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

Step 1 — What "angular velocity" even means

WHAT. A gyroscope reports one number per axis: the angular velocity $ω$ . This is the speed of turning, measured in radians per second (rad/s). One radian is just an angle-size: turning through one radian sweeps an arc as long as the radius. A full circle is $2 π \approx 6.28$ radians.

WHY start here. You cannot talk about "error in the angle" until you agree on what the gyro actually hands you. It does not hand you the angle $θ$ . It hands you how fast $θ$ is changing. That distinction is the seed of every drift problem.

PICTURE. Below, the red arrow is a spinning object. $ω$ is how many radians the arrow sweeps each second. The angle $θ$ (black wedge) is where the arrow is; $ω$ is how fast it moves.

Figure — Gyroscope — angular velocity measurement, bias, noise

Step 2 — Recovering the angle: why we must integrate

WHAT. Since $ω = d θ / d t$ , the angle at time $t$ is the running total ("area under the curve") of $ω$ from the start until now: $θ (t) = where we started θ_{0} + sum of all the little turns \int_{0}^{t} ω (τ) d τ$

WHY integrate and not something else? Integration is the tool that reverses a derivative — it answers the question "if I know the rate at every instant, what is the accumulated total?" A derivative asks "how fast?"; an integral asks "how much altogether?" We want the total turn, so we integrate. The symbol $\int_{0}^{t} \dots d τ$ literally means "chop time $0 \to t$ into slivers of width $d τ$ , multiply each sliver's rate $ω$ by its width to get a tiny turn, and add them all up." ( $τ$ is just a dummy name for the running time inside the sum, so it isn't confused with the end-time $t$ .)

PICTURE. The angle is the shaded area under the $ω$ -vs-time curve. Each thin red column is one sliver $ω (τ) d τ$ .

Step 3 — The real reading is not $ω$ : bias and noise enter

WHAT. A perfect gyro would report $\tilde{ω} = ω_{true}$ (the tilde $~$ just means "the measured version"). A real one reports three things stacked together: $what we read \tilde{ω} (t) = the truth we want ω_{true} (t) + bias: a slow offset b (t) + noise: fast jitter n (t)$

WHY these two extras? Every electronic sensor has (1) a standing offset — it reads something even when nothing is moving, like a bathroom scale that shows $1 kg$ empty — call it bias $b$ ; and (2) thermal jitter — random flicker from heat-agitated electrons — call it noise $n$ . They are physically different beasts, and we keep them separate on purpose.

PICTURE. Same true rate, three signals: the clean truth (black), the truth shifted up by a steady bias (red offset), and the truth buried in fuzzy noise (grey fuzz).

Step 4 — Freeze the truth to isolate the error

WHAT. To study only the damage the errors do, set the true rate to zero: $ω_{true} = 0$ . The gyro is sitting perfectly still. Now the reading is pure error: $\tilde{ω} (t) = b (t) + n (t)$ Integrate it to get the angle the gyro thinks it has turned (it should think zero): $\hat{θ} (t) = \int_{0}^{t} \tilde{ω} d τ = bias drift \int_{0}^{t} b d τ + noise drift \int_{0}^{t} n d τ$

WHY set truth to zero? It's a controlled experiment. If the object isn't really turning, then any angle the gyro reports is 100% error — nothing else to blame. This cleanly splits the drift into two independent pieces we can study one at a time. (The $^$ hat means "our estimate.")

PICTURE. A stationary object (not turning at all) with a gyro strapped on; the dial slowly creeps away from zero even though nothing moved.

Step 5 — Bias integrated → a straight ramp (grows like $t$ )

WHAT. Take the bias term alone and pretend it's constant, $b (t) = b$ . A constant pulled out of an integral gives: $θ_{bias} (t) = \int_{0}^{t} b d τ = b = t \int_{0}^{t} d τ = slope b \cdot time t$

WHY does a constant become a straight line? Integrating a flat, unchanging value just multiplies it by how long you've been adding it. Add $b$ every second for $t$ seconds and you have $b \cdot t$ . On a graph that is a straight line through the origin with slope $b$ — an unstoppable ramp. There is no cancellation, because the offset always pushes the same direction.

