Imu Static Calibration

IMU Static Calibration (gnc-autonomy/estimation-filtering/imu-static-calibration)

Use when the task is the static calibration of an IMU from laboratory test data: level the sensor in the +g and the -g orientation on each body axis for the six-position accelerometer test, take the mean specific force of each static hold, and solve the per-axis bias and scale factor from the paired +/- 1 g references, with the full scale-plus-misalignment sensitivity matrix available from a least-squares fit over all six positions against the known 1 g directions; then spin the sensor about each body axis at commanded rotation rates for the rate-table gyro test, regress the measured rate on the commanded rate, and read the per-axis gyro bias as the intercept and the scale factor as the slope of the linear fit. This leaf implements both reductions in pure Python (math and random only), deterministic for a fixed seed. It pairs with the estimation-filtering pack and navigation siblings, whose filters consume calibrated measurements, and it hands its coefficient set and residual fit errors to the navigation error assessment that follows calibration.

Domain quick reference

• Measurement model of a static hold: m_k = b + A*(f_k*G0), where f_k is the hold direction from the pinned fixture POSITIONS (g units), b the bias vector in m/s^2 and A the dimensionless 3x3 sensitivity matrix whose diagonal entries A_ii = s_i are the scale factors and whose off-diagonal entries are the cross-axis (misalignment) sensitivities.
• Fixture order: POSITIONS = ((1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)), so the +g hold of axis i sits at index 2i and the -g hold at 2i+1, in the order +x, -x, +y, -y, +z, -z. G0 = 9.80665 m/s^2 converts the 1 g references into specific-force units.
• Pair reduction per axis i: with plus = means[2i][i] and minus = means[2i+1][i], the +g and -g hold means of channel i, the scale factor is s_i = (plus - minus)/(2*G0) and the bias b_i = (plus + minus)/2 in m/s^2.
• Least-squares fit over all six positions: for each output channel c the design rows (f_k[0]G0, f_k[1]G0, f_k[2]G0, 1) over the six holds are regressed through the normal equations (X^T X) q = X^T y solved by Gaussian elimination with partial pivoting. The coefficient triple is row c of A (A_cc is the scale factor) and the fourth coefficient is bias_c. Because the fixture sums each f_j to zero over the six holds, X^T X = diag(2G0^2, 2G0^2, 2G0^2, 6) exactly, so the fit is always well posed: the LS diagonal equals the pair closed form within float noise and the LS off-diagonals are the cross-axis sensitivities.
• Rate-table regression per axis: measured_rate = s*commanded_rate + b, so the ordinary least-squares slope over the commanded rates is the dimensionless scale factor and the intercept is the gyro bias in deg/s.
• The diagonal pair model predicts channel i of hold k as b_i + s_i*f_k[i]*G0; its residual rms over all 18 measurements carries the cross-axis leakage the diagonal model cannot absorb, and the misalignment-aware LS residual rms drops once the off-diagonal terms absorb that leakage.
• Units: accelerometer bias in m/s^2, gyro bias in deg/s, both scale factor sets dimensionless, residual fit errors in the units of the fitted channels.
• ARP4754A frames the air-vehicle system context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Record the six-position-test static-hold campaign: level the sensor with each body axis held up and then down, and take the mean specific force of every static hold against the known plus/minus 1 g references (simulate_six_position_means generates a deterministic synthetic campaign from injected truth for validation, with per-sample Gaussian noise of standard deviation sigma and the caller-supplied integer seed).
2. Run the pair reduction to the accelerometer-bias-scale pair closed form with fit_accel_pair_bias_scale: the per-axis bias in m/s^2 and scale factor from the paired +g and -g holds, plus the diagonal model residual rms over all 18 measurements.
3. Fit the misalignment matrix by least squares over all six positions with fit_accel_misalignment_ls: the dimensionless 3x3 sensitivity matrix (scale factors on the diagonal, cross-axis sensitivities off-diagonal), the LS bias vector and the LS residual rms, which sits below the pair residual once the leakage terms are in the model.
4. Collect the rate-table-calibration runs: spin the sensor about each body axis at the commanded rotation rates and record the measured gyro rates (simulate_rate_table generates a deterministic synthetic run set from injected truth, noise sigma in deg/s).
5. Regress the gyro axes with fit_gyro_rate_table: per-axis scale factor (slope) and bias in deg/s (intercept) of the measured-on- commanded linear fit, with the per-axis residual rms in deg/s.
6. Read off the residual fit errors of both reductions and confirm the calibration: compare each residual rms against the noise level of the test campaign and run the contract test scripts/test_imu_static_calibration.py, whose assertions pin the worked example, the noise-free identities and every ValueError rejection.

