_{true} \neq = 0$ , signal ko error se alag karo	Ex 6
H	Full realistic case	true rate + bias + noise sab ek saath	Ex 7
I	Real-world word problem	drone, unit soup, decision	Ex 8
J	Exam-style twist	Allan slope / averaging trap	Ex 9

Figure — Gyroscope — angular velocity measurement, bias, noise

Upar wali figure is poore page ka map hai: horizontal axis elapsed time $t$ seconds mein hai aur vertical axis angle error degrees mein. Seedhi red line bias error hai ( $\propto t$ ), curved blue line noise error hai ( $\propto t$ ). Ye ek baar cross karti hain, yellow dot par. Time axis par D, F, E ke chhote labelled markers literally upar wali matrix cells hain: D $t = 1$ s par (small- $t$ ), F crossover $t^{⋆}$ par, E kaafi door daayein (large- $t$ ). Cells A/B/C dono curves par hi baith jaati hain (pure ramp / pure wander), aur G–J unke upar bani variations hain. Cross ke baayein, noise badi dikhti hai; cross ke daayein, bias hamesha ke liye jeet jaata hai. Yeh picture apne dimaag mein rakho.

Worked Examples

Cell A — Positive bias ek ramp mein integrate hota hai

Worked example Example 1 — ek steady tilt

Ek stationary gyro ka bias $b = + 0.03 °/ s$ hai aur noise negligible hai. Aap raw reading ko $t = 90 s$ ke liye integrate karte ho. Heading error kya hai?

Forecast: Guess karo — $1°$ se bada ya chhota? $b$ ke sign jaisa hai ya ulta?

Model ko rest par likhein. $ω_{true} = 0$ ke saath, reading $\tilde{ω} = b$ hai. Yeh step kyun? Rest par gyro sirf apna bias hi report karta hai — yahi bias ki definition hai.
Integrate karo. $\hat{θ} = \int_{0}^{t} b d τ = b t$ . Yeh step kyun? Ek constant ko time par integrate karo toh bas (constant × time) milta hai — height $b$ aur width $t$ ke rectangle ka area.
Numbers daalo (one-unit rule: bias seconds use karta hai). $Δ θ = 0.03 \times 90 = 2.7°$ . Yeh step kyun? Dono numbers already $°/ s$ aur $s$ mein hain, toh $°/ s \times s = °$ — units cleanly cancel hoti hain.

Verify: Height $0.03$ , width $90$ → rectangle area $2.7$ . Sign $+$ hai (bias ke direction mein drift karta hai). Units: $\frac{°}{s} \cdot s = °$ . ✓

Cell B — Negative bias: error ka sign flip ho jaata hai

Cell C — Zero bias, sirf noise (Angle Random Walk)

Worked example Example 3 — pure jitter

Ek bahut achhi tarah calibrated gyro ka $b \approx 0$ hai lekin $ARW = 0.4 °/ h$ . $t = 1 h$ ke baad expected angle uncertainty kya hai? Aur $9 h$ ke baad?

Forecast: $1$ h se $9$ h jaate hue — kya error $9 \times$ badhega? $3 \times$ ? Kuch aur?

Noise growth law use karo (one-unit rule: noise hours use karta hai — aur yahan $t$ already hours mein hai). $σ_{θ} = ARW \cdot t_{h}$ . Yeh step kyun? Bina bias ke, sirf error integrated white noise hai. Parent ne dikhaya tha ki density $Q$ ki integrated white noise ki variance $Q t$ hoti hai, toh standard deviation $Q t = ARW t$ hoti hai — ek $t$ spread.
$t = 1$ h par. $σ_{θ} = 0.4 \times 1 = 0.4°$ . Yeh step kyun? $ARW$ define hi per $hour$ hai, toh $t$ hours mein daalne par koi conversion nahi chahiye.
$t = 9$ h par. $σ_{θ} = 0.4 \times 9 = 0.4 \times 3 = 1.2°$ . Yeh step kyun? $9 = 3$ hota hai, $9$ nahi. Random increments partly cancel ho jaate hain, toh $9 \times$ lamba run sirf $3 \times$ badi spread deta hai.