PICTURE. Constant bias (flat red line, left) integrates into a straight rising ramp (right). The area under a flat line is a triangle-free rectangle that just keeps widening.

Step 6 — Noise integrated → a random walk (grows like $t$ )

WHAT. Now the noise term alone. White noise $n$ is random and zero-mean, so it can't just pile up in one direction — each moment it pushes a random way. Its integral is a random walk (a "drunkard's walk"): a path built from random steps. The spread of where you end up grows not like $t$ , but like $t$ .

WHY $t$ and not $t$ ? Because random steps partially cancel. If you flip a coin and step left/right, after $N$ steps you are typically about $N$ steps from home — not $N$ , because roughly half the lefts undo the rights. More time = more steps, but the cancellation keeps the wandering slow. The exact statement uses the noise power spectral density $Q$ (how much jitter-power per Hz the gyro puts out, units $(rad/s)^{2} / Hz$ ): $Var [θ_{noise} (t)] = Q t \Rightarrow σ_{θ} (t) = Q t$ Here $Var$ is the spread squared and $σ_{θ}$ is the standard deviation (typical error size). Its square-root of time is the fingerprint of a random walk.

WHAT the delta function means. The derivation below uses the symbol $δ (τ - τ^{'})$ , the Dirac delta. Think of it as an infinitely tall, infinitely thin spike sitting at $τ = τ^{'}$ , whose area is exactly $1$ . It is the mathematician's way of saying "nonzero only at one instant." For white noise it encodes one physical fact: the jitter at two different times is completely unrelated — knowing the noise now tells you nothing about the noise a millisecond later. So $E [n (τ) n (τ^{'})]$ (the average of the product of the noise at two times) is zero unless $τ = τ^{'}$ , and the spike's area sets the strength $Q$ .

WHY does Var grow linearly? Write $θ = \int_{0}^{t} n d τ$ , square it (a double integral over two dummy times $τ, τ^{'}$ ), and use the delta rule: $Var [θ] = \int_{0}^{t} \int_{0}^{t} noise correlation Q δ (τ - τ^{'}) d τ d τ^{'} = Q = t \int_{0}^{t} d τ = Q t .$ The delta spike is nonzero only along the diagonal $τ = τ^{'}$ , so one of the two integrals collapses (it "eats" the spike and returns $1$ ), leaving a single $\int_{0}^{t} d τ = t$ .

PICTURE. Many random-walk paths (grey) fan out from zero; the red envelope is $\pm Q t$ — a sideways parabola, wide but not a straight ramp.

Step 7 — Put them side by side: line vs curve

WHAT. On one graph, plot both errors versus time: the bias ramp $b t$ (straight) and the ARW envelope $ARW t$ (bending over). For small $t$ the $t$ term is actually the taller one (since $t > t$ when $t < 1$ ); but the straight line eventually overtakes any square-root curve and never looks back.

WHY it matters. This crossover tells you which enemy to fight. Over seconds-to-minutes, whether bias or noise dominates depends on the actual numbers — but because the ramp keeps climbing and the curve flattens, bias always wins in the end. That's the engineering headline: calibrate the bias first.

PICTURE. Red straight ramp (bias) crossing the black bending curve (noise). Mark the crossover.

Step 8 — The degenerate cases (never leave a scenario unshown)

Three edge cases complete the picture:

Case A — zero bias, zero noise. A perfect gyro. Then $\hat{θ} (t) = 0$ forever: flat lines, no drift. This is the target Step 1 established.

Case B — noise only, $b = 0$ . No ramp. The error is a pure random walk hovering around zero, spread $Q t$ . It wanders but has no preferred direction — averaging many runs gives zero. This is the only regime where "average longer to improve" actually helps.

Case C — bias drifts too, $\dot{b} = n_{b}$ . In reality bias itself slowly wanders (temperature, ageing). We model this by saying the bias is driven by its own white noise $n_{b}$ , whose strength is a power spectral density $Q_{b}$ (units $(rad/s^{2})^{2} / Hz$ , i.e. how hard the slope itself is randomly kicked, per Hz).

