Foundations — Gyroscope — angular velocity measurement, bias, noise

The parent note Gyroscope — angular velocity, bias, noise throws a lot of notation at you fast: $\omega$, $\int$, $\dot{b}$, $\delta$, PSD $Q$, $\sqrt{t}$. Below we earn each one, in the order they depend on each other. Nothing is used before it is drawn.

1. Angle $\theta$ — "how far turned"

The picture. Draw an arrow (the "nose" of a drone). Now rotate it. The wedge swept between where it started and where it points now is $\theta$.

Why the topic needs it. Navigation ultimately wants the answer to "which way am I facing?" — that answer is an angle. Everything the gyro does is in service of getting $\theta$.

2. Time $t$ and the idea of a rate

A rate answers "how much per second?". Speed is a rate of position (metres per second). We are about to meet the rate of angle.

3. Angular velocity $\omega$ — the star of the show

The picture. If $\theta$ is where the nose points, then $\omega$ is how quickly the nose is swinging. A fast spin = big $\omega$; sitting still = $\omega =0$.

Why a derivative and not something simpler? Because the turning speed can change moment to moment. A ratio like $\Delta \theta /\Delta t$ only gives the average over a chunk of time; the derivative $\frac{d\theta }{dt}$ gives the instantaneous rate — exactly what a gyro reports at each tick.

4. Integration $\int$ — adding up all the little turns

Since the gyro only gives $\omega$ (how fast), we recover $\theta$ (how far) by accumulating $\omega$ over time. That accumulation is integration.

The picture. Draw $\omega$ against time. The area under that curve up to time $t$ is $\theta (t)$. Wide-and-tall region → lots of angle accumulated.

Why integration is where trouble lives. If every $\omega$ reading is slightly too big by a fixed amount, the sum piles up that error slice after slice — the error accumulates. This is the seed of drift, the parent note's central villain.

5. Bias $b$ — the offset that shouldn't be there

Why the topic needs it. When you integrate a non-zero $b$, you accumulate a fake angle that grows steadily — the parent note's linear drift $bt$.

6. The dot $\dot{b}$ — "rate of change of"

7. Noise $n$ — the shaky jitter

The picture. Bias shifts the whole reading up by a fixed step; noise makes it fuzzy around that shifted value.

8. Averages, variance, and the wiggle-size $\sigma$

To talk about random things we need a way to measure "how big is the wiggle."

Why squared then square-rooted? Squaring makes over-shoots and under-shoots both count as positive spread; the final square root brings the number back to sensible units ($°$ instead of ${°}^2$).

9. The delta spike $\delta$ and power spectral density $Q$

The parent's ARW derivation uses two more symbols. Here they are, plainly.

10. Square root ​ and log-log plots (for the Allan chart)

How the foundations feed the topic

Every arrow says "you must understand the left box before the right box makes sense." The whole chain ends at drift, which is what the parent note spends its energy fighting.

Equipment checklist

Cover the answer and test yourself. If any line is fuzzy, re-read its section above before opening the parent note.

Ready? Now the parent note's model $\tilde{\omega }={\omega }_{\text{true}}+b+n$ and its drift laws will read like plain English. Related tools that build on these foundations: Coriolis Force, Sagnac Effect, Random Walk & Wiener Process, Allan Variance Analysis, Kalman Filter, Accelerometer — specific force, gravity, Inertial Measurement Unit (IMU), Attitude Estimation / Dead Reckoning.