Verify: Ratio $\frac{1.2}{0.4} = 3 = 9$ . ✓ Yeh random walk ki pehchaan hai — time $k^{2}$ guna badha, error $k$ guna badha.

Cells D & E — small- $t$ aur large- $t$ limits

Worked example Example 4 —

$t$ chhota vs bada hone par kaun jeetega? Ek gyro ka $b = 0.01 °/ s$ aur $ARW = 0.5 °/ h$ hai. Bias error aur noise error ko compare karo (D) $t = 1 s$ par aur (E) $t = 3600 s$ (1 h) par.

Forecast: $1$ second par, kaun sa term bada hai — ramp ya wander? Kya yeh $1$ hour par bhi sach rahega?

One-unit rule pehle apply karo. Bias error $= b t_{s}$ (seconds). Noise error $= ARW t_{h}$ jahan $t_{h} = t_{s} /3600$ (hours). Dono target times ek baar convert karo: $t = 1$ s $\Rightarrow t_{h} = 1/3600$ ; $t = 3600$ s $\Rightarrow t_{h} = 1$ . Yeh step kyun? Aap do numbers compare nahi kar sakte jinke $t$ alag units mein hain — ek baar convert karo, dono saath lo.
Cell D, $t = 1$ s.
- Bias: $0.01 \times 1 = 0.01°$ .
- Noise: $0.5 \times 1/3600 = 0.5 \times \frac{1}{60} = \frac{1}{120} \approx 0.00833°$ . Yeh step kyun? Ab dono degrees mein hain, toh seedha compare kar sakte hain: bias $0.01°$ noise $0.00833°$ se thoda bada hai. Toh $1$ s par bhi bias (barely) aage hai — $t = 0$ ke paas kaun jeetega yeh automatic nahi hai; yeh coefficients par depend karta hai, aur isliye hum Ex 5 mein crossover calculate karte hain.
Cell E, $t = 3600$ s (1 h).
- Bias: $0.01 \times 3600 = 36°$ .
- Noise: $0.5 \times 1 = 0.5°$ . Yeh step kyun? Lambe time par ramp $t$ , $t$ wander ko crush kar deta hai — bias $72 \times$ bada ho jaata hai.

Verify: D: $0.01 > 0.00833$ , bias barely jeetega. E: $36/0.5 = 72$ , bias hugely dominate karta hai. Units poore mein degrees hain. ✓ General limit law: jab $t \to \infty$ , bias hamesha jeetega kyunki $t$ beats $t$ ; $t \to 0$ ke paas winner actual coefficients par depend karta hai (dekho Allan Variance Analysis).

Cell F — exact crossover time

Worked example Example 5 — dono lines kahan cross karti hain

Ex 4 wala gyro: $b = 0.01 °/ s$ , $ARW = 0.5 °/ h$ . Elapsed time $t^{⋆}$ kya hai jab bias error aur noise error equal ho jaayein?

Forecast: Ex 4 se, bias $1$ s par pehle se aage tha. Toh kya tumhe expect hai ki $t^{⋆}$ , $1$ s se neeche hoga?

Dono errors equal set karo (seconds mein kaam karo, noise time inline convert karo). $b t = ARW t /3600$ (dono sides degrees mein, $t$ seconds mein). Yeh step kyun? "Crossover" ka literal matlab hai ki dono error curves ki height ek jaisi hai — yeh ek equation hai ek unknown $t$ mein.
Dono sides square karo — aur note karo squaring kya introduce kar sakti hai. $b^{2} t^{2} = ARW^{2} \frac{t}{3600}$ . Yeh step kyun? Squaring awkward square root hatati hai. Caution: squaring extraneous roots create kar sakti hai (solutions jo squared equation satisfy karti hain lekin original nahi). Toh hume end mein sirf $t > 0$ wale roots rakhne chahiye — negative $t$ physically meaningless hai (time ulta nahi chalta) aur original right-hand side $t /3600$ ko undefined bana dega. Hum yeh check karenge.
Ek $t$ cancel karo aur saare roots identify karo. Likhein $b^{2} t^{2} - \frac{ARW ^{2}}{3600} t = 0$ , yaani $t (b^{2} t - \frac{ARW ^{2}}{3600}) = 0$ . Do roots hain $t = 0$ aur $t^{⋆} = \frac{ARW ^{2}}{3600 b ^{2}}$ . Yeh step kyun? Factoring har solution explicitly dikhata hai. $t = 0$ trivial crossing hai (dono errors start mein $0$ hote hain, meaningful crossover nahi); yahan koi negative root nahi hai, toh kuch bhi extraneous nahi bachta — lekin hum ne check kiya, jaise rule kehta hai.
Meaningful root mein numbers daalo. $t^{⋆} = \frac{0. 5 ^{2}}{3600 \times 0.0 1 ^{2}} = \frac{0.25}{0.36} \approx 0.694 s$ . Yeh step kyun? $3600 \times 0.0001 = 0.36$ ; phir $0.25/0.36$ .

