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

Recall the one thing that anchors everything here: a rate gyro measures angular velocity $ω$ (rad/s), and orientation $θ$ only appears after you integrate $ω$ over time — which is exactly where bias and noise leak in.

Definition Notation used on this page (read first)

$ω_{true}$ — the actual rotation rate the world is turning at.
$\tilde{ω}$ — the tilde means "measured": it is the raw number the gyro chip reports, which equals the truth plus imperfections. The tilde is the standard symbol for "what the sensor said" as opposed to "what is true".
$b (t)$ — bias, a slow offset the reading has even when nothing rotates.
$n (t)$ — white noise, fast zero-mean jitter on the reading.
$n_{b} (t)$ — a second, separate tiny white noise that drives the bias to wander. It is not the reading-noise $n$ ; it is the little random push that makes $b$ itself change over time, written $\dot{b} = n_{b}$ (bias rate = a random nudge).
$S$ — scale factor: the gain that converts true rate into reported rate, ideally exactly $1$ .

The fuller measurement model with scale factor is $\tilde{ω} = S ω_{true} + b + n, \dot{b} = n_{b} .$

Two figures below make the three error families concrete — study them before the reveals.

Bias ramp vs. random walk. Look at how a constant offset (coral) climbs a straight line while integrated noise (lavender) wanders as $t$ :

Figure — Gyroscope — angular velocity measurement, bias, noise

The Allan-deviation "bathtub." The falling mint branch is averageable white noise; the flat butter floor is the bias-instability limit you cannot beat; the rising coral branch is drift winning:

True or false — justify

A gyroscope directly outputs your orientation angle.

False. It outputs angular rate

ω

; angle is

θ_{0} + \int ω d t

, and that integration is what lets tiny errors accumulate into drift.

Bias error and noise error grow at the same speed over time.

False. Constant bias integrates to a ramp (

\propto t

); zero-mean white noise integrates to a random walk (

\propto t

), which grows much more slowly.

Averaging the gyro output for a longer time always reduces the error.

False. It only helps in the white-noise (

- 1/2

slope) region of the Allan curve; past the minimum, bias instability dominates and longer averaging makes things worse.

The bias term

b

is a fixed constant you can measure once and subtract forever.

False. Bias slowly wanders with temperature and time (

\dot{b} = n_{b}

, the tiny random driving noise), so it must be continuously re-estimated rather than subtracted once.

A gyro reading of zero guarantees the device is perfectly still.

False. At rest the output is

b (t) + n (t)

, not zero; a momentary zero could just be noise crossing through the bias offset.

Scale-factor error and bias are the same kind of error.

False. Bias

b

adds a constant offset even at zero rate; scale-factor error means

S \neq = 1

, so it grows with the rate itself — it only shows up while you are actually rotating and is proportional to

ω_{true}

Angle Random Walk (ARW) describes the wander of the angle estimate, even though it comes from rate noise.

True. White noise on the rate, once integrated, becomes a Wiener process in the angle whose standard deviation grows like

ARW \cdot t

A more expensive gyro with lower ARW automatically has lower bias instability too.

False. ARW (high-frequency noise) and bias instability (the low-frequency flicker floor) are separate processes; a gyro can be quiet yet still drift, or noisy yet stable.

The Coriolis force a MEMS gyro exploits is a "real" push you could feel in an inertial frame.

False. It is a frame-dependent effect: in the rotating body frame the vibrating mass appears shoved sideways, and that apparent deflection is what encodes

ω

. See Coriolis Force.

An optical gyro (RLG/FOG) and a MEMS gyro measure fundamentally different quantities.

False. Both measure the same angular rate

ω

; only the physics differs — MEMS uses Coriolis deflection, optical uses the Sagnac phase.

Doubling the pure-integration time doubles the ARW angle uncertainty.

False. Uncertainty scales as

t

, so doubling time multiplies it by

2 \approx 1.41

, not by 2 — the random increments partially cancel.

Spot the error

"The gyro is quiet, so I'll just integrate it alone for the whole flight."

The error is trusting integration long-term: even a quiet gyro's small bias ramps up (

b t

) without bound, so you must reset with GPS/mag/accel via a filter.

"White noise has a mean, so integrating it gives a steady drift like bias does."

White noise is zero-mean; its integral is a random walk that wanders both ways, not a steady one-directional ramp. Only the nonzero bias produces a directed drift.

"I zeroed the reading at rest, so the gyro is fully calibrated."

