Specific force from the accelerometers (gravity-compensated linear acceleration) and angular rate from the gyroscopes (rotation rate of the body). Both are expressed in the body frame.
It integrates angular rate to track attitude, rotates the gravity-compensated specific force into NED, and integrates twice to produce velocity and position. The mechanization is where small sensor errors become big navigation errors.
Accelerometer error → velocity error (one integration) → position error (two integrations). A constant accelerometer bias produces linear velocity error and quadratic position error.
Gyro error → attitude error → wrong projection of gravity into NED → secondary horizontal acceleration error → velocity error → position error. The gyro path is indirect, but it is what drives the Schuler oscillation.
Bias is a constant offset on a sensor measurement. A constant accelerometer bias \(b_a\) produces velocity error \(v_e(t) = b_a t\) (linear) and position error \(p_e(t) = \tfrac{1}{2} b_a t^2\) (quadratic).
Sample-to-sample variation around the true value. White-noise velocity error integrates into a random-walk position error whose RMS grows like \(\sqrt{t}\) rather than \(t\), which is why noise is a much weaker error source than a bias of similar magnitude over long durations.
A discrete-time model where the bias takes a small Gaussian step at every sample: \(b_{k+1} = b_k + w_k\) with \(w_k \sim \mathcal{N}(0, q\,\Delta t)\). The bias has no preferred value; its variance grows linearly with time. Good fit for many gyro errors.
A first-order Gauss-Markov process adds a decay term: \(x_{k+1} = e^{-\Delta t / T} x_k + w_k\). The correlation time \(T\) sets a finite memory, so the error does not run away to infinity. For \(\Delta t \ll T\) it behaves like a slow random walk; for \(\Delta t \gg T\) it looks like white noise. It is the workhorse model for accelerometer biases.
The standard tool for fingerprinting noise processes inside an IMU. It is a ground test, not a flight test: a long static IMU run (typically about 6 hours) measured before any aircraft leaves the ramp. By computing Allan deviation \(\sigma(\tau)\) over a range of cluster times \(\tau\) and reading slopes off the resulting log-log plot, you can separate white noise, bias instability, and rate random walk.
Slope \(-1/2\): white noise — angle random walk (ARW) on gyros, velocity random walk (VRW) on accelerometers. Slope \(0\): bias instability — the low-frequency wander floor. Slope \(+1/2\): rate random walk — long-term bias drift. Each underlying noise process leaves a distinct fingerprint as a slope segment on the log-log plot.
White-noise (slope \(-1/2\)) ARW/VRW values populate the position-velocity entries of \(\mathbf{Q}\). The bias-instability floor and rate-random-walk slope decide whether to model an axis bias as a random walk (long correlation time) or a Gauss-Markov process (finite correlation time), and at what time constant. So Allan variance is the bridge from a vendor datasheet number to a working filter.
\(\mathbf{x} = [\mathbf{p}^\top, \mathbf{v}^\top, \mathbf{a}^\top]^\top\) with position, velocity, and an attitude representation each of dimension 3. The estimator pairs this with a \(9\times 9\) covariance \(\mathbf{P}\).
State: \(\mathbf{x}_{k+1} = f(\mathbf{x}_k, \mathbf{u}_k) + \mathbf{w}_k\) with IMU input \(\mathbf{u}_k\) and process noise \(\mathbf{w}_k\). Covariance: \(\mathbf{P}_{k+1} = \mathbf{F}_k \mathbf{P}_k \mathbf{F}_k^\top + \mathbf{Q}_k\), where \(\mathbf{F}_k\) is the Jacobian of \(f\). Together they say: uncertainty grows over time when only the IMU is feeding the algorithm.
\(T_{\text{Schuler}} = 2\pi \sqrt{R_e / g} \approx 84.4\) minutes. It is the natural frequency of an inertial system tuned to Earth's radius. Specifically: an attitude error tilts gravity into the horizontal plane, which produces a horizontal acceleration error that integrates into velocity and position errors, which feed back to correct attitude — and the closed loop oscillates at the Schuler frequency.
Horizontal inertial error has both a long-term drift component (linear in time) and a Schuler oscillation (sinusoidal at 84.4-minute period). Fitting a slope on a window shorter than one full Schuler cycle gives a slope that depends on which phase of the oscillation you happened to start and stop on. A full cycle averages the oscillation out and lets you isolate the underlying drift.
Process-noise covariance: it captures everything the motion model does not. Sensor noise, unmodeled biases, mechanization simplifications all enter \(\mathbf{Q}_k\). If you set \(\mathbf{Q}\) too small the filter will be over-confident and ignore good measurements; if you set it too large the filter will follow noise. Tuning \(\mathbf{Q}\) against truth data is a meaningful part of any real navigation deployment.
The slope of horizontal position error versus time over an unaided inertial run. Flight-test reports state it in nautical miles per hour (NM/hr) so it is directly comparable to operational accuracy requirements. The F-47 ANS Block 10 capstone uses a 1.0 NM/hr requirement on inertial-only drift over a window of at least 85 minutes.
Each fix bounds the position error to roughly the fix accuracy and, via the filter's cross-covariance, also bounds the velocity and attitude errors. Between fixes the quadratic growth resumes, but it restarts from the corrected state. The time-averaged error is dramatically smaller than free inertial coast, which is the entire motivation for fusing IMU with an aiding sensor (the topic of Blocks 3 through 7).