Verify: Dono roots $\geq 0$ hain, toh koi extraneous solution discard karne ki zaroorat nahi. $t^{⋆} \approx 0.694 s < 1 s$ , Ex 4 ke saath consistent hai jahan bias $1$ s par pehle se aage tha. $t^{⋆}$ se neeche noise lead karta hai, upar bias — scenario-map figure mein yellow dot se match karta hai. ✓

Cell G — real rotation plus bias: signal ko corruption se alag karo

Worked example Example 6 — ek genuine turn, bias se contaminated

Ek turntable truly $ω_{true} = + 5 °/ s$ par $t = 20 s$ ke liye rotate karta hai. Gyro ka bias $b = + 0.1 °/ s$ hai aur noise negligible hai. Gyro kya angle report karega, aur true angle kya hai?

Forecast: $b > 0$ hone par, kya reported angle true se bada hoga ya chhota?

True angle. $θ_{true} = ω_{true} t = 5 \times 20 = 100°$ . Yeh step kyun? Constant true rate ke saath, orientation sirf rate × time hai (sab seconds mein, toh units degrees dete hain).
Reported angle. Reading $\tilde{ω} = ω_{true} + b$ hai, toh $\hat{θ} = (ω_{true} + b) t = (5 + 0.1) \times 20 = 102°$ . Yeh step kyun? Bias true signal ke upar add hota hai; integration dono ko pass karta hai.
Error. $Δ θ = \hat{θ} - θ_{true} = 102 - 100 = 2° = b t$ . Yeh step kyun? True part exactly subtract ho jaata hai, sirf bias ramp bachta hai — confirm karta hai ki error $b$ par depend karta hai, is baat par nahi ki aap actually kitni tezi se ghoomai.

Verify: $b t = 0.1 \times 20 = 2°$ , $102 - 100$ se match karta hai. ✓ Cell G ki key insight: error true rate se independent hai; ek fast turn aur ek slow turn same time mein ek jaisa bias error accumulate karte hain.

Cell H — poora realistic case: true rate + bias + noise

Worked example Example 7 — sab kuch ek saath

Ab complete model $\tilde{ω} = ω_{true} + b + n$ use karo. Ek turntable truly $ω_{true} = + 5 °/ s$ par $t = 20 s$ ke liye rotate karta hai, bias $b = + 0.1 °/ s$ ke saath AUR noise $ARW = 0.6 °/ h$ ke saath. True angle, expected reported angle, deterministic error, aur uske upar $1 σ$ noise spread do.

Forecast: Kya noise expected reported angle badlega, ya sirf uski spread?

Har term ko alag integrate karo (integration linear hai). $\hat{θ} = \int_{0}^{t} (ω_{true} + b + n) d τ = signal ω_{true} t + bias ramp b t + random walk \int_{0}^{t} n d τ$ . Yeh step kyun? Teeno effects interfere nahi karte; integral split karne se har ek ko us law se handle kar sakte hain jo hum jaante hain.
Signal (seconds). $ω_{true} t = 5 \times 20 = 100°$ . Yeh step kyun? Yahi woh answer hai jo hum actually chahte hain — true orientation.
Bias ramp / expected error (seconds). $b t = 0.1 \times 20 = 2°$ . Yeh step kyun? Cell G jaisa hi — bias ek deterministic $+ 2°$ add karta hai, toh expected reading $100 + 2 = 102°$ hai.
Noise spread $1 σ$ (one-unit rule: $20$ s $= 20/3600$ h convert karo). $σ_{θ} = ARW t_{h} = 0.6 20/3600 \approx 0.0447°$ . Yeh step kyun? $E [\int n d τ] = 0$ , toh noise mean mein koi shift contribute nahi karta — sirf $σ_{θ}$ size ka ek $\pm$ wobble.