Zeroing at rest only removes the bias at that instant; it does nothing for scale-factor error (

S \neq = 1

), which only reveals itself under actual rotation and stays hidden while still.

"Allan deviation's rising (

+ 1/2

) branch means the sensor is getting noisier with time."

The rise reflects low-frequency flicker/bias instability / random walk taking over, not high-frequency noise; averaging longer stops helping once you pass the bathtub minimum.

0.02 °/ s

bias is negligible — it's a tiny number."

Integrated, that "tiny" rate becomes

b t

; over an hour it is

72°

of heading error, so small rates are the #1 navigation enemy.

"To fix drift I should make the gyro's sample rate faster."

Faster sampling captures more high-frequency detail but does nothing about the bias ramp; the cure is estimating and subtracting the drifting bias, e.g. with a Kalman Filter.

"The bottom of the Allan bathtub is the noise I can average away to zero."

The bathtub floor is the bias instability set by flicker noise — precisely the part you canNOT average away; it is the practical stability limit of the device.

"A 3-axis gyro alone can give me full position and heading."

A gyro gives orientation rate only; position needs acceleration integration from an accelerometer, and both together form an Inertial Measurement Unit (IMU) feeding dead reckoning.

Why questions

Why does bias grow linearly with time but noise only as

t

A constant integrated is a ramp (

\int b d t = b t

), while independent random noise increments partially cancel when summed, so their spread grows as

t

(a Wiener process — see Random Walk & Wiener Process).

Why is integration, not the sensor itself, the source of "drift"?

The raw rate error stays bounded; it is the accumulation

\int (b + n) d t

that lets even small errors pile up without limit into a growing orientation error.

Why does the tilde

\tilde{ω}

appear in the model instead of just

ω

The tilde flags the measured value — what the chip reports — so we can write it as truth plus imperfections (

S ω_{true} + b + n

) and then invert the model to recover the truth.

Why do we introduce a separate noise

n_{b}

for the bias instead of reusing

n

n

is the fast jitter on the reading, whereas

n_{b}

is the slow random push driving the bias to wander (

\dot{b} = n_{b}

); they act at different timescales and need different math, so mixing them would erase the distinction.

Why does the Allan deviation use a log-log plot against averaging time

τ

Different noise processes appear as straight lines of characteristic slope (

- 1/2

for ARW,

+ 1/2

for random walk), so log-log cleanly separates and identifies them — see Allan Variance Analysis.

Why can't averaging remove bias instability the way it removes white noise?

Bias instability comes from flicker (

1/ f

) noise concentrated at slow frequencies; averaging suppresses fast fluctuations but leaves slow ones untouched, so the flicker floor survives.

Why calibrate bias before worrying about noise in a short-duration application?

Over seconds the bias ramp (

\propto t

) already exceeds the noise wander (

\propto t

), so bias is the dominant villain first — the 80/20 fix.

Why does a Kalman filter re-estimate bias continuously rather than once?

Because

\dot{b} = n_{b}

means bias itself drifts with temperature and time; a one-time correction goes stale, so the filter keeps updating it from GPS/mag/accel.

Edge cases

If the true rate is exactly zero, does the integrated angle stay zero?

No — with

ω_{true} = 0

the estimate becomes

\int (b + n) d t

, which still ramps from bias and wanders from noise, so the angle drifts even at perfect rest.

With

ω_{true} = 0

, does scale-factor error contribute anything?

No — scale-factor error is

S ω_{true}

, which vanishes when the true rate is zero, so it is invisible at rest and only bites during high-rate maneuvers.

What happens to the ARW angle uncertainty at

t = 0

It is exactly zero, since

σ_{θ} = ARW \cdot t \to 0

; uncertainty only accumulates as you integrate over elapsed time.

If bias were truly perfectly constant and known, would there be no drift at all?

The bias ramp would vanish once subtracted, but the noise-driven random walk (

\propto t

) would remain, so some slow angle wander persists.

At the exact Allan-deviation minimum, is averaging helping or hurting?

Neither — it is the turnover point where the falling white-noise branch and the rising flicker/bias-instability branch balance, giving the best achievable stability.

In the limit of extremely long averaging time, which process dominates the error?

The random-walk / bias-instability branch, whose Allan deviation rises with

τ

, so very long averaging always worsens rather than improves the estimate.

If two gyros have equal ARW but one has double the bias, are they equally good for a 2-hour flight?

No — over hours the linear bias ramp dominates, so the higher-bias unit drifts far more despite identical noise specs.