Worked example

Injected truth: accelerometer bias b = (0.0120, -0.0080, 0.0200) m/s^2 (about +1.22, -0.82 and +2.04 mg), scale s = (1.0020, 0.9980, 1.0010), cross-axis truth rows (0.0012, -0.0009), (0.0015, 0.0006) and (-0.0011, 0.0008), noise sigma 5.0e-4 m/s^2, seed 7. Gyro bias b = (0.0500, -0.0300, 0.0200) deg/s (180, -108 and 72 deg/h), scale s = (0.9985, 1.0010, 0.9995), noise sigma 5.0e-3 deg/s, seed 11. All values below are the real module outputs.

• Simulated hold means (m/s^2, +g then -g in the order +x, -x, +y, -y, +z, -z): hold 0 (+x) (9.838135, 0.006966, 0.009100); hold 1 (-x) (-9.814421, -0.023175, 0.030681); hold 2 (+y) (0.024324, 9.779249, 0.028364); hold 3 (-y) (0.000356, -9.794839, 0.012247); hold 4 (+z) (0.002341, -0.001688, 9.836710); hold 5 (-z) (0.021075, -0.014730, -9.797329).
• Pair reduction: bias = (0.01185726, -0.00779527, 0.01969062) m/s^2 and scale = (1.00200151, 0.99800075, 1.00105737), recovered within (0.000143, 0.000205, 0.000309) m/s^2 and (0.000002, 0.000001, 0.000057) of the injected truth. The diagonal model leaves the cross-axis leakage in the residual: rms 8.713e-3 m/s^2.
• Least-squares fit over all six positions: sensitivity matrix A = ((1.00200151, 0.00122200, -0.00095519), (0.00153675, 0.99800075, 0.00066492), (-0.00110033, 0.00082171, 1.00105737)) with bias = (0.01196855, -0.00803632, 0.01996211) m/s^2 and residual rms 2.375e-4 m/s^2, below the pair residual as the off-diagonal terms absorb the leakage. The LS scale diagonal equals the pair scale within 1e-6; cross-axis recovery errors ((0.000022, 0.000055), (0.000037, 0.000065), (0.000000, 0.000022)) sit within 2e-4 of the injected truth.
• Rate-table gyro: commanded rates per axis (deg/s) axis 0 (50, 30, 10, -10, -30, -50), axis 1 (40, 25, 5, -5, -25, -40), axis 2 (45, 20, 15, -15, -20, -45); measured axis 0 (49.96888, 30.00689, 10.03997, -9.93757, -29.91164, -49.87533), axis 1 (40.01239, 25.00049, 4.96892, -5.04146, -25.05147, -40.06763), axis 2 (45.00517, 20.01491, 15.01199, -14.97387, -19.95715, -44.95894).
• Gyro regression: bias = (0.048534, -0.029794, 0.023685) deg/s and scale = (0.998506, 1.001011, 0.999550), recovered within (0.001466, 0.000206, 0.003685) deg/s and (0.000006, 0.000011, 0.000050) of the injected truth; per-axis residual rms (0.004153, 0.004688, 0.005127) deg/s sits at the injected noise sigma.
• Read-off: the six holds recover the accelerometer bias to about 0.03 mg and the scale factors to 6e-5, and the full least-squares fit pulls the residual rms from 8.7e-3 down to 2.4e-4 m/s^2 once the cross-axis terms (1 mg-class sensitivities) are in the model; the rate table recovers the gyro bias to 0.0037 deg/s (about 13 deg/h on the worst axis) and the scale factors to 5e-5 with residuals at the injected rate noise.