Verify: Expected reading $= 100 + 2 = 102°$ (noise mean $0$ hai, toh yeh move nahi karta). Noise scatter $\approx 0.0447°$ , $2°$ bias ke compare mein bahut chhota hai — ek baar phir confirm karta hai ki $20$ s par bias noise ko dominate karta hai. Units: signal, ramp, aur spread sab degrees mein hain. ✓ Yeh poora realistic picture hai: ek wanted signal $+$ ek growing bias ramp $+$ ek chhota random wobble.

Cell I — poora real-world word problem

Worked example Example 8 — GPS dropout

Ek delivery drone GPS kho deta hai aur $3 min$ ke liye gyro-only heading par fly karta hai. Uska gyro: bias $b = 0.02 °/ s$ , $ARW = 0.6 °/ h$ . Mission rule: abort karo agar heading error plausibly $5°$ se zyada ho sake. Kya yeh abort karega?

Forecast: $3$ minute chhota lagta hai. Sirf bias se, kya yeh $5°$ se kam rahega?

Time ek baar convert karo (one-unit rule). $t = 3 min = 180 s$ ; noise term ke liye $t_{h} = 180/3600 = 0.05 h$ . Aage dono carry karo. Yeh step kyun? Bias seconds use karta hai, ARW hours use karta hai — dono pehle se prepare karo taaki koi step unhe mix na kare.
Bias error (deterministic). $Δ θ_{bias} = b t_{s} = 0.02 \times 180 = 3.6°$ . Yeh step kyun? Constant bias → ramp, bilkul Ex 1 ki logic. Yeh ek fixed offset hai, certainty ke saath present.
Noise spread $1 σ$ (random). $σ_{θ} = ARW t_{h} = 0.6 \times 0.05 = 0.6 \times 0.2236 \approx 0.134°$ . Yeh step kyun? Noise $t$ contribution hai aur random hai, toh yeh standard deviation hai, fixed number nahi. Ek $1 σ$ noise sirf $0.134°$ hai.
Risk policy explicitly state karo, phir apply karo. Heading error ek fixed bias ramp of $3.6°$ plus ek random noise term of mean $0$ aur standard deviation $0.134°$ hai. Hum ek conservative $3 σ$ worst-case bound adopt karte hain (outcomes ke $\approx 99.7%$ ko cover karta hai — safety abort decision ke liye standard): $Δ θ_{max} = certain b t_{s} + worst-case noise 3 σ_{θ} = 3.6 + 3 \times 0.134 = 3.6 + 0.402 \approx 4.00°.$ Yeh step kyun? Abort rule kehta hai "plausibly exceed $5°$ " — "plausibly" ka matlab hai ki hume worst realistic case bound karna hai, average nahi. Certain bias ramp par $3 σ$ noise add karna woh bound hai. (Agar policy "on average" kehti, toh hum mean $3.6°$ use karte; agar "always/absolute worst" kehti, toh Gaussian par hard cap nahi hota, toh $3 σ$ accepted engineering convention hai.)

Verify: Worst-case $\approx 4.00° < 5°$ → abort mat karo, lekin sirf $1.0°$ margin hai, aur yeh almost poora bias hai. Agar bias $0.028°/ s$ bhi hota toh $3 σ$ bound $5°$ breach kar deta. Real system mein fix: ek accelerometer/magnetometer fuse karo heading reset karne ke liye (ek Inertial Measurement Unit (IMU) yeh Kalman Filter ke zariye karta hai). Units sab degrees mein resolve hote hain. ✓

Cell J — exam-style twist (Allan-slope trap)

Worked example Example 9 — "zyada lamba average karna help karta hai na?"

