robotics//sensor//sensor calibration//noise characterization//Allan variance

The Allan variance is a measure of how much the average of a signal changes from one block of time to the next, as a function of the block length, and it is the standard tool for reading the noise of gyroscopes, accelerometers and clocks from a static recording. Record the sensor motionless for hours at constant temperature, cut the recording into blocks of duration \(\tau\), average each block, and see how much consecutive averages differ:

The Allan variance is a measure of how much the average of a signal changes from one block of time to the next, as a function of the block length, and it is the standard tool for reading the noise of gyroscopes, accelerometers and clocks from a static recording. Record the sensor motionless for hours at constant temperature, cut the recording into blocks of duration τ\tauτ, average each block, and see how much consecutive averages differ:

σA2(τ)=12 E ⁣[(yˉk+1(τ)−yˉk(τ))2].\sigma_A^2(\tau)=\tfrac12\,\mathbb E\!\left[\left(\bar y_{k+1}(\tau)-\bar y_k(\tau)\right)^2\right].σA2​(τ)=21​E[(yˉ​k+1​(τ)−yˉ​k​(τ))2].

Here yˉk(τ)\bar y_k(\tau)yˉ​k​(τ) is the average of block kkk. Repeating this for block lengths from milliseconds to an hour and plotting the square root (the Allan deviation) on log-log axes gives a curve in which each kind of noise leaves its own slope.

At short τ\tauτ white noise dominates and the curve falls with slope −1/2-1/2−1/2, because longer averages beat it down. Its value read on that line at τ=1\tau=1τ=1 s is the angle random walk coefficient NNN of a gyroscope, the one that makes an integrated angle wander as NtN\sqrt tNt​ (dead reckoning).

Then the curve flattens into a minimum, the bias instability: the best stability averaging can ever reach, set by flicker noise, which has more power the slower it is. It ranges from a few to tens of degrees per hour in consumer MEMS gyroscopes, about 1 °/h in tactical grade and 0.01 °/h in navigation grade, with prices from euros to tens of thousands (IMU).

Beyond the minimum the curve rises again, because over long blocks the bias itself has moved (a random walk of the rate, slope +1/2+1/2+1/2). That rising branch is why a calibration at rest goes stale and why estimators carry the bias as a state (sensor bias).

Noise has species, and only the white one is averaged away.

The Allan curve says which species a sensor carries and at which time scales, so it tells the designer how long averaging helps, what the process noise of the bias state should be, and whether a cheaper part would do.

Datasheets quote the same two numbers (noise density and bias instability), and the procedure for gyroscopes is standardized in IEEE Std 952. Measuring the curve on the actual part, mounted and at operating temperature, is the honest version, since the datasheet describes a typical chip on a quiet bench (noise characterization).