Verification

• Confirm simulate_six_position_means with the worked-example truth returns the six hold means above within 1e-5 m/s^2, and that a same-seed repeat returns bit-identical tuples.
• Confirm fit_accel_pair_bias_scale returns the pair bias and scale above within 1e-5, and the diagonal-model rms_residual 8.713e-3 within 1e-3 m/s^2.
• Confirm fit_accel_misalignment_ls returns the matrix rows and bias above within 1e-4 and 1e-5, with the LS residual rms 2.375e-4 within 1e-4 and strictly below the pair residual.
• Confirm the noise-free identities: with sigma = 0.0 the pair and LS fits recover the injected bias, scale and cross-axis truth within 1e-9, and the LS diagonal equals the pair scale within 1e-9.
• Confirm simulate_rate_table returns the measured axis rates above within 1e-4 deg/s, and fit_gyro_rate_table the gyro bias within 1e-4 deg/s, scale within 1e-5 and rms_per_axis within 2e-3 of the injected noise sigma.
• Confirm every non-physical input raises ValueError: means of 5 holds or rows wider than 3 channels, negative or non-finite sigma, non-positive scale factors, a single commanded rate per axis, measured and commanded lists of unequal length, a constant commanded rate (singular regression), and non-finite means or rates.
• Run the contract test offline: python3 scripts/test_imu_static_calibration.py (31 tests, deterministic, in under 20 seconds).

Related leaves

• gnc-autonomy/estimation-filtering/complementary-filter: online gyro bias estimation inside a running attitude observer, the operational counterpart of this leaf's lab regression.
• gnc-autonomy/navigation/inertial-navigation: consumes the calibrated bias and drift inputs this leaf produces as coefficients.
• gnc-autonomy/navigation/kalman-filter-design: filter design that consumes the calibrated measurements and their residual fit errors.
• gnc-autonomy/estimation-filtering/extended-kalman-filter and the other recursive filters of this pack: downstream consumers of calibrated measurements.
• space-systems/adcs/gyro-allan-variance: noise characterization of the same sensor class, a separate step from static calibration.
• space-systems/adcs/magnetometer-calibration: the magnetometer counterpart calibrated against field magnitudes alone.
• flight-test-operations/planning/flight-test-instrumentation: the pre-test release and currency gate that decides whether an instrumented channel is current and fit to fly.

Pitfalls

• Claiming noise characterization: a single unforced rate time series with no commanded rate reference cannot separate bias and scale factor; the rate-table runs need the commanded rates as the regression regressor.
• Fitting the pair model when misalignment is present: the diagonal pairs leave the cross-axis leakage in the residual (8.7e-3 m/s^2 on the worked example); the six-position least-squares fit is required to absorb it (2.4e-4 m/s^2).
• Confusing the pair closed form with the full fit: the pair scale s_i = (plus - minus)/(2*G0) uses only channel i of the two on-axis holds, so it is exact against cross-axis contamination of the on-axis channel but reports no misalignment terms at all.
• Treating the gyro residual rms as an error of the fit rather than of the data: the per-axis residual rms sits at the injected noise sigma of the rate table (about 5e-3 deg/s on the example), so a residual far above the noise level signals a rate reference problem, not a regression problem.
• Simulating with sigma but no seed: the reduction must be deterministic, so every simulation call fixes random.Random(seed) with a caller-supplied integer seed.
• Regressing an axis from a single commanded rate: the normal matrix is singular, so each axis needs at least two distinct commanded rates; a constant commanded rate fails the same singularity guard.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_imu_static_calibration.py

The test covers the six-position worked example (hold means within 1e-5 m/s^2 of the real module outputs), the pair reduction bias and scale against the injected truth, the diagonal-model residual rms at 8.713e-3 m/s^2, the misalignment least-squares matrix and bias with the LS residual rms 2.375e-4 m/s^2 strictly below the pair residual, the LS diagonal equal to the pair scale within 1e-6, cross-axis recovery within 2e-4, the noise-free recovery identities within 1e-9, the rate-table measured runs and gyro regression with residual rms at the injected noise, bit-identical same-seed determinism, the shared normal-equations solver on closed-form lines, the pinned G0 and fixture order, stdlib-only imports, and ValueError rejection of every non-physical input listed in the spec (malformed means, negative sigma, non-positive scale, under-sampled or constant commanded rates, length mismatches, singular regressions, non-finite data).

Compliance

• Standards referenced, not reproduced: ARP4754A frames the air-vehicle system context; the calibration relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.