Position Error Calibration

Position Error Calibration (flight-test-operations/planning/position-error-calibration)

Use when the task is the airspeed position error calibration (PEC) flight test: planning the test points for the tower fly-by, trailing cone, and GPS ground speed doublet reference methods, and reducing the measured runs into the position error correction curve and the PEC table of indicated versus calibrated airspeed. This leaf implements the compressible calibrated airspeed relations and the PEC reduction in pure Python, stdlib only. It pairs with flight-test-operations/planning/flight-test-data-reduction, whose channel reduction consumes the PEC table when it computes the corrected airspeed of each recorded run, with flight-test-operations/envelope/ v-speeds, whose speed rules consume the calibrated airspeed set, and with flight-test-operations/envelope/high-angle-of-attack-testing, which runs the separate angle of attack sensor position error method against its own tower fly-by reference.

Domain quick reference

• Calibrated airspeed from impact pressure (compressible, ISA sea level standard): V_cas = a0 * sqrt(5 * ((q_c/p0 + 1)^(2/7) - 1)), with the module constants a0 = 340.294 m/s and p0 = 101325 Pa.
• Impact pressure from calibrated airspeed: q_c = p0 * ((1 + 0.2 * (V_cas/a0)^2)^3.5 - 1). The pair is inverse, so V_cas equals V_ias exactly when the position error is zero.
• Position error: dVp = V_cas - V_ias; the correction is added to the indicated airspeed and is positive when the static source makes the airspeed indicator read low.
• Tower fly-by method: the aircraft flies level at a surveyed geometric height H_g above the tower while the altimeter at the standard setting records the pressure altitude H_p. The height error of the static source is dh = H_g - H_p. The static pressure error follows the altimeter scale (hydrostatic) relation dp_s = rho * g0 * dh, with rho = p(H_p)/(R*T) evaluated at the measured temperature, and the same dp_s displaces the impact pressure of the airspeed indicator by -dp_s, so V_cas = V_isa(qc(V_ias) + dp_s) with the exact compressible airspeed indicator law. Simplified relation used by the reduced function: the pass speed is taken at the module reference fly-by speed (100 m/s, mid range of a PEC sweep) unless the scheduled pass indicated airspeed is passed explicitly; the result carries the sign of the height error, a low altimeter reading gives a positive correction.
• Trailing cone method: the reference static pressure comes from a cone trailed behind the aircraft clear of the fuselage flow; the reduction to dVp follows the same static pressure error chain as the tower fly-by at the flown indicated airspeed.
• GPS ground speed doublet: two runs on reciprocal headings at the same indicated airspeed give ground speeds V1g and V2g; with a steady wind the true airspeed is V_tas = (V1g + V2g)/2, the calibrated airspeed is V_cas = V_tas * sqrt(rho/rho0) with the density ratio at the test altitude, and dVp = V_cas - V_ias.
• PEC curve: the calibrated points (V_ias, dVp) become the knots of a piecewise linear position error correction curve; repeat passes at a scheduled speed are combined by their least squares mean, and the residual RMS of the observations about the curve is the data quality metric.
• PEC table: for each indicated airspeed the table row carries dVp from the curve and V_cas = V_ias + dVp; the table feeds the data reduction of every later flight.
• Data quality verdict: coverage is the fraction of planned test point speeds that lie inside the calibrated span; the verdict is adequate when coverage is at least 0.95 and the residual RMS is at most 1.0 m/s.
• Units are SI: speeds m/s, pressures Pa, heights m, temperatures K. FAR-25 and CS-25 set the airspeed instrument accuracy context for the certification flight test; the relations above are standard engineering methodology, summary-only per standards-map.yaml.

Workflow

1. Schedule the PEC test points across the speed range with test-point-matrix-design: choose the reference methods (tower fly-by, trailing cone, GPS ground speed doublet), the indicated airspeeds per point, and repeat passes for the data quality check.
2. Convert the recorded impact pressure channel to calibrated airspeed with calibrated_airspeed, or convert a scheduled calibrated airspeed to its impact pressure with impact_pressure_from_cas for the test card.
3. Reduce a tower fly-by pass: call tower_flyby_position_error with the surveyed geometric height, the altimeter pressure altitude, and the measured temperature (and the pass indicated airspeed when it was recorded); the returned dVp belongs to that pass point.
4. Reduce a GPS ground speed doublet: gps_doublet_tas on the two reciprocal ground speeds, tas_to_cas with the density ratio of the test altitude, then position_error against the indicated airspeed held during the doublet.
5. Combine the calibrated points from every method into one set of (V_ias, dVp) observations and fit the correction curve with fit_pec_curve; inspect the residual RMS it reports.
6. Build the PEC table with pec_table over the scheduled indicated airspeeds; each row carries the correction and the calibrated airspeed that the data reduction will use.
7. Run pec_verdict with the planned point list and the methods flown to get the coverage, residual RMS, method list, and the adequate or review verdict.
8. Hand the PEC table to flight-test-data-reduction for the channel reduction of the campaign and to v-speeds for the speed rule assessment on calibrated values.
9. Confirm the deterministic checks with the contract test scripts/test_position_error_calibration.py.

