Inertial Error Integration#
This demo makes the inertial error-integration chain concrete. An error process is injected at one end of the chain (acceleration or velocity) and integrated forward to see how it shows up as a navigation error. The error process is selectable from the four canonical models:
Constant bias — a deterministic offset, e.g. an uncalibrated accelerometer null.
White noise — uncorrelated samples each step, the model for sensor measurement noise.
Random walk — the error itself wanders; its variance grows linearly with time. The natural model for many gyro biases.
Gauss-Markov — mean-reverting noise with a finite memory τ. It bridges white noise (τ → 0) and random walk (τ → ∞), and it is the standard model for accelerometer biases that have no preferred value but do have a finite correlation time.
The teaching point is that the further “upstream” the sensor error sits in the integration chain, the higher the time-order of the resulting position error. A constant velocity bias gives linear position growth; a constant acceleration bias gives quadratic position growth. White noise integrates into a random walk (\(\sigma \propto \sqrt{t}\)); random walk integrates into something faster yet (\(\sigma \propto t^{3/2}\)). Gauss-Markov sits between the two depending on τ.
Interactive demo#
Walkthrough#
The demo opens on white noise injected at the acceleration level with \(\sigma = 0.01\) m/s². The single dark trace is one realization; the lighter shaded band is the empirical \((1-\alpha)\) envelope from a 30-realization ensemble. Try the following:
Switch to “Constant bias” and stay at acceleration injection. Drag the bias slider — the velocity panel grows linearly and the position panel grows quadratically. With \(b = 0.02\) m/s² and \(T = 600\) s, the position error reaches \(\tfrac{1}{2} b T^2 = 3600\) m. This is the worked example from the Block 2 reading.
Switch the injection point to “velocity” (still constant bias). Now the acceleration panel is empty and the position panel grows linearly at rate \(b\) — one fewer integration than the acceleration case, one fewer power of \(t\).
Switch to “White noise” + acceleration. Toggle “show ensemble” on. The 30 light realizations show the spread; the dark band is the \((1-\alpha)\) empirical bound from those realizations. Move the α slider — the band scales by \(k = \sqrt{2}\,\text{erfinv}(1-\alpha)\). Click “95%” — band widens to \(\pm 1.96\sigma\).
Switch to “Random walk” at acceleration. The acceleration trace itself wanders (variance grows as \(q^2 t\)). The integrated panels grow much faster than the white-noise case at matched amplitude — position \(\sigma \propto t^{5/2}\).
Switch to “Gauss-Markov”. Slide \(\tau\) from short (\(\tau \approx \Delta t\)) to long (\(\tau \to T\)). Short \(\tau\): looks like white noise (variance is bounded by \(\sigma_{ss}^2\)). Long \(\tau\): looks like a slow random walk over the run length. This is the bridge model the Block 2 reading talks about.
Open the cheat sheet at the bottom of the demo to see the closed-form variance growth for each model and integration depth. The metrics row shows the empirical \(k\sigma\) from the 30-run ensemble alongside the theoretical \(k\sigma\) from those formulas — they should agree closely when the simulation is long enough for the ensemble to converge.
Key observations#
Integration order matters more than amplitude. A 0.02 m/s² accel bias is sneakier than a 0.1 m/s velocity bias even though the latter sounds bigger, because the former hits position with two integrations instead of one.
Confidence interval ≠ accuracy. A tight band around a single realization tells you about ensemble spread, not whether the realization is biased. Re-seed a few times to confirm individual realizations move while the band stays put.
Random walk is qualitatively different from white noise integrated. White-noise velocity integrates into a position random walk (\(\sigma \propto \sqrt{t}\)). A random-walk velocity, on the other hand, has variance that itself grows linearly, and its integral grows like \(t^{3/2}\). That extra factor of \(t\) is what makes random walk so painful for inertial systems running open-loop.
Gauss-Markov saturates. Unlike random walk, the Gauss-Markov error stays bounded around its steady-state \(\sigma_{ss}\). Its integrated descendants still grow with time, but the source of growth is the integral of a bounded process, not an unbounded one. That’s why most accelerometer specs use Gauss-Markov for bias modeling instead of pure random walk.