Gyroscope — angular velocity measurement, bias, noise

WHAT is a gyroscope really measuring?

WHY does angular velocity matter for navigation? Because orientation (attitude) is the time-integral of angular velocity. If you know where you started and you know $\omega(t)$ perfectly, you know your orientation forever: $\theta(t) = \theta_0 + \int_0^t \omega(\tau)\, d\tau$

HOW do physical gyros sense $\omega$? Two common principles:

• MEMS Coriolis gyro (phones, drones): a tiny proof mass vibrates. When the device rotates, the Coriolis force $\vec{F}_c = -2m\,\boldsymbol{\omega}\times\vec{v}$ pushes the mass sideways. That sideways deflection is measured capacitively and is $\propto \omega$.
• Optical (RLG / FOG): uses the Sagnac effect — counter-propagating light beams in a rotating loop accumulate a phase difference $\propto \omega$.

Deriving the gyro measurement model FROM SCRATCH

We never trust a raw sensor. We build a model of what the sensor actually outputs, then invert it.

Step 1 — Start with the ideal. A perfect gyro outputs the true rate: $\tilde\omega = \omega_{\text{true}}$ Why this step? Establish the target before adding imperfections.

Step 2 — Add a slowly-varying offset (bias). Real electronics have an offset even at zero rotation. Call it $b(t)$: $\tilde\omega = \omega_{\text{true}} + b(t)$ Why this step? Measured output at rest is not zero — that constant-ish part is bias.

Step 3 — Add random measurement noise. Thermal/electronic fluctuations add a jitter $n(t)$, modeled as white noise: $\boxed{\;\tilde\omega(t) = \omega_{\text{true}}(t) + b(t) + n(t)\;}$ Why this step? No real reading is smooth; there is high-frequency randomness on top.

Step 4 — Model the bias itself. Bias isn't perfectly constant; it wanders. The standard model is a random walk driven by its own white noise $n_b$: $\dot b(t) = n_b(t)$ Why this step? Over minutes/hours the offset slowly drifts — this is captured by integrating a small white noise.

So the full gyro error model is two coupled equations: a fast noise $n$ and a slow drifting bias $b$.

HOW errors turn into drift (the derivation that matters)

Suppose the true rate is zero ($\omega_{\text{true}}=0$) and we integrate the reading to get an angle estimate $\hat\theta$: $\hat\theta(t) = \int_0^t \tilde\omega\, d\tau = \int_0^t b\,d\tau + \int_0^t n\,d\tau$

Bias contribution. If $b$ is roughly constant: $\theta_{\text{bias}}(t) = b\,t \quad\Rightarrow\quad \text{error grows LINEARLY in time}$ Why? A constant offset integrated gives a ramp. A $0.01°/\text{s}$ bias → $36°$ error after one hour. This is why bias is the #1 enemy.

Noise contribution (Angle Random Walk). White noise integrated becomes a random walk. Its standard deviation grows like $\sqrt{t}$: $\sigma_\theta(t) = \text{ARW}\cdot\sqrt{t}$

Summary of the two growth laws:

The Allan Deviation — reading a gyro's spec sheet

• Slope $-1/2$ region → dominated by white noise / ARW. Averaging longer helps (error falls).
• Minimum of the curve → best achievable stability.
• Slope $+1/2$ region → dominated by bias instability / random walk. Averaging longer HURTS (drift wins).

The bottom of the "bathtub" gives the bias instability, the floor you cannot average away.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Coriolis Force — the force MEMS gyros exploit.
• Sagnac Effect — optical gyro operating principle.
• Accelerometer — specific force, gravity — complementary IMU sensor.
• Inertial Measurement Unit (IMU) — gyro + accel package.
• Kalman Filter — estimates and removes drifting bias by sensor fusion.
• Attitude Estimation / Dead Reckoning — where gyro drift is felt.
• Allan Variance Analysis — characterizing sensor noise.
• Random Walk & Wiener Process — math behind $\sqrt t$ growth.

Concept Map

Hinglish (regional understanding)