Ek exam kehta hai: "Ek gyro ki Allan deviation $0.3 °/ h$ hai averaging time $τ = 10 s$ par, $- 1/2$ (ARW) slope par baithi hai. $τ = 40 s$ par $σ_{A}$ predict karo. Phir explain karo yahi prediction $τ = 4000 s$ par kyun fail hoti hai."

Forecast: $- 1/2$ slope par, $τ$ ko chaar guna karne se $σ_{A}$ kitna badlega?

$- 1/2$ slope law use karo. Log-log plot par, slope $- 1/2$ ka matlab hai $σ_{A} \propto τ^{- 1/2}$ . Yeh step kyun? ARW region mein, zyada lamba averaging error ko $1/ τ$ ki tarah reduce karta hai — yahi negative slope encode karta hai.
Known point se scale karo. $\frac{σ _{A} ( 40 )}{σ _{A} ( 10 )} = (\frac{40}{10})^{- 1/2} = 4^{- 1/2} = \frac{1}{2}$ . Yeh step kyun? Power law par ratios ke liye sirf $τ$ 's ka ratio chahiye — $τ$ units cancel ho jaate hain, toh yahan seconds-to-hours conversion nahi chahiye.
Calculate karo. $σ_{A} (40) = 0.3 \times \frac{1}{2} = 0.15 °/ h$ . Yeh step kyun? Aadha error, jaisa slope ne promise kiya tha.
$τ = 4000$ s par trap. $- 1/2$ region aur long $τ$ ke beech, curve pehle flatten hokar slope- $0$ floor ban jaata hai — woh flat minimum bias instability hai jo aap average karke kabhi nahi hata sakte. Sirf flat region ke baad yeh slope $+ 1/2$ ki taraf UP hota hai, jo rate random walk hai (dekho Random Walk & Wiener Process). Dono taraf se, $- 1/2$ law apply karna $0.3 \times (4000/10)^{- 1/2} = 0.3/20 = 0.015 °/ h$ predict karega, lekin real value zyada hai, kam nahi. Yeh step kyun? Ek baar ARW region chhod diya, toh zyada lamba average karna help karna band kar deta hai (flat floor) aur eventually HURT karta hai (rate random walk, slope $+ 1/2$ ) — $- 1/2$ line ko extrapolate karna classic galti hai.

Verify: $σ_{A} (40) = 0.15 °/ h$ , as expected half. Naive extrapolation $0.015 °/ h$ deta hai, jo problem batata hai galat hai kyunki process ne regime change kar liya. ✓

Recall Quick self-test (answers cover karo)

Bias $0.03°/ s$ ko $90$ s ke liye use karne par kya error aata hai? ::: $2.7°$ (Cell A, ramp $b t$ ). Wahi lekin bias $- 0.03°/ s$ ho? ::: $- 2.7°$ — same size, ulti direction (Cell B). ARW $0.4°/ h$ : $9$ h par error $1$ h se kitne guna badhega? ::: $9 = 3 \times$ , toh $1.2°$ (Cell C). Jab $t \to \infty$ , kaun sa error hamesha jeetega, bias ya noise? ::: Bias, kyunki $t$ beats $t$ (Cell E). Full model mein, kya noise expected reported angle shift karta hai? ::: Nahi — noise zero-mean hai; sirf bias mean shift karta hai, noise spread add karta hai (Cell H). Drone abort problem mein, random noise ke liye humne kya bound use ki? ::: $3 σ$ worst-case bound certain bias ramp ke upar add ki (Cell I). Allan $- 1/2$ slope par, $τ$ chaar guna karne par $σ_{A}$ ka kya hota hai? ::: Halves it ( $4^{- 1/2}$ ) (Cell J). Allan curve ka FLAT (slope-0) region kya represent karta hai? ::: Bias instability — woh floor jo average karke nahi hata sakte (Cell J).

Related building blocks: Random Walk & Wiener Process (jahan se $t$ aata hai), Coriolis Force aur Sagnac Effect (kaise $ω$ sense hota hai), Attitude Estimation / Dead Reckoning (jahan ye errors bite karte hain).