Why small gyro errors can grow into position errors

The running noise test asks whether a fitted model can predict later positions while its IMU inputs contain random errors. An IMU (inertial measurement unit) measures angular motion and specific force. An angular error can tilt the reconstructed direction of gravity, creating an acceleration error that accumulates into a position error.

A simple calculation gives a useful scale. Assume a level platform, constant gravity, small angular errors, a perfectly known initial state and independent continuous white noise. There is no later position correction. This example omits changing bias and receiver error; it is not a reconstruction of the turning, climbing aircraft.

For gyro noise amplitude \(q\), gravity \(g\) and elapsed time \(T\), the position variance on each horizontal axis is

\[ \operatorname{Var}[p(T)]=g^2q^2\int_0^T\frac{(T-s)^4}{4}\,ds =\frac{g^2q^2T^5}{20}. \]

Using the stress test's declared gyro amplitude, 0.02°/√hour, and \(g=9.80665\,\mathrm{m/s^2}\), this gives 40.8 m standard deviation per horizontal axis after 400 seconds. At one-second horizons from 1 through 400 seconds, the square root of the expected mean horizontal squared error is 23.7 m. That quantity is not the expected value of a realized RMS. The corresponding accelerometer-only endpoint scale, at the declared \(0.0002\,\mathrm{m/s/\sqrt{s}}\), is 0.924 m per axis.

The example shows why a noisy control can miss a 10 m horizontal RMS target even when arithmetic and initialization are accurate. It does not prove a lower bound for every estimator, predict the outcome of a particular test case, or measure the real instrument's noise. The assumed amplitudes remain stress settings, not hardware specifications.

Later errors also share the same earlier noise history. They are correlated; 400 later epochs do not provide 400 independent error samples. A useful prediction-error model must account for that accumulation and correlation. The current accuracy limit and all failures remain unchanged. Any revised method needs separate qualification under its stated stochastic assumptions.

The reusable analytic kernel computes both marginal and cross-time covariance. Ten local software checks include independent numerical integration of the impulse responses, symmetry, zero horizon, invalid inputs and temporal correlation. Reproduce them with:

~/venv/bin/python analysis/tests/run_numerical.py --workers 1 --threads 1 --reserved-threads 3 -- ~/venv/bin/python -m pytest -q analysis/tests/test_inertial_error_budget.py

Noise-test plan · Saved rate variation · Prediction error assumptions.