Worked example

• Compressible identity: impact_pressure_from_cas(100.0) returns 6258.4 Pa and calibrated_airspeed of that pressure returns 99.99999999999982 m/s, so position_error(100.0, V_cas) is zero by construction; V_cas equals V_ias whenever the position error is zero.
• Tower fly-by: the aircraft passes at geometric height 500 m and the altimeter reads 490 m (10 m low) at 288.15 K. The hydrostatic altimeter scale gives dp_s = rho * g0 * 10 m = 113.3 Pa, which displaces the impact pressure the airspeed indicator sees; at the reference pass speed of 100 m/s the reduction returns dVp = +0.88 m/s (the indicator reads low, the correction is added). The same pass with the altimeter 10 m high returns dVp = -0.89 m/s, and a zero height error returns 0.0. Passing the actual pass speed refines the scale: 90 m/s gives 0.99 m/s, 120 m/s gives 0.72 m/s.
• GPS ground speed doublet: reciprocal runs give V1g = 98 m/s and V2g = 102 m/s, so gps_doublet_tas returns V_tas = 100 m/s. At a density ratio rho/rho0 = 0.9 the calibrated airspeed is tas_to_cas(100.0, 0.9) = 94.87 m/s; held at V_ias = 100 m/s the point carries dVp = -5.13 m/s, the indicator reads high at this speed.
• PEC curve: five points (60, 1.2), (80, 0.9), (100, 0.6), (120, 0.2), (140, -0.3) fit to a curve whose knots reproduce every point with residual RMS 2.5e-17 (zero to float precision); the segment slopes are -0.015, -0.015, -0.02, -0.025 m/s per m/s. The table row at 70 m/s interpolates dVp = 1.05 m/s and V_cas = 71.05 m/s.
• Verdict: seven planned points spanning 55 to 145 m/s with a calibrated span of 60 to 140 m/s give coverage 5/7 = 0.714, below the 0.95 threshold, so the verdict is review until the span covers the planned points.

Pitfalls

• Confusing the sign of the tower fly-by correction: with the altimeter 10 m low the indicator reads low and dVp = +0.88 m/s (correction added); a 10 m high altimeter returns dVp = -0.89 m/s.
• Applying the fly-by correction at the wrong pass speed: the scale depends on the reference pass speed (0.99 m/s at 90 m/s, 0.72 m/s at 120 m/s), so the pass speed must match the reduction point.
• Forgetting the density ratio in the GPS doublet leg: at rho/rho0 = 0.9, tas_to_cas(100.0, 0.9) = 94.87 m/s, so a point held at V_ias = 100 m/s carries dVp = -5.13 m/s, not zero.
• Claiming coverage without the span rule: five calibrated points covering seven planned points (0.714) is below the 0.95 threshold, and the verdict stays review until the calibrated span covers the planned points.
• Feeding non-physical inputs: negative speeds, empty lists, a density ratio of zero or less, non-monotonic table speeds, negative geometric height, and a non-positive temperature all raise ValueError.
• Reading the PEC curve as the table: the curve knots reproduce every calibration point (residual RMS 2.5e-17), while the table interpolates rows only for strictly increasing speeds (70 m/s gives dVp = 1.05 m/s and V_cas = 71.05 m/s).

Verification

• Confirm calibrated_airspeed(impact_pressure_from_cas(100.0)) returns 100 m/s and that the round trip holds at every test speed.
• Confirm gps_doublet_tas(98.0, 102.0) returns 100.0 and tas_to_cas(100.0, 0.9) returns 94.87 m/s.
• Confirm tower_flyby_position_error(500.0, 490.0, 288.15) returns about +0.88 m/s, that the sign follows the height error, and that a zero height error returns zero.
• Confirm fit_pec_curve reproduces its knots with residual RMS near zero and that repeat passes collapse to their mean with the scatter reported in the residual RMS.
• Confirm pec_table interpolates the curve and returns (v_ias, dVp, v_ias + dVp) rows for strictly increasing speeds.
• Confirm ValueError rejection of negative speeds, empty lists, a density ratio of zero or less, non-monotonic table speeds, negative geometric height, and a non-positive temperature.
• Run the contract test offline: python3 scripts/test_position_error_calibration.py (34 tests, deterministic).

Related leaves

• flight-test-operations/planning/flight-test-data-reduction: consumes the PEC table when it applies the calibration corrections and computes the corrected airspeed of each recorded run.
• flight-test-operations/envelope/v-speeds: consumes the calibrated airspeed set for the certified speed rules of the program.
• flight-test-operations/envelope/high-angle-of-attack-testing: runs the angle of attack sensor position error calibration against its own tower fly-by or trailing cone reference at high angles, the companion method to this leaf's airspeed PEC.
• flight-test-operations/planning/test-point-matrix-design: lays out the condition sweeps that schedule the PEC points across the speed range.

Behavior contract (gate 3)

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

python3 scripts/test_position_error_calibration.py

The test covers the compressible calibrated airspeed identities and the zero position error identity, the GPS ground speed doublet worked example (98/102 m/s to 100 m/s true airspeed, 94.87 m/s calibrated at a 0.9 density ratio), the tower fly-by height error reduction with its sign, magnitude, pass speed sensitivity, and zero error behavior, the piecewise linear PEC fit that reproduces its knots with near zero residual RMS, repeat pass averaging, table interpolation, the coverage and verdict math, and ValueError rejection of negative speeds, empty lists, a non-positive density ratio, non-monotonic table speeds, a negative geometric height, and a non-positive temperature.

Compliance

• Standards referenced, not reproduced: FAR-25 and CS-25 airspeed instrument requirements frame the certification context by name; the airspeed relations and the PEC reduction above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.