Layer 1 — the bias becomes a random walk. This is exactly Step 6 again: integrate white noise $n_{b}$ once and you get a random walk whose spread grows like $t$ : $b (t) = \int_{0}^{t} n_{b} d τ \Rightarrow Var [b (t)] = Q_{b} t \Rightarrow σ_{b} (t) = Q_{b} t .$

Layer 2 — feed that wandering bias into the angle. The angle error is $θ_{bias} (t) = \int_{0}^{t} b (τ) d τ$ , so we integrate the random walk one more time — white noise integrated twice. Here is the WHY, made concrete: two nested integrals of the delta-correlated noise give $Var [θ_{bias} (t)] = \int_{0}^{t} \int_{0}^{t} Q_{b} min (τ, s) d τ d s = \frac{Q _{b} t ^{3}}{3} .$ Take the square root to get the typical size: $typical angle error from drifting bias σ_{θ, bias} (t) = a constant \frac{Q _{b}}{3} grows faster than t t^{3/2}$

WHY $t^{3/2}$ intuitively? Each extra integration multiplies the growth by roughly one power of $t$ : white noise integrated once wanders like $t^{1/2}$ (Step 6); integrated a second time it wanders like $t^{1/2} \cdot t = t^{3/2}$ . So a drifting bias corrupts the angle worse than the fast noise's $t$ , yet slower than a fixed bias's clean $t$ ramp — it sits between the two, built by the same delta-integral machinery, just applied twice. And because $\dot{b} = n_{b} \neq = 0$ , you can never subtract a one-time bias and be done: it must be continuously re-estimated (a Kalman filter fusing GPS/accel/mag).

PICTURE. Three stacked panels: (A) flat perfect line, (B) zero-mean wander $\propto t$ , (C) a ramp whose slope itself slowly drifts, its envelope opening like $t^{3/2}$ .

Recall The three growth laws in one place

White noise on the rate → angle error $\propto t$ (integrate white noise once). Constant bias → angle error $\propto t$ (integrate a constant). Random-walk (drifting) bias → angle error $\propto t^{3/2}$ (integrate white noise twice).

The one-picture summary

Everything on this page in a single frame: a still gyro (true rate zero) fed through the integral, splitting into a straight red ramp (bias, $\propto t$ ) and a bending noise envelope ( $\propto t$ ), which cross at a marked time.

Recall Feynman: the whole walkthrough in plain words

A gyro is a "how-fast-am-I-spinning" meter, not a "which-way-am-I-facing" meter. To learn which way you face, you add up all the how-fasts over time — that's integrating. Now here's the catch: the meter always reads a teensy bit off even when you sit perfectly still. There are two kinds of "off." The first is a steady lean, always the same direction — that's bias, like a scale that reads 1 kg empty. Add a steady lean over and over and it grows like a straight ramp: double the time, double the error. The second is nervous shaking that jumps randomly every instant — that's noise. Because it jumps both ways, half of it cancels itself out, so its damage grows only like the square-root of time (four times longer = only twice the wander). Put them on one graph: the ramp is a straight line, the noise is a curve that bends over and flattens. The straight line always wins the long race. So if you fix only one thing, fix the bias — and since even the bias slowly wanders with temperature, you can't fix it once and forget it; you keep re-checking with GPS or a compass forever. (All this maths lives in radians; we only flip to degrees when we print a number.)

Recall Quick self-check

Q: A gyro measures angle or angular velocity? A: Angular velocity — angle comes only after integrating.

Q: Bias error grows like…? A: $t$ (a straight ramp).

Q: Noise (ARW) error grows like…? A: $t$ (a random walk).

Q: A drifting bias corrupts the angle like…? A: $t^{3/2}$ (integrate white noise twice).

Q: Why $t$ and not $t$ for noise? A: Random steps partially cancel, so the spread grows slower than the count.

Q: Why can't you subtract bias just once? A: Because $\dot{b} = n_{b} \neq = 0$ — bias itself slowly drifts.

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